Arithmetic with function(s)

`fn = fncmb(function,operation)`

f = fncmb(function,function)

fncmb(function,matrix,function)

fncmb(function,matrix,function,matrix)

f = fncmb(function,op,function)

The intent is to make it easy to carry out the standard linear operations of scaling and adding within a spline space without having to deal explicitly with the relevant parts of the function(s) involved.

`fn = fncmb(function,operation)`

returns (a
description of) the function obtained by applying to the values of the function in
`function`

the operation specified by
`operation`

. The nature of the operation depends on whether
`operation`

is a *scalar*, a
*vector*, a *matrix*, or a
*character vector*, as follows.

| Multiply the function by that scalar. |

| Add that vector to the function's values; this requires the function to be vector-valued. |

| Apply that matrix to the function's coefficients. |

| Apply the function specified by that character vector to the function's coefficients. |

The remaining options only work for *univariate* functions. See
Limitations for more information.

`f = fncmb(function,function) `

returns (a
description of) the pointwise sum of the two functions. The two functions must be of
the same form. This particular case of just two input arguments is not included in
the above table since it only works for univariate functions.

`fncmb(function,matrix,function) `

is the
same as `fncmb(fncmb(function,matrix),function)`

.

`fncmb(function,matrix,function,matrix) `

is
the same as
`fncmb((fncmb(function,matrix),fncmb(function,matrix)))`

.

`f = fncmb(function,op,function) `

returns
the ppform of the spline obtained by the pointwise combining of the two functions,
as specified by the character vector `op`

. The argument
`op`

can be one of the character vectors
`'+'`

, `'-'`

, `'*'`

. If the
second function is to be a constant, it is sufficient simply to supply here that
constant.

`fncmb(fn,3.5)`

multiplies (the coefficients of) the function in
`fn`

by 3.5.

`fncmb(f,3,g,-4)`

returns the linear combination, with weights 3
and –4, of the function in `f`

and the function in
`g`

.

`fncmb(f,3,g)`

adds 3 times the function in `f`

to the function in `g`

.

If the function *f* in `f`

happens to be
scalar-valued, then `f3=fncmb(f,[1;2;3])`

contains the description
of the function whose value at *x* is the 3-vector
(*f*(*x*),
2*f*(*x*),
3*f*(*x*)). Note that, by the convention
throughout this toolbox, the subsequent statement fnval(*f*3,
*x*) returns a 1-*column*-matrix.

If `f`

describes a surface in R^{3},
i.e., the function in `f`

is 3-vector-valued bivariate, then
`f2 = fncmb(f,[1 0 0;0 0 1])`

describes the projection of that
surface to the (*x*, *z*)-plane.

The following commands produce the picture of a ... spirochete?

c = rsmak('circle'); fnplt(fncmb(c,diag([1.5,1]))); axis equal, hold on sc = fncmb(c,.4); fnplt(fncmb(sc,-[.2;-.5])) fnplt(fncmb(sc,-[.2,-.5])) hold off, axis off

If `t`

is a knot sequence of length `n+k`

and
`a`

is a matrix with `n`

columns, then
`fncmb(spmak(t,eye(n)),a)`

is the same as
`spmak(t,a)`

.

`fncmb(spmak([0:4],1),'+',ppmak([-1 5],[1 -1]))`

is the
piecewise-polynomial with breaks -`1:5`

that, on the interval [0 ..
4], agrees with the function *x*|→
*B*(*x*|0,1,2,3,4) + *x* (but
has no active break at 0 or 1, hence differs from this function outside the interval
[0 .. 4]).

`fncmb(spmak([0:4],1),'-',0)`

has the same effect as
`fn2fm(spmak([0:4],1),'pp')`

.

Assuming that `sp`

describes the B-form of a spline of order
<`k`

, the output of

fn2fm(fncmb(sp,'+',ppmak(fnbrk(sp,'interv'),zeros(1,k))),'B-')

describes the B-form of the same spline, but with its order raised to `k`

.

`fncmb`

only works for *univariate* functions,
except for the case `fncmb(function,operation)`

, i.e., when there
is just one function in the input.

Further, if two functions are involved, then they must be of the same type. This
means that they must either both be in B-form or both be in ppform, and, moreover,
have the same knots or breaks, the same order, and the same target. The only
exception to this is the command of the form
`fncmb(function,op,function)`

.

The coefficients are extracted (via `fnbrk`

) and operated on by
the specified matrix or operation (and, possibly, added), then recombined with the
rest of the function description (via `ppmak`

,
`spmak,rpmak,rsmak,stmak`

). To be sure, when the function is
rational, the matrix is only applied to the coefficients of the numerator. Again, if
we are to translate the function values by a given vector and the function is in
ppform, then only the coefficients corresponding to constant terms are so
translated.

If there are two functions input, then they must be of the same type (see
Limitations, below) *except* for the following.

`fncmb(f1,op,f2)`

returns the ppform of the function

$$x|\to f1(x)\text{}op\text{}f2(x)$$

with `op`

one of `'+', '-'`

,```
'*'
```

, and `f1`

, `f2`

of arbitrary
polynomial form. If, in addition, `f2`

is a scalar or vector, it is
taken to be the function that is constantly equal to that scalar or vector.