Numerical approximation
- 1. UNIVERSITY OF GUYANA FACULTY OF TECHNOLOGY Department of Civil With Environmental Engineering EMT 3200- ENGINEERING MATHEMATICS V TOPIC: NUMERICAL APPROXIMATION NUMERICAL APPROXIMATION 1
- 2. GROUP MEMBERS Orin Edwards 1013799 Mazule Hutson 1014749 Malik Lewis 1014806 Silos Singh 1011551 Akeem St. Louis 1012535 Haresh Jaipershad 1011056 Joemoal Williams 1014049 Parmendra Persaud 1010286 2NUMERICAL APPROXIMATION
- 3. OUTLINE OF PRESENTATION Introduction Accuracy and Precision Mistakes Errors Types of Errors Error Propagation Euler’s Method References 3NUMERICAL APPROXIMATION
- 4. INTRODUCTION • Numerical analysis is the area of mathematics and computer science that creates, analyzes, and implements algorithms for solving numerically, the problems of continuous mathematics. • Such problems originate generally from real-world applications of algebra, geometry and calculus, and they involve variables which vary continuously. • Numerical Approximation is an inexact representation of a numerical value that is still close enough to be useful. 4NUMERICAL APPROXIMATION
- 5. ACCURACY • Accuracy refers to the closeness of a measured value to a standard or known value. • For example, in lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg. • Your measurement is inaccurate as it is not close to the known value. 5NUMERICAL APPROXIMATION
- 6. PRECISION • Precision refers to the closeness of two or more measurements to each other. • Using the same example, you weigh a given substance five times and get 3.2 kg • Precision is independent of accuracy. You can be very precise but inaccurate as well as accurate but imprecise. • The term error represents the imprecision and inaccuracy of a numerical computation. 6NUMERICAL APPROXIMATION
- 7. 7NUMERICAL APPROXIMATION ACCURACY & PRECISION Figure 1 shows some examples of accuracy and precision
- 8. MISTAKES • According to the Cambridge Dictionary, a mistake is “an action, decision, or judgment that produces an unwanted or unintentional result”. • In numerical analysis, a mistake is not an error. • A mistake is due to blunder and it may be negligible, with little or no effect on the accuracy of the calculation or it may be so serious as to render the calculated results quite wrong. 8NUMERICAL APPROXIMATION
- 9. MISTAKES Some common mistakes include: • Misreading of repeated digits (e.g., reading 26638 as 26338). •Transposition of digits (e.g., reading 1832 as 1382). • Incorrectly positioning a decimal point; (e.g., placing a decimal point at 422.438 as 4224.38). • Overlooking signs (especially near sign changes). • Mistakes in reading the instrument, recording and tabulating data. •Misreading of tables (for example, referring to a wrong line or a wrong column). 9NUMERICAL APPROXIMATION
- 10. MISTAKES 10NUMERICAL APPROXIMATION Picture showing error in reading from a thermometer
- 11. MISTAKES Some ways to minimize mistakes include: • Have an awareness of common mistakes • Ensure signs are clearly written • Double check calculations 11NUMERICAL APPROXIMATION
- 12. ERRORS • The numerical errors are generated with the use of approximations to represent mathematical operations and quantities. • An error is the representation of the inaccuracy and vagueness of predictions. • For the types of errors, the relationship between the true value or true result and the approximate value is given by: True Value = Approximation + error 12NUMERICAL APPROXIMATION
- 13. ABSOLUTE ERROR • Is the amount of physical error in a measurement. • Let’s say a meter stick is used to measure a given distance. The error is rather hastily made, but it is good to ±1mm. • This is the absolute error of the measurement. That is, absolute error = ±1mm (0.001m) 13NUMERICAL APPROXIMATION
- 14. RELATIVE ERROR • This gives an indication of how good a measurement is relative to the size of the thing being measured. • Let’s say that two students measure two objects with a meter stick. One student measures the height of a room and gets a value of 3.215 meters ±1mm (0.001m). Another student measures the height of a small cylinder and measures 0.075 meters ±1mm (0.001m). • Clearly, the overall accuracy of the ceiling height is much better than that of the 7.5 cm cylinder. 14NUMERICAL APPROXIMATION
- 15. RELATIVE ERROR The comparative accuracy of these measurements can be determined by looking at their relative errors. • Relative error = absolute error / value of thing measured • Relative error (ceiling height) = 0.001m/ 3.125m •100 = 0.0003% • Relative error (cylinder height) = 0.001m/ 0.075m •100 = 0.01% The relative error in the ceiling height is considerably smaller than the relative error in the cylinder height even though the amount of absolute error is the same in each case. 15NUMERICAL APPROXIMATION
- 16. ROUND OFF ERROR • The round-off error is used because representing every number as a real number isn't possible. So rounding is introduced to adjust for this situation. • A round-off error represents the numerical amount between what a figure actually is versus its closest real number value, depending on how the round is applied. • For instance, rounding to the nearest whole number means you round up or down to what is the closest whole figure. In terms of numerical analysis the round-off error is an attempt to identify what the rounding distance is when it comes up in algorithms. It's also known as a quantization error. 16NUMERICAL APPROXIMATION
- 17. TRUNCATION ERROR • A truncation error occurs when approximation is involved in numerical analysis. • The error factor is related to how much the approximate value is at variance from the actual value in a formula or math result. • For example, consider the speed of light in a vacuum. The official value is 299,792,458 m/s. In scientific notation, it is expressed as 2.99792458 x108. Truncating it to 2 decimal places yields 2.99 x 108. •The truncation error is the difference between the actual value and the truncated value. This is equal to 0.00792458 x 108 or 7.92458 x 105 . 17NUMERICAL APPROXIMATION
- 18. NUMBER REPRESENTATION ERROR • Number representation errors are errors that occur when numbers with no exact value are approximated. • For example, numbers with infinite decimal places or numbers that don’t terminate in binary form e.g. 0.1 18NUMERICAL APPROXIMATION
- 19. APPROXIMATION ERROR • The approximation error is the discrepancy between present approximated value and the previous approximated value to it. 𝐸𝑎 = Present Approximation – Previous Approximation 19NUMERICAL APPROXIMATION
- 20. ERROR PROPAGATION • Propagation of Error is defined as the effects on a function by a variable's uncertainty. • It is a calculus derived statistical calculation designed to combine uncertainties from multiple variables, in order to provide an accurate measurement of uncertainty. 20NUMERICAL APPROXIMATION
- 21. ADDITION OF MEASURED QUANTITIES •If you have measured values for the quantities X, Y, and Z, with uncertainties dX, dY, and dZ, and your final result, R, is the sum or difference of these quantities, then the uncertainty dR is: 21NUMERICAL APPROXIMATION 𝑅 = 𝑋 + 𝑌 − 𝑍 𝛿𝑅 = 𝛿𝑋 + 𝛿𝑌 + 𝛿𝑍 𝛿𝑅 = 𝛿𝑋 2 + √ 𝛿𝑌 2 +√ 𝛿𝑍 2
- 22. MULTIPLICATION OF MEASURED QUANTITIES •In the same way as for sums and differences, we can also state the result for the case of multiplication and division: 22NUMERICAL APPROXIMATION 𝑅 = 𝑋. 𝑌 𝑍 𝛿𝑅 𝑅 ≈ 𝛿𝑋 𝑋 + 𝛿𝑌 𝑌 + 𝛿𝑍 𝑍 𝛿𝑅 = 𝑅 . √ 𝛿𝑋 𝑋 ² + √ 𝛿𝑌 𝑌 ² + √ 𝛿𝑍 𝑍 ²
- 23. MULTIPLICATION WITH A CONSTANT •If the uncertainty of an observable X is known, and it is to be multiplied by a constant that is know exactly. •The error can be found by multiplying the error in X by the absolute value of the constant. 23NUMERICAL APPROXIMATION 𝑅 = 𝑐 . 𝑋 𝛿𝑅 = 𝑐 . 𝛿𝑋
- 24. POLYNOMIAL FUNCTIONS •If there is a dependence of the result on the measured quantity X that is not described by simple multiplications or additions, we state the general answer for R as a general function of one or more variables below. 24NUMERICAL APPROXIMATION
- 25. POLYNOMIAL FUNCTIONS 25NUMERICAL APPROXIMATION •But first cover the special case that R is polynomial functions of one variable X. 𝑅 = 𝑋 𝑛 𝛿𝑅 = 𝑛 . 𝛿𝑋 𝑋 . 𝑅
- 26. GENERAL FUNCTIONS •We can express the uncertainty in R for general functions of one or more observables. •If R is a function of X and Y, written as R(X,Y), then the uncertainty in R is obtained by taking the partial derivatives of R with respect to each variable, multiplication with the uncertainty in that variable, and addition of these individual terms in quadrature. 26NUMERICAL APPROXIMATION 𝑅 = 𝑅 (𝑋, 𝑌, … ) 𝛿𝑅 = 𝜕𝑅 𝜕𝑋 . 𝛿𝑋 2 + 𝜕𝑅 𝜕𝑌 . 𝛿𝑌 2 + …
- 27. EULER’S METHOD • For most first-order differential equations, it simply is not possible to find analytic solutions, since they will not fall into the few classes for which solution techniques are available. So our final approach to analyzing first-order differential equations is to look at the possibility of constructing a numerical approximation to the unique solution to the initial-value problem. •It is important to emphasize that the Euler method does not generate a formula for the solution to the differential equation. Rather it generates a sequence of approximations to the value of the solution at specified points. The idea behind Euler’s method therefore is to use the tangent line to the solution curve through (X0, Y0) to obtain such an approximation. NUMERICAL APPROXIMATION 27
- 28. EULER’S METHOD Figure 2. Euler’s method for approximating the solution to the initial-value problem y’= dy/dx = f(x,y), y(X0) = y0. 28NUMERICAL APPROXIMATION
- 29. EULER’S METHOD 29NUMERICAL APPROXIMATION Figure 3. The exact solution to the initial-value problem considered in Figure 2 and the two approximations obtained using Euler’s method.
- 30. Using the Formula for EULER’S METHOD • This formula gives a reasonably good approximation if we take plenty of terms, and if the value of h is reasonably small. For Euler's Method, we take the first 2 terms only. • We start with some known value of Y, which we could call Y0. It has this value when x = x0 • The result of using this formula is the value of Y one h step to the right of the current value. Call it Y1. So we have: Y1=Y0+ hf(X0,Y0) 30NUMERICAL APPROXIMATION
- 31. Using the Formula for EULER’S METHOD Y1 is the next generated solution Y0 is the current solution h is the interval between steps f(x0,Y0) is the value of the derivative at the starting point (x0,Y0) This process is continued for as many trials as needed. 31NUMERICAL APPROXIMATION
- 32. EXAMPLE 1 Question: Euler's Method- Approximate y(0.4) using E.M. dy/dx= X+ 2y, y(0)=0 Step Size: h= 0.1 32NUMERICAL APPROXIMATION
- 33. EXAMPLE 1 Formulas: Xn+ 1= Xn + h Yn+ 1= Yn + hf(Xn,Yn) 33NUMERICAL APPROXIMATION
- 34. EXAMPLE 1 Solution: N.B: Y(0)=0 tells us our initial conditions for X and Y. So Since Y(X0) = Y0 then X0= 0 and Y0= 0 So X0= 0 Y0= 0 X1= 0+0.1= 0.1 Y1= 0+0.1 ( 0 + 2(0) )= 0 34NUMERICAL APPROXIMATION
- 35. EXAMPLE 1 X2= 0.1+0.1= 0.2 Y2= 0+0.1 ( 0.1 + 2(0) )= 0.01 X3= 0.2+0.1= 0.3 Y3= 0.01+ 0.1 ( 0.2 + 2(0.01) )= 0.032 X4= 0.3+0.1= 0.4 Y4= 0.032+ 0.1 (0.3+ 2(0.032) )= 0.0684 Answer: Y(0.4) approximate to 0.0684 35NUMERICAL APPROXIMATION
- 36. EXAMPLE 2 Question: Use Euler's method with step size 0.3 to compute the approximate y- value y(0.9) of the solution of the initial value problem y'= X2 , y(0)= 1. 36NUMERICAL APPROXIMATION
- 37. EXAMPLE 2 Formulas: Xn+ 1= Xn + h Yn+ 1= Yn + hf(Xn,Yn) 37NUMERICAL APPROXIMATION
- 38. EXAMPLE 2 Solution: X0= 0 Y0= 1 X1= 0+0.3= 0.3 Y1= 1+ 0.3(0)2= 1 X2= 0.3+0.3= 0.6 Y2= 1+ 0.3 (0.3)2= 1.027 X3= 0.6+0.3= 0.9 Y3= 1.027+ 0.3 (0.6)2= 1.135 Answer: Y(0.9) approxmiate to 1.135 38NUMERICAL APPROXIMATION
- 39. References http://homepage.divms.uiowa.edu/~atkinson/NA_Overview.pdf Date Retrieved: 2017-02-27 at 3:12pm https://www.ncsu.edu/labwrite/Experimental%20Design/accuracyprecision.htm Date Retrieved: 2017- 02-27 at 3:24pm https://www.slideshare.net/Mileacre/numerical-approximation-4118002 Date Retrieved: 2017-02-27 at 4:14pm https://www.techwalla.com/articles/types-of-errors-in-numerical-analysis Date Retrieved: 2017-02-27 at 4: 26pm http://www2.phy.ilstu.edu/~wenning/slh/Absolute%20Relative%20Error.pdf Date Retrieved: 2017-02-27 at 4:43pm http://www.intmath.com/differential-equations/11-eulers-method-des.php Date Retrieved: 2017-02-27 at 6:05pm 39NUMERICAL APPROXIMATION
- 40. END OF PRESENTATION NUMERICAL APPROXIMATION 40