Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Differentiate function

`fprime = fnder(f,dorder) `

fnder(f)

`fprime = fnder(f,dorder) `

is the description of the `dorder`

th 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 *t*_{1} is its leftmost knot,
then, up to rounding errors, `fnint(fnder(f))`

contains
the B-form of *f* – *f*(*t*_{1}).
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.

Was this topic helpful?