logm (MATLAB Functions)

Matrix logarithm

Syntax

```
Y = logm(X)
[Y,esterr] = logm(X)
```

Description

Y = logm(X) returns the matrix logarithm: the inverse function of expm(X). Complex results are produced if X has negative eigenvalues. A warning message is printed if the computed expm(Y) is not close to X.

[Y,esterr] = logm(X) does not print any warning message, but returns an estimate of the relative residual, norm(expm(Y)-X)/norm(X).

Remarks

If X is real symmetric or complex Hermitian, then so is logm(X).

Some matrices, like X = [0 1; 0 0], do not have any logarithms, real or complex, and logm cannot be expected to produce one.

Limitations

For most matrices:

```
logm(expm(X)) = X = expm(logm(X))
```

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

Examples

Suppose A is the 3-by-3 matrix

        1        1        0
        0        0        2
        0        0       -1

and X = expm(A) is

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.

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 may give poor results or break down completely when the matrix has repeated eigenvalues. A warning message is printed when the results may be inaccurate.

See Also

expm, funm, sqrtm

References

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

[2] 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.

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