F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07QUF (ZSPCON)

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

F07QUF (ZSPCON) estimates the condition number of a complex symmetric matrix $A$, where $A$ has been factorized by F07QRF (ZSPTRF), using packed storage.

## 2  Specification

 SUBROUTINE F07QUF ( UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
 INTEGER N, IPIV(*), INFO REAL (KIND=nag_wp) ANORM, RCOND COMPLEX (KIND=nag_wp) AP(*), WORK(2*N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zspcon.

## 3  Description

F07QUF (ZSPCON) estimates the condition number (in the $1$-norm) of a complex symmetric matrix $A$:
 $κ1A=A1A-11 .$
Since $A$ is symmetric, ${\kappa }_{1}\left(A\right)={\kappa }_{\infty }\left(A\right)={‖A‖}_{\infty }{‖{A}^{-1}‖}_{\infty }$.
Because ${\kappa }_{1}\left(A\right)$ is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of ${\kappa }_{1}\left(A\right)$.
The routine should be preceded by a call to F06UGF to compute ${‖A‖}_{1}$ and a call to F07QRF (ZSPTRF) to compute the Bunch–Kaufman factorization of $A$. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$.

## 4  References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: specifies how $A$ has been factorized.
${\mathbf{UPLO}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     AP($*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the dimension of the array AP must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}×\left({\mathbf{N}}+1\right)/2\right)$.
On entry: the factorization of $A$ stored in packed form, as returned by F07QRF (ZSPTRF).
4:     IPIV($*$) – INTEGER arrayInput
Note: the dimension of the array IPIV must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: details of the interchanges and the block structure of $D$, as returned by F07QRF (ZSPTRF).
5:     ANORM – REAL (KIND=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling F06UGF with its parameter ${\mathbf{NORM}}=\text{'1'}$. ANORM must be computed either before calling F07QRF (ZSPTRF) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{ANORM}}\ge 0.0$.
6:     RCOND – REAL (KIND=nag_wp)Output
On exit: an estimate of the reciprocal of the condition number of $A$. RCOND is set to zero if exact singularity is detected or the estimate underflows. If RCOND is less than machine precision, $A$ is singular to working precision.
7:     WORK($2×{\mathbf{N}}$) – COMPLEX (KIND=nag_wp) arrayWorkspace
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  Accuracy

The computed estimate RCOND is never less than the true value $\rho$, and in practice is nearly always less than $10\rho$, although examples can be constructed where RCOND is much larger.

A call to F07QUF (ZSPCON) involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8{n}^{2}$ real floating point operations but takes considerably longer than a call to F07QSF (ZSPTRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The real analogue of this routine is F07PGF (DSPCON).

## 9  Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i .$
Here $A$ is symmetric, stored in packed form, and must first be factorized by F07QRF (ZSPTRF). The true condition number in the $1$-norm is $32.92$.

### 9.1  Program Text

Program Text (f07qufe.f90)

### 9.2  Program Data

Program Data (f07qufe.d)

### 9.3  Program Results

Program Results (f07qufe.r)