This document discusses numerical approximation and related concepts such as accuracy, precision, types of errors, and Euler's method. It provides examples of applying Euler's method to solve initial value problems by taking successive approximations with a fixed step size. The document outlines these topics and provides examples to illustrate key concepts in numerical analysis such as error propagation and different types of errors that can occur when performing calculations. It concludes by listing references used in the presentation.

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