Spline Toolbox™

Evaluate function

Syntax

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

Description

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

Roughly speaking, the output v is obtained by replacing each entry of x by the value of 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 :

• If is scalar-valued, then v is of the same size as x.
• If 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 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 with m>1, with [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 at x(:,j2,...,js), - except that

• (1) d1, ..., dr is ignored in case 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 at (x1(j1), ..., xm(jm)), - except that d1, ..., dr is ignored in case is scalar-valued, i.e., both r and n1 are 1.

If has a jump discontinuity at x, then the value , 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 , i.e., the limit from the left, is returned.

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

fnval(...,'l') treats as continuous from the left. This means that if has a jump discontinuity at x, then the value , 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 is returned.

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

Examples

The statement fnval(csapi(x,y),xx) has the same effect as the statement csapi(x,y,xx).

Algorithm

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.

See Also

fnbrk, ppmak, rsmak, spmak, stmak

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