f01 Chapter Contents
f01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_matop_complex_gen_matrix_cond_std (f01kac)

## 1  Purpose

nag_matop_complex_gen_matrix_cond_std (f01kac) computes an estimate of the absolute condition number of a matrix function $f$ of a complex $n$ by $n$ matrix $A$ in the $1$-norm, where $f$ is either the exponential, logarithm, sine, cosine, hyperbolic sine (sinh) or hyperbolic cosine (cosh). The evaluation of the matrix function, $f\left(A\right)$, is also returned.

## 2  Specification

 #include #include
 void nag_matop_complex_gen_matrix_cond_std (Nag_MatFunType fun, Integer n, Complex a[], Integer pda, double *conda, double *norma, double *normfa, NagError *fail)

## 3  Description

The absolute condition number of $f$ at $A$, ${\mathrm{cond}}_{\mathrm{abs}}\left(f,A\right)$ is given by the norm of the Fréchet derivative of $f$, $L\left(A\right)$, which is defined by
 $LX := maxE≠0 LX,E E ,$
where $L\left(X,E\right)$ is the Fréchet derivative in the direction $E$. $L\left(X,E\right)$ is linear in $E$ and can therefore be written as
 $vec LX,E = KX vecE ,$
where the $\mathrm{vec}$ operator stacks the columns of a matrix into one vector, so that $K\left(X\right)$ is ${n}^{2}×{n}^{2}$. nag_matop_complex_gen_matrix_cond_std (f01kac) computes an estimate $\gamma$ such that $\gamma \le {‖K\left(X\right)‖}_{1}$, where ${‖K\left(X\right)‖}_{1}\in \left[{n}^{-1}{‖L\left(X\right)‖}_{1},n{‖L\left(X\right)‖}_{1}\right]$. The relative condition number can then be computed via
 $cond rel f,A = cond abs f,A A1 fA 1 .$
The algorithm used to find $\gamma$ is detailed in Section 3.4 of Higham (2008).

## 4  References

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

## 5  Arguments

1:    $\mathbf{fun}$Nag_MatFunTypeInput
On entry: indicates which matrix function will be used.
${\mathbf{fun}}=\mathrm{Nag_Exp}$
The matrix exponential, ${e}^{A}$, will be used.
${\mathbf{fun}}=\mathrm{Nag_Sin}$
The matrix sine, $\mathrm{sin}\left(A\right)$, will be used.
${\mathbf{fun}}=\mathrm{Nag_Cos}$
The matrix cosine, $\mathrm{cos}\left(A\right)$, will be used.
${\mathbf{fun}}=\mathrm{Nag_Sinh}$
The hyperbolic matrix sine, $\mathrm{sinh}\left(A\right)$, will be used.
${\mathbf{fun}}=\mathrm{Nag_Cosh}$
The hyperbolic matrix cosine, $\mathrm{cosh}\left(A\right)$, will be used.
${\mathbf{fun}}=\mathrm{Nag_Loga}$
The matrix logarithm, $\mathrm{log}\left(A\right)$, will be used.
Constraint: ${\mathbf{fun}}=\mathrm{Nag_Exp}$, $\mathrm{Nag_Sin}$, $\mathrm{Nag_Cos}$, $\mathrm{Nag_Sinh}$, $\mathrm{Nag_Cosh}$ or $\mathrm{Nag_Loga}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{a}\left[\mathit{dim}\right]$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 ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: the $n$ by $n$ matrix, $f\left(A\right)$.
4:    $\mathbf{pda}$IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
5:    $\mathbf{conda}$double *Output
On exit: an estimate of the absolute condition number of $f$ at $A$.
6:    $\mathbf{norma}$double *Output
On exit: the $1$-norm of $A$.
7:    $\mathbf{normfa}$double *Output
On exit: the $1$-norm of $f\left(A\right)$.
8:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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.
An internal error occurred when estimating the norm of the Fréchet derivative of $f$ at $A$. Please contact NAG.
An internal error occurred when evaluating the matrix function $f\left(A\right)$. You can investigate further by calling nag_matop_complex_gen_matrix_exp (f01fcc), nag_matop_complex_gen_matrix_log (f01fjc) or nag_matop_complex_gen_matrix_fun_std (f01fkc) with the matrix $A$.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

nag_matop_complex_gen_matrix_cond_std (f01kac) uses the norm estimation function nag_linsys_complex_gen_norm_rcomm (f04zdc) to estimate a quantity $\gamma$, where $\gamma \le {‖K\left(X\right)‖}_{1}$ and ${‖K\left(X\right)‖}_{1}\in \left[{n}^{-1}{‖L\left(X\right)‖}_{1},n{‖L\left(X\right)‖}_{1}\right]$. For further details on the accuracy of norm estimation, see the documentation for nag_linsys_complex_gen_norm_rcomm (f04zdc).

## 8  Parallelism and Performance

nag_matop_complex_gen_matrix_cond_std (f01kac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation.
nag_matop_complex_gen_matrix_cond_std (f01kac) 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.

Approximately $6{n}^{2}$ of complex allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routines nag_matop_complex_gen_matrix_exp (f01fcc), nag_matop_complex_gen_matrix_log (f01fjc) or nag_matop_complex_gen_matrix_fun_std (f01fkc).
nag_matop_complex_gen_matrix_cond_std (f01kac) returns the matrix function $f\left(A\right)$. This is computed using nag_matop_complex_gen_matrix_exp (f01fcc) if ${\mathbf{fun}}=\mathrm{Nag_Exp}$, nag_matop_complex_gen_matrix_log (f01fjc) if ${\mathbf{fun}}=\mathrm{Nag_Loga}$ and nag_matop_complex_gen_matrix_fun_std (f01fkc) otherwise. If only $f\left(A\right)$ is required, without an estimate of the condition number, then it is far more efficient to use nag_matop_complex_gen_matrix_exp (f01fcc), nag_matop_complex_gen_matrix_log (f01fjc) or nag_matop_complex_gen_matrix_fun_std (f01fkc) directly.
nag_matop_real_gen_matrix_cond_std (f01jac) can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh at a real matrix.

## 10  Example

This example estimates the absolute and relative condition numbers of the matrix sinh function for
 $A = 0.0+1.0i -1.0+0.0i 1.0+0.0i 2.0+0.0i 2.0+1.0i 0.0-1.0i 0.0+0.0i 1.0+0.0i 0.0+1.0i 0.0+0.0i 1.0+1.0i 0.0+2.0i 1.0+0.0i 2.0+0.0i -2.0+3.0i 0.0+1.0i .$

### 10.1  Program Text

Program Text (f01kace.c)

### 10.2  Program Data

Program Data (f01kace.d)

### 10.3  Program Results

Program Results (f01kace.r)