intgrf = fnint(f,value)
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
fnint does not work for rational splines nor for
functions in stform.
fnint(f) is the same as
Indefinite integration of a multivariate function, in
coordinate directions only, is available via
dorder having nonpositive entries.
diff(fnval(fnint(f),[a b])) provides the definite
integral over the interval [
b] of the
function described by
f is in ppform, or in B-form with its last knot of
sufficiently high multiplicity, then, up to rounding errors,
fnder(fnint(f)) are the same.
f is in ppform and
fa is the value of the
f at the left end of its basic interval, then, up to
the same, unless the function described by
f has jump
f contains the B-form of f, and
t1 is its leftmost knot, then, up to
fnint(fnder(f)) contains the B-form of
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
Here is an illustration of this last fact. The spline in
sp = spmak([0 0
1], 1) is, on its basic interval
1], the straight line that is 1 at 0
and 0 at 1. Now integrate its derivative:
fnint(fnder(sp)). As you can check, the spline in
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 the B-form, the formula [PGS; (X.22)] for integration is used.