NAG C Library Function Document

are based on a Schur decomposition, the inverse scaling and squaring method and Padé approximants. Further details can be found in Al–Mohy and Higham (2011) and Al–Mohy et al. (2012).

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

Al–Mohy A H, Higham N J and Relton S D (2012) Computing the Fréchet derivative of the matrix logarithm and estimating the condition number SIAM J. Sci. Comput. 35(4) C394–C410

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

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $a [\dim]$ – doubleInput/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 $a [(j - 1) \times pda + i - 1]$ .

On entry: the $n$ by $n$ matrix $A$ .

On exit: the $n$ by $n$ principal matrix logarithm, $\log (A)$ .
3: $pda$ – IntegerInput: On entry: the stride separating matrix row elements in the array a.

Constraint: $pda \geq n$ .
4: $condla$ – double *Output: On exit: with $fail . code =$ NE_NOERROR or NW_SOME_PRECISION_LOSS, an estimate of the relative condition number of the matrix logarithm, $κ_{\log} (A)$ . Alternatively, if $fail . code =$ NE_RCOND, contains the absolute condition number of the matrix logarithm.
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_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_NEGATIVE_EIGVAL: $A$ has eigenvalues on the negative real line. The principal logarithm is not defined in this case; nag_matop_complex_gen_matrix_cond_log (f01kjc) can be used to return a complex, non-principal log.
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_RCOND: The relative condition number is infinite. The absolute condition number was returned instead.
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

nag_matop_real_gen_matrix_cond_log (f01jjc) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04ydc) to produce an estimate

γ

of a quantity

K \in [n^{- 1} {‖L (A)‖}_{1}, n {‖L (A)‖}_{1}]

, such that

γ \leq K

. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_real_gen_norm_rcomm (f04ydc).

For a normal matrix

A

(for which

A^{T} A = A A^{T}

), the Schur decomposition is diagonal and the computation of the matrix logarithm 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. 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. See Al–Mohy and Higham (2011) and Section 11.2 of Higham (2008) for details and further discussion.

8

Parallelism and Performance

nag_matop_real_gen_matrix_cond_log (f01jjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_matop_real_gen_matrix_cond_log (f01jjc) 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

nag_matop_real_gen_matrix_cond_std (f01jac) uses a similar algorithm to nag_matop_real_gen_matrix_cond_log (f01jjc) to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of

‖A‖ / ‖\log (A)‖

). However, the required Fréchet derivatives are computed in a more efficient and stable manner by nag_matop_real_gen_matrix_cond_log (f01jjc) and so its use is recommended over nag_matop_real_gen_matrix_cond_std (f01jac).

The amount of real allocatable memory required by the algorithm is typically of the order

10 n^{2}

The cost of the algorithm is

O (n^{3})

floating-point operations; see Al–Mohy et al. (2012).

If the matrix logarithm alone is required, without an estimate of the condition number, then nag_matop_real_gen_matrix_log (f01ejc) should be used. If the Fréchet derivative of the matrix logarithm is required then nag_matop_real_gen_matrix_frcht_log (f01jkc) should be used. If

A

has negative real eigenvalues then nag_matop_complex_gen_matrix_cond_log (f01kjc) can be used to return a complex, non-principal matrix logarithm and its condition number.

10

Example

This example estimates the relative condition number of the matrix logarithm

\log (A)

, where

A = (\begin{array}{r} 4 & - 1 & 0 & 1 \\ 2 & 5 & - 2 & 2 \\ 1 & 1 & 3 & - 1 \\ 2 & 0 & 2 & 8 \end{array}) .

NAG C Library Function Document

nag_matop_real_gen_matrix_cond_log (f01jjc)

▸▿ 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