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f01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_matop_real_gen_matrix_cond_std (f01jac)

## 1  Purpose

nag_matop_real_gen_matrix_cond_std (f01jac) computes an estimate of the absolute condition number of a matrix function $f$ at a real $n$ 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_real_gen_matrix_cond_std (Nag_MatFunType fun, Integer n, double 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_real_gen_matrix_cond_std (f01jac) 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:     funNag_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:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     a[$\mathit{dim}$]doubleInput/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:     pdaIntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
On exit: an estimate of the absolute condition number of $f$ at $A$.
On exit: the $1$-norm of $A$.
On exit: the $1$-norm of $f\left(A\right)$.
8:     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_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_real_gen_matrix_exp (f01ecc)nag_matop_real_gen_matrix_log (f01ejc) or nag_matop_real_gen_matrix_fun_std (f01ekc) with the matrix $A$.

## 7  Accuracy

nag_matop_real_gen_matrix_cond_std (f01jac) uses the norm estimation function nag_linsys_real_gen_norm_rcomm (f04ydc) 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_real_gen_norm_rcomm (f04ydc).

## 8  Parallelism and Performance

nag_matop_real_gen_matrix_cond_std (f01jac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_matop_real_gen_matrix_cond_std (f01jac) 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.
In these implementations, this may make calls to the user supplied functions from within an OpenMP parallel region. Thus OpenMP directives within the user functions should be avoided, unless you are using the same OpenMP runtime library (which normally means using the same compiler) as that used to build your NAG Library implementation, as listed in the Installers' Note.

The matrix function is computed using one of three underlying matrix function routines:
Approximately $6{n}^{2}$ of real allocatable memory is required by the routine, in addition to the memory used by these underlying matrix function routines.
If only $f\left(A\right)$ is required, without an estimate of the condition number, then it is far more efficient to use the appropriate matrix function routine listed above.
nag_matop_complex_gen_matrix_cond_std (f01kac) can be used to find the condition number of the exponential, logarithm, sine, cosine, sinh or cosh matrix functions at a complex matrix.

## 10  Example

This example estimates the absolute and relative condition numbers of the matrix sinh function where
 $A = 2 1 3 1 3 -1 0 2 1 0 3 1 1 2 0 3 .$

### 10.1  Program Text

Program Text (f01jace.c)

### 10.2  Program Data

Program Data (f01jace.d)

### 10.3  Program Results

Program Results (f01jace.r)