Matrix logarithm

`L = logm(A)`

[L, exitflag] = logm(A)

`L = logm(A)`

is the principal
matrix logarithm of A, the inverse of `expm(A)`

. `L`

is
the unique logarithm for which every eigenvalue has imaginary part
lying strictly between –*π* and *π*.
If `A`

is singular or has any eigenvalues on the
negative real axis, the principal logarithm is undefined. In this
case, `logm`

computes a nonprincipal logarithm and
returns a warning message.

`[L, exitflag] = logm(A)`

returns
a scalar `exitflag`

that describes the exit condition
of `logm`

:

If

`exitflag = 0`

, the algorithm was successfully completed.If

`exitflag = 1`

, too many matrix square roots had to be computed. However, the computed value of`L`

might still be accurate.

The input `A`

can have class `double`

or `single`

.

For most matrices:

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.

Suppose `A`

is the 3-by-3 matrix

1 1 0 0 0 2 0 0 -1

and `Y =`

`expm(A)`

is

Y = 2.7183 1.7183 1.0862 0 1.0000 1.2642 0 0 0.3679

Then `A = logm(Y)`

produces the original matrix `A`

.

Y = 1.0000 1.0000 0.0000 0 0 2.0000 0 0 -1.0000

But `log(A)`

involves taking the logarithm
of zero, and so produces

ans = 0.0000 0 -35.5119 -Inf -Inf 0.6931 -Inf -Inf 0.0000 + 3.1416i

[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] Cheng, S. H., N. J. Higham, C. S. Kenney,
and A. J. Laub, "Approximating the logarithm of a matrix to
specified accuracy," *SIAM J. Matrix Anal. Appl.*,
Vol. 22, Number 4, pp. 1112-1125, 2001.

[3] Higham, N. J., "Evaluating Pade
approximants of the matrix logarithm," *SIAM J.
Matrix Anal. Appl.*, Vol. 22, Number 4, pp. 1126-1135,
2001.

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

[5] Moler, C. B. and C. F. Van Loan, "Nineteen
Dubious Ways to Compute the Exponential of a Matrix," *SIAM
Review* 20, 1978, pp. 801-836.

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