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20.3 Functions of a Matrix

— Loadable Function: expm (a)

Return the exponential of a matrix, defined as the infinite Taylor series

          expm(a) = I + a + a^2/2! + a^3/3! + ...

The Taylor series is not the way to compute the matrix exponential; see Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978. This routine uses Ward's diagonal Pade' approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal Pade' approximations are rational polynomials of matrices

          D (a)   N (a)

whose Taylor series matches the first 2q+1 terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the Pade' approximation when Dq(a) is ill-conditioned.

— Function File: logm (a)

Compute the matrix logarithm of the square matrix a. Note that this is currently implemented in terms of an eigenvalue expansion and needs to be improved to be more robust.

— Loadable Function: [result, error_estimate] = sqrtm (a)

Compute the matrix square root of the square matrix a.

Ref: Nicholas J. Higham. A new sqrtm for MATLAB. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.

— Function File: kron (a, b)

Form the kronecker product of two matrices, defined block by block as

          x = [a(i, j) b]

For example,

          kron (1:4, ones (3, 1))
                =>  1  2  3  4
                    1  2  3  4
                    1  2  3  4

— Loadable Function: x = syl (a, b, c)

Solve the Sylvester equation

          A X + X B + C = 0

using standard Lapack subroutines. For example,

          syl ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
               => [ -0.50000, -0.66667; -0.66667, -0.50000 ]