Scilab for very beginners
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This document provides an introduction to using Scilab for beginners. It describes the key windows in the Scilab environment, including the console, editor, and graphics window. It also covers basic calculations, variables, plotting, and functions. The document aims to guide new users through exploring Scilab's essential features in a simple manner.
![Scilab
for
very
beginners
-‐
14/33
Requesting
the
assignment
of
a
variable
Assignment
is
made
easily
using
the
“
=
“
operator.
Display
Writing
Typing
the
name
of
a
variable
displays
its
value,
except
when
there
is
“
;
“
at
the
end
of
the
command
line.
Brackets
Matrices
are
defined
using
square
brackets
(see
page
23).
As
mentioned
before,
matrix
computation
is
the
basis
of
calculations
in
Scilab.
A
space
or
comma
is
used
to
separate
columns
and
semicolons
are
used
to
separate
rows.
To
define
a
column
vector
and
obtain
a
column
display:
-->v=[3;-2;5]
v =
3.
- 2.
5.
To
define
a
row
vector
and
obtain
a
row
display:
-->v=[3,-2,5]
v =
3. - 2. 5.
To
define
a
matrix
and
obtain
a
tabular
display:
-->m=[1 2 3;4 5 6;7 8 9]
m =
1. 2. 3.
4. 5. 6.
7. 8. 9.
Mention
This
command
can
also
be
typed
under
the
form:
v=[3 -2 5]
Mention
This
command
can
also
be
typed
under
the
form:
m=[1,2,3;4,5,6;7,8,9]](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-14-320.jpg)
![Scilab
for
very
beginners
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16/33
or
which
decreases
in
steps
of
4
from
20
to
2:
-->u=20:-4:2
u =
20. 16. 12. 8. 4.
For
The
easiest
loop
structure
for
a
fixed
number
of
iterations
is
written
with
for … end.
Example:
Calculation
of
20
terms
of
a
sequence
defined
by
recurrence
by:
푢! = 4
푢!!! = 푢! + 2푛 + 3
Algorithm
Scilab
Editor
Put
4
in
u(1)
For
n
from
1
to
20
u(n+1)
takes
the
value
u(n)+2n+3
and
u(n)
Display
n
and
u(n)
End
for
u(1)=4;
for n=1:20
u(n+1)= u(n)+2*n+3;
disp([n,u(n)])
end
While
To
stop
a
loop
when
a
given
goal
is
reached,
while … end
is
used.
I
planted
a
Christmas
tree
in
2005
measuring
1.20
m.
It
grows
by
30
cm
per
year.
I
decided
to
cut
it
when
it
exceeds
7
m.
In
what
year
will
I
cut
the
tree?
Algorithm
Scilab
Editor
Put
1.2
in
h
(h
=
tree
height)
Put
2005
in
y
(y
=
year)
While
h<7
h
takes
the
value
h+0.3
(the
tree
grows)
y
takes
the
value
y+1
(one
year
passes)
End
while
Display
y
(the
final
year)
h=1.2;
y=2005;
while h<7
h=h+0.3;
y=y+1;
end
disp("I will cut the..
tree in "+string(y))
Mention
When
a
command
is
too
long
to
be
written
in
one
line,
the
editor
will
wrap
lines.
One
can
also
wrap
lines
using
“
..
“
(two
dots)
before
going
to
the
next
line.](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-16-320.jpg)
![Tests
Comparison
operators
Useful
tests
include
comparing
numbers
or
determining
whether
a
statement
is
true
or
false.
Below
the
corresponding
commands:
Equal
Different
Less
Greater
Less
or
equal
Greater
or
equal
== <> < > <= >=
True
False
And
Or
No
%T %F & | ~
Scilab
for
very
beginners
-‐
17/33
To
compare
two
vectors
(or
two
matrices),
the
tests
“
==
“
and
“
<>
“
will
compare
term
by
term.
For
example:
-->X=[1,2,5]; Y=[5,3,5];
-->X==Y
ans =
F F T
To
test
if
two
vectors
are
equal,
isequal
is
used
and
~isequal
is
used
if
they
are
different:
-->isequal(X,Y)
ans =
F
-->~isequal(X,Y)
ans =
T
If…then
The
basic
conditional
statements
are
the
following:
• if … then … else … end,
• if … then … elseif … then … else … end.
if … then
must
be
written
on
the
same
line
and
likewise
with
elseif … then.
Mention
Be
cautious
with
calculation
accuracy.
Calculations
are
approached
and
“
==
“
will
sometimes
give
wrong
results
(see
Calculation
accuracy,
page
30).](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-17-320.jpg)
![Scilab
for
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beginners
-‐
18/33
Example:
Alice
throws
three
dice.
• If
she
gets
three
6’s
she
wins
$20,
• If
she
gets
three
identical
numbers
different
from
6
she
wins
$10,
• If
she
gets
two
identical
numbers
she
wins
$5,
• Otherwise
she
wins
nothing.
Simulate
a
trial
and
calculate
Alice’s
winnings
using
the
functions:
• grand
(see
page
22),
• unique(D)which
keeps
only
once
the
values
that
appears
several
times
in
D,
• length(unique(D))
which
returns
the
size
of
the
obtained
vector,
that
is
to
say
1
if
three
dice
are
equal,
and
2
if
two
dice
are
equal.
Algorithm
Scilab
Editor
Put
the
three
values
in
D
If
Alice
gets
three
6,
then
Alice
wins
20
dollars
Else
if
she
receives
three
identical
values,
then
Alice
wins
10
dollars
Else,
if
she
receives
two
identical
values,
then
Alice
wins
5
dollars
Otherwise
Alice
wins
nothing
End
if
Display
Alice's
winnings
D=grand(1,3,"uin",1,6);
if D==[6,6,6] then
W=20;
elseif length(unique(D))==1 then
W=10;
elseif length (unique(D))==2 then
W=5;
else
W=0;
end
disp("Alice wins "+..
string(W)+ " dollars")
2-‐D
and
3-‐D
plots
plot
command
is
used
to
create
plots
in
the
plane.
Color
and
appearance
can
be
specified
by
putting
indications
of
color
and
points
style
within
quotes:
• Colors
"b"
=
blue
(by
default),
"k"
=
black,
"r"
=
red,
"g"
=
green,
"c"
=
cyan,
"m"
=
magenta,
"y"
=
yellow,
"w"
=
white.
• Point
styles
Joined
(by
default),
or
".",
"+",
"o",
"x",
"*".
Other
options
are
available
with:
"s",
"d",
"v",
"<",
and
">".](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-18-320.jpg)
![Scilab
for
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beginners
-‐
19/33
Basic
plots
To
plot
a
point
Plot
the
point
A(1
;
2)
with
a
red
point.
Scilab
Editor
Graphics
Window
plot(1,2,".r")
To
plot
a
segment
Plot
the
segment
[AB]
in
blue
(by
default)
with
A(1
;
2)
and
B(3
;
5).
Scilab
Editor
Graphics
Window
plot([1,3],[2,5])
Plots
of
plane
curves
defined
by
functions
y=f(x)
For
a
function
푥 → 푓(푥)
the
values
of
x
are
specified
using
the
linspace command
by
writing:
x=linspace(a,b,n);
in
which
a
is
the
smallest
value
of
the
variable
푥,
b
the
highest
value
of 푥,
and
n
the
number
of
values
calculated
between
a
and
b.
Do
not
forget
the
“
;
“
sympbol
otherwise
all
n
values
of
푥
will
be
displayed.
For
example,
consider
two
functions
푓
and
푔
defined
over
the
interval
[-‐2
;
5]
by:
푓 푥 = (푥! + 2푥)푒!!,
and
푔 푥 = sin !
!](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-19-320.jpg)
![Bivariate
statistical
data
Bivariate
statistical
data
are
given
in
the
form
of
two
vectors:
let’s
call
them
X
and
Y.
plot(X,Y,"<")
Scilab
for
very
beginners
-‐
21/33
will
plot
a
scatter
plot
of
푀(푋!; 푌!)
with
blue
triangles.
Scilab
Editor
Graphics
Window
X=[1,3,3,7,7,9,10];
Y=[8,7,5,5,4,2,2];
clf; plot(X,Y,"<")
Plots
in
3
dimensions
Scilab
can
be
used
to
plot
surfaces
and
curves
in
space,
with
many
options
for
the
treatment
of
hidden
faces,
face
colors,
points
of
view,
etc.
The
following
examples
will
illustrate
3-‐D
plots.
The
surf
function
can
be
used
for
plotting
surfaces.
This
function
has
three
input
variables,
x,
y
and
z.
x
and
y
are
respectively
vectors
of
size
푚
and
푛
corresponding
to
points
on
the
axes
(푂푥)
and
(푂푦).
z
is
a
matrix
of
dimension
푛×푚
with
element
푧!"
corresponding
to
the
height
of
the
point
with
X-‐coordinate
푥!
and
Y-‐coordinate
푦!.
To
plot
the
surface
defined
by
a
function
of
the
form
푧 = 푓(푥, 푦),
it
is
necessary
to
:
• Define
function
푓
• Calculate
z=feval(x,y,f)'
feval(x,y,f)
returns
the
푚×푛 matrix
whose
푖푗
element
is
푓(푥!, 푦!)
which
will
be
transposed
by
using
the
single
quote
symbol
“
'
“
• Execute
surf(x,y,z).
To
plot
the
surface
푧 = 2푥! + 푦!
(elliptic
paraboloid):
Scilab
Editor
Graphics
Window
function z=f(x,y)
z=2*x^2+y^2;
endfunction
x=linspace(-1,1,100);
y=linspace(-2,2,200);
z=feval(x,y,f)';
clf
surf(x,y,z)](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-21-320.jpg)
![Scilab
for
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beginners
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23/33
Statistics
All
the
standard
statistical
functions
are
listed
on
page
32.
Keep
particularly
in
mind:
• The
function
bar(x,n,color)
which
draws
bar
graphs:
Scilab
Editor
Graphics
Window
x=[1:10];
n=[8,6,13,10,6,4,16,7,8,5];
clf; bar(x,n)
• For
a
bar
graph
representing
two
sets
side
by
side:
consider
the
series
of
values
X
and
the
two
series
of
numbers
n1
and
n2.
For
plotting,
n1
and
n2
must
be
column
vectors
that
is
why
transposes
are
used
in
the
example
below:
Scilab
Editor
Graphics
Window
X=[1,2,5];n1=[5,10,5];n2=[6,8,7];
bar(X,[n1',n2'])
The
color
optional
argument
defines
the
color
as
in
the
plot
function.
Additional
information
on
matrices
and
vectors
Accessing
elements
Square
braces
are
used
to
define
matrices.
A
space
or
a
comma
is
used
to
switch
from
one
column
to
another
and
semicolons
are
used
to
switch
from
one
line
to
another.
-->m=[1 2 3;4 5 6]
m =
1. 2. 3.
4. 5. 6.
Parentheses
are
used
to
access
elements
or
to
modify
them.
Mention
This
command
can
also
be
typed
under
the
form:
m=[1,2,3;4,5,6]](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-23-320.jpg)
![Scilab
for
very
beginners
-‐
24/33
-->m(2,3)
ans =
6.
-->m(2,3)=23
m =
1. 2. 3.
4. 5. 23.
The
“
:
“
operator
is
used
to
designate
all
the
rows
or
all
columns
of
a
matrix.
To
view
the
second
row
of
the
matrix
m,
type:
-->m(2,:)
ans =
4. 5. 23.
And
the
third
column:
-->m(:,3)
ans =
3.
23.
To
obtain
the
transpose
of
a
matrix
or
a
vector
use
the
single
quote
symbol
“
'
“:
-->m'
ans =
1. 4.
2. 5.
3. 23.
Operations
The
operations
“
*
“,
“
/
“
are
matrix
operations.
To
make
element
wise
operations,
we
need
to
put
a
dot
before
the
operating
sign:
“
.*
“,
“
./
“.
-->A=[1,2,3;4,5,6]
A =
1. 2. 3.
4. 5. 6.
-->B=[1;1;2]](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-24-320.jpg)
![Scilab
for
very
beginners
-‐
26/33
-->C.*C
ans =
1. 4. 9. 16.
It
is
also
possible
to
write
C^2
because,
for
vectors,
exponents
are
interpreted
as
an
operation
element
wise.
This
is
not
the
case
with
matrices.
-->1/C
ans =
0.0333333
0.0666667
0.1
0.1333333
In
this
particular
case
with
vectors,
we
find
the
vector
X
for
which
C*X
=
1
-->(1)./C
ans =
1. 0.5 0.3333333
0.25
Reverse
element
wise
As
previously,
C^(-1)
is
possible.
The
parentheses
around
1
are
necessary
so
that
the
point
is
not
considered
as
a
comma
as
part
of
the
number
1.
It
is
also
possible
to
write
1 ./C
with
a
space
between
1
and
“
.
“
Solving
linear
systems
To
solve
the
linear
system AX
=
Y,
in
which
A
is
a
square
matrix,
use
the
backslash
“
“
X
=
A
Y.
Be
cautious,
the
operation
Y
/
A
will
give
(only
if
the
dimensions
are
correct)
another
result,
that
is
to
say
the
matrix
Z
for
which
Z
A
=
Y.
To
solve
the
system:
1 2 3
4 5 6
×푋 = 11
-->A=[1 2 3;4 5 6];
-->Y=[1;1];
-->X=AY
X =
- 0.5
0.
0.5
-->A*X
ans =
1.
1.](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-26-320.jpg)
![Scilab
for
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beginners
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27/33
Some
useful
functions
Sort
The
gsort
function
is
used
to
sort
the
elements
of
a
vector
in
ascending
or
descending
order.
-->v=[2,6,9,6,-4,0,2]
v =
2. 6. 9. 6. - 4. 0. 2.
--> gsort(v,"g","i")
ans =
- 4. 0. 2. 2. 6. 6. 9.
--> gsort(v)
ans =
9. 6. 6. 2. 2. 0. - 4.
Length
The
length
function
returns
the
number
of
coordinates
for
a
vector.
The
size
function
returns
the
dimensions
(rows,
columns)
for
a
matrix.
-->U=[1:10]
U =
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
-->length(U)
ans =
10.
-->m=[1 2 3;4 5 6];
-->size(U)
ans =
2. 3.](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-27-320.jpg)
![Scilab
for
very
beginners
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28/33
Sum
and
product
The
sum
and
prod
functions
respectively
calculate
the
sum
and
the
product
of
their
argument
elements.
-->U=[1:10];
-->sum(U)
ans =
55.
-->prod(U)
ans =
3628800.
Unique
The
unique
function
keeps
only
once
the
elements
of
a
vector
(even
if
they
are
repeated
several
times)
and
sorts
them
in
ascending
order.
-->v=[2,6,9,6,-4,0,2]
v =
2. 6. 9. 6. - 4. 0. 2.
-->unique(v)
ans =
- 4. 0. 2. 6. 9.
Find
The
find
function
searches
for
elements
in
a
vector
or
a
matrix
and
returns
a
vector
containing
the
corresponding
indices.
To
find
all
the
elements
of
the
vector
푤
smaller
than
5:
-->w=[1,5,3,8,14,7,3,2,12,6]; find(w<5)
ans =
1. 3. 7. 8.
The
resulting
vector
(1,3,7,8)
indicates
that
elements
푤!,푤!,푤!
and
푤!
are
smaller
than
5.](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-28-320.jpg)
![Scilab
for
very
beginners
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29/33
To
find
all
the
elements
of
the
vector
푤
equal
to
3:
-->w=[1,5,3,8,14,7,3,2,12,6]; find(w==3)
ans =
3. 7.
The
resulting
vector
(3,7)
indicates
that
elements
푤!
and
푤!
are
equal
to
3.
Accuracy
computation
Computation
Numbers
have
an
absolute
value
between
about
2.2×10-‐308
and
1.8×10+308.
The
number
%eps
equals
to
2.220446049D-16
gives
the
smallest
relative
precision
that
can
be
obtained
in
computations,
which
therefore
gives
about
16
decimal
digits.
Example
1:
Computation
of
sin(π)
-->sin(%pi)
ans =
1.225D-16
The
value
of
푠푖푛 휋
above
is
not
0,
but
it
is
considered
as
zero.
Indeed,
compared
to
the
maximum
value
of
the
sine
function
(i.e.
1),
it
is
equal
to
0
with
a
value
less
than
%eps.
Example
2:
Testing
if
the
triangle
with
sides
ퟑ,
1
et
2
is
a
right-‐angled
triangle:
-->a=sqrt(3)
a =
1.7320508
-->b=1
b =
1.
-->c=2
c =
2.
-->a^2+b^2==c^2
ans =
F
The
program
responds
false
because
the
value
of
a^2
+
b
^
2
is
approximate](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-29-320.jpg)
![Scilab
for
very
beginners
-‐
31/33
The
solution
y
is
a
matrix
of
dimension :
푦! 1 푦! 2 ⋮ 푦! 푇
푦! 1 푦! 2 ⋮ 푦! 푇
⋮ ⋮ ⋮
푦! 1 푦! 2 … 푦! 푇
Example:
Solving
the
differential
equation
풚" = −ퟒ풚
풚 ퟎ = ퟑ, 풚! ퟎ = ퟎ
This
equation
of
order
2
is
reduced
to
a
system
of
two
equations
of
order
1
as
defined
by:
푌 =
푌(1)
푌(2) =
푦
푦′ , 푌푝푟푖푚 =
푌푝푟푖푚(1)
푌푝푟푖푚(2) =
푦′
푦"
and
푌푝푟푖푚 1 = 2
푌푝푟푖푚 2 = −4×푌(1)
Comment
Scilab
Editor
We
define
the
function
returning
output
vector
y’
from
input
variables
t
and
y
(y
is
a
vector).
We
define
the
values
of
t
for
plotting.
(the
solver
chooses
itself
the
good
values
of
t
for
the
internal
computation
of
the
solution).
We
define
the
initial
conditions.
We
execute
ode.
We
plot
the
integral
curve
of
y
with
respect
to
t.
function yprim=f(t,y)
yprim(1)=y(2);
yprim(2)=-4*y(1) ;
endfunction
t0=0; tmax=5;
t=t0:0.05:tmax;
y0=3; yprim0=0;
y=ode([y0;yprim0],t0,t,f);
clf; plot(t,y(1,:))
€
n × T](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-31-320.jpg)
![Scilab
for
very
beginners
-‐
32/33
Chapter
3
–
Useful
Scilab
functions
Analysis
• sqrt(x)
returns
the
square
root
of
푥
with
푥
real
positive
or
zero,
and
the
complex
root
of
real
positive
part
otherwise.
• log(x)
returns
the
natural
logarithm
of
x
with
x
real
or
complex
number.
• exp(x)
returns
the
exponential
of
푥
with
푥
real
or
complex
number.
• abs(x)
returns
the
absolute
value
of
푥
real
(or
the
module
if
푥
is
complex).
• int(x)
returns
the
truncation
of
푥
real
(the
integer
before
the
decimal).
• floor(x)
returns
the
integer
part
of
푥
real
(the
integer
푛
for
which
푛 ≤ 푥 < 푛 + 1).
• ceil(x)
for
푥
real
returns
the
integer
푛
for
which
푛 − 1 < 푥 ≤ 푛.
Probability
and
statistics
• factorial(n)
returns
the
factorial
of
n
with
n positive
or
zero
integer.
• grand(1,p,"uin",m,n) returns
a
vector
of p
random
integer
sequences
taken
between
m
and
n with p positive
integer,
m
and
n
integers
and
푚 ≤ 푛.
• grand(1,p,"unf",a,b) returns
a
vector
of p
random
real
sequences
taken
between
a
and
b with p positive
integer,
a
and
b
real
and
푎 ≤ 푏.
• sum(n)
returns
the
sum
of
the
values
of
vector
푛
(used
to
calculate
a
total).
• cumsum(n)
returns
the
vector
of
increasing
cumulative
values
of
vector
푛
(used
to
calculate
the
increasing
cumulative
numbers).
• length(v)
returns
the
number
of
coordinates
of
vector
푣.
• gsort(v)
returns
the
vector
of
numbers
or
strings
푣
sorted
in
descending
order.
• gsort(v,"g","i")
returns
the
vector
of
numbers
or
strings
푣
sorted
in
ascending
order.
• mean(v)
returns
the
average
of
the
vector
of
numbers
푣.
• stdev(v)
returns
the
standard
deviation
of
numbers
푣
vector.
• bar(v,n,couleur)
draws
the
bar
graph
with
푣
as
X-‐coordinate,
푛
as
Y-‐coordinate,
푣
and
푛
being
same
size
line
vectors.
By
default,
bar(n)
draws
the
bar
graph
of
푛
in
blue
with
1,2,3…
as
X-‐coordinates.
• bar(v,[n1’,n2’])
draws
a
double
bar
graph
with
푣
as
X-‐coordinate,
n1
as
Y-‐
coordinate
in
blue
and
n2
as
Y-‐coordinate
in
green,
with
푣,
n1
and
n2
being
same
size
line
vectors.
• rand(n,p)
with
푛
and
푝
positive
integers,
returns
a
matrix
푛×푝
of
numbers
randomly
taken
between
0
and
1.
• rand()
returns
a
real
number
randomly
taken
between
0
and
1.
• floor(x)
returns
the
integer
part
of
푥
real
number.
In
particular,
if
푝
is
real
between
0
and
1,
floor(rand()+p)
will
be
1
with
푝
probability
and
0
with
1 − 푝
probability.](https://image.slidesharecdn.com/scilabbeginners-140919072711-phpapp02/85/Scilab-for-very-beginners-32-320.jpg)
Scilab for very beginners
- 1. Scilab for very beginners Scilab Enterprises S.A.S -‐ 143 bis rue Yves Le Coz -‐ 78000 Versailles (France) -‐ www.scilab-‐enterprises.com
- 2. This document has been co-‐written by Scilab Enterprises and Christine Gomez, mathematics teacher at Lycée Descartes (Descartes High School) in Antony, Hauts-‐de-‐Seine (France). © 2013 Scilab Enterprises. All rights reserved. Scilab for very beginners -‐ 2/33
- 3. Scilab for very beginners -‐ 3/33 Table of content Introduction About this document 4 Install Scilab 4 Mailing list 4 Complementary resources 4 Chapter 1 – Become familiar with Scilab The general environment and the console 5 Simple numerical calculations 6 The menu bar 7 The editor 8 The graphics window 9 Windows management and workspace customization 11 Chapter 2 -‐ Programming Variables, assignment and display 12 Loops 16 Tests 17 2 and 3D plots 18 Supplements on matrices and vectors 23 Calculation accuracy 29 Solving differential equations 30 Chapter 3 – Useful Scilab functions In analysis 32 In probability and statistics 32 To display and plot 33 Utilities 33
- 4. Scilab for very beginners -‐ 4/33 Introduction About this document The purpose of this document is to guide you step by step in exploring the various basic features of Scilab for a user who has never used numerical computation software. This presentation is voluntarily limited to the essential to allow easier handling of Scilab. Computations, graphs and illustrations are made with Scilab 5.4.0. You can reproduce all those commands from this version. Install Scilab Scilab is numerical computation software that anybody can freely download. Available under Windows, Linux and Mac OS X, Scilab can be downloaded at the following address: http://www.scilab.org/ You can be notified of new releases of Scilab software by subscribing to our channel notification at the following address: http://lists.scilab.org/mailman/listinfo/release Mailing list To facilitate the exchange between Scilab users, dedicated mailing lists exist (list in French, list for the education world, international list in English). The principle is simple: registrants can communicate with each other by e-‐mail (questions, answers, sharing of documents, feedbacks...). To browse the available lists and to subscribe, go to the following address: http://www.scilab.org/communities/user_zone/mailing_list Complementary resources If you have an Internet connection, we invite you to visit Scilab website where you will find a section dedicated to the use of Scilab (http://www.scilab.org/support/documentation), with links and relevant documents which can be freely downloaded and printed.
- 5. Scilab for very beginners -‐ 5/33 Chapter 1 – Become familiar with Scilab The useful workspace in Scilab consists of several windows: • The console for making calculations, • The editor for writing programs, • The graphics windows for displaying graphics, • The embedded help. The general environment and the console After double-‐clicking the icon to launch Scilab, Scilab environment by default consists of the following docked windows – console, files and variables browsers, command history (see “Windows management and workspace customization”, page 11): In the console after the prompt “ --> “, just type a command and press the Enter key (Windows and Linux) or Return key (Mac OS X) on the keyboard to obtain the corresponding result. --> 57/4 ans = 14.25 --> (2+9)^5 ans = 161051. Mention Before the result, ans is displayed for “answer”. It is possible to come back at any moment with the keyboard's arrow keys ← ↑ → ↓ or with the mouse. The left and right keys are used to change the instructions and the up and down keys are used to come back on a previously executed command.
- 6. Simple numerical calculations All computations done with Scilab are numerical. Scilab performs computations with matrices (see chapter 2, page 23). Operations are written with “ Scilab for very beginners -‐ 6/33 + “ for addition, “ – “ for subtraction, “ * “ for multiplication, “ / “ for division, “ ^ “ for exponents. For example: -->2+3.4 ans = 5.4 The case is sensitive. It is thus necessary to respect uppercase and lowercase for the calculations to be performed properly. For example, with sqrt command (which calculates the square root): -->sqrt(9) ans = 3. while: -->SQRT(9) !--error 4 Undefined variable: SQRT Particular numbers %e and %pi represent respectively e and π: --> %e %e = 2.7182818 --> %pi %pi = 3.1415927 %i represents the i of complexes in input and is displayed i in output: --> 2+3*%i ans = 2. + 3.i For not displaying the results In adding a semi colon “ ; “ at the end of a command line, the calculation is done but the result is not displayed. -->(1+sqrt(5))/2; --> (1+sqrt(5))/2 ans = 1.618034
- 7. To remind the name of a function The names of commonly used functions are summarized in Chapter 3 of this document (page 32). For example: Scilab for very beginners -‐ 7/33 --> exp(10)/factorial(10) ans = 0.0060699 The tab key →│ on the keyboard can be used to complete the name of a function or a variable by giving its first few letters. For example, after typing in the console the command: -->fact and then pressing the tab key, a window is displayed with all the functions and variables names beginning with fact, such as factorial and factor. Just double click on the required function or select it with the mouse or with the keys ↑ ↓ and press Enter (Windows and Linux) or Return (Mac OS X) to insert it in the command line. The menu bar The menus listed below are particularly useful. Applications • The command history allows you to find all the commands from previous sessions to the current session. • The variables browser allows you to find all variables previously used during the current session. Edit Preferences (in Scilab menu under Mac OS X) allows you to set and customize colors, fonts and font size in the console and in the editor, which is very useful for screen projection. Clicking on Clear Console clears the entire content of the console. In this case, the command history is still available and calculations made during the session remain in memory. Commands that have been erased are still available through the keyboard’s arrow keys. Control To interrupt a running program, you can: • Type pause in the program or click on Control> Interrupt in the menu bar (Ctrl X under Windows and Linux or Command X under Mac OS X), if the program is already running. In all cases, the prompt “ --> “ will turn into “ -1-> “, then into “ -2-> “…, if the operation is repeated. • To return to the time prior to the program interruption, type resume in the console or click on Control > Resume. • To quit for good a calculation without any possibility of return, type abort in the console or click on Control > Abort in the menu bar. Mention All available functions are also listed in the embedded help by clicking in the menu bar on ? > Scilab Help.
- 8. The editor Typing directly into the console has two disadvantages: it is not possible to save the commands and it is not easy to edit multiple lines of instruction. The editor is the appropriate tool to run multiple instructions. Opening the editor To open the editor from the console, click on the first icon in the toolbar or on Applications Scilab for very beginners -‐ 8/33 > SciNotes in the menu bar. The editor opens with a default file named “ Untitled 1 “. Writing in the editor Typing in the editor is like as in any word processor. In the text editor, opening and closing parentheses, end loops, function and test commands are added automatically. However, these features can be disabled in Options > Auto-‐completion on menu, in clicking on the two below entries enabled by default: • (,[,… • if,function,… While in principle each instruction should be entered on a separate line, it is possible to type multiple statements on a same line separating each statement with a semicolon “ ; “. A space at the start of the line called indentation is automatic when a loop or a test is started. In the following example, we calculate 10 terms of the sequence (푢!) defined by: 푢! = 1 푢!!! = 2푢! − 3 Mention -‐ Comments preceded with “ // “ will not be taken into account in the calculations. -‐ To change the font, click on Options > Preferences. -‐ When writing a program, indentation is automatic. If this is not the case, click on Format > Correct indentation to restore it (Ctrl I under Windows and Linux or Command I under Mac OS X).
- 9. Scilab for very beginners -‐ 9/33 Saving Any file can be saved by clicking on File > Save as. The extension “ .sce “ at the end of a file name will launch automatically Scilab when opening it (except under Linux and Mac OS X). Copying into the console, executing a program In clicking on Execute in the menu bar, three options are available: • Execute “ …file with no echo “ (Ctrl Shift E under Windows and Linux, Cmd Shift E under Mac OS X): the file is executed without writing the program in the console (saving the file first is mandatory). • Execute “ … file with echo “ (Ctrl L under Windows and Linux, Cmd L under Mac OS X): rewrite the file into the console and executes it. • Execute “ …until the caret, with echo “ (Ctrl E under Windows and Linux, Cmd E under Mac OS X): rewrite the selection chosen with the mouse into the console and executes it or execute the file data until the caret position defined by the user. Standard copy/paste can also be used. The graphics window Opening a graphics window A graphics window opens automatically when making any plot. It is possible to plot curves, surfaces, sequences of points (see chapter 2, page 18). To obtain an example of curve, type in the console: -->plot Mention -‐ To erase a previous plot, type clf (“ clear figure “). -‐ To open another graphics window, type scf; (“ set current figure “). If several graphic windows are open, you can choose in which the plot will be drawn by typing scf(n); n for the graphics window number (mentioned on the top left).
- 10. Modifying a plot The magnifying glass allows zooming. To zoom in two dimensions, click on the tool and with the mouse create a rectangle which will constitute the new enlarged view. To zoom in three dimensions, click on the tool and create a parallelepiped which will constitute the new enlarged view. It is also possible to zoom in using the mouse wheel. To return to the initial screen, click on the other magnifying glass . The icon enables rotation of the figure (particularly useful in 3-‐D) with right click actions which are guided by messages in the bottom of the graphics window. For more precise modifications, click on Scilab for very beginners -‐ 10/33 Edit > Figure properties or Axes properties and let yourselves be guided (this option is not yet available under Mac OS X). Online help To access the online help, click on ? > Scilab Help in the menu bar, or type in the console: -->help To get help with any function, type help in the console followed by the name of the appropriate function. For example: -->help sin displays help for sin (sine) function. Mention Examples of use can be executed in Scilab and edited in SciNotes in using the associated buttons in the example framework.
- 11. Windows management and workspace customization As in the default Scilab environment, where the console, files and variables browsers and command history are all together docked windows, all other windows in Scilab can be repositioned in a single one. For example, the user can choose to position the editor in the default environment of Scilab. To dock a window in another one, first identify the blue horizontal bar under Windows, or black under Mac OS X and Linux, at the top of the window in the toolbar containing a question mark on the right. • Under Windows and Linux, click on this bar with the left mouse button and, while maintaining the click, move the mouse pointer in the desired window. • Under Mac OS X, click on this bar and while maintaining the click, move it in the desired Scilab for very beginners -‐ 11/33 window. A rectangle appears indicating the future positioning of the window. When the position is the one you want, release the mouse button. To cancel and bring out the window, click on the small arrow on the right of the same bar.
- 12. n × 1 and a number is itself a matrix of dimension 1 × 1. Variables do not need to be declared in advance, but any variable must have a value. For example, obtaining the value of a variable which has not been given a value produces an error: Scilab for very beginners -‐ 12/33 Chapter 2 -‐ Programming In the examples given in this document, any line preceded by “ --> “ is a command, the other lines are the returns from commands (calculation results, error messages…). Do not write “ --> “ in the editor. They are introduced here to make the distinction between command lines and calculation results as it is displayed in the console after copying/pasting. When presented in a table (without “ --> “ and no calculation result), the commands are depicted exactly as they should be typed in the editor. Variables, assignment and display Variables Scilab is not a computer algebra system. It calculates only with numbers. All calculations are done with matrices, although this may go unnoticed. Even if the concept of matrices is unknown, vectors and sequences of numbers can explain it, as they are, in fact, matrices of dimension 1 × n or -->a !--error 4 Undefined variable : a If a value is assigned to the variable a, there is no longer an error: --> a=%pi/4 a = 0.7853982 --> a a = 0.7853982 Variables may take any name that is not already defined by the system: --> Piby2=%pi/2 Piby2 = 1.5707963 Mention Like the Scilab functions, a variable name must not have accents or special characters.
- 13. The result of a calculation that is not assigned to a variable is automatically assigned to the variable called ans: Scilab for very beginners -‐ 13/33 -->3*(4-2) ans = 6. -->ans ans = 6. Functions Functions are the easiest and most natural way to make computations from variables and for obtaining results from variables. A function definition begins with function and ends with endfunction. For example, to convert euros (e) in dollars (d) with an exchange rate (t), the dollars function is defined. The variables are e and t and the image is d. -->function d=dollars(e,t); d=e*t; endfunction -->dollars(200,1.4) ans = 280. Very usually numerical functions are functions of one real variable. For example, two functions f and g are defined using the following commands: -->function y=f(x); y=36/(8+exp(-x)); endfunction -->function y=g(x); y=4*x/9+4; endfunction The defined functions can then be used to calculate values: --> f(10) ans = 4.4999745 --> g(12.5) ans = 9.5555556 Mention The variables x and y are dummy variables, their names can be reused when defining other functions or in Scilab.
- 14. Scilab for very beginners -‐ 14/33 Requesting the assignment of a variable Assignment is made easily using the “ = “ operator. Display Writing Typing the name of a variable displays its value, except when there is “ ; “ at the end of the command line. Brackets Matrices are defined using square brackets (see page 23). As mentioned before, matrix computation is the basis of calculations in Scilab. A space or comma is used to separate columns and semicolons are used to separate rows. To define a column vector and obtain a column display: -->v=[3;-2;5] v = 3. - 2. 5. To define a row vector and obtain a row display: -->v=[3,-2,5] v = 3. - 2. 5. To define a matrix and obtain a tabular display: -->m=[1 2 3;4 5 6;7 8 9] m = 1. 2. 3. 4. 5. 6. 7. 8. 9. Mention This command can also be typed under the form: v=[3 -2 5] Mention This command can also be typed under the form: m=[1,2,3;4,5,6;7,8,9]
- 15. Scilab for very beginners -‐ 15/33 disp function disp must always be used with parentheses. With the vector v previously defined: -->v(2) ans = - 2. -->disp(v(2)) - 2. To display a string (usually a sentence), put it between quotes: -->disp("Bob won") Bob won To display a combination of words and values use the string command which converts values to character strings using a “ + “ between the different parts: -->d=500; -->disp("Bob won "+string(d)+" dollars") Bob won 500 dollars If the sentence contains an single quote, the latter needs to be doubled in the string to be displayed properly: -->disp("It''s fair") It's fair Loops Incrementation The “ : “ operator allows to define vectors of numbers whose coordinates are in arithmetic sequence. We give: << beginning value: step: ending value>>. It is possible that “ending value” is not reached. If the step is not mentioned, its default value is 1. For example, to define a row vector of integers which increments in steps of 1 from 3 and 10: -->3:10 ans = 3. 4. 5. 6. 7. 8. 9. 10. or which increments in steps of 2 from 1 to 10: -->1:2:10 ans = 1. 3. 5. 7. 9.
- 16. Scilab for very beginners -‐ 16/33 or which decreases in steps of 4 from 20 to 2: -->u=20:-4:2 u = 20. 16. 12. 8. 4. For The easiest loop structure for a fixed number of iterations is written with for … end. Example: Calculation of 20 terms of a sequence defined by recurrence by: 푢! = 4 푢!!! = 푢! + 2푛 + 3 Algorithm Scilab Editor Put 4 in u(1) For n from 1 to 20 u(n+1) takes the value u(n)+2n+3 and u(n) Display n and u(n) End for u(1)=4; for n=1:20 u(n+1)= u(n)+2*n+3; disp([n,u(n)]) end While To stop a loop when a given goal is reached, while … end is used. I planted a Christmas tree in 2005 measuring 1.20 m. It grows by 30 cm per year. I decided to cut it when it exceeds 7 m. In what year will I cut the tree? Algorithm Scilab Editor Put 1.2 in h (h = tree height) Put 2005 in y (y = year) While h<7 h takes the value h+0.3 (the tree grows) y takes the value y+1 (one year passes) End while Display y (the final year) h=1.2; y=2005; while h<7 h=h+0.3; y=y+1; end disp("I will cut the.. tree in "+string(y)) Mention When a command is too long to be written in one line, the editor will wrap lines. One can also wrap lines using “ .. “ (two dots) before going to the next line.
- 17. Tests Comparison operators Useful tests include comparing numbers or determining whether a statement is true or false. Below the corresponding commands: Equal Different Less Greater Less or equal Greater or equal == <> < > <= >= True False And Or No %T %F & | ~ Scilab for very beginners -‐ 17/33 To compare two vectors (or two matrices), the tests “ == “ and “ <> “ will compare term by term. For example: -->X=[1,2,5]; Y=[5,3,5]; -->X==Y ans = F F T To test if two vectors are equal, isequal is used and ~isequal is used if they are different: -->isequal(X,Y) ans = F -->~isequal(X,Y) ans = T If…then The basic conditional statements are the following: • if … then … else … end, • if … then … elseif … then … else … end. if … then must be written on the same line and likewise with elseif … then. Mention Be cautious with calculation accuracy. Calculations are approached and “ == “ will sometimes give wrong results (see Calculation accuracy, page 30).
- 18. Scilab for very beginners -‐ 18/33 Example: Alice throws three dice. • If she gets three 6’s she wins $20, • If she gets three identical numbers different from 6 she wins $10, • If she gets two identical numbers she wins $5, • Otherwise she wins nothing. Simulate a trial and calculate Alice’s winnings using the functions: • grand (see page 22), • unique(D)which keeps only once the values that appears several times in D, • length(unique(D)) which returns the size of the obtained vector, that is to say 1 if three dice are equal, and 2 if two dice are equal. Algorithm Scilab Editor Put the three values in D If Alice gets three 6, then Alice wins 20 dollars Else if she receives three identical values, then Alice wins 10 dollars Else, if she receives two identical values, then Alice wins 5 dollars Otherwise Alice wins nothing End if Display Alice's winnings D=grand(1,3,"uin",1,6); if D==[6,6,6] then W=20; elseif length(unique(D))==1 then W=10; elseif length (unique(D))==2 then W=5; else W=0; end disp("Alice wins "+.. string(W)+ " dollars") 2-‐D and 3-‐D plots plot command is used to create plots in the plane. Color and appearance can be specified by putting indications of color and points style within quotes: • Colors "b" = blue (by default), "k" = black, "r" = red, "g" = green, "c" = cyan, "m" = magenta, "y" = yellow, "w" = white. • Point styles Joined (by default), or ".", "+", "o", "x", "*". Other options are available with: "s", "d", "v", "<", and ">".
- 19. Scilab for very beginners -‐ 19/33 Basic plots To plot a point Plot the point A(1 ; 2) with a red point. Scilab Editor Graphics Window plot(1,2,".r") To plot a segment Plot the segment [AB] in blue (by default) with A(1 ; 2) and B(3 ; 5). Scilab Editor Graphics Window plot([1,3],[2,5]) Plots of plane curves defined by functions y=f(x) For a function 푥 → 푓(푥) the values of x are specified using the linspace command by writing: x=linspace(a,b,n); in which a is the smallest value of the variable 푥, b the highest value of 푥, and n the number of values calculated between a and b. Do not forget the “ ; “ sympbol otherwise all n values of 푥 will be displayed. For example, consider two functions 푓 and 푔 defined over the interval [-‐2 ; 5] by: 푓 푥 = (푥! + 2푥)푒!!, and 푔 푥 = sin ! !
- 20. With the program below, we plot the curve defined by푓, in blue by default: Scilab Editor Graphics Window Scilab for very beginners -‐ 20/33 function y=f(x) y=(x^2+2*x)*exp(-x) endfunction x=linspace(-2,5,50); plot(x,f) By adding the program below, we obtain the plot of two curves, that of f in red and that of g in green. The previous plot was cleared through the clf command (“ clear figure “). Scilab Editor Graphics Window function y=g(x) y=sin(x/2) endfunction x=linspace(-2,5,50); clf plot(x,f,"r",x,g,"g") Plots of sequences of points Terms of a sequence The most common case is to plot the sequences of points 푀(푛, 푢 푛 ) after calculating the coordinates 푢(푛) of a vector 푢. plot(u,"*r") specifies the style and color of the points in quotes: red and non-‐connected stars. By default, points are plotted in blue and are connected. Scilab Editor Graphics Window for n=1:50 u(n)=(-0.8)^n; end clf; plot(u,"*r")
- 21. Bivariate statistical data Bivariate statistical data are given in the form of two vectors: let’s call them X and Y. plot(X,Y,"<") Scilab for very beginners -‐ 21/33 will plot a scatter plot of 푀(푋!; 푌!) with blue triangles. Scilab Editor Graphics Window X=[1,3,3,7,7,9,10]; Y=[8,7,5,5,4,2,2]; clf; plot(X,Y,"<") Plots in 3 dimensions Scilab can be used to plot surfaces and curves in space, with many options for the treatment of hidden faces, face colors, points of view, etc. The following examples will illustrate 3-‐D plots. The surf function can be used for plotting surfaces. This function has three input variables, x, y and z. x and y are respectively vectors of size 푚 and 푛 corresponding to points on the axes (푂푥) and (푂푦). z is a matrix of dimension 푛×푚 with element 푧!" corresponding to the height of the point with X-‐coordinate 푥! and Y-‐coordinate 푦!. To plot the surface defined by a function of the form 푧 = 푓(푥, 푦), it is necessary to : • Define function 푓 • Calculate z=feval(x,y,f)' feval(x,y,f) returns the 푚×푛 matrix whose 푖푗 element is 푓(푥!, 푦!) which will be transposed by using the single quote symbol “ ' “ • Execute surf(x,y,z). To plot the surface 푧 = 2푥! + 푦! (elliptic paraboloid): Scilab Editor Graphics Window function z=f(x,y) z=2*x^2+y^2; endfunction x=linspace(-1,1,100); y=linspace(-2,2,200); z=feval(x,y,f)'; clf surf(x,y,z)
- 22. Scilab for very beginners -‐ 22/33 Curves in space may be plotted using the param3d function. param3d has three arguments, x, y and z, each vectors has the same dimension and corresponds to the points (푥!, 푦!, 푧!) on the curve. To plot the helix defined by (푥 = cos 푡 , 푦 = sin 푡 , 푧 = 푡): Scilab Editor Graphics Window t=linspace(0,4*%pi,100); param3d(cos(t),sin(t),t) Simulations and statistics Several functions are available in Scilab to perform simulations quickly and efficiently. Random sequences • grand(1,p,"uin",m,n) returns a vector of 푝 random integer sequences between 푚 and 푛 with 푝 positive integer, 푚 and 푛 integers and 푚 ≤ 푛. -->t= grand(1,4,"uin",1,6) t = 3. 1. 3. 6. • grand(1,p,"unf",a,b) returns a vector of 푝 random real sequences between 푎 and 푏 with 푝 positive integer, 푎 and 푏 real and 푎 ≤ 푏. -->tr= grand(1,2,"unf",-1,1) tr = - 0.7460264 0.9377355
- 23. Scilab for very beginners -‐ 23/33 Statistics All the standard statistical functions are listed on page 32. Keep particularly in mind: • The function bar(x,n,color) which draws bar graphs: Scilab Editor Graphics Window x=[1:10]; n=[8,6,13,10,6,4,16,7,8,5]; clf; bar(x,n) • For a bar graph representing two sets side by side: consider the series of values X and the two series of numbers n1 and n2. For plotting, n1 and n2 must be column vectors that is why transposes are used in the example below: Scilab Editor Graphics Window X=[1,2,5];n1=[5,10,5];n2=[6,8,7]; bar(X,[n1',n2']) The color optional argument defines the color as in the plot function. Additional information on matrices and vectors Accessing elements Square braces are used to define matrices. A space or a comma is used to switch from one column to another and semicolons are used to switch from one line to another. -->m=[1 2 3;4 5 6] m = 1. 2. 3. 4. 5. 6. Parentheses are used to access elements or to modify them. Mention This command can also be typed under the form: m=[1,2,3;4,5,6]
- 24. Scilab for very beginners -‐ 24/33 -->m(2,3) ans = 6. -->m(2,3)=23 m = 1. 2. 3. 4. 5. 23. The “ : “ operator is used to designate all the rows or all columns of a matrix. To view the second row of the matrix m, type: -->m(2,:) ans = 4. 5. 23. And the third column: -->m(:,3) ans = 3. 23. To obtain the transpose of a matrix or a vector use the single quote symbol “ ' “: -->m' ans = 1. 4. 2. 5. 3. 23. Operations The operations “ * “, “ / “ are matrix operations. To make element wise operations, we need to put a dot before the operating sign: “ .* “, “ ./ “. -->A=[1,2,3;4,5,6] A = 1. 2. 3. 4. 5. 6. -->B=[1;1;2]
- 25. Scilab for very beginners -‐ 25/33 B = 1. 1. 2. -->A*B ans = 9. 21. Matrix multiplication -->A*A !--error 10 Inconsistent multiplication. Dimensions are not consistent -->A.*A ans = 1. 4. 9. 16. 25. 36. Element wise multiplication -->2*(A+2) ans = 6. 8. 10. 12. 14. 16. The operation is performed on each element because 2 is a number -->A/A ans = 1. 1.518D-16 3.795D-15 1. Gives the matrix X for which X*A = A The exact result is: 1. 0 0 1. For reasons of calculation accuracy, the result can be slightly different depending on your version of Scilab and your operating system (see calculation accuracy, page 29). -->A./A ans = 1. 1. 1. 1. 1. 1. Gives the matrix divided element wise In the case of vectors: -->C=1:4 C = 1. 2. 3. 4. -->C*C !--error 10 Inconsistent multiplication. Dimensions are not consistent
- 26. Scilab for very beginners -‐ 26/33 -->C.*C ans = 1. 4. 9. 16. It is also possible to write C^2 because, for vectors, exponents are interpreted as an operation element wise. This is not the case with matrices. -->1/C ans = 0.0333333 0.0666667 0.1 0.1333333 In this particular case with vectors, we find the vector X for which C*X = 1 -->(1)./C ans = 1. 0.5 0.3333333 0.25 Reverse element wise As previously, C^(-1) is possible. The parentheses around 1 are necessary so that the point is not considered as a comma as part of the number 1. It is also possible to write 1 ./C with a space between 1 and “ . “ Solving linear systems To solve the linear system AX = Y, in which A is a square matrix, use the backslash “ “ X = A Y. Be cautious, the operation Y / A will give (only if the dimensions are correct) another result, that is to say the matrix Z for which Z A = Y. To solve the system: 1 2 3 4 5 6 ×푋 = 11 -->A=[1 2 3;4 5 6]; -->Y=[1;1]; -->X=AY X = - 0.5 0. 0.5 -->A*X ans = 1. 1.
- 27. Scilab for very beginners -‐ 27/33 Some useful functions Sort The gsort function is used to sort the elements of a vector in ascending or descending order. -->v=[2,6,9,6,-4,0,2] v = 2. 6. 9. 6. - 4. 0. 2. --> gsort(v,"g","i") ans = - 4. 0. 2. 2. 6. 6. 9. --> gsort(v) ans = 9. 6. 6. 2. 2. 0. - 4. Length The length function returns the number of coordinates for a vector. The size function returns the dimensions (rows, columns) for a matrix. -->U=[1:10] U = 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. -->length(U) ans = 10. -->m=[1 2 3;4 5 6]; -->size(U) ans = 2. 3.
- 28. Scilab for very beginners -‐ 28/33 Sum and product The sum and prod functions respectively calculate the sum and the product of their argument elements. -->U=[1:10]; -->sum(U) ans = 55. -->prod(U) ans = 3628800. Unique The unique function keeps only once the elements of a vector (even if they are repeated several times) and sorts them in ascending order. -->v=[2,6,9,6,-4,0,2] v = 2. 6. 9. 6. - 4. 0. 2. -->unique(v) ans = - 4. 0. 2. 6. 9. Find The find function searches for elements in a vector or a matrix and returns a vector containing the corresponding indices. To find all the elements of the vector 푤 smaller than 5: -->w=[1,5,3,8,14,7,3,2,12,6]; find(w<5) ans = 1. 3. 7. 8. The resulting vector (1,3,7,8) indicates that elements 푤!,푤!,푤! and 푤! are smaller than 5.
- 29. Scilab for very beginners -‐ 29/33 To find all the elements of the vector 푤 equal to 3: -->w=[1,5,3,8,14,7,3,2,12,6]; find(w==3) ans = 3. 7. The resulting vector (3,7) indicates that elements 푤! and 푤! are equal to 3. Accuracy computation Computation Numbers have an absolute value between about 2.2×10-‐308 and 1.8×10+308. The number %eps equals to 2.220446049D-16 gives the smallest relative precision that can be obtained in computations, which therefore gives about 16 decimal digits. Example 1: Computation of sin(π) -->sin(%pi) ans = 1.225D-16 The value of 푠푖푛 휋 above is not 0, but it is considered as zero. Indeed, compared to the maximum value of the sine function (i.e. 1), it is equal to 0 with a value less than %eps. Example 2: Testing if the triangle with sides ퟑ, 1 et 2 is a right-‐angled triangle: -->a=sqrt(3) a = 1.7320508 -->b=1 b = 1. -->c=2 c = 2. -->a^2+b^2==c^2 ans = F The program responds false because the value of a^2 + b ^ 2 is approximate
- 30. Scilab for very beginners -‐ 30/33 -->abs(a^2+b^2-c^2)<%eps ans = F The program responds false because absolute precision is called for -->abs(a^2+b^2-c^2)/c^2<%eps ans = T The program responds true because relative precision is called for Display Results are displayed, by default, with 10 characters, including the decimal point and the sign. The format function is used to display more digits. To display 20 digits, type format(20). Reconsidering 푎 = 3: -->a^2 ans = 3. Here there are 7 decimal places, and we do not see the error. -->format(20) -->a^2 ans = 2.99999999999999956 Here there are 17 decimal places, and we can see the error Solving differential equations To find the solutions of an explicit system of differential equations, simply reduce to differential equations of order 1. 푦! 푡 = 푓 푡, 푦 푡 in which 푡! and 푡 represent time and 푦! and 푦(푡) are vectors, 푦 푡! = 푦! then apply the ode function: y=ode(y0,t0,t,f), with: • y0: initial condition, vector of dimension 푛, • t0: initial time, • t: vector of dimension T corresponding to the times in which the solution is computed. This vector must begin with t0. • f: function defining the system under the form: function yprim=f(t,y) yprim(1)=... yprim(2)=... .... yprim(n)=... endfunction
- 31. Scilab for very beginners -‐ 31/33 The solution y is a matrix of dimension : 푦! 1 푦! 2 ⋮ 푦! 푇 푦! 1 푦! 2 ⋮ 푦! 푇 ⋮ ⋮ ⋮ 푦! 1 푦! 2 … 푦! 푇 Example: Solving the differential equation 풚" = −ퟒ풚 풚 ퟎ = ퟑ, 풚! ퟎ = ퟎ This equation of order 2 is reduced to a system of two equations of order 1 as defined by: 푌 = 푌(1) 푌(2) = 푦 푦′ , 푌푝푟푖푚 = 푌푝푟푖푚(1) 푌푝푟푖푚(2) = 푦′ 푦" and 푌푝푟푖푚 1 = 2 푌푝푟푖푚 2 = −4×푌(1) Comment Scilab Editor We define the function returning output vector y’ from input variables t and y (y is a vector). We define the values of t for plotting. (the solver chooses itself the good values of t for the internal computation of the solution). We define the initial conditions. We execute ode. We plot the integral curve of y with respect to t. function yprim=f(t,y) yprim(1)=y(2); yprim(2)=-4*y(1) ; endfunction t0=0; tmax=5; t=t0:0.05:tmax; y0=3; yprim0=0; y=ode([y0;yprim0],t0,t,f); clf; plot(t,y(1,:)) € n × T
- 32. Scilab for very beginners -‐ 32/33 Chapter 3 – Useful Scilab functions Analysis • sqrt(x) returns the square root of 푥 with 푥 real positive or zero, and the complex root of real positive part otherwise. • log(x) returns the natural logarithm of x with x real or complex number. • exp(x) returns the exponential of 푥 with 푥 real or complex number. • abs(x) returns the absolute value of 푥 real (or the module if 푥 is complex). • int(x) returns the truncation of 푥 real (the integer before the decimal). • floor(x) returns the integer part of 푥 real (the integer 푛 for which 푛 ≤ 푥 < 푛 + 1). • ceil(x) for 푥 real returns the integer 푛 for which 푛 − 1 < 푥 ≤ 푛. Probability and statistics • factorial(n) returns the factorial of n with n positive or zero integer. • grand(1,p,"uin",m,n) returns a vector of p random integer sequences taken between m and n with p positive integer, m and n integers and 푚 ≤ 푛. • grand(1,p,"unf",a,b) returns a vector of p random real sequences taken between a and b with p positive integer, a and b real and 푎 ≤ 푏. • sum(n) returns the sum of the values of vector 푛 (used to calculate a total). • cumsum(n) returns the vector of increasing cumulative values of vector 푛 (used to calculate the increasing cumulative numbers). • length(v) returns the number of coordinates of vector 푣. • gsort(v) returns the vector of numbers or strings 푣 sorted in descending order. • gsort(v,"g","i") returns the vector of numbers or strings 푣 sorted in ascending order. • mean(v) returns the average of the vector of numbers 푣. • stdev(v) returns the standard deviation of numbers 푣 vector. • bar(v,n,couleur) draws the bar graph with 푣 as X-‐coordinate, 푛 as Y-‐coordinate, 푣 and 푛 being same size line vectors. By default, bar(n) draws the bar graph of 푛 in blue with 1,2,3… as X-‐coordinates. • bar(v,[n1’,n2’]) draws a double bar graph with 푣 as X-‐coordinate, n1 as Y-‐ coordinate in blue and n2 as Y-‐coordinate in green, with 푣, n1 and n2 being same size line vectors. • rand(n,p) with 푛 and 푝 positive integers, returns a matrix 푛×푝 of numbers randomly taken between 0 and 1. • rand() returns a real number randomly taken between 0 and 1. • floor(x) returns the integer part of 푥 real number. In particular, if 푝 is real between 0 and 1, floor(rand()+p) will be 1 with 푝 probability and 0 with 1 − 푝 probability.
- 33. Scilab for very beginners -‐ 33/33 Display and plot • clf means “ clear figure “ and clears the current figure on the graphics window. • plot allows to draw curves and scatter plots in 2 dimensions. • linspace(a,b,n), with 푎 and 푏 real and 푛 integer, defines a vector of 푛 values regularly spaced between 푎 and 푏. • scf allows to open or to select other graphics windows than the current one. • surf allows 3-‐D surface plots. • bar(X,Y) in which X and Y are vectors, draws the bar graph of the series of values for X which has for numbers the values of Y. • plot(X,Y,"*") draws the scatter plot of coordinates (X(i),Y(i)) as stars. The color can be specified. • plot(Y,"+") draws the scatter plot of coordinates (i,Y(i)) as cross. • disp("Sentence") displays what is written in double quotes. • disp(A) in which A is a matrix of numbers, displays the table of the values of A. • disp("Sentence"+string(x)) displays the sentence and the value of number 푥. • xclick returns the coordinates of the point clicked in a graphics window. Utilities • unique(v) returns the vector 푣 with a unique occurrence of its repeated components. • sum(v) returns the sum of all the elements of the vector or the matrix 푣. • prod(v) returns the product of all the elements of the vector or the matrix 푣. • find(<test about v>) returns the indices of the elements of vector 푣 satisfying the test. • disp(x,y,…) displays the values of its arguments in the console. • string(x) converts number 푥 to a string. • format(n) in which 푛 is an integer greater than or equal to 2, sets the display to 푛 characters, including the sign and the decimal dot. • zeros(n,p) defines a 푛×푝 matrix that only contains zeros. • feval(x,y,f) in which 푥 and 푦 are respectively vectors of size 푚 and 푛, defines the matrix 푚×푛 whose element (푖, 푗) is 푓(푥 푖 , 푦 푗 ). • help function opens the help browser on the right function page. • tic starts a clock. • toc stops the clock.