nag_dpbtrs (f07hec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dpbtrs (f07hec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dpbtrs (f07hec) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,
AX=B ,
where A has been factorized by nag_dpbtrf (f07hdc).

2  Specification

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

3  Description

nag_dpbtrs (f07hec) is used to solve a real symmetric positive definite band system of linear equations AX=B, the function must be preceded by a call to nag_dpbtrf (f07hdc) which computes the Cholesky factorization of A. The solution X is computed by forward and backward substitution.
If uplo=Nag_Upper, A=UTU, where U is upper triangular; the solution X is computed by solving UTY=B and then UX=Y.
If uplo=Nag_Lower, A=LLT, where L is lower triangular; the solution X is computed by solving LY=B and then LTX=Y.

4  References

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 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: specifies how A has been factorized.
uplo=Nag_Upper
A=UTU, where U is upper triangular.
uplo=Nag_Lower
A=LLT, where L is lower triangular.
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 or subdiagonals of the matrix A.
Constraint: kd0.
5:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
6:     ab[dim]const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the Cholesky factor of A, as returned by nag_dpbtrf (f07hdc).
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: 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: 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.
Constraints:
  • 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

NE_ALLOC_FAIL
Dynamic memory allocation failed.
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, nrhs=value.
Constraint: nrhs0.
On entry, pdab=value.
Constraint: pdab>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
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.
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.

7  Accuracy

For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where ck+1 is a modest linear function of k+1, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x ck+1condA,xε
where condA,x=A-1Ax/xcondA=A-1AκA. Note that condA,x can be much smaller than condA.
Forward and backward error bounds can be computed by calling nag_dpbrfs (f07hhc), and an estimate for κA (=κ1A) can be obtained by calling nag_dpbcon (f07hgc).

8  Further Comments

The total number of floating point operations is approximately 4nkr, assuming nk.
This function may be followed by a call to nag_dpbrfs (f07hhc) to refine the solution and return an error estimate.
The complex analogue of this function is nag_zpbtrs (f07hsc).

9  Example

This example solves the system of equations AX=B, where
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 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .
Here A is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_dpbtrf (f07hdc).

9.1  Program Text

Program Text (f07hece.c)

9.2  Program Data

Program Data (f07hece.d)

9.3  Program Results

Program Results (f07hece.r)


nag_dpbtrs (f07hec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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