Numerical Analysis/Differentiation/Examples

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When deriving a finite difference approximation of the th derivative of a function , we wish to find and such that

or, equivalently,

where is the error, the difference between the correct answer and the approximation, expressed using Big-O notation. Because may be presumed to be small, a larger value for is better than a smaller value.

A general method for finding the coefficients is to generate the Taylor expansion of and choose and such that and the remainder term are the only non-zero terms. If there are no such coefficients, a smaller value for must be chosen.

For a function of variables , the procedure is similar, except are replaced by points in and the multivariate extension of Taylor's theorem is used.


Single-Variable[edit]

In all single-variable examples, and are unknown, and is small. Additionally, let be 5 times continuously differentiable on .

First Derivative[edit]

Find such that best approximates .

Let be 42 times continuously differentiable on . Find the largest such that

In other words, find the order of the error of the method.

Second Derivative[edit]

Find such that best approximates .

Multivariate[edit]

In all two-variable examples, and are unknown, and is small.

Non-Mixed Derivatives[edit]

Because of the nature of partial derivatives, some of them may be calculated using single-variable methods. This is done by holding constant all but one variable to form a new function of one variable. For example if , then .

Find an approximation of

Mixed Derivatives[edit]

Mixed derivatives may require the multivariate extension of Taylor's theorem.

Let be 42 times continuously differentiable on and let be defined as

Find the largest such that

In other words, find the order of the error of the approximation.

Example Code[edit]

Implementing these methods is reasonably simple in programming languages that support higher-order functions. For example, the method from the first example may be implemented in C++ using function pointers, as follows:

//  Returns an approximation of the derivative of f at x.
double derivative (double (*f)(double), double x, double h =0.01) {
  return (f(x + h) - f(x - h)) / (2 * h);
}