Numerical Integration and Differentiation - Numerical Structural Analysis - page 119

Next, expanding the Taylor series for a function y=f ( x ) at x= ( x i -h ) gives
the following equation:

2

yh yh ′′

3

′′′

( ) =−′ +

'

i i

yx hyyh − − + (3.7)

i i i

2 ! 3 !

If Equation 3.7 is subtracted from Equation 3.6, a first derivative relation-
ship is as follows:

( ) −− ( ) −− ′′′

yx hyxh
h

+

2

yh

i i i

y ′ = +

(3.8)

i

2 6

This may be written as follows with successive values of y and the higher
order terms omitted:

y − y

i + 1 i − 1

y ′ =

i

2 h

This equation is known as the central-difference approximation of y′ i at x i
with errors, order of h 2 . If Equation 3.7 is added to Equation 3.6, a second
derivative relationship is as follows:

( ) −+ − ( ) −

yx hy yx h
h

+ 2

2

yh ′′′ ′

(3.9)

i i i i

y ′′= +

i

2

12

This may be written as follows with successive values of y and the higher
order terms omitted:

′′= −+

y 2 y y

i + 1 i i − 1

y

i

2

h

This equation is known as the central-difference approximation of y i ″ at x i
with errors, order of h 2 . Next, expanding the Taylor series for a function
y=f ( x ) at x= ( x i + 2 h ) gives the following equation:

′′ () + ′′′ () + ′′′ ′ () +

2 3 4

yx hyyh yhyhy h
i

2
2

2
3

2

( ) =+′

i i i

+ 2 2 + (3.10)

i i

! ! 4 !