Documentation

fnval

Evaluate spline function

v = fnval(f,x)
fnval(x,f)
fnval(...,'l')

Description

v = fnval(f,x) and v = fnval(x,f) both provide the value f(x) at the points in x of the spline function f whose description is contained in f.

Roughly speaking, the output v is obtained by replacing each entry of x by the value of f at that entry. This is literally true in case the function in f is scalar-valued and univariate, and is the intent in all other cases, except that, for a d-valued m-variate function, this means replacing m-vectors by d-vectors. The full details are as follows.

For a univariate f :

• If f is scalar-valued, then v is of the same size as x.

• If f is [d1,...,dr]-valued, and x has size [n1,...,ns], then v has size [d1,...,dr, n1,...,ns], with v(:,...,:, j1,...,js) the value of f at x(j1,...,js), – except that

(1) n1 is ignored if it is 1 and s is 2, i.e., if x is a row vector; and

(2) MATLAB® ignores any trailing singleton dimensions of x.

For an m-variate f with m>1, with f [d1,...,dr]-valued, x may be either an array, or else a cell array {x1,...,xm}.

• If x is an array, of size [n1,...,ns] say, then n1 must equal m, and v has size [d1,...,dr, n2,...,ns], with v(:,...,:, j2,...,js) the value of f at x(:,j2,...,js), – except that

(1) d1, ..., dr is ignored in case f is scalar-valued, i.e., both r and n1 are 1;

(2) MATLAB ignores any trailing singleton dimensions of x.

• If x is a cell array, then it must be of the form {x1,...,xm}, with xj a vector, of length nj, and, in that case, v has size [d1,...,dr, n1,...,nm], with v(:,...,:, j1,...,jm) the value of f at (x1(j1), ..., xm(jm)), – except that d1, ..., dr is ignored in case f is scalar-valued, i.e., both r and n1 are 1.

If f has a jump discontinuity at x, then the value f(x +), i.e., the limit from the right, is returned, except when x equals the right end of the basic interval of the form; for such x, the value f(x–), i.e., the limit from the left, is returned.

fnval(x,f) is the same as fnval(f,x).

fnval(...,'l') treats f as continuous from the left. This means that if f has a jump discontinuity at x, then the value f(x–), i.e., the limit from the left, is returned, except when x equals the left end of the basic interval; for such x, the value f(x +) is returned.

If the function is multivariate, then the above statements concerning continuity from the left and right apply coordinatewise.

Examples

Evaluate Functions at Specified Points

Interpolate some data and plot and evaluate the resulting functions.

Define some data.

x = [0.074 0.31 0.38 0.53 0.57 0.58 0.59 0.61 0.61 0.65 0.71 0.81 0.97];
y = [0.91 0.96 0.77 0.5 0.5 0.51 0.51 0.53 0.53 0.57 0.62 0.61 0.31];

Interpolate the data and plot the resulting function, f.

f = csapi( x, y )
fnplt( f )

Find the value of the function f at x = 0.5.

fnval( f, 0.5 )

Find the value of the function f at 0, 0.1, ..., 1.

fnval( f, 0:0.1:1 )

Create a function f2 that represents a surface.

x = 0.0001+(-4:0.2:4);
y = -3:0.2:3;
[yy, xx] = meshgrid( y, x );
r = pi*sqrt( xx.^2+yy.^2 );
z = sin( r )./r;
f2 = csapi( {x,y}, z );

Plot the function f2.

fnplt( f2 )
axis( [-5, 5, -5, 5, -0.5, 1] );

Find the value of the function f2 at x = -2 and y = 3.

fnval( f2, [-2; 3] )

Algorithms

For each entry of x, the relevant break- or knot-interval is determined and the relevant information assembled. Depending on whether f is in ppform or in B-form, nested multiplication or the B-spline recurrence (see, e.g., [PGS; X.(3)]) is then used vector-fashion for the simultaneous evaluation at all entries of x. Evaluation of a multivariate polynomial spline function takes full advantage of the tensor product structure.

Evaluation of a rational spline follows up evaluation of the corresponding vector-valued spline by division of all but its last component by its last component.

Evaluation of a function in stform makes essential use of stcol, and tries to keep the matrices involved to reasonable size.