F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HGF (DPBCON)

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

F07HGF (DPBCON) estimates the condition number of a real symmetric positive definite band matrix $A$, where $A$ has been factorized by F07HDF (DPBTRF).

## 2  Specification

 SUBROUTINE F07HGF ( UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, IWORK, INFO)
 INTEGER N, KD, LDAB, IWORK(N), INFO REAL (KIND=nag_wp) AB(LDAB,*), ANORM, RCOND, WORK(3*N) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dpbcon.

## 3  Description

F07HGF (DPBCON) estimates the condition number (in the $1$-norm) of a real symmetric positive definite band 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 F06REF to compute ${‖A‖}_{1}$ and a call to F07HDF (DPBTRF) to compute the Cholesky 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.
${\mathbf{UPLO}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{UPLO}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, 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:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{KD}}\ge 0$.
4:     AB(LDAB,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the Cholesky factor of $A$, as returned by F07HDF (DPBTRF).
5:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HGF (DPBCON) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
6:     ANORM – REAL (KIND=nag_wp)Input
On entry: the $1$-norm of the original matrix $A$, which may be computed by calling F06REF with its parameter ${\mathbf{NORM}}=\text{'1'}$. ANORM must be computed either before calling F07HDF (DPBTRF) 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($3×{\mathbf{N}}$) – 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 F07HGF (DPBCON) 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 $4nk$ floating point operations (assuming $n\gg k$) but takes considerably longer than a call to F07HEF (DPBTRS) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this routine is F07HUF (ZPBCON).

## 9  Example

This example estimates the condition number in the $1$-norm (or $\infty$-norm) of the matrix $A$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 .$
Here $A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by F07HDF (DPBTRF). The true condition number in the $1$-norm is $74.15$.

### 9.1  Program Text

Program Text (f07hgfe.f90)

### 9.2  Program Data

Program Data (f07hgfe.d)

### 9.3  Program Results

Program Results (f07hgfe.r)