f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_matop_complex_gen_matrix_log (f01fjc)

## 1  Purpose

nag_matop_complex_gen_matrix_log (f01fjc) computes the principal matrix logarithm, $\mathrm{log}\left(A\right)$, of a complex $n$ by $n$ matrix $A$, with no eigenvalues on the closed negative real line.

## 2  Specification

 #include #include
 void nag_matop_complex_gen_matrix_log (Nag_OrderType order, Integer n, Complex a[], Integer pda, NagError *fail)

## 3  Description

Any nonsingular matrix $A$ 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 $\left\{z:-\pi <\mathrm{Im}\left(z\right)<\pi \right\}$. 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.
$\mathrm{log}\left(A\right)$ 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:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ principal matrix logarithm, $\mathrm{log}\left(A\right)$, unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_EIGENVALUES, in which case a non-principal logarithm is returned.
4:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:     failNagError *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.
On entry, argument $⟨\mathit{\text{value}}⟩$ 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, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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.
NE_SINGULAR
$A$ is singular so the logarithm cannot be computed.
NW_SOME_PRECISION_LOSS
$\mathrm{log}\left(A\right)$ has been computed using an IEEE double precision Padé approximant, although the arithmetic precision is higher than IEEE double precision.

## 7  Accuracy

For a normal matrix $A$ (for which ${A}^{\mathrm{H}}A=A{A}^{\mathrm{H}}$), the Schur decomposition is diagonal and the algorithm 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. See Al–Mohy and Higham (2011) and Section 9.4 of Higham (2008) for details and further discussion.
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.
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.

The cost of the algorithm is $O\left({n}^{3}\right)$ floating-point operations (see Al–Mohy and Higham (2011)). The Complex allocatable memory required is approximately $3×{n}^{2}$.
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
 $A = 1.0+2.0i 0.0+1.0i 1.0+0.0i 3.0+2.0i 0.0+3.0i -2.0+0.0i 0.0+0.0i 1.0+0.0i 1.0+0.0i -2.0+0.0i 3.0+2.0i 0.0+3.0i 2.0+0.0i 0.0+1.0i 0.0+1.0i 2.0+3.0i .$

### 10.1  Program Text

Program Text (f01fjce.c)

### 10.2  Program Data

Program Data (f01fjce.d)

### 10.3  Program Results

Program Results (f01fjce.r)