This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.


Differentiate function


fprime = fnder(f,dorder)


fprime = fnder(f,dorder) is the description of the dorderth derivative of the function whose description is contained in f. The default value of dorder is 1. For negative dorder, the particular |dorder|th indefinite integral is returned that vanishes |dorder|-fold at the left endpoint of the basic interval.

The output is of the same form as the input, i.e., they are both ppforms or both B-forms or both stforms. fnder does not work for rational splines; for them, use fntlr instead. fnder works for stforms only in a limited way: if the type is tp00, then dorder can be [1,0] or [0,1].

fnder(f) is the same as fnder(f,1).

If the function in f is multivariate, say m-variate, then dorder must be given, and must be of length m.


If f is in ppform, or in B-form with its last knot of sufficiently high multiplicity, then, up to rounding errors, f and fnder(fnint(f)) are the same.

If f is in ppform and fa is the value of the function in f at the left end of its basic interval, then, up to rounding errors, f and fnint(fnder(f),fa) are the same, unless the function described by f has jump discontinuities.

If f contains the B-form of f, and t1 is its leftmost knot, then, up to rounding errors, fnint(fnder(f)) contains the B-form of ff(t1). However, its leftmost knot will have lost one multiplicity (if it had multiplicity > 1 to begin with). Also, its rightmost knot will have full multiplicity even if the rightmost knot for the B-form of f in f doesn't.

Here is an illustration of this last fact. The spline in sp = spmak([0 0 1], 1) is, on its basic interval [0..1], the straight line that is 1 at 0 and 0 at 1. Now integrate its derivative: spdi = fnint(fnder(sp)). As you can check, the spline in spdi has the same basic interval, but, on that interval, it agrees with the straight line that is 0 at 0 and –1 at 1.

See the examples “Intro to B-form” and “Intro to ppform” for examples.


For differentiation of either polynomial form, the derivatives are found in the piecewise-polynomial sense. This means that, in effect, each polynomial piece is differentiated separately, and jump discontinuities between polynomial pieces are ignored during differentiation.

For the B-form, the formulas [PGS; (X.10)] for differentiation are used.

For the stform, differentiation relies on knowing a formula for the relevant derivative of the basis function of the particular type.

See Also

| | |