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

x \leftarrow α A^{- 1} x or x \leftarrow α A^{- T} x,

where

A

is an

n \times 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

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

or equivalently

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

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

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

Nag_Lower

3: $trans$ – Nag_TransType Input

On entry: specifies the operation to be performed.

$trans = Nag_NoTrans$: $x \leftarrow α A^{- 1} x$ .
$trans = Nag_Trans$ or $Nag_ConjTrans$: $x \leftarrow α A^{- T} x$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

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

Nag_UnitDiag

5: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $k$ – Integer Input

On entry:

k

, the number of subdiagonals or superdiagonals of the matrix

A

Constraint:

k \geq 0

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

\max (1, pdab \times n)

On entry: the

n \times 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

A_{i j}

, depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ab [k + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k)$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k)$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ab [k + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k), \dots, i$ .

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:

pdab \geq k + 1

10: $x [\dim]$ – double Input/Output

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

\max (1, 1 + (n - 1) | incx |)

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:

incx \neq 0

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: $incx \neq 0$ .

On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdab = ⟨ value ⟩$ , $k = ⟨ value ⟩$ .
Constraint: $pdab \geq k + 1$ .
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

A x = y

, where

A

is the

4 \times 4

triangular matrix given with one subdiagonal given by

A = (\begin{matrix} - 4.16 \\ - 2.25 & 4.78 \\ 5.86 & 6.32 \\ - 4.82 & 0.16 \end{matrix})

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.

f16pk: FL CL CPP AD

NAG CL Interfacef16pkc (dtbsv)

▸▿ 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
f16pkc (dtbsv)