nag_dpbsv (f07hac) (PDF version)
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f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbsv (f07hac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbsv (f07hac) computes the solution to a real system of linear equations
AX=B ,
where A is an n by n symmetric positive definite band matrix of bandwidth 2 kd + 1  and X and B are n by r matrices.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_dpbsv (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, Integer nrhs, double ab[], Integer pdab, double b[], Integer pdb, NagError *fail)

3  Description

nag_dpbsv (f07hac) uses the Cholesky decomposition to factor A as A=UTU if uplo=Nag_Upper or A=LLT if uplo=Nag_Lower, where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations AX=B.

4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 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: if uplo=Nag_Upper, the upper triangle of A is stored.
If uplo=Nag_Lower, the lower triangle of A is stored.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the number of linear equations, i.e., 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:     nrhsIntegerInput
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint: nrhs0.
6:     ab[dim]doubleInput/Output
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 band matrix A.
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.
On exit: if fail.code= NE_NOERROR, the triangular factor U or L from the Cholesky factorization A=UTU or A=LLT of the band matrix A, in the same storage format as A.
7:     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.
8:     b[dim]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×nrhs when order=Nag_ColMajor;
  • max1,n×pdb when order=Nag_RowMajor.
The i,jth element of the matrix B is stored in
  • b[j-1×pdb+i-1] when order=Nag_ColMajor;
  • b[i-1×pdb+j-1] when order=Nag_RowMajor.
On entry: the n by r right-hand side matrix B.
On exit: if fail.code= NE_NOERROR, the n by r solution matrix X.
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
  • if order=Nag_ColMajor, pdbmax1,n;
  • if order=Nag_RowMajor, pdbmax1,nrhs.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, kd=value.
Constraint: kd0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdb=value.
Constraint: pdb>0.
On entry, pdab=value and kd=value.
Constraint: pdabkd+1.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
On entry, pdb=value and nrhs=value.
Constraint: pdbmax1,nrhs.
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.
The leading minor of order value of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

7  Accuracy

The computed solution for a single right-hand side, x^ , satisfies an equation of the form
A+E x^=b ,
E1 = Oε A1
and ε  is the machine precision. An approximate error bound for the computed solution is given by
x^-x1 x1 κA E1 A1 ,
where κA = A-11 A1 , the condition number of A  with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
nag_dpbsvx (f07hbc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_real_sym_posdef_band_lin_solve (f04bfc) solves Ax=b  and returns a forward error bound and condition estimate. nag_real_sym_posdef_band_lin_solve (f04bfc) calls nag_dpbsv (f07hac) to solve the equations.

8  Further Comments

When nk , the total number of floating point operations is approximately nk+12+4nkr , where k  is the number of superdiagonals and r  is the number of right-hand sides.
The complex analogue of this function is nag_zpbsv (f07hnc).

9  Example

This example solves the equations
Ax=b ,
where A  is the symmetric positive definite band matrix
A = 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17   and   b = 22.09 9.31 -5.24 11.83 .
Details of the Cholesky factorization of A  are also output.

9.1  Program Text

Program Text (f07hace.c)

9.2  Program Data

Program Data (f07hace.d)

9.3  Program Results

Program Results (f07hace.r)

nag_dpbsv (f07hac) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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