# NAG FL Interfacef07vgf (dtbcon)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

f07vgf estimates the condition number of a real triangular band matrix.

## 2Specification

Fortran Interface
 Subroutine f07vgf ( norm, uplo, diag, n, kd, ab, ldab, work, info)
 Integer, Intent (In) :: n, kd, ldab Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: ab(ldab,*) Real (Kind=nag_wp), Intent (Out) :: rcond, work(3*n) Character (1), Intent (In) :: norm, uplo, diag
C Header Interface
#include <nag.h>
 void f07vgf_ (const char *norm, const char *uplo, const char *diag, const Integer *n, const Integer *kd, const double ab[], const Integer *ldab, double *rcond, double work[], Integer iwork[], Integer *info, const Charlen length_norm, const Charlen length_uplo, const Charlen length_diag)
The routine may be called by the names f07vgf, nagf_lapacklin_dtbcon or its LAPACK name dtbcon.

## 3Description

f07vgf estimates the condition number of a real triangular band matrix $A$, in either the $1$-norm or the $\infty$-norm:
 $κ1(A)=‖A‖1‖A-1‖1 or κ∞(A)=‖A‖∞‖A-1‖∞ .$
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 computes ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$ exactly, and 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: $\mathbf{uplo}$Character(1) Input
On entry: specifies whether $A$ is upper or lower triangular.
${\mathbf{uplo}}=\text{'U'}$
$A$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3: $\mathbf{diag}$Character(1) Input
On entry: indicates whether $A$ is a nonunit or unit triangular matrix.
${\mathbf{diag}}=\text{'N'}$
$A$ is a nonunit triangular matrix.
${\mathbf{diag}}=\text{'U'}$
$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$.
Constraint: ${\mathbf{diag}}=\text{'N'}$ or $\text{'U'}$.
4: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{kd}$Integer Input
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{uplo}}=\text{'L'}$.
Constraint: ${\mathbf{kd}}\ge 0$.
6: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Real (Kind=nag_wp) array Input
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 $n×n$ triangular band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If ${\mathbf{diag}}=\text{'U'}$, the diagonal elements of $A$ are assumed to be $1$, and are not referenced.
7: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07vgf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
8: $\mathbf{rcond}$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.
9: $\mathbf{work}\left(3×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
10: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Workspace
11: $\mathbf{info}$Integer Output
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

Background information to multithreading can be found in the Multithreading documentation.
f07vgf 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.

## 9Further Comments

A call to f07vgf 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 $2nk$ floating-point operations (assuming $n\gg k$) but takes considerably longer than a call to f07vef 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 f07vuf.

## 10Example

This example estimates the condition number in the $1$-norm of the matrix $A$, where
 $A= ( -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16 ) .$
Here $A$ is treated as a lower triangular band matrix with one subdiagonal. The true condition number in the $1$-norm is $69.62$.

### 10.1Program Text

Program Text (f07vgfe.f90)

### 10.2Program Data

Program Data (f07vgfe.d)

### 10.3Program Results

Program Results (f07vgfe.r)