NAG Library Function Document

nag_matop_complex_gen_matrix_log (f01fjc)

has infinitely many logarithms. For a matrix with no eigenvalues on the closed negative real line, the principal logarithm is the unique logarithm whose spectrum lies in the strip

\{z : - π < Im (z) < π\}

\log (A)

is computed using the Schur–Parlett algorithm for the matrix logarithm described in Higham (2008) and Davies and Higham (2003).

4 References

Davies P I and Higham N J (2003) A Schur–Parlett algorithm for computing matrix functions. SIAM J. Matrix Anal. Appl. 25(2) 464–485

Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

5 Arguments

1: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

or Nag_ColMajor.

2: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: a[ $pda \times n$ ] – ComplexInput/Output

Note: the

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

n

n

matrix

A

On exit: the

n

n

principal matrix logarithm,

\log (A)

4: pda – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, n)$ ;
if $order = Nag_RowMajor$ , $pda \geq n$ .

5: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Allocation of memory failed. The Complex allocatable memory required is approximately $4 \times n^{2}$ .
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_EIGENVALUES: $A$ was found to have eigenvalues on the closed, negative real line. The principal logarithm cannot be calculated in this case.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.; An unexpected internal error occured when ordering the eigenvalues of $A$ . Please contact NAG.; Computation of the square root of a submatrix failed.
Note: this failure should not occur and suggests that the function has been called incorrectly.; The function was unable to compute the Schur decomposition of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.; There was an error whilst reordering the Schur form of $A$ .
Note: this failure should not occur and suggests that the function has been called incorrectly.; There was a problem obtaining the weights and nodes from the Gaussian quadrature function nag_quad_1d_gauss_wgen (d01tcc). This failure should not occur, nag_quad_1d_gauss_wgen (d01tcc) returns $fail .errnum = 〈value〉$ ; please send this information to NAG for further advice.
NE_SINGULAR: The linear equations to be solved are nearly singular and the Padé approximant may have no correct figures.
Note: this failure should not occur and suggests that the function has been called incorrectly.

7 Accuracy

For a normal matrix

A

(for which

A^{H} A = A A^{H}

), the Schur decomposition is diagonal and the algorithm reduces to evaluating the logarithm of the eigenvalues of

A

and then constructing

\log (A)

using the Schur vectors. See Section 9.4 of Higham (2008) for details and further discussion.

For discussion of the condition of the matrix logarithm see Section 11.2 of Higham (2008) floating-point operations.

8 Further Comments

Up to

4 \times n^{2}

of Complex allocatable memory may be required.

The cost of the algorithm is

O (n^{3})

. The exact cost depends on the eigenvalue distribution of

A

; see Algorithm 11.11 of Higham (2008).

nag_matop_real_gen_matrix_log (f01ejc) can be used to find the principal logarithm of a real matrix.

9 Example

This example finds the principal matrix logarithm of the matrix

A = (\begin{array}{r} 1.0 + 2.0 i & 0.0 + 1.0 i & 1.0 + 0.0 i & 3.0 + 2.0 i \\ 0.0 + 3.0 i & - 2.0 + 0.0 i & 0.0 + 0.0 i & 1.0 + 0.0 i \\ 1.0 + 0.0 i & - 2.0 + 0.0 i & 3.0 + 2.0 i & 0.0 + 3.0 i \\ 2.0 + 0.0 i & 0.0 + 1.0 i & 0.0 + 1.0 i & 2.0 + 3.0 i \end{array}) .

NAG Library Function Documentnag_matop_complex_gen_matrix_log (f01fjc)

+− Contents

1 Purpose

2 Specification

3 Description