nag_dpbtrs (f07hec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG 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  Parallelism and Performance

nag_dpbtrs (f07hec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dpbtrs (f07hec) 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 Users' Note for your implementation for any additional implementation-specific information.

9  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).

10  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).

10.1  Program Text

Program Text (f07hece.c)

10.2  Program Data

Program Data (f07hece.d)

10.3  Program Results

Program Results (f07hece.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014