1

Numerical Differentiation

2

Taylor’s Theorem

h=(xi+1- xi)

High-Accuracy Differentiation Formulas

2ii1i

i hOh2

xf-

h

xfxfxf

ixx

idx

xdfxf

one more sample xi+2 = xi+1 + h

hOh

xfxfxf i1i

i

hOh

xfx2fxfxf

2

i1i2ii

2

2

i1i2ii1ii hOh

2h

xfxfxf-

h

xfxfxf

2

one extra sample xi+1 = xi + h

first-order accurate

second-order accurate

Big O notation describes the limiting behavior of a

function when the argument tends towards a particular

value or infinity, usually in terms of simpler functions.

Forward difference

xi1 xi xi+1

x

h

Backward difference

xi1 xi xi+1

x

h

Centered difference

xi1 xi xi+1

x

2h

First Derivatives

• Forward difference

• Backward difference

• Central difference

)x(f

i-2 i-1 i i+1 i+2

1i1i

1i1i

1i1i

1i1i

1ii

1ii

1ii

1ii

i1i

i1i

i1i

i1i

xx

yy

xx

)x(f)x(f)x(f

xx

yy

xx

)x(f)x(f)x(f

xx

yy

xx

)x(f)x(f)x(f

x

y

8

Forward Finite-Divided Difference Formulas

second-order accurate forward

difference formula for f’(x)

9

Backward Finite-Divided Difference Formulas

10

Centered Finite-Divided Difference Formulas

11

Example

hOh

xfxfxf i1i

i

12

13

Example of High-Accuracy Differentiation

14

Miscellaneous problems #1

15

16

Miscellaneous problems #2

17

18

Miscellaneous problems #3

19

20

Homework

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