Numerical differentiation: numdiff

### 5.19.6 Numerical differentiation: numdiff

The numdiff command finds numerical approximations to derivatives.

- numdiff takes three mandatory arguments and one optional argument.
  - X=[α0,α1,…,αn], Y=[β0,β1,…,βn], two lists of real numbers, where n≥ 1.
  - x0, a real number.
  - Optionally, m, an integer or a sequence of integers (by default 1).
- numdiff(X,Y,x0 ⟨,m⟩) returns an approximation of the mth derivative of a function f at x0, or a sequence of derivatives of order given by the sequence m, where f has values given by f(αk)=βk, k=0,1,…,n.

numdiff uses Fornberg’s algorithm (1988) improved by Sadiq and Viswanath (2014). The complexity of the algorithm is O(n2m) in both time and space.

Note that α0,α1,…,αn do not have to be equally spaced, but they must be mutually different and input in ascending order. There are no restrictions on the choice of x0.

Rounding errors are usually not damaging for m≤ 4. For higher m, consider providing exact input data.

#### Examples

- Let f(x)=sin(x)e−x, x∈[0,1]. Sample this function at the points in

  X=[0,0.1,0.2,0.4,0.5,0.7,0.8,1]

  to approximate f″(1/π).
  Input:

  f:=unapply(sin(x)*exp(-x),x):;
  X:=[0,0.1,0.2,0.4,0.5,0.7,0.8,1]:;
  Y:=apply(f,X):;

  Now you can approximate the second derivative of f at the point x0=1/π.
  Input:

  x0:=1/pi:;
  d:=numdiff(X,Y,x0,2)

  Output:

  −1.38167652799

  Finally, compute the relative error of the obtained approximation.
  Input:
  abs(d-f’’(x0))/abs(f’’(x0))*100
  Output:

  2.82975186496×10−5

  The result is expressed in percentages.
- Use a sequence of values for the parameter m to find a list of approximations of the respective derivatives at x0. This is faster than calling numdiff to approximate one derivative at a time.
  Specifically, approximate the first, second and third derivative of the function

  f(x)=1− |
  1
  1+x2
  | , x∈[0,1],

  at the point x0=0.57 by sampling f at 21 equidistant points in the segment [0,1].
  Input:

  f:=unapply(1-1/(1+x^2),x)
  X:=[(0.05*k)$(k=0..20)]:; Y:=apply(f,X):;
  numdiff(X,Y,0.57,1,2,3)

  Output:

  ⎡ ⎣ | 0.649439427528,0.0217571104587,−2.99724196738 | ⎤ ⎦

  Actual values: f′(x0)=0.649439427528, f″(x0)=0.0217571104558 and f‴(x0)=−2.99724196764.

numdiff can be used for generating custom finite-difference stencils for approximation of derivatives.

#### Example

Let X=[−1,0,2,4], Y=[a,b,c,d] and x0=1. To obtain an approximation formula for the second derivative:
Input:

numdiff([-1,0,2,4],[a,b,c,d],1,2)

Output:

2
5
| a− |
b
2
| + |
d
10

The approximation is always a linear combination of elements in Y, regardless of X, x0 and m.

Given the lists X=[α0,α1,…,αn] and Y=[β0,β1,…,βn], the Lagrange polynomial passing through points (αk,βk) where k=0,1,…,n can be obtained by setting m=0 and entering a symbol for x0.

#### Example

Let X=[−2,0,1] and Y=[2,4,1]:
Input:

expand(numdiff([-2,0,1],[2,4,1],x,0))

Output:

− |
4
3
| x2− |
5
3
| x+4

The same result is obtained by entering lagrange([-2,0,1],[2,4,1],x).