Numerical Analysis - Errors

8/13/2019 Numerical Analysis - Errors

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Model:Involves simplification or omission in themodel equation

Data:Errors in measurement or estimation of raw data

Numerical Method: Errors based on some

approximation

Representation of Numbers: for example, pcannot be represented exactly by a finite number of

digits

Arithmetic: Mistakes in carrying out operations suchas addition or multiplication


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Number Representation:Numerical calculation caninvolve numbers that cannot be represented exactly by

a finite number of digits. For instance 2/3 (base 10), or

irrational numbers (p)

Round-off Error:Reducing the number to asignificant digit

Truncation Error: Reducing an infinite series oriterative process


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Absolute Error:Absolute difference between the exactnumberxand the approximate numberX

Relative Error:Ratio of the absolute error and the

absolute exact number

Implicit Error: Errors of the original numbers or factors

Accumulated Error: Total cumulative error at any givenstep or iteration


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Consider two numbers

Under Addition/Subtraction:The magnitude ofthe propagated error is not more than the sum of the

initial absolute errors

Under Multiplication:The maximum relative errorpropagated is approximately the sum of the initial

relative errors


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Addition of a small but non-zero number may have no

effect5.18x102 + 4.37x10-1 = 5.18x102 + 0.00437x102 = 5.18437x102~ 5.18x102

The multiplicative inverse may not exist; a*(1/a) 1a = 3.00x100 1/ais 3.33x10-1 a*(1/a) is 9.99 * 10-1

The associative property may not apply; (a+b)+c a+(b+c)

a= 6.31x101, b= 4.24x100, c= 2.47x101,

then

(a+b)+c = (6.31x101 + 0.424x101) + 2.47x101 6.73x101 + .0247x101 6.75x101,

whereas

a+(b+c) = 6.31x101 + (4.24x100 + 2.47x100) 6.31x101 + 4.49x100 6.31x101+4.49x100

6.31x101+ 0.449x101 6.76x101

Subtracting a number from another nearly equal

number may result in loss of significance or cancellation

error.1 - cos(0.05) = 1-0.99875 1.00x100 - 0.999x100 1.00x10-3


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Representing a function as an infinite series of terms

involving simpler functions

Taylor series: where z is between a and x

=

1!

2! 2

! ( )

+

1 !( )+

() ()

= ()


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For the number 5/3 = 1.66666 determine the magnitude of round-off

error when it is represented by a number obtained from the decimal formby:

1. Chopping to 3S

2. Chopping to 3D

3. Rounding to 3S

4. Rounding to 3D

Evaluate the following operations as accurately as possible, including all

errors

1. 8.24 + 5.33

2. 124.53124.523. 4.27 x 3.13

4. 9.48 x 0.5136.72

5. 0.25 x 2.84 0.64

6. 1.732.16 +0.08 + 1.00 -2.23 -0.97 +3.02


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Modify the previous assignment using a Taylor series approximation about

a = 0 for cosand sin. Choose where to truncate the series and comparethe results with those from your previous program using the intrinsic

functions. How big are the discrepancies?

Numerical Analysis - Errors

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