GNU Octave Manual Version 3by John W. Eaton, David Bateman, Søren Hauberg Paperback (6"x9"), 568 pages ISBN 095461206X RRP £24.95 ($39.95) |

## 18.4 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

-1 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.

See also expm, logm, funm

__Loadable Function:__**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 ]

ISBN 095461206X | GNU Octave Manual Version 3 | See the print edition |