NAG CL Interface
f16pkc (dtbsv)

1 Purpose

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

2 Specification

#include <nag.h>
void  f16pkc (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)
The function may be called by the names: f16pkc, nag_blast_dtbsv or nag_dtbsv.

3 Description

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

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf

5 Arguments

1: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: uplo Nag_UploType Input
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: trans Nag_TransType Input
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: diag Nag_DiagType Input
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: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
6: k Integer Input
On entry: k, the number of subdiagonals or superdiagonals of the matrix A.
Constraint: k0.
7: alpha double Input
On entry: the scalar α.
8: ab[dim] const double Input
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: pdab Integer Input
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] double Input/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: incx Integer Input
On entry: the increment in the subscripts of x between successive elements of x.
Constraint: incx0.
12: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8 Parallelism and Performance

f16pkc is not threaded in any implementation.

9 Further Comments

None.

10 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,-14.14 T .  
A is stored in array ab using banded storage format and y is stored in array x. f16pkc returns the solution in x.

10.1 Program Text

Program Text (f16pkce.c)

10.2 Program Data

Program Data (f16pkce.d)

10.3 Program Results

Program Results (f16pkce.r)