NUMERICAL ERROR Student Notes

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NUMERICAL ERROR Student 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.

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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
• 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

Accuracy and Precision

increasing accuracy

increasing precision

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

Zero order approximation

This is good if the function is a constant.

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