F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01KJF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F01KJF computes an estimate of the relative condition number ${\kappa }_{\mathrm{log}}\left(A\right)$ of the logarithm of a complex $n$ by $n$ matrix $A$, in the $1$-norm. The principal matrix logarithm $\mathrm{log}\left(A\right)$ is also returned.

## 2  Specification

 SUBROUTINE F01KJF ( N, A, LDA, CONDLA, IFAIL)
 INTEGER N, LDA, IFAIL REAL (KIND=nag_wp) CONDLA COMPLEX (KIND=nag_wp) A(LDA,*)

## 3  Description

For a matrix with no eigenvalues on the closed negative real line, the principal matrix logarithm $\mathrm{log}\left(A\right)$ is the unique logarithm whose spectrum lies in the strip $\left\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \right\}$.
The Fréchet derivative of the matrix logarithm of $A$ is the unique linear mapping $E⟼L\left(A,E\right)$ such that for any matrix $E$
 $logA+E - logA - LA,E = oE .$
The derivative describes the first order effect of perturbations in $A$ on the logarithm $\mathrm{log}\left(A\right)$.
The relative condition number of the matrix logarithm can be defined by
 $κlogA = LA A logA ,$
where $‖L\left(A\right)‖$ is the norm of the Fréchet derivative of the matrix logarithm at $A$.
To obtain the estimate of ${\kappa }_{\mathrm{log}}\left(A\right)$, F01KJF first estimates $‖L\left(A\right)‖$ by computing an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, such that $\gamma \le K$.
The algorithms used to compute ${\kappa }_{\mathrm{log}}\left(A\right)$ and $\mathrm{log}\left(A\right)$ 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).
If $A$ is nonsingular but has negative real eigenvalues, the principal logarithm is not defined, but F01KJF will return a non-principal logarithm and its condition number.

## 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 MIMS EPrint 2012.72
Higham N J (2008) Functions of Matrices: Theory and Computation SIAM, Philadelphia, PA, USA

## 5  Parameters

1:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least ${\mathbf{N}}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$. Alternatively, if ${\mathbf{IFAIL}}={\mathbf{2}}$, a non-principal logarithm is returned.
3:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01KJF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
4:     $\mathrm{CONDLA}$ – REAL (KIND=nag_wp)Output
On exit: with ${\mathbf{IFAIL}}={\mathbf{0}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$, an estimate of the relative condition number of the matrix logarithm, ${\kappa }_{\mathrm{log}}\left(A\right)$. Alternatively, if ${\mathbf{IFAIL}}={\mathbf{4}}$, contains the absolute condition number of the matrix logarithm.
5:     $\mathrm{IFAIL}$ – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
$A$ is singular so the logarithm cannot be computed.
${\mathbf{IFAIL}}=2$
$A$ has eigenvalues on the negative real line. The principal logarithm is not defined in this case, so a non-principal logarithm was returned.
${\mathbf{IFAIL}}=3$
$\mathrm{log}\left(A\right)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.
${\mathbf{IFAIL}}=4$
The relative condition number is infinite. The absolute condition number was returned instead.
${\mathbf{IFAIL}}=5$
An unexpected internal error occurred. This failure should not occur and suggests that the routine has been called incorrectly.
${\mathbf{IFAIL}}=-1$
On entry, ${\mathbf{N}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{N}}\ge 0$.
${\mathbf{IFAIL}}=-3$
On entry, ${\mathbf{LDA}}=〈\mathit{\text{value}}〉$ and ${\mathbf{N}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}$.
${\mathbf{IFAIL}}=-99$
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

## 7  Accuracy

F01KJF uses the norm estimation routine F04ZDF to produce an estimate $\gamma$ of a quantity $K\in \left[{n}^{-1}{‖L\left(A\right)‖}_{1},n{‖L\left(A\right)‖}_{1}\right]$, such that $\gamma \le K$. For further details on the accuracy of norm estimation, see the documentation for F04ZDF.
For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), 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 $\mathrm{log}\left(A\right)$ 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 $\mathrm{log}\left(A\right)$ 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

F01KJF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F01KJF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

F01KAF uses a similar algorithm to F01KJF to compute an estimate of the absolute condition number (which is related to the relative condition number by a factor of $‖A‖/‖\mathrm{log}\left(A\right)‖$). However, the required Fréchet derivatives are computed in a more efficient and stable manner by F01KJF and so its use is recommended over F01KAF.
The amount of complex allocatable memory required by the algorithm is typically of the order $10{n}^{2}$.
The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point operations; see Al–Mohy et al. (2012).
If the matrix logarithm alone is required, without an estimate of the condition number, then F01FJF should be used. If the Fréchet derivative of the matrix logarithm is required then F01KKF should be used. The real analogue of this routine is F01JJF.

## 10  Example

This example estimates the relative condition number of the matrix logarithm $\mathrm{log}\left(A\right)$, where
 $A = 3+2i 1 1 1+2i 0+2i -4 0 0 1 -2 3+2i 0+i 1 i 1 2+3i .$

### 10.1  Program Text

Program Text (f01kjfe.f90)

### 10.2  Program Data

Program Data (f01kjfe.d)

### 10.3  Program Results

Program Results (f01kjfe.r)