void	f16yjc (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)

The function may be called by the names: f16yjc, nag_blast_dtrsm or nag_dtrsm.

3 Description

f16yjc performs one of the matrix-matrix operations

\begin{array}{l} B \leftarrow α A^{- 1} B, & B \leftarrow α A^{- T} B, \\ B \leftarrow α B A^{- 1} or & B \leftarrow α B A^{- T}, \end{array}

where

A

is a real triangular matrix,

B

is an

m \times n

real matrix, and

α

is a real scalar.

A^{- T}

denotes

{(A^{T})}^{- 1}

or equivalently

{(A^{- 1})}^{T}

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

Nag_ColMajor

2: $side$ – Nag_SideType Input

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

Nag_RightSide

3: $uplo$ – 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

Nag_Lower

4: $trans$ – Nag_TransType Input

On entry: specifies the operation to be performed.

$trans = Nag_Trans$ or $Nag_ConjTrans$ and $side = Nag_LeftSide$: $B \leftarrow α A^{- T} B$ .
$trans = Nag_NoTrans$ and $side = Nag_LeftSide$: $B \leftarrow α A^{- 1} B$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$ and $side = Nag_RightSide$: $B \leftarrow α B A^{- T}$ .
$trans = Nag_NoTrans$ and $side = Nag_RightSide$: $B \leftarrow α B A^{- 1}$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

5: $diag$ – 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

Nag_UnitDiag

6: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

B

; the order of

A

side = Nag_LeftSide

Constraint:

m \geq 0

7: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

B

; the order of

A

side = Nag_RightSide

Constraint:

n \geq 0

8: $alpha$ – double Input

On entry: the scalar

α

9: $a [\dim]$ – const double Input

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times m)$ when $side = Nag_LeftSide$ ;
$\max (1, pda \times n)$ when $side = Nag_RightSide$ .

On entry: the triangular matrix

A

;

A

m \times m

side = Nag_LeftSide

, or

n \times n

side = Nag_RightSide

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

A

is upper triangular and the elements of the array corresponding to the lower triangular part of

A

are not referenced.

uplo = Nag_Lower

A

is lower triangular and the elements of the array corresponding to the upper triangular part of

A

are not referenced.

diag = Nag_UnitDiag

, the diagonal elements of

A

are assumed to be

1

, and are not referenced.

10: $pda$ – Integer Input

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 $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .

11: $b [\dim]$ – double Input/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdb)$ when $order = Nag_RowMajor$ .

order = Nag_ColMajor

B_{i j}

is stored in

b [(j - 1) \times pdb + i - 1]

order = Nag_RowMajor

B_{i j}

is stored in

b [(i - 1) \times pdb + j - 1]

On entry: the

m \times n

matrix

B

alpha = 0

, b need not be set.

On exit: the updated matrix

B

12: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, n)$ .

13: $fail$ – 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_ENUM_INT_2: On entry, $side = ⟨ value ⟩$ , $pda = ⟨ value ⟩$ , $m = ⟨ value ⟩$ .
Constraint: if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ .

On entry, $side = ⟨ value ⟩$ , $pda = ⟨ value ⟩$ , $n = ⟨ value ⟩$ .
Constraint: if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdb = ⟨ value ⟩$ , $m = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, m)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
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 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

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 f16yjc. Such tests must be performed before calling this function.

10 Example

Premultiply real

4 \times 2

matrix

B

by inverse of lower triangular

4 \times 4

matrix

A

B \leftarrow A^{- 1} B

(or solve

A X = B

and return result in

B

), where

A = (\begin{matrix} 4.30 \\ - 3.96 & - 4.87 \\ 0.40 & 0.31 & - 8.02 \\ - 0.27 & 0.07 & - 5.95 & 0.12 \end{matrix})

and

B = (\begin{matrix} - 12.90 & - 21.50 \\ 16.75 & 14.93 \\ - 17.55 & 6.33 \\ - 11.04 & 8.09 \end{matrix}) .

f16yj: FL CL CPP AD

NAG CL Interfacef16yjc (dtrsm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f16yjc (dtrsm)