NUMERICAL CALCULUS PACKAGE SYMBOL

ND

gives a numerical approximation to the derivative of expr with respect to x at the point

.

gives a numerical approximation to the

derivative of expr.

MORE INFORMATION

To use , you first need to load the Numerical Calculus Package using .

The expression expr must be numeric when its argument x is numeric.

is equivalent to .

is unable to recognize small numbers that should in fact be zero. Chop may be needed to eliminate these spurious residuals.

The following options can be given:

Method	EulerSum	method to use
Scale	1	size at which variations are expected
Terms	7	number of terms to be used
WorkingPrecision	MachinePrecision	precision to use in internal computations

Possible settings for Method include:

	EulerSum	use Richardson's extrapolation to the limit
	NIntegrate	use Cauchy's integral formula

With Method, needs to evaluate expr at .

If expr is not analytic in the neighborhood of , then the default method must be used.

The option Scale->s is used to capture the scale of variation when using Method.

When the value of the derivative depends on the direction, the default is to the right. Other directions can be chosen with the option Scale->s, where the direction is s.

The option Terms->n gives the number of terms to use for extrapolation when using Method.

With Method->NIntegrate, the expression expr must be analytic in a neighborhood of the point .

The option Scale->r specifies the radius of the contour of integration to use with Method->NIntegrate.

EXAMPLES

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Basic Examples (1)

Needs["NumericalCalculus`"]

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Scope (1)

The expression and evaluation point may be complex:

Generalizations & Extensions (1)

is threaded element-wise:

Options (7)

Use the default Method

when expr is not analytic in the neighborhood of

:

Check:

An incorrect answer is obtained with Method->NIntegrate:

Here is a derivative where the default method works poorly:

The correct answer is:

In this case the expression is analytic, so Method->NIntegrate will work well:

Use Scale->s to capture the region of variation:

The scale of variation is around

:

A value of Scale->s which is too large can be compensated by increasing the number of terms:

Use Scale to specify directional derivatives. The left and right derivatives of the nonanalytic function

:

Check:

Complex directions may also be specified:

Check:

Use the option Scale to avoid regions of non-analyticity when the method used is NIntegrate:

Shrinking the radius avoids the essential singularity at x=1:

Check:

Increasing the number of terms may improve accuracy. Here is a somewhat inaccurate approximation:

Check:

Increasing the number of terms produces a more accurate answer:

Increasing the number of terms further can produce nonsense due to numerical instability:

Combining an increase in the number of terms with a higher working precision often will reduce the error:

High-order derivatives with Method

experience significant subtractive cancellation:

Using a higher working precision and additional terms produces an accurate answer:

For this problem, Method->NIntegrate with default options produces a correct answer:

Higher-order derivatives will again experience numerical instability:

Increasing WorkingPrecision will improve the accuracy:

An alternative is to increase the radius of the contour of integration:

Applications (1)

is useful for differentiating functions which are only defined numerically. Here is such a function:

Here is the derivative of

with respect to b evaluated at

:

can be used as an aid in developing and testing a more robust function for finding the derivative:

Check:

Properties & Relations (3)

The option Method->NIntegrate uses Cauchy's integral formula to compute derivatives:

The equivalent computation can be performed using NResidue:

Mathematica has built-in code to compute derivatives of numerical functions:

The built-in numerical derivative code can be used. However, it is unable to capture the rapid oscillations:

The correct answer:

Using

with the appropriate options can compute an accurate derivative:

With Method

,

must be able to evaluate expr at the point

:

Adding an additional definition for f allows

to compute the derivative:

Check:

In this case, Method->NIntegrate produces a more accurate answer:

Neat Examples (1)

Some fractional and complex derivatives can be computed: