f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dgbcon (f07bgc)

## 1  Purpose

nag_dgbcon (f07bgc) estimates the condition number of a real band matrix $A$, where $A$ has been factorized by nag_dgbtrf (f07bdc).

## 2  Specification

 #include #include
 void nag_dgbcon (Nag_OrderType order, Nag_NormType norm, Integer n, Integer kl, Integer ku, const double ab[], Integer pdab, const Integer ipiv[], double anorm, double *rcond, NagError *fail)

## 3  Description

nag_dgbcon (f07bgc) 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 function actually returns an estimate of the reciprocal of the condition number.
The function should be preceded by a call to nag_dgb_norm (f16rbc) to compute ${‖A‖}_{1}$ or ${‖A‖}_{\infty }$, and a call to nag_dgbtrf (f07bdc) to compute the $LU$ factorization of $A$. The function 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  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     normNag_NormTypeInput
On entry: indicates whether ${\kappa }_{1}\left(A\right)$ or ${\kappa }_{\infty }\left(A\right)$ is estimated.
${\mathbf{norm}}=\mathrm{Nag_OneNorm}$
${\kappa }_{1}\left(A\right)$ is estimated.
${\mathbf{norm}}=\mathrm{Nag_InfNorm}$
${\kappa }_{\infty }\left(A\right)$ is estimated.
Constraint: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$ or $\mathrm{Nag_InfNorm}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     klIntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
5:     kuIntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
6:     ab[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the $LU$ factorization of $A$, as returned by nag_dgbtrf (f07bdc).
7:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
8:     ipiv[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, 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 nag_dgbtrf (f07bdc).
9:     anormdoubleInput
On entry: if ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, the $1$-norm of the original matrix $A$.
If ${\mathbf{norm}}=\mathrm{Nag_InfNorm}$, the $\infty$-norm of the original matrix $A$.
anorm may be computed by calling nag_dgb_norm (f16rbc) with the same value for the argument norm.
anorm must be computed either before calling nag_dgbtrf (f07bdc) or else from a copy of the original matrix $A$ (see Section 9).
Constraint: ${\mathbf{anorm}}\ge 0.0$.
10:   rconddouble *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.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ku}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}>0$.
NE_INT_3
On entry, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{anorm}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{anorm}}\ge 0.0$.

## 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 nag_dgbcon (f07bgc) 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 nag_dgbtrs (f07bec) with one right-hand side, because extra care is taken to avoid overflow when $A$ is approximately singular.
The complex analogue of this function is nag_zgbcon (f07buc).

## 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 nag_dgbtrf (f07bdc). The true condition number in the $1$-norm is $56.40$.

### 9.1  Program Text

Program Text (f07bgce.c)

### 9.2  Program Data

Program Data (f07bgce.d)

### 9.3  Program Results

Program Results (f07bgce.r)