MATLAB Function Reference |

**Syntax**

**Description**

```
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 - and . If `A`

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

computes a non-principal 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`

, one or more Taylor series evaluations did not converge. However, the computed value of`L`

might still be accurate.

The input `A`

can have class `double`

or `single`

.

**Remarks**

If `A`

is real symmetric or complex Hermitian, then so is `logm(A)`

.

Some matrices, like `A = [0 1; 0 0]`

, do not have any logarithms, real or complex, so `logm`

cannot be expected to produce one.

**Limitations**

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**

Suppose `A`

is the 3-by-3 matrix

Then `A = logm(Y)`

produces the original matrix `A`

.

But `log(A)`

involves taking the logarithm of zero, and so produces

**Algorithm**

The algorithm `logm`

uses is described in [1].

**See Also**

**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] 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, 1979,pp. 801-836.

loglog | logspace |

© 1994-2005 The MathWorks, Inc.