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funm - Evaluate general matrix function

Syntax

F = funm(A,fun) F = funm(A, fun, options) F=funm(A, fun, options, p1, p2,...) [F, exitflag] = funm(...) [F, exitflag, output] = funm(...)

Description

F = funm(A,fun) evaluates the user-defined function fun at the square matrix argument A. F = fun(x, k) must accept a vector x and an integer k, and return a vector f of the same size of x, where f(i) is the kth derivative of the function fun evaluated at x(i). The function represented by fun must have a Taylor series with an infinite radius of convergence, except for fun = @log, which is treated as a special case.

You can also use funm to evaluate the special functions listed in the following table at the matrix A.

Function	Syntax for Evaluating Function at Matrix A
`exp`	`funm(A, @exp)`
`log`	`funm(A, @log)`
`sin`	`funm(A, @sin)`
`cos`	`funm(A, @cos)`
`sinh`	`funm(A, @sinh)`
`cosh`	`funm(A, @cosh)`

For matrix square roots, use sqrtm(A) instead. For matrix exponentials, which of expm(A) or funm(A, @exp) is the more accurate depends on the matrix A.

The function represented by fun must have a Taylor series with an infinite radius of convergence. The exception is @log, which is treated as a special case. Parametrizing Functions, in the online MATLAB Mathematics documentation, explains how to provide additional parameters to the function fun, if necessary.

F = funm(A, fun, options) sets the algorithm's parameters to the values in the structure options.

The following table lists the fields of options.

Field	Description	Values
`options.Display`	Level of display	`'off'` (default), `'on'`, `'verbose'`
`options.TolBlk`	Tolerance for blocking Schur form	Positive scalar. The default is `0.1`.
`options.TolTay`	Termination tolerance for evaluating the Taylor series of diagonal blocks	Positive scalar. The default is `eps`.
`options.MaxTerms`	Maximum number of Tayor series terms	Positive integer. The default is `250`.
`options.MaxSqrt`	When computing a logarithm, maximum number of square roots computed in inverse scaling and squaring method.	Positive integer. The default is `100`.
`options.Ord`	Specifies the ordering of the Schur form `T`.	A vector of length `length(A)`. `options.Ord(i)` is the index of the block into which `T(i,i)` is placed. The default is `[]`.

F=funm(A, fun, options, p1, p2,...) passes extra inputs p1, p2,... to the function.

[F, exitflag] = funm(...) returns a scalar exitflag that describes the exit condition of funm. exitflag can have the following values:

0 — The algorithm was successful.
1 — One or more Taylor series evaluations did not converge, or, in the case of a logarithm, too many square roots are needed. However, the computed value of F might still be accurate. This is different from R13 and earlier versions that returned an expensive and often inaccurate error estimate as the second output argument.

[F, exitflag, output] = funm(...) returns a structure output with the following fields:

Field	Description
`output.terms`	Vector for which `output.terms(i)` is the number of Taylor series terms used when evaluating the ith block, or, in the case of the logarithm, the number of square roots of matrices of dimension greater than 2.
`output.ind`	Cell array for which the `(i,j)` block of the reordered Schur factor `T` is `T(output.ind{i}, output.ind{j})`.
`output.ord`	Ordering of the Schur form, as passed to `ordschur`
`output.T`	Reordered Schur form

If the Schur form is diagonal then output = struct('terms',ones(n,1),'ind',{1:n}).

Examples

Example 1

The following command computes the matrix sine of the 3-by-3 magic matrix.

F=funm(magic(3), @sin)

F =

   -0.3850    1.0191    0.0162
    0.6179    0.2168   -0.1844
    0.4173   -0.5856    0.8185

Example 2

The statements

S = funm(X,@sin);
C = funm(X,@cos);

produce the same results to within roundoff error as

E = expm(i*X);
C = real(E);
S = imag(E);

In either case, the results satisfy S*S+C*C = I, where I = eye(size(X)).

Example 3

To compute the function exp(x) + cos(x) at A with one call to funm, use

F = funm(A,@fun_expcos)

where fun_expcos is the following M-file function.

function f = fun_expcos(x, k)
% Return kth derivative of exp + cos at X.
        g = mod(ceil(k/2),2);
        if mod(k,2)
           f = exp(x) + sin(x)*(-1)^g;
        else
           f = exp(x) + cos(x)*(-1)^g;
        end

Algorithm

The algorithm funm uses is described in [1].

References

[1] Davies, P. I. and N. J. Higham, "A Schur-Parlett algorithm for computing matrix functions," SIAM J. Matrix Anal. Appl., Vol. 25, Number 2, pp. 464-485, 2003.

[2] Golub, G. H. and C. F. Van Loan, Matrix Computation, Third Edition, Johns Hopkins University Press, 1996, p. 384.

[3] Moler, C. B. and C. F. Van Loan, "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later" SIAM Review 20, Vol. 45, Number 1, pp. 1-47, 2003.