# NAG Library Routine Document

## 1Purpose

f07buf (zgbcon) estimates the condition number of a complex band matrix $A$, where $A$ has been factorized by f07brf (zgbtrf).

## 2Specification

Fortran Interface
 Subroutine f07buf ( norm, n, kl, ku, ab, ldab, ipiv, work, info)
 Integer, Intent (In) :: n, kl, ku, ldab, ipiv(*) Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: anorm Real (Kind=nag_wp), Intent (Out) :: rcond, rwork(n) Complex (Kind=nag_wp), Intent (In) :: ab(ldab,*) Complex (Kind=nag_wp), Intent (Out) :: work(2*n) Character (1), Intent (In) :: norm
#include <nagmk26.h>
 void f07buf_ (const char *norm, const Integer *n, const Integer *kl, const Integer *ku, const Complex ab[], const Integer *ldab, const Integer ipiv[], const double *anorm, double *rcond, Complex work[], double rwork[], Integer *info, const Charlen length_norm)
The routine may be called by its LAPACK name zgbcon.

## 3Description

f07buf (zgbcon) estimates the condition number of a complex band matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1A=A1A-11 or κ∞A=A∞A-1∞ .$
Note that ${\kappa }_{\infty }\left(A\right)={\kappa }_{1}\left({A}^{\mathrm{H}}\right)$.
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 f06ubf to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to f07brf (zgbtrf) 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 ${‖{A}^{-1}‖}_{\infty }$.

## 4References

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

## 5Arguments

1:     $\mathbf{norm}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kl}$ – IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4:     $\mathbf{ku}$ – IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Complex (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 f07brf (zgbtrf).
6:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f07buf (zgbcon) is called.
Constraint: ${\mathbf{ldab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
7:     $\mathbf{ipiv}\left(*\right)$ – 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 f07brf (zgbtrf).
8:     $\mathbf{anorm}$ – 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 f06ubf with the same value for the argument norm.
anorm must be computed either before calling f07brf (zgbtrf) or else from a copy of the original matrix $A$ (see Section 10).
Constraint: ${\mathbf{anorm}}\ge 0.0$.
9:     $\mathbf{rcond}$ – 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:   $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
11:   $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
12:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

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.

## 8Parallelism and Performance

f07buf (zgbcon) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

A call to f07buf (zgbcon) involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{H}}x=b$; the number is usually $5$ and never more than $11$. Each solution involves approximately $8n\left(2{k}_{l}+{k}_{u}\right)$ real floating-point operations (assuming $n\gg {k}_{l}$ and $n\gg {k}_{u}$) but takes considerably longer than a call to f07bsf (zgbtrs) 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 f07bgf (dgbcon).

## 10Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i .$

### 10.1Program Text

Program Text (f07bufe.f90)

### 10.2Program Data

Program Data (f07bufe.d)

### 10.3Program Results

Program Results (f07bufe.r)