nag_dtbsv (f16pkc) (PDF version)
f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dtbsv (f16pkc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtbsv (f16pkc) solves a system of equations given as a real triangular band matrix.

2  Specification

#include <nag.h>
#include <nagf16.h>
void  nag_dtbsv (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Integer k, double alpha, const double ab[], Integer pdab, double x[], Integer incx, NagError *fail)

3  Description

nag_dtbsv (f16pkc) performs one of the matrix-vector operations
xαA-1x  or  xαA-Tx,
where A is an n by n real triangular band matrix with k subdiagonals or superdiagonals, x is an n-element real vector and α is a real scalar. A-T denotes AT-1 or equivalently A-1T.
No test for singularity or near-singularity of A is included in this function. Such tests must be performed before calling this function.

4  References

The BLAS Technical Forum Standard (2001) http://www.netlib.org/blas/blast-forum

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: specifies the operation to be performed.
trans=Nag_NoTrans
xαA-1x.
trans=Nag_Trans or Nag_ConjTrans
xαA-Tx.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
4:     diagNag_DiagTypeInput
On entry: specifies whether A has nonunit or unit diagonal elements.
diag=Nag_NonUnitDiag
The diagonal elements are stored explicitly.
diag=Nag_UnitDiag
The diagonal elements are assumed to be 1 and are not referenced.
Constraint: diag=Nag_NonUnitDiag or Nag_UnitDiag.
5:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
6:     kIntegerInput
On entry: k, the number of subdiagonals or superdiagonals of the matrix A.
Constraint: k0.
7:     alphadoubleInput
On entry: the scalar α.
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[k+i-j+j-1×pdab], for j=1,,n and i=max1,j-k,,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+k;
  • 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+k;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ab[k+j-i+i-1×pdab], for i=1,,n and j=max1,i-k,,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: pdabk+1.
10:   x[dim]doubleInput/Output
Note: the dimension, dim, of the array x must be at least max1,1+n-1incx.
On entry: the right-hand side vector b.
On exit: the solution vector x.
11:   incxIntegerInput
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
12:   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, incx=value.
Constraint: incx0.
On entry, k=value.
Constraint: k0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pdab=value, k=value.
Constraint: pdabk+1.

7  Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of The BLAS Technical Forum Standard (2001)).

8  Further Comments

None.

9  Example

This example solves the real triangular band system of linear equations Ax=y , where A  is the 4 by 4 triangular matrix given with one subdiagonal given by
A = -4.16 -2.25 4.78 5.86 6.32 -4.82 0.16
and where
y = -16.64,-13.78,13.10,-14.14T .
A  is stored in array ab using banded storage format and y is stored in array x. nag_dtbsv (f16pkc) returns the solution in x.

9.1  Program Text

Program Text (f16pkce.c)

9.2  Program Data

Program Data (f16pkce.d)

9.3  Program Results

Program Results (f16pkce.r)


nag_dtbsv (f16pkc) (PDF version)
f16 Chapter Contents
f16 Chapter Introduction
NAG C Library Manual

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