nag_dpbequ (f07hfc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbequ (f07hfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbequ (f07hfc) computes a diagonal scaling matrix S  intended to equilibrate a real n  by n  symmetric positive definite band matrix A , with bandwidth 2kd+1 , and reduce its condition number.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpbequ (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const double ab[], Integer pdab, double s[], double *scond, double *amax, NagError *fail)

3  Description

nag_dpbequ (f07hfc) computes a diagonal scaling matrix S  chosen so that
sj=1 / ajj .
This means that the matrix B  given by
B=SAS ,
has diagonal elements equal to unity. This in turn means that the condition number of B , κ2B , is within a factor n  of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

4  References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: indicates whether the upper or lower triangular part of A is stored in the array ab, as follows:
uplo=Nag_Upper
The upper triangle of A is stored.
uplo=Nag_Lower
The lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     kdIntegerInput
On entry: kd, the number of superdiagonals of the matrix A if uplo=Nag_Upper, or the number of subdiagonals if uplo=Nag_Lower.
Constraint: kd0.
5:     ab[dim]const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the upper or lower triangle of the symmetric positive definite band matrix A whose scaling factors are to be computed.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of Aij, depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ab[kd+i-j+j-1×pdab], for j=1,,n and i=max1,j-kd,,j;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ab[i-j+j-1×pdab], for j=1,,n and i=j,,minn,j+kd;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ab[j-i+i-1×pdab], for i=1,,n and j=i,,minn,i+kd;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[kd+j-i+i-1×pdab], for i=1,,n and j=max1,i-kd,,i.
Only the elements of the array ab corresponding to the diagonal elements of A are referenced. (Row kd+1 of ab when uplo=Nag_Upper, row 1 of ab when uplo=Nag_Lower.)
6:     pdabIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array ab.
Constraint: pdabkd+1.
7:     s[n]doubleOutput
On exit: if fail.code= NE_NOERROR, s contains the diagonal elements of the scaling matrix S.
8:     sconddouble *Output
On exit: if fail.code= NE_NOERROR, scond contains the ratio of the smallest value of s to the largest value of s. If scond0.1 and amax is neither too large nor too small, it is not worth scaling by S.
9:     amaxdouble *Output
On exit: maxaij. If amax is very close to overflow or underflow, the matrix A should be scaled.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, pdab=value.
Constraint: pdab>0.
NE_INT_2
On entry, pdab=value and kd=value.
Constraint: pdabkd+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_MAT_NOT_POS_DEF
The valueth diagonal element of A is not positive (and hence A cannot be positive definite).

7  Accuracy

The computed scale factors will be close to the exact scale factors.

8  Further Comments

The complex analogue of this function is nag_zpbequ (f07htc).

9  Example

This example equilibrates the symmetric positive definite matrix A  given by
A = 5.49 -2.68×1010 -0 -0 2.68×1010 -5.63×1020 -2.39×1010 -0 0 -2.39×1010 -2.60 -2.22 0 -0 -2.22 -5.17 .
Details of the scaling factors and the scaled matrix are output.

9.1  Program Text

Program Text (f07hfce.c)

9.2  Program Data

Program Data (f07hfce.d)

9.3  Program Results

Program Results (f07hfce.r)


nag_dpbequ (f07hfc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012