# NAG Library Function Document

## 1Purpose

nag_matop_real_gen_matrix_cond_num (f01jbc) 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. Numerical differentiation is used to evaluate the derivatives of $f$ when they are required.

## 2Specification

 #include #include
void  nag_matop_real_gen_matrix_cond_num (Integer n, double a[], Integer pda,
 void (*f)(Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm),
Nag_Comm *comm, Integer *iflag, double *conda, double *norma, double *normfa, NagError *fail)

## 3Description

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_num (f01jbc) 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).
The function $f$ is supplied via function f which evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.

## 4References

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

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:    $\mathbf{a}\left[\mathit{dim}\right]$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, $f\left(A\right)$.
3:    $\mathbf{pda}$IntegerInput
On entry: the stride separating matrix row elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
4:    $\mathbf{f}$function, supplied by the userExternal Function
The function f evaluates $f\left({z}_{i}\right)$ at a number of points ${z}_{i}$.
The specification of f is:
 void f (Integer *iflag, Integer nz, const Complex z[], Complex fz[], Nag_Comm *comm)
1:    $\mathbf{iflag}$Integer *Input/Output
On entry: iflag will be zero.
On exit: iflag should either be unchanged from its entry value of zero, or may be set nonzero to indicate that there is a problem in evaluating the function $f\left(z\right)$; for instance $f\left(z\right)$ may not be defined. If iflag is returned as nonzero then nag_matop_real_gen_matrix_cond_num (f01jbc) will terminate the computation, with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
2:    $\mathbf{nz}$IntegerInput
On entry: ${n}_{z}$, the number of function values required.
3:    $\mathbf{z}\left[{\mathbf{nz}}\right]$const ComplexInput
On entry: the ${n}_{z}$ points ${z}_{1},{z}_{2},\dots ,{z}_{{n}_{z}}$ at which the function $f$ is to be evaluated.
4:    $\mathbf{fz}\left[{\mathbf{nz}}\right]$ComplexOutput
On exit: the ${n}_{z}$ function values. ${\mathbf{fz}}\left[\mathit{i}-1\right]$ should return the value $f\left({z}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{n}_{z}$. If ${z}_{i}$ lies on the real line, then so must $f\left({z}_{i}\right)$.
5:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_matop_real_gen_matrix_cond_num (f01jbc) you may allocate memory and initialize these pointers with various quantities for use by f when called from nag_matop_real_gen_matrix_cond_num (f01jbc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_matop_real_gen_matrix_cond_num (f01jbc). If your code inadvertently does return any NaNs or infinities, nag_matop_real_gen_matrix_cond_num (f01jbc) is likely to produce unexpected results.
5:    $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
6:    $\mathbf{iflag}$Integer *Output
On exit: ${\mathbf{iflag}}=0$, unless iflag has been set nonzero inside f, in which case iflag will be the value set and fail will be set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
7:    $\mathbf{conda}$double *Output
On exit: an estimate of the absolute condition number of $f$ at $A$.
8:    $\mathbf{norma}$double *Output
On exit: the $1$-norm of $A$.
9:    $\mathbf{normfa}$double *Output
On exit: the $1$-norm of $f\left(A\right)$.
10:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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_real_gen_matrix_fun_num (f01elc) with the matrix $A$ and the function $f$.
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_USER_STOP
iflag has been set nonzero by the user-supplied function.

## 7Accuracy

nag_matop_real_gen_matrix_cond_num (f01jbc) 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).

## 8Parallelism and Performance

nag_matop_real_gen_matrix_cond_num (f01jbc) 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. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.
nag_matop_real_gen_matrix_cond_num (f01jbc) 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.

The matrix function is computed using the underlying matrix function routine nag_matop_real_gen_matrix_fun_num (f01elc). Approximately $6{n}^{2}$ of real allocatable memory is required by the routine, in addition to the memory used by the underlying matrix function routine.
If only $f\left(A\right)$ is required, without an estimate of the condition number, then it is far more efficient to use the underlying matrix function routine.
The complex analogue of this function is nag_matop_complex_gen_matrix_cond_num (f01kbc).

## 10Example

This example estimates the absolute and relative condition numbers of the matrix function $\mathrm{cos}2A$ where
 $A= -1 -1 -2 1 0 1 -1 0 -1 -2 1 -1 0 -1 0 -1 .$

### 10.1Program Text

Program Text (f01jbce.c)

### 10.2Program Data

Program Data (f01jbce.d)

### 10.3Program Results

Program Results (f01jbce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017