F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07BGF (DGBCON)

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

F07BGF (DGBCON) estimates the condition number of a real band matrix $A$, where $A$ has been factorized by F07BDF (DGBTRF).

## 2  Specification

 SUBROUTINE F07BGF ( NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
 INTEGER N, KL, KU, LDAB, IPIV(*), IWORK(N), INFO REAL (KIND=nag_wp) AB(LDAB,*), ANORM, RCOND, WORK(3*N) CHARACTER(1) NORM
The routine may be called by its LAPACK name dgbcon.

## 3  Description

F07BGF (DGBCON) estimates the condition number of a real band matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1A=A1A-11 or κ∞A=A∞A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{T}}\right)$.
Because the condition number is infinite if $A$ is singular, the routine actually returns an estimate of the reciprocal of the condition number.
The routine should be preceded by a call to F06RBF to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to F07BDF (DGBTRF) to compute the $LU$ factorization of $A$. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate ${‖{A}^{-1}‖}_{1}$ or ${‖{A}^{-1}‖}_{\infty }$.

## 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:     NORM – CHARACTER(1)Input
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{NORM}}=\text{'1'}$ or $\text{'O'}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{NORM}}=\text{'I'}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{NORM}}=\text{'1'}$, $\text{'O'}$ or $\text{'I'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     KL – INTEGERInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KL}}\ge 0$.
4:     KU – INTEGERInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KU}}\ge 0$.
5:     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 $LU$ factorization of $A$, as returned by F07BDF (DGBTRF).
6:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07BGF (DGBCON) is called.
Constraint: ${\mathbf{LDAB}}\ge 2×{\mathbf{KL}}+{\mathbf{KU}}+1$.
7:     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: the pivot indices, as returned by F07BDF (DGBTRF).
8:     ANORM – REAL (KIND=nag_wp)Input
On entry: if ${\mathbf{NORM}}=\text{'1'}$ or $\text{'O'}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{NORM}}=\text{'I'}$, the $\infty$-norm of the original matrix $A$.
ANORM may be computed by calling F06RBF with the same value for the parameter NORM.
ANORM must be computed either before calling F07BDF (DGBTRF) or else from a copy of the original matrix $A$ (see Section 9).
Constraint: ${\mathbf{ANORM}}\ge 0.0$.
9:     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.
10:   WORK($3×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
11:   IWORK(N) – INTEGER arrayWorkspace
12:   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 F07BGF (DGBCON) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2n\left(2{k}_{l}+{k}_{u}\right)$ floating point operations (assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$) but takes considerably longer than a call to F07BEF (DGBTRS) 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 F07BUF (ZGBCON).

## 9  Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 .$
Here $A$ is nonsymmetric and is treated as a band matrix, which must first be factorized by F07BDF (DGBTRF). The true condition number in the $1$-norm is $56.40$.

### 9.1  Program Text

Program Text (f07bgfe.f90)

### 9.2  Program Data

Program Data (f07bgfe.d)

### 9.3  Program Results

Program Results (f07bgfe.r)