Mathematical Tools for Physics

11—Numerical Analysis 323

The previous example of the derivative ofsinxatx= 0. 2 withh= 0. 1 gives, using this formula:

1

2. 4

[0. 0499792 − 27 × 0 .1494381 + 27× 0. 2474040 − 0 .3428978] = 0. 9800661 ,

and the error is less by a factor of about 7.

You can find higher derivatives the same way.

f(h) =f(0) +hf′(0) +

1

2

h^2 f′′(0) +

1

6

h^3 f′′′(0) +

1

24

h^4 f′′′′(0)

f(h) +f(−h) = 2f(0) +h^2 f′′(0) +

1

12

h^4 f′′′′(0) +···

f′′(0) =

f(−h)− 2 f(0) +f(h)

h^2

−

1

12

h^2 f′′′′(0) +··· (12)

Notice that the numerical approximation forf′′(0)is even inhbecause the second derivative is unchanged ifx
is changed to−x.

You can get any of these expressions for higher derivatives recursively, though finding the error estimates
requires the series method. The above expression forf′′(0)can be viewed as a combination of first derivative
formulas:

f′′(0)≈

[

f′(h/2)−f′(−h/2)

]

/h

≈

1

h

[

f(h)−f(0)

h

−

f(0)−f(−h)

h

]

=

[

f(h)− 2 f(0) +f(−h)

]

/h^2. (13)

Similarly, the third and higher derivatives can be computed. The numbers that appear in these numerical deriva-
tives are simply the binomial coefficients, Eq. (2.13).