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\}

. If

A

is nonsingular but has eigenvalues on the negative real line, the principal logarithm is not defined, but nag_matop_complex_gen_matrix_log (f01fjc) will return a non-principal logarithm.

\log (A)

is computed using the inverse scaling and squaring algorithm for the matrix logarithm described in Al–Mohy and Higham (2011).

4

References

Al–Mohy A H and Higham N J (2011) Improved inverse scaling and squaring algorithms for the matrix logarithm SIAM J. Sci. Comput. 34(4) C152–C169

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a [\dim]$ – ComplexInput/Output

Note: the dimension, dim, of the array a must be at least

pda \times n

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)

, unless

fail . code =

NE_EIGENVALUES, in which case a non-principal logarithm is returned.

4: $pda$ – IntegerInput

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

Constraint:

pda \geq n

5: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_EIGENVALUES: $A$ was found to have eigenvalues on the negative real line. The principal logarithm is not defined in this case, so a non-principal logarithm was returned.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pda \geq n$ .
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR: $A$ is singular so the logarithm cannot be computed.
NW_SOME_PRECISION_LOSS: $\log (A)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

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. This should give a very accurate result. In general, however, no error bounds are available for the algorithm. See Al–Mohy and Higham (2011) and Section 9.4 of Higham (2008) for details and further discussion.

The sensitivity of the computation of

\log (A)

is worst when

A

has an eigenvalue of very small modulus or has a complex conjugate pair of eigenvalues lying close to the negative real axis.

If estimates of the condition number of the matrix logarithm are required then nag_matop_complex_gen_matrix_cond_log (f01kjc) should be used.

8

Parallelism and Performance

nag_matop_complex_gen_matrix_log (f01fjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_complex_gen_matrix_log (f01fjc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The cost of the algorithm is

O (n^{3})

floating-point operations (see Al–Mohy and Higham (2011)). The Complex allocatable memory required is approximately

3 \times n^{2}

If the Fréchet derivative of the matrix logarithm is required then nag_matop_complex_gen_matrix_frcht_log (f01kkc) should be used.

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

10

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 Document

nag_matop_complex_gen_matrix_log (f01fjc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results