NAG Library Function Document

nag_dtrsm (f16yjc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_dtrsm (f16yjc) solves a system of equations given as a real triangular matrix with multiple right-hand sides.

2
Specification

#include <nag.h>
#include <nagf16.h>
void  nag_dtrsm (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer m, Integer n, double alpha, const double a[], Integer pda, double b[], Integer pdb, NagError *fail)

3
Description

nag_dtrsm (f16yjc) performs one of the matrix-matrix operations
BαA-1B, BαA-TB, BαBA-1  or BαBA-T,  
where A is a real triangular matrix, B is an m by n real matrix, and α is a real scalar. A-T denotes AT-1  or equivalently A-1T .

4
References

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

5
Arguments

1:     order Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     side Nag_SideTypeInput
On entry: specifies whether B is operated on from the left or the right.
side=Nag_LeftSide
B is pre-multiplied from the left.
side=Nag_RightSide
B is post-multiplied from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     uplo Nag_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.
4:     trans Nag_TransTypeInput
On entry: specifies the operation to be performed.
trans=Nag_Trans or Nag_ConjTrans and side=Nag_LeftSide
BαA-TB.
trans=Nag_NoTrans and side=Nag_LeftSide
BαA-1B.
trans=Nag_Trans or Nag_ConjTrans and side=Nag_RightSide
BαBA-T.
trans=Nag_NoTrans and side=Nag_RightSide
BαBA-1.
Constraint: trans=Nag_NoTrans, Nag_Trans or Nag_ConjTrans.
5:     diag Nag_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.
6:     m IntegerInput
On entry: m, the number of rows of the matrix B; the order of A if side=Nag_LeftSide.
Constraint: m0.
7:     n IntegerInput
On entry: n, the number of columns of the matrix B; the order of A if side=Nag_RightSide.
Constraint: n0.
8:     alpha doubleInput
On entry: the scalar α.
9:     a[dim] const doubleInput
Note: the dimension, dim, of the array a must be at least
  • max1,pda×m when side=Nag_LeftSide;
  • max1,pda×n when side=Nag_RightSide.
On entry: the triangular matrix A; A is m by m if side=Nag_LeftSide, or n by n if side=Nag_RightSide.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, A is upper triangular and the elements of the array corresponding to the lower triangular part of A are not referenced.
If uplo=Nag_Lower, A is lower triangular and the elements of the array corresponding to the upper triangular part of A are not referenced.
If diag=Nag_UnitDiag, the diagonal elements of A are assumed to be 1, and are not referenced.
10:   pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraints:
  • if side=Nag_LeftSide, pdamax1,m;
  • if side=Nag_RightSide, pdamax1,n.
11:   b[dim] doubleInput/Output
Note: the dimension, dim, of the array b must be at least
  • max1,pdb×n when order=Nag_ColMajor;
  • max1,m×pdb when order=Nag_RowMajor.
If order=Nag_ColMajor, Bij is stored in b[j-1×pdb+i-1].
If order=Nag_RowMajor, Bij is stored in b[i-1×pdb+j-1].
On entry: the m by n matrix B.
If alpha=0, b need not be set.
On exit: the updated matrix B.
12:   pdb IntegerInput
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,m;
  • if order=Nag_RowMajor, pdbmax1,n.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_2
On entry, side=value, pda=value, m=value.
Constraint: if side=Nag_LeftSide, pdamax1,m.
On entry, side=value, pda=value, n=value.
Constraint: if side=Nag_RightSide, pdamax1,n.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pdb=value, m=value.
Constraint: pdbmax1,m.
On entry, pdb=value and n=value.
Constraint: pdbmax1,n.
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_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_dtrsm (f16yjc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

No test for singularity or near-singularity of A is included in nag_dtrsm (f16yjc). Such tests must be performed before calling this function.

10
Example

Premultiply real 4 by 2 matrix B by inverse of lower triangular 4 by 4 matrix A, BA-1B (or solve AX=B and return result in B), where
A = 4.30 -3.96 -4.87 0.40 0.31 -8.02 -0.27 0.07 -5.95 0.12  
and
B = -12.90 -21.50 16.75 14.93 -17.55 6.33 -11.04 8.09 .  

10.1
Program Text

Program Text (f16yjce.c)

10.2
Program Data

Program Data (f16yjce.d)

10.3
Program Results

Program Results (f16yjce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017