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### NUMERICAL ERRORStudent Notes

ENGR 351

Numerical Methods for Engineers

Southern Illinois University Carbondale

College of Engineering

Dr. L.R. Chevalier

Objectives

- To understand error terms
- Become familiar with notation and techniques used in this course

Approximation and ErrorsSignificant Figures

- 4 significant figures
- 1.845
- 0.01845
- 0.0001845
- 43,500 ? confidence
- 4.35 x 104 3 significant figures
- 4.350 x 104 4 significant figures
- 4.3500 x 104 5 significant figures

Accuracy and Precision

- Accuracy - how closely a computed or measured value agrees with the true value
- Precision - how closely individual computed or measured values agree with each other
- number of significant figures
- spread in repeated measurements or computations

Error Definitions

- Numerical error - use of approximations to represent exact mathematical operations and quantities
- true value = approximation + error
- error, et=true value - approximation
- subscript trepresents the true error
- shortcoming....gives no sense of magnitude
- normalize by true value to get true relative error

Error definitions cont.

- True relative percent error

Example

- Consider a problem where the true answer is 7.91712. If you report the value as 7.92, answer the following questions.
- How many significant figures did you use?
- What is the true error?
- What is the true relative percent error?

Error definitions cont.

- May not know the true answer apriori

- This leads us to develop an iterative approach to numerical methods

Error definitions cont.

- Usually not concerned with sign, but with tolerance
- Want to assure a result is correct to nsignificant figures

Example

Consider a series expansion to estimate trigonometric functions

Estimate sin(p/ 2) to three significant figures. Calculate et and ea.

STRATEGY

Strategy

Stop when ea ≤ es

Error Definitions cont.

- Round off error - originate from the fact that computers retain only a fixed number of significant figures
- Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure

Error Definitions cont.

- Round off error - originate from the fact that computers retain only a fixed number of significant figures
- Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure

To gain insight consider the mathematical

formulation that is used widely in numerical

methods - TAYLOR SERIES

TAYLOR SERIES

- Provides a means to predict a function value at one point in terms of the function value at and its derivative at another point

Taylor Series Expansion

First order approximation

slope multiplied by distance

Still a straight line but capable of predicting an increase or decrease - LINEAR

Taylor Series Expansion

Second order approximation - captures some of the curvature

Example

Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1). Calculate et after each step.

Note:

f(1) = 0.2

STRATEGY

Strategy

- Estimate the function using only the first term
- Use x = 0 to estimate f(1), which is the y-value when x = 1
- Calculate error, et
- Estimate the function using the first and second term
- Calculate the error, et
- Progressively add terms

Objectives

- To understand error terms
- Become familiar with notation and techniques used in this course

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