![A symbolic system like Mathematica, is able to carry out this algorithm not
only with numbers but also with symbols. It means that the unknown elements of
ya ¼ yðxaÞ can be considered as unknown symbols. These symbols will appear in
every evaluated yi value, as well as in yb ¼ yðxbÞ too.
Let us consider a simple illustrative example. The differential equation is:
d2
dx2 yðxÞ
1 x
5
yðxÞ ¼ x.
The given boundary values are:
yð1Þ ¼ 2
and
yð3Þ ¼ 1
After introducing new variables, we get a first-order system,
y1ðxÞ ¼ yðxÞ
and
y2ðxÞ ¼
d
dx
yðxÞ
the matrix form of the differential equation is:
d
dx y1ðxÞ; d
dx y2ðxÞ ¼
0 1
1 x=5 0
y1ðxÞ; y2ðxÞ
½ þ 0; x
½ .
Employing Mathematica’s notation:
A[x_]:={{0,1},{1-1/5 x,0}};
b[x_]:={0,x};
x0=1;
y0={2.,s}
The unknown initial value is s. The order of the system M = 2. Let us consider
the number of the integration steps as N = 10, so the step size is h = 0.2.
ysol=RKSymbolic[x0,y0,A,b,2,10,0.2];
The result is a list of list data structure containing the corresponding (x, y) pairs,
where the y values depend on s.
ysol[[2]][[1]]
{{1,2.},{1.2,2.05533+0.200987 s},{1.4,2.22611+0.407722
s},
{1.6,2.52165+0.625515 s},
{1.8,2.95394+0.859296s}, {2.,3.53729+1.11368s},](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-26-320.jpg)
![{2.2,4.28801+1.39298 s},
{2.4,5.22402+1.70123 s},{2.6,6.36438+2.0421 s},
{2.8,7.72874+2.41888 s},{3.,9.33669+2.8343 s}}
Consequently, we have got a symbolic result using traditional numerical Runge–
Kutta algorithm.
In order to compute the proper value of the unknown initial value, s, the
boundary condition can be applied at x ¼ 3. In our case y1ð3Þ ¼ 1.
eq=ysol[[1]][[1]]==-1
9.33669+2.8343 s==-1
Let us solve this equation numerically, and assign the solution to the symbol s:
sol=Solve[eq,s]
{{s - -3.647}}
s=s/.sol
{-3.647}
s=s[[1]]
-3.647
Then, we get the numerical solution for the problem:
ysol[[2]][[1]]
{{1,2.},{1.2,1.32234},{1.4,0.739147},{1.6,0.240397},
{1.8,-0.179911}, {2.,-0.524285},{2.2,-0.792178},
{2.4,-0.980351},{2.6,-1.08317},{2.8,-1.09291},
{3.,-1.}}
The truncation error can be decreased by using smaller step size h, and the
round-off error can be controlled by the employed number of digits.](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-27-320.jpg)
![where M(y) is
1 y 0
y2
4 0 1
0 1 y
0
@
1
A
x0
x1
x2
0
@
1
A ¼
0:
Since x0
is really 1, any solution to this homogeneous system must be nontrivial.
Thus
detðMðyÞÞ ¼ 1 þ 4y2
y4
¼ 0:
Solving this gives us y; we have eliminated x. This function is built into
Mathematica,
Resultant[p,g,x]
1 - 4y2
+ y4
For the other variable
Resultant[p,g,y]
1 - 4x2
+ x4
The solutions of these two polynomials are the solutions of the system
p x; y
ð Þ; g x; y
ð Þ
f g.
Roots[ - 1 + 4y2
- y4
= = 0,y]
y =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 -
ffiffiffi
3
p
p
jjy = -
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 -
ffiffiffi
3
p
p
jjy =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 +
ffiffiffi
3
p
p
jjy = -
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 +
ffiffiffi
3
p
p
Roots[1 - 4x2
+ x4
= = 0,x]
x =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 -
ffiffiffi
3
p
p
jjx = -
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 -
ffiffiffi
3
p
p
jjx =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 +
ffiffiffi
3
p
p
jjx = -
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 +
ffiffiffi
3
p
p
The main handicap of the Sylvester resultant is that it can be directly employed only
for systems of two polynomials.
1.2.2 Dixon Resultant
Let us introduce a new variable b and define the following polynomial,
d x; y; b
ð Þ ¼
p x; y
ð Þg x; b
ð Þ p x; b
ð Þg x; y
ð Þ
y b
¼ b 4x þ x3
þ y bxy:
(One can show that the numerator above is always divisible by y – b). We call this
polynomial the Dixon polynomial.
1.2 Resultant Methods 5](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-31-320.jpg)
![It is easy to see that plugging in any common root of p x; y
ð Þ; g x; y
ð Þ
f g forces the
Dixon polynomial to be 0, for any value of b. The Dixon polynomial can be written as,
d x; y; b
ð Þ ¼ b0
4x þ x3
þ y
þ b1
1 xy
ð Þ:
Then the following homogeneous linear system should have solutions for every b
4x þ x3
þ y ¼ 0;
1 xy ¼ 0
or, where x is considered as parameter
4x þ x3
1
1 x
y0
y1
¼ 0;
therefore
det
4x þ x3
1
1 x
¼ 1 þ 4x2
x4
;
must be zero. The matrix
4x þ x3
1
1 x
is called the Dixon matrix, and its determinant is called as Dixon resultant.
[Historical note: the argument above, for two variables, was first used by
Bezout.]
Let us employ Mathematica
Resultant0
Dixon0
DixonPolynomial[{p,g},{y},{b}]
b - 4x + x3
+ y - bxy
DixonMatrix[{p,g},{y},{b}]==MatrixForm
- 4x + x3
1
1 - x
DixonResultant[{p,g},{y},{b}]
- 1 + 4x2
- x4
Similarly, for the other variable, we get
DixonResultant[{p,g},{x},{a}]
- 1 + 4y2
- y4
6 1 Solution of Algebraic Polynomial Systems](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-32-320.jpg)
![Here a and b are dummy formal variables (symbolic variables), without assigned
values.
The Dixon resultant method can be generalized to polynomial systems of more
than two polynomials. For example,
P = x + y + z;
G = x - 2y + z3
;
S = x2
- 2y3
+ z;
To eliminate variables x and y, we introduce dummy variables X and Y, then
DixonResultant[{P,G,S},{x,y},{X,Y}]==Expand
324z + 144z2
+ 24z3
+ 144z4
- 72z5
+ 36z6
+ 72z7
- 24z9
or
%=12==Expand
27z + 12z2
+ 2z3
+ 12z4
- 6z5
+ 3z6
+ 6z7
- 2z9
(% refers to the last previous output.)
Remark 1 For other multivariate resultant methods such as Sturmfels’ approach,
see Awange and Paláncz (2016).
Remark 2 For three or more variables, the discussion above of the Dixon resultant
has been simplified. Sometimes the Dixon matrix is not square, and sometimes
when it is square the determinant is identically 0. Then the method would seem to
fail. Kappur et al. (1994) showed how to proceed and define the Dixon resultant
using maximal minors, see Exercises 1.6.4 Dixon KSY solution.
Remark 3 If the system contains parameters, then the resultant will contain those
parameters.
Remark 4 If there are n equations, the Dixon resultant will eliminate n − 1
variables.
1.3 Gröbner Basis
This technique introduced by Buchberger and named after his Ph.D. supervisor
Gröbner, is more general and mostly more efficient than the resultant methods,
unless parameters are present. To have an idea how this method works, first, let us
see how the greatest common divisor of polynomials can be defined.
1.2 Resultant Methods 7](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-33-320.jpg)
![1.3.1 Greatest Common Divisor of Polynomials
Greatest common divisor, GCD, is a familiar concept from arithmetic, as in GCD
(12, 30) = 6. The same concept applies to polynomials, of any number of variables.
Let us consider two univariate polynomials s(x) and v(x) with the same variable x
s = 8 + 22x + 21x2
+ 8x3
+ x4
;
v = 6 + 11x + 6x2
+ x3
;
The greatest common divisor (GCD) of these polynomials
gcd = PolynomialGCD[s,v]
2 + 3x + x2
Let us divide s(x) with this GCD
{{Cs},Rs} = PolynomialReduce[s,gcd,x]
{{4 + 5x + x2
},0}
The remainder is zero and we can check
Csgcd
(2 + 3x + x2
)(4 + 5x + x2
)
Expand[%]
8 + 22x + 21x2
+ 8x3
+ x4
Let us carry out these operations with v(x), too
{{Cv},Rv} = PolynomialReduce[v,gcd,x]
{{3 + x},0}
and
Csgcd
(3 + x)(2 + 3x + x2
)
Expand[%]
6 + 11x + 6x2
+ x3
This means that the original polynomials s(x) and v(x) can be expressed as the linear
combination of the GCD, like.
8 1 Solution of Algebraic Polynomial Systems](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-34-320.jpg)
![sðxÞ ¼ CsðxÞ gcd(xÞ þ 0 gcd(xÞ
and
vðxÞ ¼ 0 gcd(xÞ þ CvðxÞ gcd(xÞ
or
s x
ð Þ
v x
ð Þ
=
Cs x
ð Þ 0
0 Cv x
ð Þ
gcdðxÞ
gcdðxÞ
!
:
Since there is only one variable, the roots of gcd(x), the GCD of these two poly-
nomials s x
ð Þ; v x
ð Þ
f g are the roots of the polynomial system. This important fact is
because any one-variable polynomial can be factored over C into linear pieces. The
roots of gcd x
ð Þ ¼ 2 þ 3 x þ x2
¼ 0 are in Fig. 1.2.
However, we normally have polynomials of two variables (x, y)
p
- 1 + xy
g
- 4 + x2
+ y2
In case of more than one variable, the greatest common divisor, though it exists,
does not play the role it did in the previous paragraph. That role is filled by the
Gröbner basis (Buchberger and Winkler 1998).
{G1,G2} = GroebnerBasis[{p,g},{x,y}]
{1 - 4y2
+ y4
,x - 4y + y3
}
As in the case of univariate polynomial, for the two variables ðx; yÞ, the original
system can be expressed as the linear combination of the polynomials of the
Gröbner basis G1 y
ð Þ; G2 x; y
ð Þ
f g, where the coefficients are also polynomials.
Fig. 1.2 Common roots of
polynomials
1.3 Gröbner Basis 9](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-35-320.jpg)
![The coefficients are the remainders.
{c1,r1} = PolynomialReduce[p,{G1,G2},{x,y}]
{{ - 1,y},0}
Then p(x, y) can be expressed as
{ - 1,y}:
G1
G2
==Simplify
{ - 1 + x y}
and
{c2,r2} = PolynomialReduce[g,{G1,G2},{x,y}]
{{ - 4 + y2
,x + 4y - y3
},0}
then g(x, y)
{ - 4 + y2
,x + 4y - y3
}:
G1
G2
==Simplify
{ - 4 + x2
+ y2
}
In matrix form
p x; y
ð Þ
g x; y
ð Þ
¼
1 y
4 þ y2
x þ 4y y3
G1 y
ð Þ
G2 x; y
ð Þ
or
p x; y
ð Þ
g x; y
ð Þ
¼
1
4 þ y2
G1 y
ð Þ þ
y
x þ 4y y3
G2 x; y
ð Þ
The roots of the system p x; y
ð Þ; g x; y
ð Þ
f g are the same as the roots of the system
G1 y
ð Þ; G2 x; y
ð Þ
f g. Note that this basis consists of special polynomials, since G1
(y) is a univariate polynomial!
Generally speaking, the original polynomial system p x; y
ð Þ; g x; y
ð Þ
f g can be
expressed as a linear combination of the basis polynomials G1 x; y
ð Þ; G2 x; y
ð Þ
f g.
There are many other basis polynomials too and the set of these basis polynomials
is called the ideal of the original polynomial. However, the Gröbner basis is a special
basis, since one of its polynomials is a univariate one. If the Gröbner basis is 1, the
polynomials have no common divisor, consequently they have no common roots.
Remark The theory of Gröbner bases is much more extensive and sophisticated than
we can go into here. Our focus is on using Gröbner bases to eliminate variables.
10 1 Solution of Algebraic Polynomial Systems](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-36-320.jpg)
![Let us employ the built-in function for the system {P, S, G} considered in
previous Sect. 1.2.2,
GroebnerBasis[{P,S,G},{x,y,z}]
{ - 27z - 12z2
- 2z3
- 12z4
+ 6z5
- 3z6
- 6z7
+ 2z9
,3y + z - z3
,3x + 2z + z3
}
where the first element of the Gröbner basis is the same provided by the Dixon
resultant.
Now, let us compute the Gröbner basis of the following system
U = x2
+ y2
= = 1
x2
+ y2
- 1
V = x2
+ y2
= = 2
x2
+ y2
- 2
GroebnerBasis[{U,V},{x,y}]
{1}
There are no common roots, see Fig. 1.3, however the upper limit of the number of
roots is 2 2 = 4.
1.3.2 Reduced Gröbner Basis
The Mathematica built in function can carry out the elimination process too,
employing the so called reduced Gröbner Basis.
To get the univariate polynomial of x, we should eliminate y and z,
grbx = GroebnerBasis[{P,S,G},{x},{y,z}]
{ - 27x + 18x2
- 342x3
+ 306x4
- 186x5
+ 229x6
- 18x7
+ 12x8
+ 8x9
}
Fig. 1.3 No common roots
of polynomials
1.3 Gröbner Basis 11](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-37-320.jpg)
![and then for the other variables,
grby = GroebnerBasis[{P,S,G},{y},{x,z}]
{ - 21y - 23y2
- 30y3
- 36y4
- 9y5
+ 6y6
- 12y7
+ 8y8
}
grbz = GroebnerBasis[{P,S,G},{z},{x,y}]
{ - 27z - 12z2
- 2z3
- 12z4
+ 6z5
- 3z6
- 6z7
+ 2z9
}
These algebraic methods are very impressive and useful, but they are limited by
the size of the system. In general, systems with more than ten unknowns cannot be
solved this way due to time and space (RAM) limitations.
1.3.3 Polynomials with Inexact Coefficients
Computing Gröbner bases with inexact coefficients is often desired in industrial
applications, but the computation with floating-point numbers is quite unstable if
performed naively (Sasaki 2014). The solution methods of the Gröbner basis are
very sensitive to round off error, therefore sometimes in case of systems that are
over-constrained or have roots with multiplicities, and are given with inexact
coefficients, using hybrid symbolic-numeric methods are required (Szanto 2011).
Lichblau (2013) discussed computation of Gröbner bases using approximate
arithmetic for coefficients and showed how certain considerations of tolerance,
corresponding roughly to accuracy and precision from numeric computation, allow
us to obtain good approximate solutions to problems that are overdetermined.
Let us consider the following polynomial system,
polys = - 4 + x2
- 1:49071xy + y2
, - 8 + x2
- 0:4xz + z2
,
- 4 + t2
- 0:894427tx + x2
, - 4 + y2
- 1:49071yz + z2
,
- 8 + t2
- 0:666667ty + y2
, - 4 + t2
- 0:894427tz + z2
;
If we try to find the Gröbner basis, we get the trivial answer {1:}, which means
there is no relationship between the polynomials.
sol = GroebnerBasis[polys,{x,y,z,t}]
{1:}
Even employing rationalization of the coefficients will not solve the problem,
n = 10;
polysR = Map[Rationalize[#,10- n
],polys]
- 4 + x2
- (149071xy)=100000 + y2
, - 8 + x2
- (2xz)=5 + z2
,
- 4 + t2
- (216200tx)=241719 + x2
, - 4 + y2
- (149071yz)=100000 + z2
,
- 8 + t2
- (666503ty)=999754 + y2
, - 4 + t2
- (216200tz)=241719 + z2
12 1 Solution of Algebraic Polynomial Systems](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-38-320.jpg)
![solR = GroebnerBasis[polysR,{x,y,z,t}]
{1:}
However, applying an approximate hybrid technique,
solA = GroebnerBasis[polys,x,y,z,t,Tolerance ! 10( - 3)
]
yields
solt = NSolve[solA[[1]],t]
{{t ! - 1:00002 - 0:0044912 i},{t ! - 1:00002 + 0:0044912 i},
{t ! 1:00002 - 0:0044912 i},{t ! 1:00002 + 0:0044912 i}}
Since we are interested in real solutions,
Map[Re[#[[2]]],Flatten[solt]]
{ - 1:00002, - 1:00002,1:00002,1:00002}
Then the other variables are
solz = NSolve[solA[[2]]=:t ! 1:00002,z]
{{z ! 2:2361}}
soly = NSolve[solA[[3]]=:t ! 1:00002,y]
{{y ! 3:00005}}
solx = NSolve[solA[[4]]=:t ! 1:00002,x]
{{x ! 2:23606}}
Let us check our result via least squares technique employing global minimization.
Our objective function is
G = Total[Map[#2
,polys]]
( - 4 + t2
- 0:894427 tx + x2
)2
+ ( - 8 + t2
- 0:666667 ty + y2
)2
+ ( - 4 + x2
- 1:49071 xy + y2
)2
+ ( - 4 + t2
- 0:894427 tz + z2
)2
+ ( - 8 + x2
- 0:4 xz + z2
)2
+ ( - 4 + y2
- 1:49071 yz + z2
)2
and
NMinimize[G,{x,y,z,t}]
2:10012 10- 10
,x ! 2:23607,y ! 3:,z ! 2:23607,t ! 1:0023
1.3 Gröbner Basis 13](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-39-320.jpg)
![to decide whether a point is on a curve or surface or not. Finding the common
points of two or more geometrical objects is the generalization of this task.
Converting explicit to implicit just means eliminating the parameter.
Application 1 Let us compute the implicit equation of a 2D circle.
The form of the explicit equation with the parameter is,
x ¼ cosðaÞ;
y ¼ sinðaÞ
and in addition we know that
sin2
ðaÞ þ cos2
ðaÞ ¼ 1:
Solution
Therefore, we have the following system of equations with unknowns (x, y, a)
x cosðaÞ;
y sinðaÞ;
1 þ sin2
ðaÞ þ cos2
ðaÞ:
and we should eliminate the variable a. Let us compute the Gröbner basis for x and
y eliminating a
GroebnerBasis[{x - cos[a],y - sin[a],sin[a]2
+ cos[a]2
- 1},
{x,y},{a,cos[a],sin[a]}]
{ - 1 + x2
+ y2
}
This elimination could easily be done with the Dixon resultant. Note that we really
have four variables x, y, cosðaÞ, and sinðaÞ and three equations. With three equa-
tions we can eliminate any two variables, so we choose the latter two.
Application 2 Now let us compute the common points of a cardioid and a circle.
The parametric equation of the cardioid is, see Fig. 1.7,
x ¼ 2 ð1 þ cosðtÞÞ cosðt),
y ¼ 2 ð1 þ cosðtÞÞ sinðt):
1.5 Applications 19](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-45-320.jpg)
![Fig. 1.8 The two
geometrical objects
Fig. 1.9 The common points
of the two geometrical objects
The reduced Gröbner basis for the x coordinate is given as
GroebnerBasis[{g1,g2},{x},{y}]
{ - 1 - 2x + x2
}
Similarly for the y coordinate
GroebnerBasis[{g1,g2},{y},{x}]
{ - 7 + 2y2
+ y4
}
or with the built-in function Solve
solp = {x,y}=:Solve[{g1 = = 0,g2 = = 0},{x,y}]==Simplify
1 -
ffiffiffi
2
p
, -
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
- 1 + 2
ffiffiffi
2
p
p
n o
, 1 -
ffiffiffi
2
p
,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
- 1 + 2
ffiffiffi
2
p
p
n o
,
n
1 +
ffiffiffi
2
p
, - i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 + 2
ffiffiffi
2
p
p
n o
, 1 +
ffiffiffi
2
p
,i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 + 2
ffiffiffi
2
p
p
n oo
There are only two real solutions! Let us visualize the common points, see Fig. 1.9.
1.5 Applications 21](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-47-320.jpg)
![Employing the necessary conditions of the minimum, differentiate the integral, we
get an algebraic polynomial system for the unknown coefficients
eq1 = D[r,c1]
4k + 2k2
+ 8c1 + 8kc1 + 8k2
c1 +
4
5
k2
c3
1 + 12c2 + 20kc2 + 16k2
c2
+
8
5
k2
c1c2 +
18
5
k2
c2
1c2 +
12
5
k2
c2
2 +
40
7
k2
c1c2
2
+
111
35
k2
c3
2
eq2 = D[r,c2]
6k + 4k2
+ 12c1 + 20kc1 + 16k2
c1 +
4
5
k2
c2
1 +
6
5
k2
c3
1 + 24c2 + 48kc2
+
168k2
c2
5
+
24
5
k2
c1c2 +
40
7
k2
c2
1c2 +
192
35
k2
c2
2
+
333
35
k2
c1c2
2 +
192
35
k2
c3
2
The Gröbner basis for c1,
GroebnerBasis[{eq1,eq2},{c1},{c2}]
{2222640000k + 25041744000k2
+ 106983636480k3
+ 216207482400k4
+ 217869466458k5
+ 105383544084k6
+ 21747027960k7
+ 982690800k8
+ 4445280000c1
+ 45638208000kc1 + 172774344960k2
c1 + 305473573440k3
c1
+ 294315313236k4
c1 + 170466205476k5
c1 + 78560129424k6
c1
+ 27948563160k7
c1 + 4880962800k8
c1 + 4834771200k2
c2
1
+ 14644375200k3
c2
1 + 3743455968k4
c2
1 - 11828581632k5
c2
1
- 15676868844k6
c2
1 - 6328977648k7
c2
1 - 849050160k8
c2
1
+ 398664000k2
c3
1 - 377496000k3
c3
1 + 1763997312k4
c3
1
+ 2443968240k5
c3
1 + 2017898358k6
c3
1 + 441062580k7
c3
1
+ 61164425k8
c3
10 + 52698240k4
c4
1 + 370528368k5
c4
1
+ 23358168k6
c4
1 + 197374240k7
c4
1 + 18557000k8
c4
1
- 4040400k4
c5
1 - 27938400k5
c5
1 - 12698784k6
c5
1 - 6985752k7
c5
1
+ 4109544k8
c5
1 - 1465920k6
c6
1 - 716616k7
c6
1 - 872028k8
c6
1
+ 55600k6
c7
1 + 18640k7
c7
1 + 58568k8
c7
1 - 3120k8
c8
1 + 100k8
c9
1}
and for c2, we get similar polynomial.
1.5 Applications 23](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-49-320.jpg)
![GroebnerBasis[{eq1,eq2},{c2},{c1}]
{ - 709927680k2
- 1419855360k3
- 904619968k4
- 194692288k5
+ 4100908k6
+ 1419855360c2 + 4259566080kc2 + 5168334976k2
c2
+ 3237393152k3
c2 + 1556328200k4
c2 + 647559304k5
c2
+ 151166960k6
c2 - 92198400c2
2 - 276595200kc2
2 + 220266368k2
c2
2
+ 901524736k3
c2
2 + 983883152k4
c2
2 + 487021584k5
c2
2
+ 107131248k6
c2
2 + 16464000c3
2 + 49392000kc3
2 + 253787072k2
c3
2
+ 425254144k3
c3
2 + 384371484k4
c3
2 + 179976412k5
c3
2 + 43090110k6
c3
2
- 6679680k2
c4
2 - 13359360k3
c4
2 - 604072k4
c4
2 + 6075608k5
c4
2
+ 5758480k6
c4
2 - 1117200k2
c5
2 - 2234400k3
c5
2 + 811440k4
c5
2
+ 1928640k5
c5
2 + 1604652k6
c5
2 + 312480k4
c6
2 + 312480k5
c6
2
+ 267470k6
c6
2 + 16800k4
c7
2 + 16800k5
c7
2 + 24612k6
c7
2 + 1560k6
c8
2
+ 75k6
c9
2}
From a practical point of view, it is more convenient to employ numerical Gröbner
basis function, as in Mathematica using k = 1,
sol = NSolve[{eq1,eq2}=:k ! 1,{c1, c2},Reals]==Flatten
{c1 ! - 0:6251338312334316,c2 ! 0:19045444692157196}
Then the temperature profile is
T = h=:sol
x - 0:625134 ( - x + x2
) + 0:190454 ( - x + x3
)
Figure 1.10 shows the dimensionless temperature profile for k = 1,
Fig. 1.10 The dimensionless
temperature profile in case of
k = 1
24 1 Solution of Algebraic Polynomial Systems](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-50-320.jpg)
![The general function for any k = j can be written as,
X[j ]: = h=:(NSolve[{eq1,eq2}=:k
! j,{c1,c2},Reals]==Flatten)
Let us test this function for k = 1
X[1]
x - 0:625134( - x + x2
) + 0:190454( - x + x3
)
We utilized the common capability of the Computer Algebra System (CAS) type
language providing symbolic computation as well as any size of digits in order to
reduce round-off error.
One can realize that this example is a nice illustration of the hybrid computation,
since our function is computed partly in numerical and partly in symbolic way.
1.5.3 Helmert Transformation
Let us consider a 2D Helmert transformation with parameters a and b,
X
Y
¼ s
cos X
ð Þ sin X
ð Þ
sin X
ð Þ cos X
ð Þ
x
y
¼
a b
b a
x
y
:
We have three control points in both systems, namely (Table 1.1).
Assuming that these values have errors in both systems (EIV model) let us
consider the adjustments as Dxi and DXi i ¼ 1; 2; 3.
In order compute these adjustments the following minimization problem should
be solved,
F ¼
X
3
i¼1
Dx2
i þ DX2
i
with the constraints,
Table 1.1 Numerical data
for the 2D Helmert
transformation problem
i xi yi Xi Yi
1 0.0 1.0 −2.1 1.1
2 1.0 0.0 1.0 2.0
3 1.0 1.0 −0.9 2.8
1.5 Applications 25](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-51-320.jpg)
![CHAPTER III.
THE SYNOPTIC GOSPELS—CONTINUED.
THE EPISTLES OF IGNATIUS—THE EPISTLE OF POLYCARP—
JUSTIN MARTYR—HEGESIPPUS—PAPIAS—THE
CLEMENTINES—THE EPISTLE TO DIOGNETUS.
Next our author examines quotations in the Epistles of Ignatius,
though he says they really appertain to a very much later period, for
they are all pronounced, by a large mass of critics, spurious
compositions. He suffered martyrdom, it is said, on the 20th
December, A.D. 115, when he was condemned to be cast to wild
beasts in the amphitheatre, not at Rome, but at Antioch, in
consequence of the fanatical excitement produced by the
earthquake which took place on the thirteenth of that month.[31] If
any of his fifteen letters, says our author, could be accepted as
genuine, the references to them might be important. Dr. Mosheim
says his whole epistles are extremely dubious. The shorter of the
two versions of Ignatius is, however, generally allowed to be
genuine. Tischendorf says its genuineness is now generally
admitted. In it we find, What would a man be profited if he should
gain the whole world and lose his own soul? which of course is a
quotation from Matt. xvi. 26.
The next document mentioned is the Epistle of Polycarp to the
Philippians, who, Irenæus says, was in his youth a disciple of the
Apostle John. He was Bishop of Smyrna, and ended his life by
martyrdom, A.D. 167. Irenæus knew Polycarp personally. It is said
that the epistle was written before A.D. 120. Our author ascribes it to
a later date, and says that there are potent reasons for considering it
spurious. As, however, Irenæus, Polycarp's disciple, believed it to be](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-56-320.jpg)
![earliest writers of the Church not long after the apostles. His
conversion took place about the year 132, and his martyrdom, A.D.
165.
In his second Apology, A.D. 139, and in his Dialogue with Tryphon
the Jew, are many quotations of passages found in the Gospels. He
quotes from all the four Evangelists, and our author's elaborate
attempt to prove the contrary is certainly not successful. His
objection, based on slight discrepancies in the words while the sense
is identical, is frivolous in the extreme. Supposing there were in
Justin's hands a primitive work which supplied the passages, and
that work was embodied in the canonical compilation, they can be
truthfully said to be quotations from the latter. The objection to his
quotations on the grounds that they are not verbatim, is neutralized
by the fact that neither are his quotations from the Old Testament
always exact.
It has been shown that if Justin did not quote from our Gospels,
there must have been in his hands, in the second century, a variety
of accounts of Christ's life, to which he, a leading Christian apologist,
attached the greatest importance; and yet, in the course of the few
following years, those accounts must have disappeared, and four
others, of which this eminent Christian apologist knew nothing, must
have taken their place. This would have been what Canon Westcott
justly calls a 'revolution,' for it would have, in a single generation,
entirely changed the records of the life of Christ publicly used by the
Christians.[32]
Justin quotes from a book entitled the Memoirs, which he says are
called Gospels, and our author tries to make out that the passage
quoted is an interpolation. It is not the only instance where the
wish, and not the proof, is father to the thought.
In Justin's work, the Apology, occur the words, And thou shalt call
his name Jesus, for he shall save his people from their sins; which
are found in the apocryphal Gospel of James, as said to the Virgin
Mary, while in Matthew's Gospel they are spoken to Joseph. It is](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-62-320.jpg)
![early date for granted, or he would not have occupied forty pages in
their examination.
The first quotation which, he says, agrees with a passage in our
Synoptics, occurs in the third Homily, p. 52: And he cried, saying,
Come unto me all ye that are weary; which agrees with Matt. xi. 28.
Because the quotation is not continued, but the following words are
an explanation of what Come unto me, c., means—that is, who
are seeking truth, and not finding it,—we are to deem it evident
that so short and fragmentary a phrase cannot prove anything. I
exclaim, Indeed! Not in a book that contains a hundred references to
the words of Jesus! Not, considering that they are especially the
words of Jesus, that no one else so said to the weary, Come unto
me! Most readers will surely think the contrary should be inferred!
Among the quotations are words resembling the text of Matthew
xxv. 26-30: Thou wicked and slothful servant: thou oughtest to
have put out my money with the exchangers, and at my coming I
should have exacted mine own.[33] If this were the only reference
to the Gospels as we have them, the quotation is sufficiently near to
make the inference certain that such writings, in some shape, must
have been in existence when the Clementine Homilies were written.
This our author acknowledges, but he says (vol. ii. p. 17): If the
variations were the exception among a mass of quotations perfectly
agreeing with the parallels in our Gospels, it might be exaggeration
to base upon such divergences a conclusion that they were derived
from a different source. The variations being the rule, instead of the
exception, these, however slight, become evidence of the use of a
different Gospel from ours.[34]
I remark, supposing this be so, that the author of these Homilies
had, in the year 160, other Gospel manuscripts before him, it is not
pretended that our Gospels contain all that was known of the
sayings of Jesus, and all the events of His public ministry. We are
told in the Fourth Gospel: There are also many other things which
Jesus did, the which, if they should be written every one, I suppose](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-64-320.jpg)
![that even the world itself could not contain the books that should be
written.[35] If the author of the Fourth Gospel did not include many
things which he knew had been previously written about, why
should we be surprised to find the authors of the Synoptic Gospels
record only portions?
We know that Paul wrote an epistle to the Church at Laodicea, which
is not preserved to us. We hold that Paul was as much an inspired
writer as any of the apostles, and instead of making all sorts of
difficulties about the books we have, we ought to be grateful that
they are extant. We read in Paul's Epistle to the Colossians, iv. 16:
And when this epistle is read among you, cause that it be read also
in the Church of the Laodiceans; and that ye likewise read the
epistle from Laodicea.
I wonder whether our author has an objection to the genuineness of
the Epistle to the Colossians, because Epictetus, who was born at
Hierapolis about A.D. 50, which was within a few miles of Colosse and
Laodicea, and who would be likely to know, at that time, what was
there going on, does not refer to Paul and the Churches there?
But it is useless to disprove the assertion that there are no
quotations from the Gospels, for we are met at every turn with the
objection that those specified are probably quotations from the
numerous lost Gospels known to have been in circulation. He says:
The great mass of intelligent critics are agreed that our Synoptics
have assumed their present form only after repeated modifications
by various editors of earlier evangelical works. The primitive Gospels
have entirely disappeared, supplanted by the later and more
amplified versions (p. 459). The first two Synoptics bear no author's
name, because they are not the work of any one man, but the
collected materials of many. The third only pretends to be a
compilation for private use, and the fourth bears no simple
signature, because it is neither the work of an apostle nor of an eye-
witness of the events it records (p. 401). I remark, if Luke's Gospel
does only pretend to be for private use, does that affect its value? If](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-65-320.jpg)
![Although we do not know how these were embodied in our New
Testament Scriptures, it is probable that they were in some way
included, or the copies of the present Gospels may not all have
uniformly borne the same titles as we know them by. In our day it is
not usual for an author's name to appear in the body of his work,
and often a title-page gives more than one title.[36] How few
persons can give the exact title of the book known as Butler's
Analogy. The value of a book does not depend essentially upon the
person who wrote it. We do not know who wrote the Book of Job,
many of the Psalms, the Epistle to the Hebrews, and other portions
of the Bible, but it would be unwise to reject their teaching on that
account.
Our author says: No reason whatever has been shown for accepting
the testimony of these Gospels as sufficient to establish the reality of
miracles (p. 249). I remark, the question is, Do they show such
insufficient testimony as to warrant the conclusion that the general
evidence based on a great variety of proofs is not to be accepted?
The Epistle to Diognetus is a short composition, which has been
ascribed to Justin Martyr, but its authorship is uncertain, and the
date of its composition. It is not quoted or mentioned by any ancient
writer. The two concluding chapters are supposed to have been
written by a different hand. To the first quarter of the second half to
the end of that century the date is variously assigned. It is written in
pure Greek, and is elegant in style. Bunsen, in his valuable book,
Hippolytus and his Age, asserts that the epistle is certainly the
work of a contemporary of Justin the Martyr; that he believes he
has proved that the first part is a portion of the lost early Letter of
Marcion, of which Tertullian speaks; and that the very beautiful and
justly admired second fragment, which in our editions of Justin's
works is given at the end of that Patristic gem, the Epistle to
Diognetus,[37] does not belong to that letter, but is the conclusion
of the great work, in ten books, by Hippolytus, The Refutation of all
Heresies. Our author, in the eighteen pages devoted to the Epistle
to Diognetus, says nothing of this, although it is both important and](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-67-320.jpg)
![suit the Jewish Nazarene sect, who, we know, had a Hebrew text of
their own, which they did not hesitate to alter and adapt to their
own views. Basilides, says our author, expressly states that he
received his knowledge of the truth from Glaucis, the interpreter of
Peter, whose disciple he claimed to be. Basilides also claimed to
have received from a certain Matthias the report of private
discourses which he had heard from the Saviour for his special
instruction. Canon Westcott writes: Since Basilides lived on the
verge of the apostolic times, it is not surprising that he made use of
other sources of Christian doctrine besides the canonical books. The
belief in Divine inspiration was still fresh and real.[38] Our author
says: It is apparent, however, that Basilides, in basing his doctrine
on these apocryphal books as inspired, and upon tradition, and in
having a special Gospel called after his own name, ignores the
canonical Gospels, offers no evidence for their existence, but proves
that he did not recognise any such works as of authority. I remark,
the question is not their authority, but, Did they exist? Basilides
wrote a book, called it a Gospel, or commentary of the Gospel, and
made as much use as suited his heretical purpose of the canonical
records, of tradition, and of other books. This seems to be what we
can arrive at. Hippolytus, writing of the Basilideans and describing
their doctrines, uses the singular pronoun he—he says, in a
passage of which our author gives an unintelligible translation. This
pronoun is an inconvenient witness. Our author wants it to be
they, in order that the disciples of Basilides living at a later period,
when the Gospels were generally recognised, may be meant, and
not Basilides, who lived A.D. 125. Hippolytus has a sentence of
Basilides, which our author translates as follows:—Jesus, however,
was generated according to these, as we have already said. But
when the generation which has already been declared had taken
place, all things regarding the Saviour, according to them, occurred
in a similar way as they have been written in the Gospel. This
means that the things referring to the Incarnation were as written in
the Gospel, not as preached, but as written; and if Basilides, as the
founder of the sect, is referred to, the statement testifies to the](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-71-320.jpg)
![existence of the Gospels in the year 125, and the doctrine of the
Incarnation being in them. But our author says the statement is not
made in connection with Basilides, but his followers; that it is made
about A.D. 225, by Hippolytus, and affords no proof that either
Basilides or his followers used the Gospels or admitted their
authority. The exclusive use, by any one, of the Gospel according to
the Hebrews, for instance, would be perfectly consistent with the
statement (p. 48). No one who considers what is known of that
Gospel, or who thinks of the use made of it in the first half of the
second century by perfectly orthodox Fathers, before we hear
anything of our Gospels, can doubt this (p. 48). I remark, that
those who adopt Tischendorf's view, that Matthew was written in
Greek, and a corrupted version in Hebrew, used in certain countries,
will not have to resort to any such explanation as our author
suggests. His examination in detail of the several quotations is
important, because it exhibits his want of appreciation of the
evidence they afford. The first passage Tischendorf points out is
found in the Stromata of Clement of Alexandria, and it is certainly
from our Gospel of Matthew,[39] however that work may have been
compiled (for it is not necessary to insist that no other records than
Matthew's own are included in the book which, we contend, was at
very early date read in the Churches, and is what we now have).
They say the Lord answered, All men cannot receive this saying. For
there are eunuchs who are indeed from birth, but others from
necessity.[40] Our author says this passage in its affinity to, and
material variation from, our First Gospel, might be quoted as
evidence for the use of the Gospel according to the Hebrews, but it
is simply preposterous to point to it as evidence for the use of
Matthew. Apologists ... seem altogether to ignore the history of the
creation of written Gospels, and to forget the very existence of the
πολλοἱ of Luke. We value his acknowledgment, and find no
difficulty, notwithstanding the silence of some apologists, in
reconciling our belief in the four Gospels with the facts or
probabilities of what can be ascertained as to their creation. We
allow that the word Luke uses (πολλοἱ) refers to many, which is](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-72-320.jpg)
![consistent with the idea that many committed to writing what they
knew, and that their records were embodied in the Synoptic Gospels.
The next passage referred to by Tischendorf is one quoted by
Epiphanius: And therefore he said, Cast not ye pearls before swine,
neither give that which is holy unto dogs.[41] It is introduced in the
section of the work of Epiphanius directed against the Basilideans.
As in dealing with all these heresies there is continual interchange of
reference to the head and later followers, there is no certainty who
is referred to in these quotations, and in this instance nothing to
indicate that the passage is ascribed to Basilides himself. His name is
mentioned in the first line of the first chapter, but not again until the
fifth chapter (p. 50).
I remark, it was the founder of the sect and not the followers who
wrote the book, and those who opposed the heresy would, although
they alluded to the sect, have regard to the founder when they
referred to the doctrines held, and quoted the written opinions which
distinguished the party on gospel matters. To make the matter as
plain as I can, I will suppose a case as an illustration of the point.
Supposing that in Pliny's letter to Trajan there were found these
words referring to the Christians: They say, the rule which should
be observed in regard to an enemy is, Love your enemies, bless
them that curse you, do good to them that hate you, and pray for
them which persecute you—would it be right to assert that the
quotation is no proof that Christ so taught, but His disciples, long
afterwards? This is something like what our author's objection,
referring to the pronouns he and they in Hippolytus, amounts to.
They does not mean he when thus used; and he, when actually
used in the first line of the first chapter, and afterwards means,
they; that is, He (Basilides) says, means They (his followers at a
later date) say.
The plural pronoun is used, indicating the sect, Basilides and his
followers. Therefore our author says there is uncertainty as to who
he is when used in the same sentence. He says Hippolytus is giving](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-73-320.jpg)
![and I will give thee a crown of life. They lived in the consciousness
of these truths, and died (Bishop Pothinus, for instance) a martyr's
death rather than deny them.
There is this remark to be made in reference to the alleged uncritical
age of the Fathers. How is it that Marcion is seen to be so critical?
He is surely after the modern model. He who wrote the Antithesis,
and, as our author says, anticipated in some of his opinions those
held by many in our own time; he who wrote,—If the God of the
Old Testament be good, prescient of the future, and able to avert
evil, why did he allow man, made in his own image, to be deceived
by the devil, and to fall from obedience of the law into sin and
death?[42] How came the devil, the origin of lying and deceit, to be
made at all?[43] surely he is an instance of a man in that age
possessing the critical faculty. He has the boldness to question, and
say,—Yea, hath God said? Anticipating the results of modern
criticism, says our author, Marcion denies the applicability to Jesus
of the so-called Messianic prophecies (p. 106).
If the research which is going on as to the Gospel of Marcion be
conducted in a proper manner, and from a proper motive, not from
antipathy to parsons and ecclesiastical assumptions, which was the
incentive of Strauss in attacking Christianity, good will come of it. As
Justin Martyr did not, as far as we know, suppose the book to be a
corrupted version of the Gospel according to Luke, Tertullian may
have been mistaken, and it may have been an independent work,
one of the many Luke refers to, the existence of which does not
necessarily invalidate the canonical ones. We may naturally suppose
that events of such marvellous speciality and importance as those
which had come to pass in those days among the Jews, would be
more or less described in letters and other writings by many persons
who were eye-witnesses. Such writings would be collected and read
when the first Christians assembled. The difference between the four
canonical Gospels and other manuscripts would consist in their being
compiled by persons competent to the task, who, like Ezra, were
instruments Divinely influenced to compile and set forth in order a](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-78-320.jpg)
![He sits on no precarious throne,
Nor borrows leave to be.
They who believe in the inspiration by the Holy Ghost of the
prophets of the Old Testament see no difficulty in regard to the
inspiration of the writers of the New. If Isaiah and Jeremiah and
Daniel had supernatural communications made to them, in order
that the Eternal Creator might be manifested, why not Paul and John
and Matthew? It is the foregone conclusion, on the part of critics,
that the miraculous is impossible, which embarrasses their
researches. One of John Stuart Mill's last sentences is: It remains a
possibility that Christ actually was what He supposed Himself to be.
If this had occurred to the great reasoner at the outset of his career
instead of the close, how much might the world have been
advantaged!
Tatian is a witness whose evidence our author next tries to set aside.
He was an Assyrian by birth, a disciple of Justin Martyr at Rome, and
afterwards, having joined the sect of the Eucratites, a conspicuous
exponent of their austere and ascetic doctrines. The only one of his
writings extant is his Oration to the Greeks, written after Justin's
death, as it refers to that event, and it is generally dated A.D. 170-
175. One point contested is Canon Westcott's affirmation that it
contains a clear reference to a parable recorded by Matthew:[44]
The kingdom of heaven is like unto treasure hidden in a field, which
a man found and hid, and for his joy he goeth and selleth all that he
hath and buyeth that field. And the supposed reference by Tatian is,
For by means of a certain hidden treasure he has taken to himself
all that we possess, for which, while we are digging, we are indeed
covered with dust, but we succeed in making it our fixed
possession.[45]
There is certainly not much similarity between the two passages,
although Tatian may be well supposed to have had the parable in his
mind when he wrote. The more important question is, Did Tatian
write A Harmony of Four Gospels, which recognises our four](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-80-320.jpg)
![Evangelists? Was his Diatessaron such a book, or was it the Gospel
according to the Hebrews? If the latter, what is the Gospel according
to the Hebrews? I say it is probable it is the corrupted Hebrew
translation of the Greek Gospel of Matthew, and this conjecture has
more in its favour than our author's hypothesis.
Dionysius of Corinth, Eusebius tells us, wrote seven epistles to
various Churches, and a letter to Chrysophora, a most faithful
sister. Only a few short fragments exist, which are all from the
epistle to Soter, Bishop of Rome, whose date in that pastorate is A.D.
168-176. In these fragments we find the following words:—For the
brethren having requested me to write epistles, I write them. And
the apostles of the devil have filled these with tares, both taking
away parts and adding others, for whom the woe is destined. It is
not surprising, then, if some have recklessly ventured to adulterate
the Scriptures of the Lord, when they have corrupted these, which
are not of such importance.[46] After quoting this passage, our
author reiterates his statement that We have seen that there has
not been a trace of any New Testament Canon in the writings of the
Fathers before and during this age. Does he suppose his readers
will have seen as he sees, or rather refuse to see what is plain
enough? He has his own opinion, but he need not assume that he
has convinced his readers that he has proved what he alleges. He
talks of Westcott's boldness, and of his imagination running away
with him, and that it is simply preposterous to suppose that this
passage refers to the New Testament. I leave Canon Westcott to
defend his own words, but I say it is not preposterous to infer that
when Dionysius speaks of the Scriptures of the Lord he means
Gospel writings, which are included in our New Testament. If it be
assumed that the defence of the authority of the New Testament
writings and of evangelical views is necessarily based on the
synodical authority of the early Church, there may be some weight
in his objections; but Christianity has a position independent of
ecclesiastical pretensions to infallibility, and the latter may be
overthrown without the great institution established by Divine mercy
for the recovery of humanity from sin and its consequences being in](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-81-320.jpg)
![the slightest degree damaged. Dr. Donaldson is quoted, who
remarks: It is not easy to settle what this term, 'Scriptures of the
Lord,' is; but my own opinion is that it most probably refers to the
Gospels, as containing the sayings and doings of the Lord. It is not
likely, as Lardner supposes, that such a term would be applied to the
whole of the New Testament.[47] The word Scripture, in Greek,
ΓραφἡΓραφἡ (Graphé), in Latin, Scriptura, has, no doubt, a meaning
which denotes an inspired writing. It is used fifty-one times in the
New Testament in the same sense, for Christ and the authors of the
New Testament regarded the Old Testament as distinguished from all
other writings, as the writing—the writing of God. By speaking of
their own books as Graphai, the apostles place them on a level with
the Old Testament, and thus assert their Divine character.[48]
Dr. Davidson speaks of the New Testament writings being ranked as
Holy Scripture by Dionysius of Corinth, A.D. 170.
Our author asserts (p. 167) that many works were regarded as
inspired by the Fathers besides those in our Canon, and mentions
especially the Gospel of Peter having been read at Rhossus. He says:
The fact that Serapion, in the third century, allowed the Gospel of
Peter to be used in the Church of Rhossus shows the consideration
in which it was held, and the incompleteness of the canonical
position of the New Testament. Now, he ought to have quoted
Serapion's own explanation, which we have preserved by Eusebius.
He says (in his treatise written to confute what was false in the
Gospel of Peter): We receive Peter and the other apostles even as
Christ; but the writings falsely called by their names, we, as
competent critics, renounce, knowing that we received not such
things. For when I was with you I supposed that all were agreed
with the true faith; and, without reading the Gospel called Peter's,
which they brought forward, I said, If this is the only thing that
seems to cause you dissension, let it be read. Serapion says he
borrowed the book and read it, and found many things agreeable to
Christ's doctrine, but some discrepant additions.](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-82-320.jpg)
![Thus the reading of the Gospel of Peter at Rhossus cannot be
instanced as a proof that other Gospels besides the canonical ones
were used as inspired books, nor can any other be mentioned as
having been thus regarded, the Gospel according to the Hebrews not
being apocryphal, but a part of the New Testament, whether we
take it to be, as our author supposes, the basis of Matthew's Gospel,
or, as we say, a corrupted version of that apostle's Greek work. To
argue that because one spurious Gospel was temporarily received
among a few persons, therefore there was no real canon of
Scripture, and we cannot be sure that any Gospel is genuine, shows
about as much common sense and logical acumen as would be
displayed by a critic eighteen centuries hence, who, discovering in
one of our newspapers an account of the conviction of a gang of
coiners, should argue that because their base half-crowns had got
into circulation, and had passed current with some persons who
might have been expected to detect the fraud, therefore there was
no such thing as a legal currency of intrinsic value among us; or if
there were, still we did not know or care to inquire into the
genuineness of the coin which we accepted and passed.[49]
Our author says (p. 16): 'The Pastor of Hermas,' which was read in
the churches, and nearly secured a permanent place in the Canon,
was quoted as inspired by Irenæus.[50]
The word Irenæus uses is Graphé, which is sometimes translated,
when found in his works, Scripture, and at other times writings, as
may best suit the argument of a critic like Dr. Davidson, who does so
adapt the translation to suit his purpose.
Whatever erroneous notions might prevail as to apocryphal writings,
the discrimination of Serapion, in regard to the Gospel of Peter,
shows that such a work as the Pastor of Hermas, in which, as
Mosheim says, the angels are made to talk more insipidly than our
scavengers and porters, would not be put on a level with the books
whose internal evidence, as well as historical pretensions, placed
them in a much superior position. The contrast is too great for such](https://image.slidesharecdn.com/3417805-250617135514-d478abe2/85/Mathematical-Geosciences-Hybrid-Symbolicnumeric-Methods-Joseph-L-Awange-83-320.jpg)
Mathematical Geosciences Hybrid Symbolicnumeric Methods Joseph L Awange
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- 5. Joseph L. Awange • Béla Paláncz Robert H. Lewis • Lajos Völgyesi Mathematical Geosciences Hybrid Symbolic-Numeric Methods 123
- 6. Joseph L. Awange Spatial Sciences Curtin University Perth, WA Australia Béla Paláncz Budapest University of Technology and Economics Budapest Hungary Robert H. Lewis Fordham University New York, NY USA Lajos Völgyesi Budapest University of Technology and Economics Budapest Hungary ISBN 978-3-319-67370-7 ISBN 978-3-319-67371-4 (eBook) https://doi.org/10.1007/978-3-319-67371-4 Library of Congress Control Number: 2017953801 © Springer International Publishing AG 2018
- 7. Foreword Hybrid symbolic-numeric computation (HSNC, for short) is a large and growing area at the boundary of mathematics and computer science, devoted to the study and implementation of methods that mix symbolic with numeric computation. As the title suggests, this is a book about some of the methods and algorithms that benefit from a mix of symbolic and numeric computation. Three major areas of computation are covered herein. The first part discusses methods for computing all solutions to a system of polynomials. Purely symbolic methods, e.g., via Gröbner bases tend to suffer from algorithmic inefficiencies, and purely numeric methods such as Newton iterations have trouble finding all solutions to such systems. One class of hybrid methods blends numerics into the purely algebraic approach, e.g., computing numeric Gröbner bases or Dixon resultants (the latter being extremely efficient, e.g., for elimination of variables). Another mixes symbolic methods into more numerical approaches, e.g., finding initializations for numeric homotopy tracking to obtain all solutions. The second part goes into the realm of “soft” optimization methods, including genetic methods, simulated annealing, and particle swarm optimization, among others. These are all popular and heavily used, especially in the context of global optimization. While often considered as “numeric” methods, they benefit from symbolic computation in several ways. One is that implementation is typically straightforward when one has access to a language that supports symbolic com- putation. Updates of state, e.g., to handle mutations and gene crossover, are easily coded. (Indeed, this sort of thing can be so deceptively simple. baked into the language so to speak, that one hardly realizes symbolic computation is happening.) Among many applications in this part there is, again, that of solving systems of equations. Also covered is mixed-integer programming (wherein some variables are discrete-valued and others continuous). This is a natural area for HSNC since it combines aspects of exact and numeric methods in the handling of both discrete and continuous variables. The third part delves into data modeling. This begins with use of radial basis functions and proceeds to machine learning, e.g., via support vector machine (SVM) methods. Symbolic regression, a methodology that combines numerics with
- 8. evolutionary programming, is also introduced for the purpose of modeling data. Another area seeing recent interest is that of robust optimization and regression, wherein one seeks results that remain relatively stable with respect to perturbations in input or random parameters used in the optimization. Several hybrid methods are presented to address problems in this realm. Stochastic modeling is also discussed. This is yet another area in which hybrid methods are quite useful. Symbolic computing languages have seen a recent trend toward ever more high level support for various mathematical abstractions. This appears for example in exact symbolic programming involving probability, geometry, tensors, engineering simulation, and many other areas. Under the hood is a considerable amount of HSNC (I write this as one who has been immersed at the R&D end of hybrid computation for two decades.) Naturally, such support makes it all the easier for one to extend hybrid methods; just consider how much less must be built from scratch to support, say, stochastic equation solving, when the language already supports symbolic probability and statistics computations. This book presents to the reader some of the major areas and methods that are being changed, by the authors and others, in furthering this interplay of symbolic and numeric computation. The term hybrid symbolic-numeric computation has been with us for over two decades now. I anticipate the day when it falls into disuse, not because the technology goes out of style, but rather that it is just an integral part of the plumbing of mathematical computation. Urbana—Champaign IL, USA July 2017 Daniel Lichtblau Ph.D., Mathematics UIUC 1991 Algebra, Applied Mathematics Wolfram Research, Champaign
- 9. Preface It will surprise no one to hear that digital computers have been used for numerical computations ever since their invention during World War II. Indeed, until around 1990, it was not widely understood that computers could do anything else. For many years, when students of mathematics, engineering, and the sciences used a computer, they wrote a program (typically in Fortran) to implement mathematical algorithms for solving equations in one variable, or systems of linear equations, or differential equations. The input was in so-called “float” numbers with 8–12 significant figures of accuracy. The output was the same type of data, and the program worked entirely with the same type of data. This is numerical computing. By roughly 1990, computer algebra software had become available. Now it was possible to enter data like x2 þ 3x þ 2 and receive output like ðx þ 2Þðx þ 1Þ. The computer is doing algebra! More precisely, it is doing symbolic computing. The program that accomplishes such computing almost certainly uses no float numbers at all. What is still not widely understood is that often it is productive to have algo- rithms that do both kinds of computation. We call these hybrid symbolic-numeric methods. Actually, such methods have been considered by some mathematicians and computer scientists since at least 1995 (ISSAC 1995 conference). In this book, the authors provide a much-needed introduction and reference for applied mathe- maticians, geoscientists, and other users of sophisticated mathematical software. No mathematics beyond the undergraduate level is needed to read this book, nor does the reader need any pure mathematics background beyond a first course in linear algebra. All methods discussed here are illustrated with copious examples. A brief list of topics covered: • Systems of polynomial equations with resultants and Gröbner bases • Simulated annealing • Genetic algorithms • Particle swarm optimization • Integer programming • Approximation with radial basis functions
- 10. • Support vector machines • Symbolic regression • Quantile regression • Robust regression • Stochastic modeling • Parallel computations Most of the methods discussed in the book will probably be implemented by the reader on a computer algebra system (CAS). The two most fully developed and widely used CAS are Mathematica and Maple. Some of the polynomial compu- tations here are better done on the specialized system Fermat. Other systems worthy of mention are Singular and SageMath. The second author is a regular user of Mathematica, who carried out the com- putations, therefore frequent mention is made of Mathematica commands. However, this book is not a reference manual for any system, and we have made an effort to keep the specialized commands to a minimum, and to use commands whose syntax makes them as self-explanatory as possible. More complete Mathematica programs to implement some of the examples are available online. Similarly, a program written in Fermat for the resultant method called Dixon-EDF is available online. The authors: July 2017 Joseph L. Awange Perth, Australia Béla Paláncz Budapest, Hungary Robert H. Lewis New York, USA Lajos Völgyesi Budapest, Hungary
- 11. Contents Part I Solution of Nonlinear Systems 1 Solution of Algebraic Polynomial Systems. . . . . . . . . . . . . . . . . . . . . 3 1.1 Zeros of Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Resultant Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Sylvester Resultant . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Dixon Resultant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Gröbner Basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Greatest Common Divisor of Polynomials. . . . . . . . . . 8 1.3.2 Reduced Gröbner Basis . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Polynomials with Inexact Coefficients . . . . . . . . . . . . . 12 1.4 Using Dixon-EDF for Symbolic Solution of Polynomial Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Explanation of Dixon-EDF . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Distance from a Point to a Standard Ellipsoid . . . . . . . 16 1.4.3 Distance from a Point to Any 3D Conic . . . . . . . . . . . 16 1.4.4 Pose Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4.5 How to Run Dixon-EDF . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.1 Common Points of Geometrical Objects . . . . . . . . . . . 18 1.5.2 Nonlinear Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.3 Helmert Transformation. . . . . . . . . . . . . . . . . . . . . . . . 25 1.6 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.1 Solving a System with Different Techniques . . . . . . . . 28 1.6.2 Planar Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.6.3 3D Resection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6.4 Pose Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
- 12. 2 Homotopy Solution of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . 41 2.1 The Concept of Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Solving Nonlinear Equation via Homotopy . . . . . . . . . . . . . . . . 43 2.3 Tracing Homotopy Path as Initial Value Problem. . . . . . . . . . . . 45 2.4 Types of Linear Homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.1 General Linear Homotopy . . . . . . . . . . . . . . . . . . . . . . 47 2.4.2 Fixed-Point Homotopy . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.3 Newton Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.4 Affine Homotopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4.5 Mixed Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 Regularization of the Homotopy Function . . . . . . . . . . . . . . . . . 49 2.6 Start System in Case of Algebraic Polynomial Systems . . . . . . . 49 2.7 Homotopy Methods in Mathematica. . . . . . . . . . . . . . . . . . . . . . 51 2.8 Parallel Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.9 General Nonlinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.10 Nonlinear Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.10.1 Quadratic Bezier Homotopy Function . . . . . . . . . . . . . 58 2.10.2 Implementation in Mathematica. . . . . . . . . . . . . . . . . . 61 2.10.3 Comparing Linear and Quadratic Homotopy . . . . . . . . 62 2.11 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.11.1 Nonlinear Heat Conduction . . . . . . . . . . . . . . . . . . . . . 65 2.11.2 Local Coordinates via GNSS. . . . . . . . . . . . . . . . . . . . 68 2.12 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.12.1 GNSS Positioning N-Point Problem . . . . . . . . . . . . . . 71 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3 Overdetermined and Underdetermined Systems . . . . . . . . . . . . . . . . 77 3.1 Concept of the Over and Underdetermined Systems. . . . . . . . . . 77 3.1.1 Overdetermined Systems . . . . . . . . . . . . . . . . . . . . . . . 77 3.1.2 Underdetermined Systems . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Gauss–Jacobi Combinatorial Solution. . . . . . . . . . . . . . . . . . . . . 80 3.3 Gauss–Jacobi Solution in Case of Nonlinear Systems. . . . . . . . . 84 3.4 Transforming Overdetermined System into a Determined System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.5 Extended Newton–Raphson Method . . . . . . . . . . . . . . . . . . . . . . 90 3.6 Solution of Underdetermined Systems . . . . . . . . . . . . . . . . . . . . 92 3.6.1 Direct Minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.6.2 Method of Lagrange Multipliers . . . . . . . . . . . . . . . . . 93 3.6.3 Method of Penalty Function . . . . . . . . . . . . . . . . . . . . 95 3.6.4 Extended Newton–Raphson. . . . . . . . . . . . . . . . . . . . . 95 3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.7.1 Geodetic Application—The Minimum Distance Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
- 13. 3.7.2 Global Navigation Satellite System (GNSS) Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.7.3 Geometric Application. . . . . . . . . . . . . . . . . . . . . . . . . 101 3.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.8.1 Solution of Overdetermined System . . . . . . . . . . . . . . 105 3.8.2 Solution of Underdetermined System . . . . . . . . . . . . . 107 Part II Optimization of Systems 4 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1 Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 Realization of the Metropolis Algorithm. . . . . . . . . . . . . . . . . . . 114 4.2.1 Representation of a State. . . . . . . . . . . . . . . . . . . . . . . 114 4.2.2 The Free Energy of a State . . . . . . . . . . . . . . . . . . . . . 115 4.2.3 Perturbation of a State. . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.4 Accepting a New State . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.5 Implementation of the Algorithm. . . . . . . . . . . . . . . . . 116 4.3 Algorithm of the Simulated Annealing . . . . . . . . . . . . . . . . . . . . 118 4.4 Implementation of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 118 4.5 Application to Computing Minimum of a Real Function . . . . . . 120 4.6 Generalization of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.7.1 A Packing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.7.2 The Traveling Salesman Problem . . . . . . . . . . . . . . . . 127 4.8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1 The Genetic Evolution Concept . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2 Mutation of the Best Individual . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3 Solving a Puzzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Application to a Real Function. . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.5 Employing Sexual Reproduction. . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5.1 Selection of Parents. . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5.2 Sexual Reproduction: Crossover and Mutation . . . . . . 152 5.6 The Basic Genetic Algorithm (BGA) . . . . . . . . . . . . . . . . . . . . . 154 5.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.7.1 Nonlinear Parameter Estimation. . . . . . . . . . . . . . . . . . 157 5.7.2 Packing Spheres with Different Sizes . . . . . . . . . . . . . 160 5.7.3 Finding All the Real Solutions of a Non-algebraic System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.8.1 Foxhole Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
- 14. 6 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.1 The Concept of Social Behavior of Groups of Animals . . . . . . . 167 6.2 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.3 The Pseudo Code of the Algorithm . . . . . . . . . . . . . . . . . . . . . . 170 6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.4.1 1D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.4.2 2D Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.4.3 Solution of Nonlinear Non-algebraic System. . . . . . . . 178 6.5 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7 Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.1 Integer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.2 Discrete Value Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.3 Simple Logical Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.4 Some Typical Problems of Binary Programming . . . . . . . . . . . . 191 7.4.1 Knapsack Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.4.2 Nonlinear Knapsack Problem . . . . . . . . . . . . . . . . . . . 192 7.4.3 Set-Covering Problem . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.5 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.5.1 Binary Countdown Method . . . . . . . . . . . . . . . . . . . . . 194 7.5.2 Branch and Bound Method . . . . . . . . . . . . . . . . . . . . . 196 7.6 Mixed–Integer Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.7.1 Integer Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 200 7.7.2 Optimal Number of Oil Wells . . . . . . . . . . . . . . . . . . . 202 7.8 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.8.1 Study of Mixed Integer Programming . . . . . . . . . . . . . 203 7.8.2 Mixed Integer Least Square. . . . . . . . . . . . . . . . . . . . . 205 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.1 Concept of Multiobjective Problem . . . . . . . . . . . . . . . . . . . . . . 207 8.1.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.1.2 Interpretation of the Solution. . . . . . . . . . . . . . . . . . . . 208 8.2 Pareto Optimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.2.1 Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.2.2 Pareto-Front and Pareto-Set . . . . . . . . . . . . . . . . . . . . . 211 8.3 Computation of Pareto Optimum . . . . . . . . . . . . . . . . . . . . . . . . 212 8.3.1 Pareto Filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.3.2 Reducing the Problem to the Case of a Single Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.3.3 Weighted Objective Functions. . . . . . . . . . . . . . . . . . . 219 8.3.4 Ideal Point in the Function Space . . . . . . . . . . . . . . . . 220
- 15. 8.3.5 Pareto Balanced Optimum. . . . . . . . . . . . . . . . . . . . . . 220 8.3.6 Non-convex Pareto-Front. . . . . . . . . . . . . . . . . . . . . . . 222 8.4 Employing Genetic Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . 223 8.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.5.1 Nonlinear Gauss-Helmert Model . . . . . . . . . . . . . . . . . 229 8.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Part III Approximation of Functions and Data 9 Approximation with Radial Bases Functions. . . . . . . . . . . . . . . . . . . 245 9.1 Basic Idea of RBF Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.2 Positive Definite RBF Function . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.3 Compactly Supported Functions . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.4 Some Positive Definite RBF Function . . . . . . . . . . . . . . . . . . . . 253 9.4.1 Laguerre-Gauss Function. . . . . . . . . . . . . . . . . . . . . . . 253 9.4.2 Generalized Multi-quadratic RBF . . . . . . . . . . . . . . . . 254 9.4.3 Wendland Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.4.4 Buchmann-Type RBF . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.5 Generic Derivatives of RBF Functions . . . . . . . . . . . . . . . . . . . . 257 9.6 Least Squares Approximation with RBF. . . . . . . . . . . . . . . . . . . 260 9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.7.1 Image Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.7.2 RBF Collocation Solution of Partial Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 9.8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 9.8.1 Nonlinear Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . 276 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10 Support Vector Machines (SVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.1 Concept of Machine Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.2 Optimal Hyperplane Classifier . . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.2.1 Linear Separability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 10.2.2 Computation of the Optimal Parameters . . . . . . . . . . . 283 10.2.3 Dual Optimization Problem . . . . . . . . . . . . . . . . . . . . . 284 10.3 Nonlinear Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 10.4 Feature Spaces and Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 10.5 Application of the Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 10.5.1 Computation Step by Step. . . . . . . . . . . . . . . . . . . . . . 289 10.5.2 Implementation of the Algorithm. . . . . . . . . . . . . . . . . 292 10.6 Two Nonlinear Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.6.1 Learning a Chess Board . . . . . . . . . . . . . . . . . . . . . . . 294 10.6.2 Two Intertwined Spirals . . . . . . . . . . . . . . . . . . . . . . . 297 10.7 Concept of SVM Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
- 16. 10.7.1 e-Insensitive Loss Function . . . . . . . . . . . . . . . . . . . . . 299 10.7.2 Concept of the Support Vector Machine Regression (SVMR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.7.3 The Algorithm of the SVMR. . . . . . . . . . . . . . . . . . . . 302 10.8 Employing Different Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 10.8.1 Gaussian Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 10.8.2 Polynomial Kernel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 10.8.3 Wavelet Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10.8.4 Universal Fourier Kernel . . . . . . . . . . . . . . . . . . . . . . . 311 10.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10.9.1 Image Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10.9.2 Maximum Flooding Level . . . . . . . . . . . . . . . . . . . . . . 315 10.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 10.10.1 Noise Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 11 Symbolic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 11.1 Concept of Symbolic Regression . . . . . . . . . . . . . . . . . . . . . . . . 321 11.2 Problem of Kepler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 11.2.1 Polynomial Regression . . . . . . . . . . . . . . . . . . . . . . . . 326 11.2.2 Neural Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 11.2.3 Support Vector Machine Regression . . . . . . . . . . . . . . 328 11.2.4 RBF Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 11.2.5 Random Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.2.6 Symbolic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 11.3.1 Correcting Gravimetric Geoid Using GPS Ellipsoidal Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 11.3.2 Geometric Transformation . . . . . . . . . . . . . . . . . . . . . . 342 11.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 11.4.1 Bremerton Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 12 Quantile Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.1 Problems with the Ordinary Least Squares . . . . . . . . . . . . . . . . . 359 12.1.1 Correlation Height and Age. . . . . . . . . . . . . . . . . . . . . 359 12.1.2 Engel’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 12.2 Concept of Quantile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 12.2.1 Quantile as a Generalization of Median. . . . . . . . . . . . 362 12.2.2 Quantile for Probability Distributions . . . . . . . . . . . . . 366 12.3 Linear Quantile Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 12.3.1 Ordinary Least Square (OLS) . . . . . . . . . . . . . . . . . . . 369 12.3.2 Median Regression (MR) . . . . . . . . . . . . . . . . . . . . . . 369 12.3.3 Quantile Regression (QR) . . . . . . . . . . . . . . . . . . . . . . 370
- 17. 12.4 Computing Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.4.1 Quantile Regression via Linear Programming . . . . . . . 376 12.4.2 Boscovich’s Problem. . . . . . . . . . . . . . . . . . . . . . . . . . 377 12.4.3 Extension to Linear Combination of Nonlinear Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 12.4.4 B-Spline Application . . . . . . . . . . . . . . . . . . . . . . . . . . 382 12.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 12.5.1 Separate Outliers in Cloud Points . . . . . . . . . . . . . . . . 387 12.5.2 Modelling Time-Series . . . . . . . . . . . . . . . . . . . . . . . . 393 12.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 12.6.1 Regression of Implicit-Functions . . . . . . . . . . . . . . . . . 400 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 13 Robust Regression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 13.1 Basic Methods in Robust Regression . . . . . . . . . . . . . . . . . . . . . 405 13.1.1 Concept of Robust Regression. . . . . . . . . . . . . . . . . . . 405 13.1.2 Maximum Likelihood Method. . . . . . . . . . . . . . . . . . . 406 13.1.3 Danish Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 13.1.4 Danish Algorithm with PCA . . . . . . . . . . . . . . . . . . . . 426 13.1.5 RANSAC Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 432 13.2 Application Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 13.2.1 Fitting a Sphere to Point Cloud Data. . . . . . . . . . . . . . 442 13.2.2 Fitting a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 13.3 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 13.3.1 Fitting a Plane to a Slope . . . . . . . . . . . . . . . . . . . . . . 502 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 14 Stochastic Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 14.1 Basic Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 14.1.1 Concept of Stochastic Processes . . . . . . . . . . . . . . . . . 517 14.1.2 Examples for Stochastic Processes. . . . . . . . . . . . . . . . 517 14.1.3 Features of Stochastic Processes . . . . . . . . . . . . . . . . . 519 14.2 Time Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 14.2.1 Concept of Time Series . . . . . . . . . . . . . . . . . . . . . . . . 521 14.2.2 Models of Time Series . . . . . . . . . . . . . . . . . . . . . . . . 521 14.3 Stochastic Differential Equations (SDE) . . . . . . . . . . . . . . . . . . . 528 14.3.1 Ito Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 14.3.2 Ito Numerical Integral . . . . . . . . . . . . . . . . . . . . . . . . . 528 14.3.3 Euler-Maruyama Method. . . . . . . . . . . . . . . . . . . . . . . 529 14.4 Numerical Solution of (SDE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 14.4.1 Single Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 14.4.2 Many Realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 14.4.3 Slice Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 14.4.4 Standard Error Band . . . . . . . . . . . . . . . . . . . . . . . . . . 532
- 18. 14.5 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 14.5.1 Measurement Values . . . . . . . . . . . . . . . . . . . . . . . . . . 533 14.5.2 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . 533 14.5.3 Maximization of the Likelihood Function . . . . . . . . . . 534 14.5.4 Simulation with the Estimated Parameters. . . . . . . . . . 535 14.5.5 Deterministic Versus Stochastic Modeling. . . . . . . . . . 536 14.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 14.6.1 Rotating Ellipsoid with a Stochastic Flattening . . . . . . 537 14.6.2 Analysis of Changes in Groundwater Radon . . . . . . . . 545 14.7 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 14.7.1 Deterministic Lorenz Attractor. . . . . . . . . . . . . . . . . . . 549 14.7.2 Stochastic Lorenz Attractor . . . . . . . . . . . . . . . . . . . . . 553 15 Parallel Computations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 15.2 Amdahl’s-Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 15.3 Implicit and Explicit Parallelism. . . . . . . . . . . . . . . . . . . . . . . . . 560 15.4 Dispatching Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 15.5 Balancing Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 15.6 Parallel Computing with GPU . . . . . . . . . . . . . . . . . . . . . . . . . . 568 15.6.1 Neural Network Computing with GPU . . . . . . . . . . . . 568 15.6.2 Image Processing with GPU . . . . . . . . . . . . . . . . . . . . 574 15.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 15.7.1 3D Ranging Using the Dixon Resultant . . . . . . . . . . . 577 15.7.2 Reducing Colors via Color Approximation . . . . . . . . . 582 15.8 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 15.8.1 Photogrammetric Positioning by Gauss-Jacobi Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
- 19. Introduction Numeric and Symbolic Methods—What are they? Basically, a numeric (or numerical) method is one that could be done with a simple handheld calculator, using basic arithmetic, square roots, some trigonometry functions, and a few other functions most people learn about in high school. Depending on the task, one may have to press the calculator buttons thousands (or even millions) of times, but theoretically, a person with a calculator and some paper could implement a numerical method. When finished, the paper would be full of arithmetic. A symbolic method involves algebra. It is a method that if a person implemented, would involve algebraic or higher rational thought. A person implementing a symbolic method will rarely need to reach for a calculator. When finished, there may be some numbers, but the paper would be full of variables like x, y, z. Students usually meet the topic of quadratic equations in junior high school. Suppose you wanted to solve the equation x2 þ 3x 2 ¼ 0. With a handheld cal- culator, one could simply do “intelligent guessing.” Let us guess, say, x =1. Plug it in, get a positive result. OK, that is too big. Try x = 0; that is too small. Go back and forth; stop when satisfied with the accuracy. It does not take long to get x = 0.56155, which might well be considered accurate enough. Furthermore, it is easy to write a computer program to implement this idea. That is a numeric method. But wait. There is another answer, which the numeric method missed, namely −3.56155. Even worse, if one were to continue this method on many problems, one would soon notice that some equations do not seem to have solutions, such as x2 2x þ 4 ¼ 0. A great deal of effort could be expended in arithmetic until finally giving up and finding no solution. The problem is cured by learning algebra and the symbolic method called the quadratic formula. Given ax2 þ bx þ c ¼ 0 the solution is x ¼ b ffiffiffiffiffiffiffiffiffiffiffi b24ac p 2a . It is now immediately obvious why some problems have no solution: it happens precisely when b2 4ac 0.
- 20. In the previous example, x2 þ 3x 2 ¼ 0, we see that the two roots are exactly ð3 ffiffiffiffiffi 17 p Þ=2. There is no approximation whatever. Should a decimal answer correct to, say, 16 digits be desired, that would be trivially obtained on any modern computer. There is more. Not only does the symbolic method concisely represent all solutions, it invites the question, can we define a new kind of number in which the negative under the square root may be allowed? The symbolic solution has led to a new concept, that of complex numbers! Symbolic methods may be hard to develop, and they may be difficult for a computer to implement, but they lead to insight. Fortunately, we are not forced into a strict either/or dichotomy. There are symbolic-numeric methods, hybrids using the strengths of both ideas. Numeric Solution In order to further illustrate numeric, symbolic, and symbolic-numeric solutions, let us consider an algebraic system of polynomial equations. For such systems, there may be no solution, one solution, or many solutions. With numerical solutions, one commonly utilizes iterative techniques starting from an initially guessed value. Let us start with a two variable system of two equations f x; y ð Þ ¼ 0 and g x; y ð Þ ¼ 0, f ¼ x 2 ð Þ2 þ y 3 ð Þ2 , g ¼ x 1 2 2 þ y 3 4 2 5. This actual problem has two real solutions, see Fig. 1. Fig. 1 Geometrical representation of a multivariate polynomial system
- 21. A numeric solution starts with the initial value and proceeds step-by-step locally. Depending on the method, we expect to converge to one of the solutions in an efficient manner. Employing the initial value (4, −1) and a multivariate Newton’s method, the solution after seven steps is (2.73186, 0.887092). Let us visualize the iteration steps, see Fig. 2. However, if the initial guess is not proper, for example (0, 0), then, we may have a problem with the convergence since the Jacobian may become singular. Symbolic Solution Let us transform the original system into another one, which has the same solutions, but for which variables can be isolated and solved one-by-one. Employing Gröbner basis, we can reduce one of the equations to a univariate polynomial, gry ¼ 2113 3120y þ 832y2 , grxy ¼ 65 þ 16x þ 24y. First, solving the quadratic equation gry, we have y ¼ 1 104 195 2 ffiffiffiffiffiffiffiffiffiffi 2639 p , y ¼ 1 104 195 þ 2 ffiffiffiffiffiffiffiffiffiffi 2639 p . Fig. 2 Local solution with initial guess and iteration steps
- 22. Then employing these roots of y, the corresponding values of x can be computed from the second polynomial of the Gröbner basis as x ¼ 1 104 130 þ 3 ffiffiffiffiffiffiffiffiffiffi 2639 p , x ¼ 1 104 130 3 ffiffiffiffiffiffiffiffiffiffi 2639 p . So, we have computed both solutions with neither guessing nor iteration. In addition, there is no round-off error. Let us visualize the two solutions, see Fig. 3: Let us summarize the main features of the symbolic and numeric computations: Numeric computations: – usually require initial values and iterations. They are sensitive to round-off errors, provide only one local solution, – can be employed for complex problems, and the computation times are short in general because the steps usually translate directly into computer machine language. Symbolic computations: – do not require initial values and iterations. They are not sensitive for round-off errors, and provide all solutions, – often cannot be employed for complex problems, and the computation time is long in general because the steps usually require computer algebra system software. Fig. 3 Global solution—all solutions without initial guess and iteration
- 23. Ideally, the best strategy is to divide the algorithm into symbolic and numeric parts in order to utilize the advantages of both techniques. Inevitably, numeric computations will always be used to a certain extent. For example, if polynomial gry above had been degree, say, five, then a numeric univariate root solver would have been necessary. Hybrid (symbolic-numeric) Solution Sometimes, we can precompute a part of a numerical algorithm in symbolic form. Here is a simple illustrative example. Consider a third polynomial and add it to our system above: h ¼ x þ 1 2 2 þ y 7 4 3 5. In that case, there is no solution, since there is no common point of the three curves representing the three equations, see Fig. 4. However, we can look for a solution of this overdetermined system in the minimal least squares sense by using the objective function G ¼ f 2 þ g2 þ h2 , Fig. 4 Now, there is no solution of the overdetermined system
- 24. or G = 5 þ 2 þ x ð Þ2 þ 3 þ y ð Þ2 2 þ 5 þ 1 2 þ x 2 þ 7 4 þ y 3 !2 þ 5 þ 1 2 þ x 2 þ 3 4 þ y 2 !2 and minimizing it. Employing Newton’s method, we get x¼ 2:28181,y¼ 0:556578. The computation time for this was 0.00181778 s. The solution of the overde- termined system can be seen in Fig. 5. Here, the gradient vector as well as the Hessian matrix is computed in numerical form in every iteration step. But we can compute the gradient in symbolic form: grad = 1 32 2x 173 þ 192 2 þ x ð Þx ð Þ þ 216xy 16 41 þ 26x ð Þy2 þ 64 1 þ 2x ð Þy3 þ 3 809 þ 740y ð Þ 137829 512 þ 555x 8 þ 27x2 8 þ 60527y 128 41xy 13x2 y 6321y2 16 þ 6xy2 þ 6x2 y2 þ 767y3 4 105y4 2 þ 6y5 : Employing this symbolic form the computation time can be reduced. The running time can be further reduced if the Hessian matrix is also computed symbolically, Fig. 5 The solution of the overdetermined system
- 25. H ¼ 173 16 þ 12x 4 þ 3x ð Þ þ 27y 4 13y2 þ 4y3 555 8 þ y 41 þ 6y ð Þ þ x 27 4 þ 2y 13 þ 6y ð Þ 555 8 þ y 41 þ 6y ð Þ þ x 27 4 þ 2y 13 þ 6y ð Þ 60527 128 41x13x2 6321y 8 þ 12xy þ 12x2 y þ 2301y2 4 210y3 þ 30y4 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 . Now, the computation time is less than half of the original one. So using symbolic forms, the computation time can be reduced considerably. This so-called hybrid computation has an additional advantage too, namely the symbolic part of the algorithm does not generate round-off errors. Another approach of applying the hybrid computation is to merge symbolic evaluation with numeric algorithm. This technique is illustrated using the following example. Let us consider a linear, nonautonomous differential equation system of n vari- ables in matrix form: d dx yðxÞ ¼ AðxÞyðxÞ þ bðxÞ, where A is a matrix of nn dimensions, yðxÞ and bðxÞ are vectors of n dimen- sions, and x is a scalar independent variable. In the case of a boundary value problem, the values of some dependent variables are not known at the beginning of the integration interval, at x ¼ xa, but they are given at the end of this interval, at x ¼ xb. The usually employed methods need subsequent integration of the system, because of their trial–error technique or they require solution of a large linear equation system, in the case of discretization methods. The technique is based on the symbolic evaluation of the well-known Runge–Kutta algorithm. This technique needs only one integration of the differential equation system and a solution of the linear equation system representing the boundary conditions at x ¼ xb. The well-known fourth-order Runge–Kutta method, in our case, can be repre- sented by the following formulas: R1i ¼ AðxiÞyðxiÞ þ bðxiÞ, R2i ¼ A xi þ h 2 yðxiÞ þ R1ih 2 þ b xi þ h 2 , R3i ¼ A xi þ h 2 yðxiÞ þ R2ih 2 þ b xi þ h 2 , R4i ¼ Aðxi þ hÞ yðxiÞ þ R3ih ð Þ þ bðxi þ hÞ and then the new value of yðxÞ can be computed as: yi þ 1 ¼ yðxiÞ þ R1i þ 2 R2i þ R3i ð Þ þ R4i ð Þh 6 .
- 26. A symbolic system like Mathematica, is able to carry out this algorithm not only with numbers but also with symbols. It means that the unknown elements of ya ¼ yðxaÞ can be considered as unknown symbols. These symbols will appear in every evaluated yi value, as well as in yb ¼ yðxbÞ too. Let us consider a simple illustrative example. The differential equation is: d2 dx2 yðxÞ 1 x 5 yðxÞ ¼ x. The given boundary values are: yð1Þ ¼ 2 and yð3Þ ¼ 1 After introducing new variables, we get a first-order system, y1ðxÞ ¼ yðxÞ and y2ðxÞ ¼ d dx yðxÞ the matrix form of the differential equation is: d dx y1ðxÞ; d dx y2ðxÞ ¼ 0 1 1 x=5 0 y1ðxÞ; y2ðxÞ ½ þ 0; x ½ . Employing Mathematica’s notation: A[x_]:={{0,1},{1-1/5 x,0}}; b[x_]:={0,x}; x0=1; y0={2.,s} The unknown initial value is s. The order of the system M = 2. Let us consider the number of the integration steps as N = 10, so the step size is h = 0.2. ysol=RKSymbolic[x0,y0,A,b,2,10,0.2]; The result is a list of list data structure containing the corresponding (x, y) pairs, where the y values depend on s. ysol[[2]][[1]] {{1,2.},{1.2,2.05533+0.200987 s},{1.4,2.22611+0.407722 s}, {1.6,2.52165+0.625515 s}, {1.8,2.95394+0.859296s}, {2.,3.53729+1.11368s},
- 27. {2.2,4.28801+1.39298 s}, {2.4,5.22402+1.70123 s},{2.6,6.36438+2.0421 s}, {2.8,7.72874+2.41888 s},{3.,9.33669+2.8343 s}} Consequently, we have got a symbolic result using traditional numerical Runge– Kutta algorithm. In order to compute the proper value of the unknown initial value, s, the boundary condition can be applied at x ¼ 3. In our case y1ð3Þ ¼ 1. eq=ysol[[1]][[1]]==-1 9.33669+2.8343 s==-1 Let us solve this equation numerically, and assign the solution to the symbol s: sol=Solve[eq,s] {{s - -3.647}} s=s/.sol {-3.647} s=s[[1]] -3.647 Then, we get the numerical solution for the problem: ysol[[2]][[1]] {{1,2.},{1.2,1.32234},{1.4,0.739147},{1.6,0.240397}, {1.8,-0.179911}, {2.,-0.524285},{2.2,-0.792178}, {2.4,-0.980351},{2.6,-1.08317},{2.8,-1.09291}, {3.,-1.}} The truncation error can be decreased by using smaller step size h, and the round-off error can be controlled by the employed number of digits.
- 28. Part I Solution of Nonlinear Systems
- 29. Chapter 1 Solution of Algebraic Polynomial Systems 1.1 Zeros of Polynomial Systems Let us consider the following polynomial p ¼ 2x þ x3 y2 þ y2 : The monomials are x3 y2 with coefficient 1, and x1 y0 with coefficient 2 and x0 y2 with coefficient 1. The degree of such a monomial is defined as the sum of the exponents of the variables. For example, the second monomial x3 y2 , has degree 3 + 2 = 5. The degree of the polynomial is the maximum degree of its constituent monomials. In this case deg ðpÞ ¼ max 1; 5; 2 ð Þ ¼ 5. Some polynomials contain parameters as well as variables. For example, the equation of a circle centered at the origin is x2 þ y2 r2 ¼ 0. Only x and y are actual variables; the r is a parameter. Now consider a polynomial system like g x; y ð Þ ¼ a1 þ a2x þ a3xy þ a4y; h x; y ð Þ ¼ b1 þ b2x2 y þ b3xy2 : The total degree of the system is defined to be deg ðgÞ deg ðhÞ ¼ 2 3 ¼ 6: Notice that we do not count the parameters in this computation. Define the roots or zeros of a polynomial system to be the set of pairs (r, s) such that g(r, s) = 0 and h(r, s) = 0.
- 30. Bézout’s Theorem: Consider two polynomial equations in two unknowns: g x; y ð Þ ¼ h x; y ð Þ ¼ 0. If this system has only finitely many zeros ðx; yÞ 2 C2 , then the number of zeros is at most deg ðgÞ deg ðhÞ. Here deg (g) and deg (h) are the total degree of g(x, y) and h(x, y). 1.2 Resultant Methods In this section we introduce two different symbolic methods: Sylvester and Dixon resultants see Dickenstein and Emiris (2005). These techniques eliminate variables and yield univariate polynomials, which then can be solved numerically. 1.2.1 Sylvester Resultant Let us consider the following system (Fig. 1.1) p ¼ xy 1; g ¼ x2 þ y2 4: Since linear systems of equations are well known, let’s try to convert this into a useful system of linear equations. With x as the “real” variable and y as a “pa- rameter,” consider x0 , x1 , and x2 as three independent symbols. The two equations in the original system give us two linear equations, and we generate a third by multiplying p by x. This yields MðyÞ x0 x1 x2 0 @ 1 A ¼ 0; Fig. 1.1 Graphical interpretation of the real roots of the system 4 1 Solution of Algebraic Polynomial Systems
- 31. where M(y) is 1 y 0 y2 4 0 1 0 1 y 0 @ 1 A x0 x1 x2 0 @ 1 A ¼ 0: Since x0 is really 1, any solution to this homogeneous system must be nontrivial. Thus detðMðyÞÞ ¼ 1 þ 4y2 y4 ¼ 0: Solving this gives us y; we have eliminated x. This function is built into Mathematica, Resultant[p,g,x] 1 - 4y2 + y4 For the other variable Resultant[p,g,y] 1 - 4x2 + x4 The solutions of these two polynomials are the solutions of the system p x; y ð Þ; g x; y ð Þ f g. Roots[ - 1 + 4y2 - y4 = = 0,y] y = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 - ffiffiffi 3 p p jjy = - ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 - ffiffiffi 3 p p jjy = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + ffiffiffi 3 p p jjy = - ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + ffiffiffi 3 p p Roots[1 - 4x2 + x4 = = 0,x] x = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 - ffiffiffi 3 p p jjx = - ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 - ffiffiffi 3 p p jjx = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + ffiffiffi 3 p p jjx = - ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + ffiffiffi 3 p p The main handicap of the Sylvester resultant is that it can be directly employed only for systems of two polynomials. 1.2.2 Dixon Resultant Let us introduce a new variable b and define the following polynomial, d x; y; b ð Þ ¼ p x; y ð Þg x; b ð Þ p x; b ð Þg x; y ð Þ y b ¼ b 4x þ x3 þ y bxy: (One can show that the numerator above is always divisible by y – b). We call this polynomial the Dixon polynomial. 1.2 Resultant Methods 5
- 32. It is easy to see that plugging in any common root of p x; y ð Þ; g x; y ð Þ f g forces the Dixon polynomial to be 0, for any value of b. The Dixon polynomial can be written as, d x; y; b ð Þ ¼ b0 4x þ x3 þ y þ b1 1 xy ð Þ: Then the following homogeneous linear system should have solutions for every b 4x þ x3 þ y ¼ 0; 1 xy ¼ 0 or, where x is considered as parameter 4x þ x3 1 1 x y0 y1 ¼ 0; therefore det 4x þ x3 1 1 x ¼ 1 þ 4x2 x4 ; must be zero. The matrix 4x þ x3 1 1 x is called the Dixon matrix, and its determinant is called as Dixon resultant. [Historical note: the argument above, for two variables, was first used by Bezout.] Let us employ Mathematica Resultant0 Dixon0 DixonPolynomial[{p,g},{y},{b}] b - 4x + x3 + y - bxy DixonMatrix[{p,g},{y},{b}]==MatrixForm - 4x + x3 1 1 - x DixonResultant[{p,g},{y},{b}] - 1 + 4x2 - x4 Similarly, for the other variable, we get DixonResultant[{p,g},{x},{a}] - 1 + 4y2 - y4 6 1 Solution of Algebraic Polynomial Systems
- 33. Here a and b are dummy formal variables (symbolic variables), without assigned values. The Dixon resultant method can be generalized to polynomial systems of more than two polynomials. For example, P = x + y + z; G = x - 2y + z3 ; S = x2 - 2y3 + z; To eliminate variables x and y, we introduce dummy variables X and Y, then DixonResultant[{P,G,S},{x,y},{X,Y}]==Expand 324z + 144z2 + 24z3 + 144z4 - 72z5 + 36z6 + 72z7 - 24z9 or %=12==Expand 27z + 12z2 + 2z3 + 12z4 - 6z5 + 3z6 + 6z7 - 2z9 (% refers to the last previous output.) Remark 1 For other multivariate resultant methods such as Sturmfels’ approach, see Awange and Paláncz (2016). Remark 2 For three or more variables, the discussion above of the Dixon resultant has been simplified. Sometimes the Dixon matrix is not square, and sometimes when it is square the determinant is identically 0. Then the method would seem to fail. Kappur et al. (1994) showed how to proceed and define the Dixon resultant using maximal minors, see Exercises 1.6.4 Dixon KSY solution. Remark 3 If the system contains parameters, then the resultant will contain those parameters. Remark 4 If there are n equations, the Dixon resultant will eliminate n − 1 variables. 1.3 Gröbner Basis This technique introduced by Buchberger and named after his Ph.D. supervisor Gröbner, is more general and mostly more efficient than the resultant methods, unless parameters are present. To have an idea how this method works, first, let us see how the greatest common divisor of polynomials can be defined. 1.2 Resultant Methods 7
- 34. 1.3.1 Greatest Common Divisor of Polynomials Greatest common divisor, GCD, is a familiar concept from arithmetic, as in GCD (12, 30) = 6. The same concept applies to polynomials, of any number of variables. Let us consider two univariate polynomials s(x) and v(x) with the same variable x s = 8 + 22x + 21x2 + 8x3 + x4 ; v = 6 + 11x + 6x2 + x3 ; The greatest common divisor (GCD) of these polynomials gcd = PolynomialGCD[s,v] 2 + 3x + x2 Let us divide s(x) with this GCD {{Cs},Rs} = PolynomialReduce[s,gcd,x] {{4 + 5x + x2 },0} The remainder is zero and we can check Csgcd (2 + 3x + x2 )(4 + 5x + x2 ) Expand[%] 8 + 22x + 21x2 + 8x3 + x4 Let us carry out these operations with v(x), too {{Cv},Rv} = PolynomialReduce[v,gcd,x] {{3 + x},0} and Csgcd (3 + x)(2 + 3x + x2 ) Expand[%] 6 + 11x + 6x2 + x3 This means that the original polynomials s(x) and v(x) can be expressed as the linear combination of the GCD, like. 8 1 Solution of Algebraic Polynomial Systems
- 35. sðxÞ ¼ CsðxÞ gcd(xÞ þ 0 gcd(xÞ and vðxÞ ¼ 0 gcd(xÞ þ CvðxÞ gcd(xÞ or s x ð Þ v x ð Þ = Cs x ð Þ 0 0 Cv x ð Þ gcdðxÞ gcdðxÞ ! : Since there is only one variable, the roots of gcd(x), the GCD of these two poly- nomials s x ð Þ; v x ð Þ f g are the roots of the polynomial system. This important fact is because any one-variable polynomial can be factored over C into linear pieces. The roots of gcd x ð Þ ¼ 2 þ 3 x þ x2 ¼ 0 are in Fig. 1.2. However, we normally have polynomials of two variables (x, y) p - 1 + xy g - 4 + x2 + y2 In case of more than one variable, the greatest common divisor, though it exists, does not play the role it did in the previous paragraph. That role is filled by the Gröbner basis (Buchberger and Winkler 1998). {G1,G2} = GroebnerBasis[{p,g},{x,y}] {1 - 4y2 + y4 ,x - 4y + y3 } As in the case of univariate polynomial, for the two variables ðx; yÞ, the original system can be expressed as the linear combination of the polynomials of the Gröbner basis G1 y ð Þ; G2 x; y ð Þ f g, where the coefficients are also polynomials. Fig. 1.2 Common roots of polynomials 1.3 Gröbner Basis 9
- 36. The coefficients are the remainders. {c1,r1} = PolynomialReduce[p,{G1,G2},{x,y}] {{ - 1,y},0} Then p(x, y) can be expressed as { - 1,y}: G1 G2 ==Simplify { - 1 + x y} and {c2,r2} = PolynomialReduce[g,{G1,G2},{x,y}] {{ - 4 + y2 ,x + 4y - y3 },0} then g(x, y) { - 4 + y2 ,x + 4y - y3 }: G1 G2 ==Simplify { - 4 + x2 + y2 } In matrix form p x; y ð Þ g x; y ð Þ ¼ 1 y 4 þ y2 x þ 4y y3 G1 y ð Þ G2 x; y ð Þ or p x; y ð Þ g x; y ð Þ ¼ 1 4 þ y2 G1 y ð Þ þ y x þ 4y y3 G2 x; y ð Þ The roots of the system p x; y ð Þ; g x; y ð Þ f g are the same as the roots of the system G1 y ð Þ; G2 x; y ð Þ f g. Note that this basis consists of special polynomials, since G1 (y) is a univariate polynomial! Generally speaking, the original polynomial system p x; y ð Þ; g x; y ð Þ f g can be expressed as a linear combination of the basis polynomials G1 x; y ð Þ; G2 x; y ð Þ f g. There are many other basis polynomials too and the set of these basis polynomials is called the ideal of the original polynomial. However, the Gröbner basis is a special basis, since one of its polynomials is a univariate one. If the Gröbner basis is 1, the polynomials have no common divisor, consequently they have no common roots. Remark The theory of Gröbner bases is much more extensive and sophisticated than we can go into here. Our focus is on using Gröbner bases to eliminate variables. 10 1 Solution of Algebraic Polynomial Systems
- 37. Let us employ the built-in function for the system {P, S, G} considered in previous Sect. 1.2.2, GroebnerBasis[{P,S,G},{x,y,z}] { - 27z - 12z2 - 2z3 - 12z4 + 6z5 - 3z6 - 6z7 + 2z9 ,3y + z - z3 ,3x + 2z + z3 } where the first element of the Gröbner basis is the same provided by the Dixon resultant. Now, let us compute the Gröbner basis of the following system U = x2 + y2 = = 1 x2 + y2 - 1 V = x2 + y2 = = 2 x2 + y2 - 2 GroebnerBasis[{U,V},{x,y}] {1} There are no common roots, see Fig. 1.3, however the upper limit of the number of roots is 2 2 = 4. 1.3.2 Reduced Gröbner Basis The Mathematica built in function can carry out the elimination process too, employing the so called reduced Gröbner Basis. To get the univariate polynomial of x, we should eliminate y and z, grbx = GroebnerBasis[{P,S,G},{x},{y,z}] { - 27x + 18x2 - 342x3 + 306x4 - 186x5 + 229x6 - 18x7 + 12x8 + 8x9 } Fig. 1.3 No common roots of polynomials 1.3 Gröbner Basis 11
- 38. and then for the other variables, grby = GroebnerBasis[{P,S,G},{y},{x,z}] { - 21y - 23y2 - 30y3 - 36y4 - 9y5 + 6y6 - 12y7 + 8y8 } grbz = GroebnerBasis[{P,S,G},{z},{x,y}] { - 27z - 12z2 - 2z3 - 12z4 + 6z5 - 3z6 - 6z7 + 2z9 } These algebraic methods are very impressive and useful, but they are limited by the size of the system. In general, systems with more than ten unknowns cannot be solved this way due to time and space (RAM) limitations. 1.3.3 Polynomials with Inexact Coefficients Computing Gröbner bases with inexact coefficients is often desired in industrial applications, but the computation with floating-point numbers is quite unstable if performed naively (Sasaki 2014). The solution methods of the Gröbner basis are very sensitive to round off error, therefore sometimes in case of systems that are over-constrained or have roots with multiplicities, and are given with inexact coefficients, using hybrid symbolic-numeric methods are required (Szanto 2011). Lichblau (2013) discussed computation of Gröbner bases using approximate arithmetic for coefficients and showed how certain considerations of tolerance, corresponding roughly to accuracy and precision from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. Let us consider the following polynomial system, polys = - 4 + x2 - 1:49071xy + y2 , - 8 + x2 - 0:4xz + z2 , - 4 + t2 - 0:894427tx + x2 , - 4 + y2 - 1:49071yz + z2 , - 8 + t2 - 0:666667ty + y2 , - 4 + t2 - 0:894427tz + z2 ; If we try to find the Gröbner basis, we get the trivial answer {1:}, which means there is no relationship between the polynomials. sol = GroebnerBasis[polys,{x,y,z,t}] {1:} Even employing rationalization of the coefficients will not solve the problem, n = 10; polysR = Map[Rationalize[#,10- n ],polys] - 4 + x2 - (149071xy)=100000 + y2 , - 8 + x2 - (2xz)=5 + z2 , - 4 + t2 - (216200tx)=241719 + x2 , - 4 + y2 - (149071yz)=100000 + z2 , - 8 + t2 - (666503ty)=999754 + y2 , - 4 + t2 - (216200tz)=241719 + z2 12 1 Solution of Algebraic Polynomial Systems
- 39. solR = GroebnerBasis[polysR,{x,y,z,t}] {1:} However, applying an approximate hybrid technique, solA = GroebnerBasis[polys,x,y,z,t,Tolerance ! 10( - 3) ] yields solt = NSolve[solA[[1]],t] {{t ! - 1:00002 - 0:0044912 i},{t ! - 1:00002 + 0:0044912 i}, {t ! 1:00002 - 0:0044912 i},{t ! 1:00002 + 0:0044912 i}} Since we are interested in real solutions, Map[Re[#[[2]]],Flatten[solt]] { - 1:00002, - 1:00002,1:00002,1:00002} Then the other variables are solz = NSolve[solA[[2]]=:t ! 1:00002,z] {{z ! 2:2361}} soly = NSolve[solA[[3]]=:t ! 1:00002,y] {{y ! 3:00005}} solx = NSolve[solA[[4]]=:t ! 1:00002,x] {{x ! 2:23606}} Let us check our result via least squares technique employing global minimization. Our objective function is G = Total[Map[#2 ,polys]] ( - 4 + t2 - 0:894427 tx + x2 )2 + ( - 8 + t2 - 0:666667 ty + y2 )2 + ( - 4 + x2 - 1:49071 xy + y2 )2 + ( - 4 + t2 - 0:894427 tz + z2 )2 + ( - 8 + x2 - 0:4 xz + z2 )2 + ( - 4 + y2 - 1:49071 yz + z2 )2 and NMinimize[G,{x,y,z,t}] 2:10012 10- 10 ,x ! 2:23607,y ! 3:,z ! 2:23607,t ! 1:0023 1.3 Gröbner Basis 13
- 40. 1.4 Using Dixon-EDF for Symbolic Solution of Polynomial Systems We have discussed the basic idea of a system of polynomial equations in the Introduction. Earlier in this chapter we introduced the ideas of resultants and Gröbner bases and did some examples. In this section we will show some much more difficult problems that reveal the great power of the Dixon resultant as extended with “Early Detection of Factors”, or Dixon-EDF. As before, we have in each case n equations in n variables x1, x2, …, xn and some parameters. We assume that the system is neither over- nor underdetermined. Usually 3 n 15, though we can work with more variables if the system is sparse enough and does not involve variables with high exponent. In most examples from actual applications, one rarely sees an exponent larger than 2. Again, by “solve the system” we mean we have eliminated all but one of the variables. We are left with one equation in one variable and the parameters. If desired, numerical values for the parameters can then be substituted, and the variable obtained by one-variable numerical solvers. The ideas in this section were developed by Lewis (2007, 2008). 1.4.1 Explanation of Dixon-EDF The basic idea of the Dixon method is to construct a square matrix M whose determinant D is a multiple of the resultant. Usually M is not unique, it is obtained as a maximal minor, in a larger matrix we shall call M + , and there are usually many maximal minors—any one of which will do. The entries in M are polynomials in parameters. The factors of D that are not the resultant are called the spurious factors, and their product is sometimes referred to as the spurious factor. The naive way to proceed is to compute D, factor it, and separate the spurious factor from the actual resultant. But there are problems. On one the hand, the determinant may be so large as for it to be impractical or even impossible to compute; even though the resultant is relatively small, the spurious factor is huge. On the other hand, the determinant may be so large that factoring it is impractical. Lewis developed three heuristic methods to overcome these problems (2008). The methods were discovered by experimentation and may apply to other resultant formulations, such as the Macaulay. The one that concerns here is called EDF. The EDF method exploits the observed fact that D usually has many factors. In other words, we try to turn the existence of spurious factors to our advantage. By elementary row and column manipulations (Gaussian elimination) we discover probable factors of D and extract them from M0 = M. This produces a smaller matrix M1, still with polynomial entries, and a list of discovered numerators and denominators. 14 1 Solution of Algebraic Polynomial Systems
- 41. Here is very simple example. M0 ¼ 9 2 4 4 numerators: denominators: Of course the determinant is trivial, but suppose we wish to keep the arithmetic very simple, and never work with numbers bigger than 9. We factor a 2 out of the second column, then a 2 from the second row. Thus: M0 ¼ 9 1 2 1 numerators: 2; 2 denominators: We change the second row by subtracting 2/9 of the first: M0 ¼ 9 1 0 7=9 numerators: 2; 2 denominators: We pull out the denominator 9 from the second row, and factor out 9 from the first column: M0 ¼ 1 1 0 7 numerators: 2; 2; 9 denominators: 9 We “clean up” by dividing out the common factor of 9 from the numerator and denominator lists; any 1 that occurs may be erased and the list compacted. Since the first column is canonically simple, we are finished with one step of the algorithm, and have produced a one-smaller M1 for the next step. M1 ¼ 7 ð Þ numerators: 2; 2 denominators: 1 The algorithm terminates by pulling out the 7: numerators: 2; 2; 7 denominators: 1 As expected (since the original matrix contained only integers) the denominator list is trivial. The product of all the entries in the numerator list is the determinant, but we never needed to deal with any number larger than 9. The EPF method is implemented in the computer algebra system Fermat by Lewis (2009). The Dixon resultant is a very attractive tool for solving systems of multivariate polynomial geodetic equations (see Paláncz et al. 2008). Comparing it to other multi-polynomial resultant like Sturmfels’s method it has advantages of (i) the small size of the Dixon matrix, (ii) faster computational speed, (iii) being robust. In the following sections we provide some examples where Dixon EDF method proved to be very effective. 1.4 Using Dixon-EDF for Symbolic Solution of Polynomial Systems 15
- 42. 1.4.2 Distance from a Point to a Standard Ellipsoid Given an ellipsoid x2 /a2 + y2 /b2 + z2 /c2 − 1 = 0 and a point (u, v, w), compute the point (x, y, z) on the ellipsoid closest to the point. We have three variables x, y, z. We derive equations using partial derivatives, so we must add two more variables to stand for @z=@x; @z=@y. There are six parameters a, b, c, u, v, w. The new variables representing @z=@x; @z=@y are artifacts. We don’t care about them. We want to know just x, y, z. One advantage of resultants is that you can’t tell a Gröbner basis algorithm not to bother with some of the variables (Fig. 1.4). This is a fairly easy problem. The resultant is degree 6 in x. With Dixon: 0.038 s, 22 MB RAM with Magma’s Gröbner basis, 1 s, 100 MB. (Similar results were obtained with Maple and Mathematica.) But we can say more. The coefficient of x6 is b2 c2 2abc2 þ a2 c2 2ab2 c þ 4a2 bc 2a3 c þ a2 b2 2a3 b þ a4 : This factors into (a − c)2 (a − b)2 , so we learn that if b = a or c = a there is a simpler solution. In fact, if c = a it drops to degree 4. As we pointed out in the Introduction, the symbolic method leads to insight! 1.4.3 Distance from a Point to Any 3D Conic Here is the image for a general ellipsoid, but we could have any 3D conic (Fig. 1.5). Given ax2 þ by2 þ cz2 þ d xy þ e xz þ f yz þ gx þ hy þ iz þ j ¼ 0 and point (u, v, w), compute point (x, y, z) with shortest distance. We have again three variables x, y, z, but now 13 parameters a, b, c,…, u, v, w. At least one artifact variable must be added. Fig. 1.4 Given u, v, w find x, y, z 16 1 Solution of Algebraic Polynomial Systems
- 43. This problem is much harder than the previous. With Dixon-EDF: 12 s, 270 MB RAM. The answer has 38,984 terms, degree 6. With Magma: killed after 24 h, 24 GB RAM. With Maple’s FGb routine: Success after 5.8 h, 52 GB RAM. The coefficient of x6 has two factors, one is af 2 def þ be2 þ cd2 4abc. If this were 0, the resultant simplifies. 1.4.4 Pose Estimation Suppose we have a quadrilateral ABCE; it does not have to be planar. The distances between each pair of vertices are known. The object moves. We observe it from point P, noting the angles spanned by each pair of vertices. The classic four point pose problem is to deduce the distances X1, X2, X3, X4 (Fig. 1.6). It is easy to derive six equations from the law of cosines: X2 1 þ X2 2 X1X2 r AB j j2 X2 1 þ X2 3 X1X3q AC j j2 X2 2 þ X2 3 X2X3 p BC j j2 X2 1 þ X2 4 X1X4 s AE j j2 X2 4 þ X2 3 X4X3 t CE j j2 X2 2 þ X2 4 X2X4 u BE j j2 r, p, q, s, t, u, are cosines. There are four variables X1, X2, X3, X4. The parameters are the lengths of AB, BC, CE, AE, AC, BE, and the six cosines. Fig. 1.5 Given u, v, w find x, y, z 1.4 Using Dixon-EDF for Symbolic Solution of Polynomial Systems 17
- 44. Using any four equations but including at least one diagonal AC or BE yields an easy system of equations, solvable by many means. Indeed, one could select, say, the first three equations and obtain a complete three variable system; see the exercises at the end of this chapter. But suppose the object could be flexible! Then we must use four equations from only the outside edges; diagonal distances might change. This turns out to be a much harder system to solve and is only doable with Dixon-EDF. This problem is similar to resection; see Sect. 1.6.3. 1.4.5 How to Run Dixon-EDF As far as we know, Dixon is implemented only in Mathematica, no other large multipurpose CAS. It is a package that must be downloaded and installed. It implements part of the KSY idea, but not EDF. Dixon-EDF is implemented in Fermat as a series of procedures; see Lewis (2009). 1.5 Applications 1.5.1 Common Points of Geometrical Objects It is well known that the visualization of curves and surfaces is easy and com- fortable via parametric explicit equations of the geometrical objects. However, the implicit form of these equations is sometimes needed. For example one would like Fig. 1.6 Pose estimation problem 18 1 Solution of Algebraic Polynomial Systems
- 45. to decide whether a point is on a curve or surface or not. Finding the common points of two or more geometrical objects is the generalization of this task. Converting explicit to implicit just means eliminating the parameter. Application 1 Let us compute the implicit equation of a 2D circle. The form of the explicit equation with the parameter is, x ¼ cosðaÞ; y ¼ sinðaÞ and in addition we know that sin2 ðaÞ þ cos2 ðaÞ ¼ 1: Solution Therefore, we have the following system of equations with unknowns (x, y, a) x cosðaÞ; y sinðaÞ; 1 þ sin2 ðaÞ þ cos2 ðaÞ: and we should eliminate the variable a. Let us compute the Gröbner basis for x and y eliminating a GroebnerBasis[{x - cos[a],y - sin[a],sin[a]2 + cos[a]2 - 1}, {x,y},{a,cos[a],sin[a]}] { - 1 + x2 + y2 } This elimination could easily be done with the Dixon resultant. Note that we really have four variables x, y, cosðaÞ, and sinðaÞ and three equations. With three equa- tions we can eliminate any two variables, so we choose the latter two. Application 2 Now let us compute the common points of a cardioid and a circle. The parametric equation of the cardioid is, see Fig. 1.7, x ¼ 2 ð1 þ cosðtÞÞ cosðt), y ¼ 2 ð1 þ cosðtÞÞ sinðt): 1.5 Applications 19
- 46. Solution As a first step, we compute the implicit form of the equation of the cardioid. x 2 cosðtÞ 2 cos2 ðtÞ; y 2 sinðtÞ 2 cosðtÞ sinðt), 1 þ 2 cos2 ðtÞ þ sin2 ðtÞ: The Gröbner basis of the system is, 4x3 þ x4 4y2 4xy2 þ 2x2 y2 þ y4 : Now let us consider the following circle, x2 þ y2 2 ¼ 0: Then, the two geometrical objects together are as shown in Fig. 1.8. The next step is the computation of the common points employing these implicit equations. Then the following system should be solved g1 = - 4x3 + x4 - 4y2 - 4xy2 + 2x2 y2 + y4 - 4x3 + x4 - 4y2 - 4xy2 + 2x2 y2 + y4 g2 = x2 + y2 - 2 - 2 + x2 + y2 Fig. 1.7 A cardioid curve 20 1 Solution of Algebraic Polynomial Systems
- 47. Fig. 1.8 The two geometrical objects Fig. 1.9 The common points of the two geometrical objects The reduced Gröbner basis for the x coordinate is given as GroebnerBasis[{g1,g2},{x},{y}] { - 1 - 2x + x2 } Similarly for the y coordinate GroebnerBasis[{g1,g2},{y},{x}] { - 7 + 2y2 + y4 } or with the built-in function Solve solp = {x,y}=:Solve[{g1 = = 0,g2 = = 0},{x,y}]==Simplify 1 - ffiffiffi 2 p , - ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi - 1 + 2 ffiffiffi 2 p p n o , 1 - ffiffiffi 2 p , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi - 1 + 2 ffiffiffi 2 p p n o , n 1 + ffiffiffi 2 p , - i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 2 ffiffiffi 2 p p n o , 1 + ffiffiffi 2 p ,i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 + 2 ffiffiffi 2 p p n oo There are only two real solutions! Let us visualize the common points, see Fig. 1.9. 1.5 Applications 21
- 48. To solve this problem with the Dixon resultant, just take the three equations defining the cardioid and the one defining the circle. First, eliminate the three variables y, cos(t), and sin(t). That yields the equation for x, - 1 - 2x + x2 . Then repeat, eliminating y, cos(t), and sin(t) to get the equation for y. 1.5.2 Nonlinear Heat Transfer The nonlinear dimensionless equation of the steady state heat transfer in 1D is, d dx k h ð Þ dh dx ¼ 0: The boundary conditions are, hð0Þ ¼ 0 and hð1Þ ¼ 1: The heat transfer coefficient depending on the temperature, kðhÞ ¼ 1 þ kh: Let us approximate the temperature profile with the following polynomial, h x ð Þ ¼ x þ c1 x2 x þ c2 x3 x ; which satisfies the boundary conditions. Let us compute the ci coefficients. Solution Substituting the temperature profile into the differential equation, we get eq ¼ k þ 2c1 2kc1 þ 6kxc1 þ kc2 1 6kxc2 1 þ 6kx2 c2 1 2kc2 þ 6xc2 þ 12kx2 c2 þ 2kc1c2 6kxc1c2 12kx2 c1c2 þ 20kx3 c1c2 þ kc2 2 12kx2 c2 2 þ 15kx4 c2 2: Using the global integral method, the square of the integral should be minimized, r ¼ Z 1 0 eq2 dx ¼ k2 þ 4kc1 þ 2k2 c1 þ 4c2 1 þ 4kc2 1 þ 4k2 c2 1 þ 1 5 k2 c4 1 þ 6kc2 þ 4k2 c2 þ 12c1c2 þ 20kc1c2 þ 16k2 c1c2 þ 4 5 k2 c2 1c2 þ 6 5 k2 c3 1c2 þ 12c2 2 þ 24kc2 2 þ 84 5 k2 c2 2 þ 12 5 k2 c1c2 2 þ 20 7 k2 c2 1c2 2 þ 64 35 k2 c3 2 þ 111 35 k2 c1c3 2 þ 48 35 k2 c4 2: 22 1 Solution of Algebraic Polynomial Systems
- 49. Employing the necessary conditions of the minimum, differentiate the integral, we get an algebraic polynomial system for the unknown coefficients eq1 = D[r,c1] 4k + 2k2 + 8c1 + 8kc1 + 8k2 c1 + 4 5 k2 c3 1 + 12c2 + 20kc2 + 16k2 c2 + 8 5 k2 c1c2 + 18 5 k2 c2 1c2 + 12 5 k2 c2 2 + 40 7 k2 c1c2 2 + 111 35 k2 c3 2 eq2 = D[r,c2] 6k + 4k2 + 12c1 + 20kc1 + 16k2 c1 + 4 5 k2 c2 1 + 6 5 k2 c3 1 + 24c2 + 48kc2 + 168k2 c2 5 + 24 5 k2 c1c2 + 40 7 k2 c2 1c2 + 192 35 k2 c2 2 + 333 35 k2 c1c2 2 + 192 35 k2 c3 2 The Gröbner basis for c1, GroebnerBasis[{eq1,eq2},{c1},{c2}] {2222640000k + 25041744000k2 + 106983636480k3 + 216207482400k4 + 217869466458k5 + 105383544084k6 + 21747027960k7 + 982690800k8 + 4445280000c1 + 45638208000kc1 + 172774344960k2 c1 + 305473573440k3 c1 + 294315313236k4 c1 + 170466205476k5 c1 + 78560129424k6 c1 + 27948563160k7 c1 + 4880962800k8 c1 + 4834771200k2 c2 1 + 14644375200k3 c2 1 + 3743455968k4 c2 1 - 11828581632k5 c2 1 - 15676868844k6 c2 1 - 6328977648k7 c2 1 - 849050160k8 c2 1 + 398664000k2 c3 1 - 377496000k3 c3 1 + 1763997312k4 c3 1 + 2443968240k5 c3 1 + 2017898358k6 c3 1 + 441062580k7 c3 1 + 61164425k8 c3 10 + 52698240k4 c4 1 + 370528368k5 c4 1 + 23358168k6 c4 1 + 197374240k7 c4 1 + 18557000k8 c4 1 - 4040400k4 c5 1 - 27938400k5 c5 1 - 12698784k6 c5 1 - 6985752k7 c5 1 + 4109544k8 c5 1 - 1465920k6 c6 1 - 716616k7 c6 1 - 872028k8 c6 1 + 55600k6 c7 1 + 18640k7 c7 1 + 58568k8 c7 1 - 3120k8 c8 1 + 100k8 c9 1} and for c2, we get similar polynomial. 1.5 Applications 23
- 50. GroebnerBasis[{eq1,eq2},{c2},{c1}] { - 709927680k2 - 1419855360k3 - 904619968k4 - 194692288k5 + 4100908k6 + 1419855360c2 + 4259566080kc2 + 5168334976k2 c2 + 3237393152k3 c2 + 1556328200k4 c2 + 647559304k5 c2 + 151166960k6 c2 - 92198400c2 2 - 276595200kc2 2 + 220266368k2 c2 2 + 901524736k3 c2 2 + 983883152k4 c2 2 + 487021584k5 c2 2 + 107131248k6 c2 2 + 16464000c3 2 + 49392000kc3 2 + 253787072k2 c3 2 + 425254144k3 c3 2 + 384371484k4 c3 2 + 179976412k5 c3 2 + 43090110k6 c3 2 - 6679680k2 c4 2 - 13359360k3 c4 2 - 604072k4 c4 2 + 6075608k5 c4 2 + 5758480k6 c4 2 - 1117200k2 c5 2 - 2234400k3 c5 2 + 811440k4 c5 2 + 1928640k5 c5 2 + 1604652k6 c5 2 + 312480k4 c6 2 + 312480k5 c6 2 + 267470k6 c6 2 + 16800k4 c7 2 + 16800k5 c7 2 + 24612k6 c7 2 + 1560k6 c8 2 + 75k6 c9 2} From a practical point of view, it is more convenient to employ numerical Gröbner basis function, as in Mathematica using k = 1, sol = NSolve[{eq1,eq2}=:k ! 1,{c1, c2},Reals]==Flatten {c1 ! - 0:6251338312334316,c2 ! 0:19045444692157196} Then the temperature profile is T = h=:sol x - 0:625134 ( - x + x2 ) + 0:190454 ( - x + x3 ) Figure 1.10 shows the dimensionless temperature profile for k = 1, Fig. 1.10 The dimensionless temperature profile in case of k = 1 24 1 Solution of Algebraic Polynomial Systems
- 51. The general function for any k = j can be written as, X[j ]: = h=:(NSolve[{eq1,eq2}=:k ! j,{c1,c2},Reals]==Flatten) Let us test this function for k = 1 X[1] x - 0:625134( - x + x2 ) + 0:190454( - x + x3 ) We utilized the common capability of the Computer Algebra System (CAS) type language providing symbolic computation as well as any size of digits in order to reduce round-off error. One can realize that this example is a nice illustration of the hybrid computation, since our function is computed partly in numerical and partly in symbolic way. 1.5.3 Helmert Transformation Let us consider a 2D Helmert transformation with parameters a and b, X Y ¼ s cos X ð Þ sin X ð Þ sin X ð Þ cos X ð Þ x y ¼ a b b a x y : We have three control points in both systems, namely (Table 1.1). Assuming that these values have errors in both systems (EIV model) let us consider the adjustments as Dxi and DXi i ¼ 1; 2; 3. In order compute these adjustments the following minimization problem should be solved, F ¼ X 3 i¼1 Dx2 i þ DX2 i with the constraints, Table 1.1 Numerical data for the 2D Helmert transformation problem i xi yi Xi Yi 1 0.0 1.0 −2.1 1.1 2 1.0 0.0 1.0 2.0 3 1.0 1.0 −0.9 2.8 1.5 Applications 25
- 52. eq1 ¼ a x1 þ Dx1 ð Þ by1 X1 þ DX1 ð Þ; eq2 ¼ b x1 þ Dx1 ð Þ þ ay1 Y1; eq3 ¼ a x2 þ Dx2 ð Þ by2 X2 þ DX2 ð Þ; eq4 ¼ b x2 þ Dx2 ð Þ þ ay2 Y2; eq5 ¼ a x3 þ Dx3 ð Þ by3 X3 þ DX3 ð Þ; eq6 ¼ b x3 þ Dx3 ð Þ þ ay3 Y3: To transform this problem into a minimization without constraints, let us employ Lagrange-multipliers, G ¼ F þ X 6 i¼1 ki eqi ¼ Dx2 1 þ Dx2 2 þ Dx2 3 þ DX2 1 þ DX2 2 þ DX2 3 þ X1 by1 þ a x1 þ Dx1 ð Þ DX1 ð Þk1 þ ay1 Y1 þ b x1 þ Dx1 ð Þ ð Þk2 þ X2 by2 þ a x2 þ Dx2 ð Þ DX2 ð Þk3 þ ay2 Y2 þ b x2 þ Dx2 ð Þ ð Þk4 þ X3 by3 þ a x3 þ Dx3 ð Þ DX3 ð Þk5 þ ay3 Y3 þ b x3 þ Dx3 ð Þ ð Þk6 : Using the necessary condition, after differentiating the objective, we get the fol- lowing algebraic polynomial system for the unknowns Dx1; DX1; Dx2; f DX2; Dx3; DX3; a; b; k1; k2; k3; k4; k5; k6g 2Dx1 þ ak1 þ bk2; 2DX1 k1; 2Dx2 þ ak3 þ bk4; 2DX2 k3; 2Dx3 þ ak5 þ bk6; 2DX3 k5; x1k1 þ Dx1k1 þ y1k2 þ x2k3 þ Dx2k3 þ y2k4 þ x3k5 þ Dx3k5 þ y3k6; y1k1 þ x1k2 þ Dx1k2 y2k3 þ x2k4 þ Dx2k4 y3k5 þ x3k6 þ Dx3k6; ax1 X1 by1 þ aDx1 DX1; bx1 þ ay1 Y1 þ bDx1; ax2 X2 by2 þ aDx2 DX2; 26 1 Solution of Algebraic Polynomial Systems
- 53. bx2 þ ay2 Y2 þ bDx2; ax3 X3 by3 þ aDx3 DX3; bx3 þ ay3 Y3 þ bDx3: Substituting the numerical values for fxi; yig and fXi; Yig from Table 1.1, we get 14 polynomials of the Gröbner basis of the problem. The rows of Table 1.2 show the exponents of the unknown variables in the different polynomials, The first base is a degree six polynomial for k6, 7630949955162482528767108340 þ 42959839227889682667793048133k6 48918108637327112393858971361k2 6 þ 10461095486070027991388157780k3 6 þ 10401874932371574116079405000k4 6 3829299680266483288767890625k5 6 þ 349089071788949996689453125k6 6 Which has two real solutions: k6 ! 2:14502; k6 ! 0:238268 f g. We consider the positive solution. (The reason will be given later.) Then the solutions can be obtained with successive elimination from the other bases, Dx1 ! 0:021889; DX1 ! 0:165307; Dx2 ! 0:021537; DX2 ! 0:079912; f Dx3 ! 0:109804; DX3 ! 0:116768; a ! 1:057144; b ! 1:957834g: Table 1.2 Numerical data for the 2D Helmert transformation problem Dx1 DX1 Dx2 DX2 Dx3 DX3 a b k1 k2 k3 k4 k5 k6 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 1 5 0 0 0 0 0 0 0 0 0 0 0 1 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 1 0 0 0 5 0 0 0 0 0 0 0 0 1 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 0 0 0 0 5 0 0 1 0 0 0 0 0 0 0 0 0 0 5 0 1 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 0 0 0 0 0 5 1.5 Applications 27
- 54. Wecouldsolvetheproblemviadirectminimization,too.Employingglobalminimization method, we can get the same solution. The minimization problem has two local mini- mums and negative k6 refers to the other local minimum, which is not the global one. 1.6 Exercises 1.6.1 Solving a System with Different Techniques Let us consider the following system, f ðx; y; zÞ ¼ x y z 1; gðx; y; zÞ ¼ x2 þ 2y2 þ 4z2 7; hðx; y; zÞ ¼ 2x2 þ y3 þ 6z 7: We do not know approximate solutions, therefore we have no idea which initial values would be proper to start with in case of numerical (iterative) solutions. Problem (a) Estimate the number of common roots (b) Find common roots via Sylvester resultant (c) Find common roots via Dixon resultant (d) Find the univariate polynomials for the unknowns (x, y, z) via Gröbner basis (e) Compute the roots of these polynomials (f) Carry out the computation with built-in function NSolve (g) Employ high precision computation. Solution Considering the degree of the polynomials of the system, the total degree of the system is d ¼ 3 2 3 ¼ 18; Therefore the upper limit of the number of the common roots is 18. Solution via Sylvester resultant f = x y z - 1 - 1 + x y z g = x2 + 2y2 + 4z2 - 7 - 7 + x2 + 2y2 + 4z2 28 1 Solution of Algebraic Polynomial Systems
- 55. Exploring the Variety of Random Documents with Different Content
- 56. CHAPTER III. THE SYNOPTIC GOSPELS—CONTINUED. THE EPISTLES OF IGNATIUS—THE EPISTLE OF POLYCARP— JUSTIN MARTYR—HEGESIPPUS—PAPIAS—THE CLEMENTINES—THE EPISTLE TO DIOGNETUS. Next our author examines quotations in the Epistles of Ignatius, though he says they really appertain to a very much later period, for they are all pronounced, by a large mass of critics, spurious compositions. He suffered martyrdom, it is said, on the 20th December, A.D. 115, when he was condemned to be cast to wild beasts in the amphitheatre, not at Rome, but at Antioch, in consequence of the fanatical excitement produced by the earthquake which took place on the thirteenth of that month.[31] If any of his fifteen letters, says our author, could be accepted as genuine, the references to them might be important. Dr. Mosheim says his whole epistles are extremely dubious. The shorter of the two versions of Ignatius is, however, generally allowed to be genuine. Tischendorf says its genuineness is now generally admitted. In it we find, What would a man be profited if he should gain the whole world and lose his own soul? which of course is a quotation from Matt. xvi. 26. The next document mentioned is the Epistle of Polycarp to the Philippians, who, Irenæus says, was in his youth a disciple of the Apostle John. He was Bishop of Smyrna, and ended his life by martyrdom, A.D. 167. Irenæus knew Polycarp personally. It is said that the epistle was written before A.D. 120. Our author ascribes it to a later date, and says that there are potent reasons for considering it spurious. As, however, Irenæus, Polycarp's disciple, believed it to be
- 57. genuine, we shall take the liberty of differing from our author, and of believing it to be so. The epistle contains the following: Remembering what the Lord said, teaching: Judge not, that ye be not judged; forgive, and it shall be forgiven you; be pitiful, that ye may be pitied; with what measure you mete it shall be measured to you again; and that blessed are the poor, and those that are persecuted for righteousness' sake, for theirs is the kingdom of God. Also: Beseeching in our prayers the all-seeing God not to lead us into temptation, as the Lord said, The spirit indeed is willing, but the flesh is weak. Also: If, therefore, we pray the Lord that he may forgive us, we ought also ourselves to forgive. Our author demurs to these being quotations from our Gospels, and says they might have been from orally current accounts of the Sermon on the Mount, or from many of the records of the teaching of Jesus in circulation. Hegisippus is the next early writer referred to. He made use of the Gospel according to the Hebrews. Jerome says (confirming Eusebius) that the Gospel according to the Hebrews is written in the Chaldaic and Syriac (Syro-Chaldaic) language, but with Hebrew characters. We have, says our author, direct intimation that Hegesippus made use of the Gospel according to the Hebrews. He was one of the contemporaries of Justin—a Palestinian Jewish Christian. In order to make himself thoroughly acquainted with the state of the Church, he travelled widely, and came to Rome when Anicitus was bishop. Subsequently he wrote a work of historical memoirs in five books, and thus became the first ecclesiastical historian of Christianity. This work is lost, but portions have been preserved by Eusebius, and one other fragment is also extant. It must have been written after the succession of Eleutherius to the Roman bishopric (A.D. 177-193), as that event is mentioned in the book. The testimony of Hegesippus is of great value, not only as a man born near the primitive Christian tradition, but also as that of an
- 58. intelligent traveller amongst many Christian communities (p. 430). Hegesippus says, in the fifth book of his Memoirs, that these words ('Good things prepared for the righteous neither eye hath seen nor ear heard, nor have they entered into the heart of man,' from 1 Cor. ii. 9) are vainly spoken, and that those who say these things give the lie to the Divine writings and to the Lord saying, 'Blessed are your eyes that see, and your ears that hear,' c. This fragment is preserved by Stephanus Gobarus, a learned monophysite of the sixth century. Nothing is more certain, says our author, than the fact that, in spite of the opportunities for collecting information afforded him by his travels through so many Christian communities, for the express purpose of such inquiry, Hegesippus did not find any New Testament Canon, or, that such a rule of faith did not exist in Rome in A.D. 160 and 170. I ask, How in the world can our author be certain of this, when only portions of Hegesippus are extant? This applies generally to his argument that the silence of the early writers is of as much importance as their supposed allusions to the Gospels. Such a mode of reasoning is aptly commented upon by the Rev. Kentish Bache, in his letter to Dr. Davidson on the Fourth Gospel. He says: When but small portions of a work have been preserved to our use, it is no wonder that these portions should make no mention of many circumstances interesting and important, which the writer must certainly have known and told of. If I tear a few leaves from the middle of my English History book, I shall find on them (the few leaves) no record of the Norman Conquest or of the Battle of Waterloo. Would it thence be a fair conclusion that these events are unhistorical and fictitious? Papias is next referred to. He was Bishop of Hierapolis, in Phrygia, in the first half of the second century, and is said to have suffered martyrdom under Marcus Aurelius, about A.D. 160-167. About the middle of the second century he wrote a work in five books, called,
- 59. Exposition of the Lord's Oracles, which is lost, excepting a few fragments preserved by Eusebius and Irenæus. We have the preface to his book, which states: I shall not hesitate to set beside my interpretations all that I rightly learnt from the Presbyters, and rightly remembered, earnestly testifying to its truth. For I have not, like the multitude, delighted in those who spoke much, but in those who taught the truth; nor in those who recorded alien commandments, but in those who recall those delivered by the Lord to faith, and which come from truth itself. If it happened that any one came who had followed the Presbyters, I inquired minutely after the words of the Presbyters—what Andrew or what Peter said, or what Philip or what Thomas or James, or what John or Matthew, or what any other of the disciples of the Lord, and what Aristion and the Presbyter John, the disciples of the Lord, say; for I held that what was to be derived from books was not so profitable as that from the living and abiding voice. It is clear (says our author) from this that even if Papias knew any of our Gospels, he attached little or no value to them, and that he knew absolutely nothing of the Canonical Scriptures of the New Testament (p. 445). I remark that it is far from clear that he attached no value to our Gospels from anything he says in the fragments extant, and of course we know nothing of those portions that are lost. We know that he was making a book, consisting of what he could gather from tradition about the truth, to set beside his interpretations about the commandments delivered by the Lord to faith. There were Gospel writings in circulation, and he was supplementing what they recorded. There is positively no evidence to make us think that our present Gospels were unknown to him. He does not, in the fragments we have, mention Paul's writings, nor the Gospel of Luke, nor the Fourth Gospel, but he does allude to a book by Matthew and another by Mark, and Eusebius tells us that Papias makes use of passages taken from Peter's first epistle and John's first epistle. So, on the whole, the testimony of Papias, instead of being against is in favour of the Synoptics, and also of the Fourth Gospel; for the silence inference applies no more to it than it does to Paul and
- 60. Luke's writings, and the statement of Eusebius about John's Epistle is not to be set aside, for if John wrote it, it will be allowed he wrote the Gospel. His evidence respecting Mark is important, for the fragments contain a statement that Mark recorded what fell from Peter, writing accurately, and taking especial care neither to omit nor to misrepresent anything; and Papias says that Peter preached with a view to the benefit of his hearers, and not to give a history of Christ's discourses. Our author's inference is that it is some other person of the name of Mark that is connected with the Second Gospel, and not the Mark that Papias refers to. This is very far- fetched and improbable, for the description tallies well with our Second Gospel, and quite admits of the supposition that Mark had every opportunity of obtaining from eye-witnesses the historical materials of his Gospel. No one supposes that every statement in the book emanated from Peter's discourses. Papias is the only early writer that our author acknowledges furnishes any evidence in favour of the Synoptic Gospels. He cannot deny that he records that Matthew composed discourses of the Lord in the Hebrew tongue, but he says that totally excludes the claim of our Greek Gospel to apostolic origin. The boldness of this assertion can only be properly met by an equally explicit denial that it does anything of the kind. If the translation be a faithful one from a Hebrew version, it is of course entitled to the epithet apostolic if the original possessed it. Our author must have some peculiar notions about verbal inspiration if this be the rule he lays down. But he altogether overlooks the supposition that Matthew's Gospel was not originally written in Hebrew, notwithstanding this statement of Papias. Tischendorf, in his book issued by the Tract Society, entitled, When were our Gospels Written? maintains that the assertion of Papias rests on a misunderstanding, and he briefly states his reasons for this view. He says: This Hebrew text must have been lost very early, for not one even of the very oldest Church fathers had ever seen or used it. There were two parties among the Judaisers—the
- 61. one the Nazarenes and the other the Ebionites. Each of these parties used a gospel according to Matthew, the one party using a Greek and the other party a Hebrew text. That they did not scruple to tamper with the text, to suit their creed, is probable from their very sectarian spirit. The text, as we have certain means of proving, rested upon our received text of Matthew, with, however, occasional departures, to suit their arbitrary views. When then it was reported, in later times, that these Nazarenes, who were one of the earliest Christian sects, possessed a Hebrew version of Matthew, what was more natural than that some person or other, thus falling in with the pretensions of this sect, should say that Matthew was originally written in Hebrew, and that the Greek was only a version from it? How far these two texts differed from each other no one cared to inquire; and with such separatists who withdrew themselves to the shores of the Dead Sea, it would not have been easy to have attempted it. Jerome, who knew Hebrew, as other Latin and Greek fathers did not, obtained in the fourth century a copy of this Hebrew Gospel of the Nazarenes, and at once asserted that he had found the original. But when he looked more closely into the matter, he confined himself to the statement that many supposed this Hebrew text was the original of Matthew's Gospel. He translated it into Latin and Greek, and added a few observations of his own on it. From these observations of Jerome, as well as from other fragments, we must conclude that this notion of Papias cannot be substantiated; but, on the contrary, this Hebrew has been drawn from the Greek text, and disfigured moreover here and there with certain arbitrary changes. The same is applicable to a Greek text of the Hebrew Gospel in use among the Ebionites. This text, from the fact that it was in Greek, was better known to the Church than the Hebrew version of the Nazarenes; but it was always regarded, from the earliest times, as only another text of Matthew's Gospel. The references to Justin Martyr occupy nearly one hundred and fifty pages of the work. He was one of the most learned and one of the
- 62. earliest writers of the Church not long after the apostles. His conversion took place about the year 132, and his martyrdom, A.D. 165. In his second Apology, A.D. 139, and in his Dialogue with Tryphon the Jew, are many quotations of passages found in the Gospels. He quotes from all the four Evangelists, and our author's elaborate attempt to prove the contrary is certainly not successful. His objection, based on slight discrepancies in the words while the sense is identical, is frivolous in the extreme. Supposing there were in Justin's hands a primitive work which supplied the passages, and that work was embodied in the canonical compilation, they can be truthfully said to be quotations from the latter. The objection to his quotations on the grounds that they are not verbatim, is neutralized by the fact that neither are his quotations from the Old Testament always exact. It has been shown that if Justin did not quote from our Gospels, there must have been in his hands, in the second century, a variety of accounts of Christ's life, to which he, a leading Christian apologist, attached the greatest importance; and yet, in the course of the few following years, those accounts must have disappeared, and four others, of which this eminent Christian apologist knew nothing, must have taken their place. This would have been what Canon Westcott justly calls a 'revolution,' for it would have, in a single generation, entirely changed the records of the life of Christ publicly used by the Christians.[32] Justin quotes from a book entitled the Memoirs, which he says are called Gospels, and our author tries to make out that the passage quoted is an interpolation. It is not the only instance where the wish, and not the proof, is father to the thought. In Justin's work, the Apology, occur the words, And thou shalt call his name Jesus, for he shall save his people from their sins; which are found in the apocryphal Gospel of James, as said to the Virgin Mary, while in Matthew's Gospel they are spoken to Joseph. It is
- 63. urged that Justin must, therefore, have quoted them from a lost Gospel; but why should it be supposed so when they are in the apocryphal Gospel of James, which, Origen says, was everywhere known about the end of the second century, and which, there is good ground for believing, was written in the early part of that century? A few other passages in Justin's work, which are not found in our Gospels, may be accounted for by supposing them to be quotations either from lost Gospels, genuine or apocryphal, or tradition may have supplied them. There is no certain inference to be arrived at. Justin tells us in his first Apology (A.D. 139), that the memoirs of the apostles called evangels were read after the prophets every Lord's Day in the assemblies of the Christians. This must have reference to the writings which alone, a few years later, were universally known as the Four Gospels, or the Acts of the Apostles. The second volume of the work opens with an examination of the evidence furnished by the apocryphal religious romance generally known by the name of 'The Clementines,' which includes the Homilies, the Recognitions, and a so-called Epitome—the Homilies and Recognitions being, he says, the one merely a version of the other, and the Epitome a blending of the other two. As there are in the Clementine Homilies upwards of a hundred quotations of expressions of Jesus, or references to His history (not less than fifty passages from the Sermon on the Mount), it is important to ascertain, if possible, when they were written, and from what writings they quote. The date cannot be determined. The range of probability is from the middle of the second century. If much later, the inquiry does not amount to much, because we know, from ample evidence, such as that of Irenæus, that the Four Gospels as we have them were in existence, and read in the Churches, in the middle of the second century. We presume, therefore, our author takes an
- 64. early date for granted, or he would not have occupied forty pages in their examination. The first quotation which, he says, agrees with a passage in our Synoptics, occurs in the third Homily, p. 52: And he cried, saying, Come unto me all ye that are weary; which agrees with Matt. xi. 28. Because the quotation is not continued, but the following words are an explanation of what Come unto me, c., means—that is, who are seeking truth, and not finding it,—we are to deem it evident that so short and fragmentary a phrase cannot prove anything. I exclaim, Indeed! Not in a book that contains a hundred references to the words of Jesus! Not, considering that they are especially the words of Jesus, that no one else so said to the weary, Come unto me! Most readers will surely think the contrary should be inferred! Among the quotations are words resembling the text of Matthew xxv. 26-30: Thou wicked and slothful servant: thou oughtest to have put out my money with the exchangers, and at my coming I should have exacted mine own.[33] If this were the only reference to the Gospels as we have them, the quotation is sufficiently near to make the inference certain that such writings, in some shape, must have been in existence when the Clementine Homilies were written. This our author acknowledges, but he says (vol. ii. p. 17): If the variations were the exception among a mass of quotations perfectly agreeing with the parallels in our Gospels, it might be exaggeration to base upon such divergences a conclusion that they were derived from a different source. The variations being the rule, instead of the exception, these, however slight, become evidence of the use of a different Gospel from ours.[34] I remark, supposing this be so, that the author of these Homilies had, in the year 160, other Gospel manuscripts before him, it is not pretended that our Gospels contain all that was known of the sayings of Jesus, and all the events of His public ministry. We are told in the Fourth Gospel: There are also many other things which Jesus did, the which, if they should be written every one, I suppose
- 65. that even the world itself could not contain the books that should be written.[35] If the author of the Fourth Gospel did not include many things which he knew had been previously written about, why should we be surprised to find the authors of the Synoptic Gospels record only portions? We know that Paul wrote an epistle to the Church at Laodicea, which is not preserved to us. We hold that Paul was as much an inspired writer as any of the apostles, and instead of making all sorts of difficulties about the books we have, we ought to be grateful that they are extant. We read in Paul's Epistle to the Colossians, iv. 16: And when this epistle is read among you, cause that it be read also in the Church of the Laodiceans; and that ye likewise read the epistle from Laodicea. I wonder whether our author has an objection to the genuineness of the Epistle to the Colossians, because Epictetus, who was born at Hierapolis about A.D. 50, which was within a few miles of Colosse and Laodicea, and who would be likely to know, at that time, what was there going on, does not refer to Paul and the Churches there? But it is useless to disprove the assertion that there are no quotations from the Gospels, for we are met at every turn with the objection that those specified are probably quotations from the numerous lost Gospels known to have been in circulation. He says: The great mass of intelligent critics are agreed that our Synoptics have assumed their present form only after repeated modifications by various editors of earlier evangelical works. The primitive Gospels have entirely disappeared, supplanted by the later and more amplified versions (p. 459). The first two Synoptics bear no author's name, because they are not the work of any one man, but the collected materials of many. The third only pretends to be a compilation for private use, and the fourth bears no simple signature, because it is neither the work of an apostle nor of an eye- witness of the events it records (p. 401). I remark, if Luke's Gospel does only pretend to be for private use, does that affect its value? If
- 66. Matthew wrote at all, and our author acknowledges he did in Hebrew, his work would be likely to be translated into Greek, either by himself or some one else, and many copies circulated. Supposing the original in Hebrew to be lost, it is not probable the Greek copies could be all collected from various places, and all altered and supplemented. How could any one do this? He might write and issue a new version, but he could not suppress the original one unless all the existing copies were under his own control. As we have a certain work preserved, and no other, pretending to be Matthew's, it is highly probable that what Matthew contributed to the Church is that Gospel. A fictitious one would be less likely to be preserved than a real one, though we are asked to believe the contrary. Our author suggests that if we had the original writings we should find them minus the miracles, which is altogether inconsistent with what he has said about the prevalence of miraculous notions among the Jews at the time. At any rate, if the books in circulation did not relate miracles, they would not be in harmony with the gospel preached by Paul, and believed by the first Christians. Supposing that there were, as Luke intimates, and as our author asserts, many original writings, what more likely than that Matthew should collect some of them, and embody them, with his own record, in one book, under his own name? It is quite true that we meet with references to apostolic writings under other titles than those in the New Testament: we read of,— The Gospel according to the Hebrews. The Gospel according to the Egyptians. The Memoirs of the Apostles. The Gospel of Matthew in Hebrew. The Gospel of the Lord. The Discourses of Peter. The Collection of Discourses.
- 67. Although we do not know how these were embodied in our New Testament Scriptures, it is probable that they were in some way included, or the copies of the present Gospels may not all have uniformly borne the same titles as we know them by. In our day it is not usual for an author's name to appear in the body of his work, and often a title-page gives more than one title.[36] How few persons can give the exact title of the book known as Butler's Analogy. The value of a book does not depend essentially upon the person who wrote it. We do not know who wrote the Book of Job, many of the Psalms, the Epistle to the Hebrews, and other portions of the Bible, but it would be unwise to reject their teaching on that account. Our author says: No reason whatever has been shown for accepting the testimony of these Gospels as sufficient to establish the reality of miracles (p. 249). I remark, the question is, Do they show such insufficient testimony as to warrant the conclusion that the general evidence based on a great variety of proofs is not to be accepted? The Epistle to Diognetus is a short composition, which has been ascribed to Justin Martyr, but its authorship is uncertain, and the date of its composition. It is not quoted or mentioned by any ancient writer. The two concluding chapters are supposed to have been written by a different hand. To the first quarter of the second half to the end of that century the date is variously assigned. It is written in pure Greek, and is elegant in style. Bunsen, in his valuable book, Hippolytus and his Age, asserts that the epistle is certainly the work of a contemporary of Justin the Martyr; that he believes he has proved that the first part is a portion of the lost early Letter of Marcion, of which Tertullian speaks; and that the very beautiful and justly admired second fragment, which in our editions of Justin's works is given at the end of that Patristic gem, the Epistle to Diognetus,[37] does not belong to that letter, but is the conclusion of the great work, in ten books, by Hippolytus, The Refutation of all Heresies. Our author, in the eighteen pages devoted to the Epistle to Diognetus, says nothing of this, although it is both important and
- 68. interesting. He says the supposed allusions in the Fourth Gospel may be all referable to Paul's epistles, that the date and author are unknown, and that the letter is of no evidential value. His two brief allusions to Bunsen's work show that the ignoring of that eminent man's opinion was not unintentional; while the absence of any reference to Bunsen's elaborate proof that Hippolytus wrote the Refutation, is also significant.
- 69. CHAPTER IV. THE SYNOPTIC GOSPELS—CONTINUED. It remains a possibility that Christ actually was what He supposed Himself to be. John Stuart Mill.
- 70. CHAPTER IV. THE SYNOPTIC GOSPELS—CONTINUED. BASILIDES—VALENTINUS—MARCION—TATIAN—DIONYSIUS OF CORINTH—MELITO OF SARDIS—CLAUDIUS APOLLINARIS —ATHENAGORAS—EPISTLE OF VIENNE AND LYONS— PTOLEMÆUS, HERACLEON, CELSUS—CANON OF MURATORI. Our author says of Basilides, He was founder of a system of Gnosticism, who lived at Alexandria about the year 125. With the exception of a very few brief fragments, none of his writings have been preserved, and all our information regarding them is derived from writers opposed to him. Eusebius states that Agrippa Castor, who had written a refutation of the doctrines of Basilides, 'Says that he had composed twenty-four books upon the gospel.' This is interpreted by Tischendorf to imply that the work was a commentary upon our four Gospels, a conclusion the audacity of which can scarcely be exceeded (p. 42). I remark that by the gospel would be meant the gospel which was preached by the apostles, and Tischendorf is not far wrong in supposing that the written records of it in the hands of the first Christians was the subject of the commentary. Our author has certainly not proved the contrary. He says: We know that Basilides made use of a Gospel, written by himself it is said, but certainly called after his own name; ... but the fragments of that work which are extant are of a character which precludes the possibility of the work being considered a Gospel. Neander affirmed the Gospel of Basilides to be the Gospel according to the Hebrews. I remark that that is not only probable, but that the Gospel to the Hebrews may have been the Hebrew translation of the Greek Gospel of Matthew, with its additions and modifications, to
- 71. suit the Jewish Nazarene sect, who, we know, had a Hebrew text of their own, which they did not hesitate to alter and adapt to their own views. Basilides, says our author, expressly states that he received his knowledge of the truth from Glaucis, the interpreter of Peter, whose disciple he claimed to be. Basilides also claimed to have received from a certain Matthias the report of private discourses which he had heard from the Saviour for his special instruction. Canon Westcott writes: Since Basilides lived on the verge of the apostolic times, it is not surprising that he made use of other sources of Christian doctrine besides the canonical books. The belief in Divine inspiration was still fresh and real.[38] Our author says: It is apparent, however, that Basilides, in basing his doctrine on these apocryphal books as inspired, and upon tradition, and in having a special Gospel called after his own name, ignores the canonical Gospels, offers no evidence for their existence, but proves that he did not recognise any such works as of authority. I remark, the question is not their authority, but, Did they exist? Basilides wrote a book, called it a Gospel, or commentary of the Gospel, and made as much use as suited his heretical purpose of the canonical records, of tradition, and of other books. This seems to be what we can arrive at. Hippolytus, writing of the Basilideans and describing their doctrines, uses the singular pronoun he—he says, in a passage of which our author gives an unintelligible translation. This pronoun is an inconvenient witness. Our author wants it to be they, in order that the disciples of Basilides living at a later period, when the Gospels were generally recognised, may be meant, and not Basilides, who lived A.D. 125. Hippolytus has a sentence of Basilides, which our author translates as follows:—Jesus, however, was generated according to these, as we have already said. But when the generation which has already been declared had taken place, all things regarding the Saviour, according to them, occurred in a similar way as they have been written in the Gospel. This means that the things referring to the Incarnation were as written in the Gospel, not as preached, but as written; and if Basilides, as the founder of the sect, is referred to, the statement testifies to the
- 72. existence of the Gospels in the year 125, and the doctrine of the Incarnation being in them. But our author says the statement is not made in connection with Basilides, but his followers; that it is made about A.D. 225, by Hippolytus, and affords no proof that either Basilides or his followers used the Gospels or admitted their authority. The exclusive use, by any one, of the Gospel according to the Hebrews, for instance, would be perfectly consistent with the statement (p. 48). No one who considers what is known of that Gospel, or who thinks of the use made of it in the first half of the second century by perfectly orthodox Fathers, before we hear anything of our Gospels, can doubt this (p. 48). I remark, that those who adopt Tischendorf's view, that Matthew was written in Greek, and a corrupted version in Hebrew, used in certain countries, will not have to resort to any such explanation as our author suggests. His examination in detail of the several quotations is important, because it exhibits his want of appreciation of the evidence they afford. The first passage Tischendorf points out is found in the Stromata of Clement of Alexandria, and it is certainly from our Gospel of Matthew,[39] however that work may have been compiled (for it is not necessary to insist that no other records than Matthew's own are included in the book which, we contend, was at very early date read in the Churches, and is what we now have). They say the Lord answered, All men cannot receive this saying. For there are eunuchs who are indeed from birth, but others from necessity.[40] Our author says this passage in its affinity to, and material variation from, our First Gospel, might be quoted as evidence for the use of the Gospel according to the Hebrews, but it is simply preposterous to point to it as evidence for the use of Matthew. Apologists ... seem altogether to ignore the history of the creation of written Gospels, and to forget the very existence of the πολλοἱ of Luke. We value his acknowledgment, and find no difficulty, notwithstanding the silence of some apologists, in reconciling our belief in the four Gospels with the facts or probabilities of what can be ascertained as to their creation. We allow that the word Luke uses (πολλοἱ) refers to many, which is
- 73. consistent with the idea that many committed to writing what they knew, and that their records were embodied in the Synoptic Gospels. The next passage referred to by Tischendorf is one quoted by Epiphanius: And therefore he said, Cast not ye pearls before swine, neither give that which is holy unto dogs.[41] It is introduced in the section of the work of Epiphanius directed against the Basilideans. As in dealing with all these heresies there is continual interchange of reference to the head and later followers, there is no certainty who is referred to in these quotations, and in this instance nothing to indicate that the passage is ascribed to Basilides himself. His name is mentioned in the first line of the first chapter, but not again until the fifth chapter (p. 50). I remark, it was the founder of the sect and not the followers who wrote the book, and those who opposed the heresy would, although they alluded to the sect, have regard to the founder when they referred to the doctrines held, and quoted the written opinions which distinguished the party on gospel matters. To make the matter as plain as I can, I will suppose a case as an illustration of the point. Supposing that in Pliny's letter to Trajan there were found these words referring to the Christians: They say, the rule which should be observed in regard to an enemy is, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which persecute you—would it be right to assert that the quotation is no proof that Christ so taught, but His disciples, long afterwards? This is something like what our author's objection, referring to the pronouns he and they in Hippolytus, amounts to. They does not mean he when thus used; and he, when actually used in the first line of the first chapter, and afterwards means, they; that is, He (Basilides) says, means They (his followers at a later date) say. The plural pronoun is used, indicating the sect, Basilides and his followers. Therefore our author says there is uncertainty as to who he is when used in the same sentence. He says Hippolytus is giving
- 74. an epitome of the views of the school with nothing more definite than a subjectless φησἱ (he says) to indicate who is referred to. None of the quotations which we have considered are directly referred to Basilides himself, but they are introduced by the utterly vague expression, 'He says' (φησἱ), without any subject accompanying the verb. The suggestion (p. 51) that Hippolytus consciously or unconsciously, in the course of transfer to his pages, corrected the text, is very unsatisfactory. An intelligent reader cannot fail to see how an obvious inference is avoided, and how ingenuity is taxed to make words square with foregone conclusions. Tischendorf asks: Who is there so sapient as to draw the line between what the master alone says, and that which the disciples state, without in the least repeating the master? (p. 59) and our author says, Tischendorf solves the difficulty by referring everything indiscriminately to the master (p. 59). To say that Tischendorf does this is reckless assertion. When our author has to account for such a passage in Basilides as, The Holy Spirit shall come upon thee, and the power of the Highest shall overshadow thee, he says it happens to agree with the words in Luke i. 55; and resorts to his usual mode of avoiding the acknowledgment that such a verbatim quotation is against his hypothesis, by saying, There is good reason for concluding that the narrative to which it belongs was contained in other Gospels. The following sentence is startling, and apt to mislead those who do not take the trouble to be sure of his meaning. He says (p. 67): Nothing, however, can be clearer than the fact that this quotation, by whomsoever made, is not taken from our Third Synoptic, inasmuch as there does not exist a single MS. which contains such a passage. What does he mean? We turn to Luke i. 35, and read: The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God. Does he mean the whole passage is not in any MS? No: he means the following, with
- 75. the slight variation at the end, is not in any MS. The Holy Spirit shall come upon thee, and the power of the Highest shall overshadow thee, therefore the thing begotten of thee shall be called holy. Only the words in italics are different in the two passages, and the meaning is the same, the only difference being that the latter does not include the words the Son of God. The remark that the quotation happens to agree with the passage in Luke i. 35, should not be unnoticed. Happens! Mark the peculiar inappropriateness of the word. It indicates our author's whereabouts, and is a beacon in the book to warn the reader. Events transpire, and they happen to agree with prophetic visions which plainly foretold them! Reason being unequal to an explanation, coincidence must be resorted to. Was it an accident that, at one particular point in history, and in one special individual, the elements of a new religious development, which, per se, were already extant, should have concentrated themselves in a new life? This, says Baur, is the wonder in the history of the origin of Christianity which no historical reflection can further analyse. Did it happen that the Messiah came as was predicted centuries before? Did Paul happen to have a vision just at the time when the whole course of his life underwent a change, and from being a chief persecutor of the faith he became a chief apostle—no less an apostle than the most prominent among the Twelve? If the Saviour did not meet him on the way to Damascus he could not be an apostle; and as he was an honest man, and no impostor, could what happened to him have been other than what he asserted? Baur was in a great difficulty about the matter, and said, No analysis, either psychological or didactic, can clear up the mystery of that act in which God revealed His Son in Paul. Jeremiah prophesied that the Jews should return to their own land after seventy years of exile, and they happened to do so! The artful way in which the evidence from the writings of Hippolytus is disposed of is one of the most notable things in the book we are reviewing. The reader's attention is taxed to keep up with the
- 76. sophistical argument, and our author finds it necessary to explain why he has been forced to go at such a length into these questions, as to risk being very wearisome to his readers (p. 73). These remarks apply to a great extent to the examination of the evidence of Valentinus, described as another Gnostic leader, who, about the year A.D. 140, came from Alexandria to Rome, and flourished till about A.D. 160. Very little remains of the writings of this Gnostic, and we gain our only knowledge of them from a few quotations in the works of Clement of Alexandria, and some doubtful fragments preserved by others (p. 56). Marcion, the son of a bishop of Pontus, became a conspicuous heretic in the second century, and there was a book called Marcion's Gospel, which has long furnished a field for criticism. He was a Pauline heretic, denouncing the Jewish party which insisted upon dragging Jewish observances into Christianity. He went to Rome about A.D. 139-142, and taught there some twenty years. His opinions were widely disseminated. His collection of apostolic writings, which is the oldest of which we have any trace, includes (says our author) a single Gospel and ten Epistles of Paul—viz., Galatians, Corinthians (2), Romans, Thessalonians (2), Ephesians (in the superscription of which there is, to the Laodiceans), Colossians, Philippians, and Philemon. The Gospel of Marcion is not extant, but it is referred to by his opponents, who affirmed that his evangelical work was an audaciously mutilated version of Luke's Gospel. Our author gives a brief account of the various opinions which have prevailed about the book during the last hundred years, and considers the discussion upon it far from closed. Is it a mutilation of Luke, or an independent work derived from the same source as his, or is it a more primitive version of that Gospel? Whence are the materials from which the portions of the text extant are derived? Tertullian and Epiphanius denounced Marcion's heresy. The former called him impious and sacrilegious, which, our author says, implies anything but fair and legitimate criticism. I remark, Did he deserve the epithets? Would
- 77. Paul, who tells the Colossians to beware lest any man spoil them through philosophy and vain deceit, after the traditions of men, after the rudiments of the world, and not after Christ, have been less emphatic in his denunciations in such a case? Marcion was more Pauline than Petrine, but would Paul have failed to censure in the strongest language such a misrepresentation of Jehovah and the Old Testament economy as Marcion disseminated? Can our author's assertion be absolutely true that Tertullian and Epiphanius were only dogmatical, and not in the least critical? How could they be otherwise than to a certain extent critical? They were not critics in the way of taking nothing for granted, after the modern fashion; but they must have weighed, compared, and tested Marcion's views while writing against them. The spirit of the age, he says, was indeed so uncritical, that not even the canonical text could awaken it into activity. This is a sentence which suggests that the position in the Church of the canonical text was so evident, that to question it was then unwarrantable, as, indeed, it has continued to be to this day. The combined internal and external evidences harmonising with the believer's consciousness, his necessities, and his aspirations, were sufficient to preclude sceptical and captious criticism. The Christian contemporaries of Irenæus, Tertullian, and Epiphanius were uncritical in that they did not doubt that the foundations of their faith were sure. The gospel which had been preached to them, which had changed the whole course of their lives, corresponded in its main features with the four books which were held in estimation by the Church at that time above all other writings; and they would not be likely to wrangle about the title instead of cultivating the faith they possessed. They could not, perhaps, prove by the rules of logic that God is, and is the rewarder of them that diligently seek him; that Christ is the brightness of the Father's glory, and the express image of his Person; but they knew that He had said,—Ye believe in God believe also in me; In my Father's house are many mansions; and, I go to prepare a place for you. Be thou faithful unto death,
- 78. and I will give thee a crown of life. They lived in the consciousness of these truths, and died (Bishop Pothinus, for instance) a martyr's death rather than deny them. There is this remark to be made in reference to the alleged uncritical age of the Fathers. How is it that Marcion is seen to be so critical? He is surely after the modern model. He who wrote the Antithesis, and, as our author says, anticipated in some of his opinions those held by many in our own time; he who wrote,—If the God of the Old Testament be good, prescient of the future, and able to avert evil, why did he allow man, made in his own image, to be deceived by the devil, and to fall from obedience of the law into sin and death?[42] How came the devil, the origin of lying and deceit, to be made at all?[43] surely he is an instance of a man in that age possessing the critical faculty. He has the boldness to question, and say,—Yea, hath God said? Anticipating the results of modern criticism, says our author, Marcion denies the applicability to Jesus of the so-called Messianic prophecies (p. 106). If the research which is going on as to the Gospel of Marcion be conducted in a proper manner, and from a proper motive, not from antipathy to parsons and ecclesiastical assumptions, which was the incentive of Strauss in attacking Christianity, good will come of it. As Justin Martyr did not, as far as we know, suppose the book to be a corrupted version of the Gospel according to Luke, Tertullian may have been mistaken, and it may have been an independent work, one of the many Luke refers to, the existence of which does not necessarily invalidate the canonical ones. We may naturally suppose that events of such marvellous speciality and importance as those which had come to pass in those days among the Jews, would be more or less described in letters and other writings by many persons who were eye-witnesses. Such writings would be collected and read when the first Christians assembled. The difference between the four canonical Gospels and other manuscripts would consist in their being compiled by persons competent to the task, who, like Ezra, were instruments Divinely influenced to compile and set forth in order a
- 79. declaration of those things, for the benefit of future ages and the religious instruction of the race. The analysis of the text of Marcion by Hahn, Ritschl, Volkmar, Helgenfeld, and others, who have examined and systemised the data of the Fathers, is supposed to be sufficient to awaken in any inquirer uncertainty, and stimulate conjecture (p. 101). I do not doubt it. German hypercriticism is able, by a process of ratiocination, to discredit any truth, even to persuade men that the Throne of the universe is vacant, and that the only altar that man has the knowledge to rear is one to the Unknown God; but
- 80. He sits on no precarious throne, Nor borrows leave to be. They who believe in the inspiration by the Holy Ghost of the prophets of the Old Testament see no difficulty in regard to the inspiration of the writers of the New. If Isaiah and Jeremiah and Daniel had supernatural communications made to them, in order that the Eternal Creator might be manifested, why not Paul and John and Matthew? It is the foregone conclusion, on the part of critics, that the miraculous is impossible, which embarrasses their researches. One of John Stuart Mill's last sentences is: It remains a possibility that Christ actually was what He supposed Himself to be. If this had occurred to the great reasoner at the outset of his career instead of the close, how much might the world have been advantaged! Tatian is a witness whose evidence our author next tries to set aside. He was an Assyrian by birth, a disciple of Justin Martyr at Rome, and afterwards, having joined the sect of the Eucratites, a conspicuous exponent of their austere and ascetic doctrines. The only one of his writings extant is his Oration to the Greeks, written after Justin's death, as it refers to that event, and it is generally dated A.D. 170- 175. One point contested is Canon Westcott's affirmation that it contains a clear reference to a parable recorded by Matthew:[44] The kingdom of heaven is like unto treasure hidden in a field, which a man found and hid, and for his joy he goeth and selleth all that he hath and buyeth that field. And the supposed reference by Tatian is, For by means of a certain hidden treasure he has taken to himself all that we possess, for which, while we are digging, we are indeed covered with dust, but we succeed in making it our fixed possession.[45] There is certainly not much similarity between the two passages, although Tatian may be well supposed to have had the parable in his mind when he wrote. The more important question is, Did Tatian write A Harmony of Four Gospels, which recognises our four
- 81. Evangelists? Was his Diatessaron such a book, or was it the Gospel according to the Hebrews? If the latter, what is the Gospel according to the Hebrews? I say it is probable it is the corrupted Hebrew translation of the Greek Gospel of Matthew, and this conjecture has more in its favour than our author's hypothesis. Dionysius of Corinth, Eusebius tells us, wrote seven epistles to various Churches, and a letter to Chrysophora, a most faithful sister. Only a few short fragments exist, which are all from the epistle to Soter, Bishop of Rome, whose date in that pastorate is A.D. 168-176. In these fragments we find the following words:—For the brethren having requested me to write epistles, I write them. And the apostles of the devil have filled these with tares, both taking away parts and adding others, for whom the woe is destined. It is not surprising, then, if some have recklessly ventured to adulterate the Scriptures of the Lord, when they have corrupted these, which are not of such importance.[46] After quoting this passage, our author reiterates his statement that We have seen that there has not been a trace of any New Testament Canon in the writings of the Fathers before and during this age. Does he suppose his readers will have seen as he sees, or rather refuse to see what is plain enough? He has his own opinion, but he need not assume that he has convinced his readers that he has proved what he alleges. He talks of Westcott's boldness, and of his imagination running away with him, and that it is simply preposterous to suppose that this passage refers to the New Testament. I leave Canon Westcott to defend his own words, but I say it is not preposterous to infer that when Dionysius speaks of the Scriptures of the Lord he means Gospel writings, which are included in our New Testament. If it be assumed that the defence of the authority of the New Testament writings and of evangelical views is necessarily based on the synodical authority of the early Church, there may be some weight in his objections; but Christianity has a position independent of ecclesiastical pretensions to infallibility, and the latter may be overthrown without the great institution established by Divine mercy for the recovery of humanity from sin and its consequences being in
- 82. the slightest degree damaged. Dr. Donaldson is quoted, who remarks: It is not easy to settle what this term, 'Scriptures of the Lord,' is; but my own opinion is that it most probably refers to the Gospels, as containing the sayings and doings of the Lord. It is not likely, as Lardner supposes, that such a term would be applied to the whole of the New Testament.[47] The word Scripture, in Greek, ΓραφἡΓραφἡ (Graphé), in Latin, Scriptura, has, no doubt, a meaning which denotes an inspired writing. It is used fifty-one times in the New Testament in the same sense, for Christ and the authors of the New Testament regarded the Old Testament as distinguished from all other writings, as the writing—the writing of God. By speaking of their own books as Graphai, the apostles place them on a level with the Old Testament, and thus assert their Divine character.[48] Dr. Davidson speaks of the New Testament writings being ranked as Holy Scripture by Dionysius of Corinth, A.D. 170. Our author asserts (p. 167) that many works were regarded as inspired by the Fathers besides those in our Canon, and mentions especially the Gospel of Peter having been read at Rhossus. He says: The fact that Serapion, in the third century, allowed the Gospel of Peter to be used in the Church of Rhossus shows the consideration in which it was held, and the incompleteness of the canonical position of the New Testament. Now, he ought to have quoted Serapion's own explanation, which we have preserved by Eusebius. He says (in his treatise written to confute what was false in the Gospel of Peter): We receive Peter and the other apostles even as Christ; but the writings falsely called by their names, we, as competent critics, renounce, knowing that we received not such things. For when I was with you I supposed that all were agreed with the true faith; and, without reading the Gospel called Peter's, which they brought forward, I said, If this is the only thing that seems to cause you dissension, let it be read. Serapion says he borrowed the book and read it, and found many things agreeable to Christ's doctrine, but some discrepant additions.
- 83. Thus the reading of the Gospel of Peter at Rhossus cannot be instanced as a proof that other Gospels besides the canonical ones were used as inspired books, nor can any other be mentioned as having been thus regarded, the Gospel according to the Hebrews not being apocryphal, but a part of the New Testament, whether we take it to be, as our author supposes, the basis of Matthew's Gospel, or, as we say, a corrupted version of that apostle's Greek work. To argue that because one spurious Gospel was temporarily received among a few persons, therefore there was no real canon of Scripture, and we cannot be sure that any Gospel is genuine, shows about as much common sense and logical acumen as would be displayed by a critic eighteen centuries hence, who, discovering in one of our newspapers an account of the conviction of a gang of coiners, should argue that because their base half-crowns had got into circulation, and had passed current with some persons who might have been expected to detect the fraud, therefore there was no such thing as a legal currency of intrinsic value among us; or if there were, still we did not know or care to inquire into the genuineness of the coin which we accepted and passed.[49] Our author says (p. 16): 'The Pastor of Hermas,' which was read in the churches, and nearly secured a permanent place in the Canon, was quoted as inspired by Irenæus.[50] The word Irenæus uses is Graphé, which is sometimes translated, when found in his works, Scripture, and at other times writings, as may best suit the argument of a critic like Dr. Davidson, who does so adapt the translation to suit his purpose. Whatever erroneous notions might prevail as to apocryphal writings, the discrimination of Serapion, in regard to the Gospel of Peter, shows that such a work as the Pastor of Hermas, in which, as Mosheim says, the angels are made to talk more insipidly than our scavengers and porters, would not be put on a level with the books whose internal evidence, as well as historical pretensions, placed them in a much superior position. The contrast is too great for such
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