Formula
1. Four-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
|
|
|
2. Four-point FDF (Forward difference formula) for second derivatives
`f^('')(x)=(2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))/(h^2)`
|
3. Four-point BDF (Backward difference formula) for second derivatives
`f^('')(x)=(-f(x-3h)+4f(x-2h)-5f(x-h)+2f(x))/(h^2)`
|
Examples
Using Four point Forward difference, Backward difference, Central difference formula numerical differentiation to find solution
| x | 1 | 1.05 | 1.10 | 1.15 | 1.20 | 1.25 | 1.30 |
| f(x) | 1 | 1.02470 | 1.04881 | 1.07238 | 1.09545 | 1.11803 | 1.14018 |
`f^'(1.10) and f^('')(1.10)`
Solution:
The value of table for `x` and `y`
| x | 1 | 1.05 | 1.1 | 1.15 | 1.2 | 1.25 | 1.3 |
|---|
| y | 1 | 1.0247 | 1.0488 | 1.0724 | 1.0954 | 1.118 | 1.1402 |
|---|
Four-point CDF (Central difference formula)
`f^'(x)=1/(12h)[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)]`
`f^'(1.10)=1/(12*0.05)[f(1.10-2*0.05)-8f(1.10-0.05)+8f(1.10+0.05)-f(1.10+2*0.05)]`
`f^'(1.10)=1/0.6[f(1)-8f(1.05)+8f(1.15)-f(1.2)]`
`f^'(1.10)=1/0.6[1-8(1.0247)+8(1.0724)-1.0954]`
`f^'(1.10)=0.4767`
Four-point FDF (Forward difference formula) for second derivatives
`f^('')(x)=(2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))/(h^2)`
`f^('')(1.10)=(2f(1.10)-5f(1.10+0.05)+4f(1.10+2*0.05)-f(1.10+3*0.05))/((0.05)^2)`
`f^('')(1.10)=(2f(1.10)-5f(1.15)+4f(1.2)-f(1.25))/(0.0025)`
`f^('')(1.10)=(2(1.0488)-5(1.0724)+4(1.0954)-(1.118))/(0.0025)`
`f^('')(1.10)=-0.204`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
