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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 error student notes

NUMERICAL ERRORStudent Notes

ENGR 351

Numerical Methods for Engineers

Southern Illinois University Carbondale

College of Engineering

Dr. L.R. Chevalier

objectives
Objectives
  • To understand error terms
  • Become familiar with notation and techniques used in this course
approximation and errors significant figures
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 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
slide6

Accuracy and Precision

increasing accuracy

increasing precision

error definitions
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
Error definitions cont.
  • True relative percent error
example
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 cont1
Error definitions cont.
  • May not know the true answer apriori
  • This leads us to develop an iterative approach to numerical methods
error definitions cont2
Error definitions cont.
  • Usually not concerned with sign, but with tolerance
  • Want to assure a result is correct to nsignificant figures
example1
Example

Consider a series expansion to estimate trigonometric functions

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

STRATEGY

strategy
Strategy

Stop when ea ≤ es

error definitions cont3
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 cont4
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
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 series1
TAYLOR SERIES

Zero order approximation

This is good if the function is a constant.

taylor series expansion
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 expansion1
Taylor Series Expansion

Second order approximation - captures some of the curvature

example2
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

strategy1
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
objectives1
Objectives
  • To understand error terms
  • Become familiar with notation and techniques used in this course
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