nag_dtbtrs (f07vec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dtbtrs (f07vec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtbtrs (f07vec) solves a real triangular band system of linear equations with multiple right-hand sides, AX=B or ATX=B.

2  Specification

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

3  Description

nag_dtbtrs (f07vec) solves a real triangular band system of linear equations AX=B or ATX=B.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

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 whether A is upper or lower triangular.
uplo=Nag_Upper
A is upper triangular.
uplo=Nag_Lower
A is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     transNag_TransTypeInput
On entry: indicates the form of the equations.
trans=Nag_NoTrans
The equations are of the form AX=B.
trans=Nag_Trans or Nag_ConjTrans
The equations are of the form ATX=B.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4:     diagNag_DiagTypeInput
On entry: indicates whether A is a nonunit or unit triangular matrix.
diag=Nag_NonUnitDiag
A is a nonunit triangular matrix.
diag=Nag_UnitDiag
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag=Nag_NonUnitDiag or Nag_UnitDiag.
5:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     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.
7:     nrhsIntegerInput
On entry: r, the number of right-hand sides.
Constraint: nrhs0.
8:     ab[dim]const doubleInput
Note: the dimension, dim, of the array ab must be at least max1,pdab×n.
On entry: the n by n triangular 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.
If diag=Nag_UnitDiag, the diagonal elements of AB are assumed to be 1, and are not referenced.
9:     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.
10:   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.
11:   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.
12:   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.
NE_SINGULAR
avalue,value is exactly zero. A is singular and the solution has not been computed.

7  Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
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
EckεA ,
ck is a modest linear function of k, 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 ckcondA,xε ,   provided   ckcondA,xε<1 ,
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAT to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_dtbrfs (f07vhc), and an estimate for κA can be obtained by calling nag_dtbcon (f07vgc) with norm=Nag_InfNorm.

8  Further Comments

The total number of floating point operations is approximately 2nkr if kn.
The complex analogue of this function is nag_ztbtrs (f07vsc).

9  Example

This example solves the system of equations AX=B, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16   and   B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .
Here A is treated as a lower triangular band matrix with one subdiagonal.

9.1  Program Text

Program Text (f07vece.c)

9.2  Program Data

Program Data (f07vece.d)

9.3  Program Results

Program Results (f07vece.r)


nag_dtbtrs (f07vec) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

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