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fnint
Integrate function
Syntax
intgrf = fnint(f,value)
fnint(f)
Description
intgrf = fnint(f,value)
is the description of an indefinite integral of
the univariate function whose description is
contained in f. The integral is normalized to have
the specified value at the left endpoint of the
function's basic
interval, with the default value being zero.
The output is of the same type as the input, i.e., they are
both ppforms or both B-forms. fnint does not work
for rational splines nor for functions in stform.
fnint(f) is the same
as fnint(f,0).
Indefinite integration of a multivariate function,
in coordinate directions only, is available via fnder(f,dorder) with dorder having
nonpositive entries.
Examples
The statement diff(fnval(fnint(f),[a b])) provides
the definite integral over the interval [a .. b]
of the function described by f.
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 the B-form, the formula [PGS; (X.22)] for integration is used.
