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Measuring Errors

Why measure errors?

1) To determine the accuracy of numerical

results.

2) To develop stopping criteria for iterative

algorithms.

Example—True Error

The derivative, f^ ′(^ x ) of a function f^ ( x ) can be

approximated by the equation,

h

f x h f x f x

If

x

f x e

1. 5

( )= 7 and^ h =^0.^3

a) Find the approximate value of f '(^2 )

b) True value of f '(^2 )

c) True error for part (a)

Example (cont.)

Solution:

a) For x^ =^2 and h =^0.^3

f f f

f ( 2. 3 )− f ( 2 )

1. 5 ( 2. 3 ) 0. 5 ( 2 ) e − e =

= =^10.^263

Relative True Error

• Defined as the ratio between the true error,

and the true value.

Relative True Error ( ∈ t^ ) =^

True Error

True Value

Example—Relative True Error

Following from the previous example for true error,

find the relative true error for

x f x e

1. 5

( )= 7 at f '(^2 )

with h =^0.^3

Et =− 0. 722

From the previous example,

True Value

TrueError ∈ t =

Relative True Error is defined as

9. 5140

− 0. 722 = (^) =− 0. 075888

as a percentage,

∈ t =− 0. 075888 × 100 % =− 7. 5888 %

Example—Approximate Error

For

x

f x e

1. 5

( )= 7 at x^ =^2 find the following,

a) f^ ′(^2 ) using h =^0.^3

b) f^ ′(^2 ) using h =^0.^15

c) approximate error for the value of f^ ′(^2 ) for part b)

Solution:

a) For

h

f x h f x f x

x = 2 and h = 0. 3

f f

f

Example (cont.)

f ( 2. 3 )− f ( 2 )

Solution: (cont.)

1. 5 ( 2. 3 ) 0. 5 ( 2 ) e − e =

2. 3

3. 107 − 19. 028

b) For x^ =^2 and h =^0.^15

1. 15

( 2 0. 15 ) ( 2 ) ( 2 )

' f f f

• − ≈

f ( 2. 15 )− f ( 2 )

Relative Approximate Error

• Defined as the ratio between the

approximate error and the present

approximation.

Relative Approximate Error (

Approximate Error

Present Approximation

∈ a ) =

Example—Relative Approximate Error

For

x

f x e

1. 5

at x^ =^2 , find the relative approximate

error using values from h^ =^0.^3 and h =^0.^15

Solution:

From Example 3, the approximate value of f ′(^2 )=^10.^263

using h^ =^0.^3 and f^ ′(^2 )=^9.^8800 using h =^0.^15

= a E (^) Present Approximation – Previous Approximation

How is Absolute Relative Error used as a

stopping criterion?

If |∈^ a^ |≤∈ s where ∈ s^ is a pre-specified tolerance, then

no further iterations are necessary and the process is

stopped.

If at least m significant digits are required to be

correct in the final answer, then

| | 0. 5 10 %

2 m a

− ∈ ≤ ×

Table of Values

For

x

f x e

1. 5

at x^ =^2 with varying step size, h

0.3 10.263 N/A 0

h f^ ′(^2 ) a

∈ m