F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07MGF (DSYCON)

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

F07MGF (DSYCON) estimates the condition number of a real symmetric indefinite matrix $A$, where $A$ has been factorized by F07MDF (DSYTRF).

## 2  Specification

 SUBROUTINE F07MGF ( UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
 INTEGER N, LDA, IPIV(*), IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), ANORM, RCOND, WORK(2*N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dsycon.

## 3  Description

F07MGF (DSYCON) estimates the condition number (in the $1$-norm) of a real symmetric indefinite 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 F06RCF to compute ${‖A‖}_{1}$ and a call to F07MDF (DSYTRF) 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:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: details of the factorization of $A$, as returned by F07MDF (DSYTRF).
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07MGF (DSYCON) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     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 F07MDF (DSYTRF).
6:     ANORM – REAL (KIND=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling F06RCF with its parameter ${\mathbf{NORM}}=\text{'1'}$. ANORM must be computed either before calling F07MDF (DSYTRF) or else from a copy of the original matrix $A$.
Constraint: ${\mathbf{ANORM}}\ge 0.0$.
7:     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.
8:     WORK($2×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
9:     IWORK(N) – INTEGER arrayWorkspace
10:   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 F07MGF (DSYCON) involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ floating point operations but takes considerably longer than a call to F07MEF (DSYTRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogues of this routine are F07MUF (ZHECON) for Hermitian matrices and F07NUF (ZSYCON) for symmetric matrices.

## 9  Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 .$
Here $A$ is symmetric indefinite and must first be factorized by F07MDF (DSYTRF). The true condition number in the $1$-norm is $75.68$.

### 9.1  Program Text

Program Text (f07mgfe.f90)

### 9.2  Program Data

Program Data (f07mgfe.d)

### 9.3  Program Results

Program Results (f07mgfe.r)