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fnder
Differentiate function
Syntax
fprime = fnder(f,dorder)
fnder(f)
Description
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.
Examples
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 f – f(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.
Algorithms
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.
