logm

Synopsis

Y = logm(X)

Description

logm(A) is the inverse function of expm(A) in that for most matrices A

logm(expm(A)) = A = expm(logm(A))

These identities may fail for some A. For example, if the computed eigenvalues of A include an exact zero, then logm(A) generates infinity. Or, if the elements of A are too large, expm(A) may overflow.

Examples

Start with the matrix X produced by the example in expm.

X =
    2.7183    1.7183    1.0862
         0    1.0000    1.2642
         0         0    0.3679

Then A = logm(X) produces the original matrix A used in the expm example.

A =
    1.0000    1.0000    0.0000
         0         0    2.0000
         0         0   -1.0000

But log(X) involves taking the logarithm of zero, and so produces

ans =
    1.0000    0.5413    0.0826
      -Inf         0    0.2345
      -Inf      -Inf   -1.0000

Algorithm

The matrix functions are evaluated using an algorithm due to Parlett, which is described in [1]. The algorithm uses the Schur factorization of the matrix and can give poor results or break down completely when the matrix has repeated eigenvalues. A warning message is printed when the results may be inaccurate.

References

[1] G. H. Golub and C. F. Van Loan, Matrix Computation, Johns Hopkins University Press, 1983, p. 384.

[2] C. B. Moler and C. F. Van Loan, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," SIAM Review 20, pp. 801-836, 1979.