void	nag_ztpsv (Nag_OrderType order, Nag_UploType uplo, Nag_TransType trans, Nag_DiagType diag, Integer n, Complex alpha, const Complex ap[], Complex x[], Integer incx, NagError *fail)

3 Description

nag_ztpsv (f16slc) performs one of the matrix-vector operations

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

where

A

is an

n

n

complex triangular matrix, stored in packed form,

x

is an

n

-element complex vector and

α

is a complex scalar.

A^{- T}

denotes

A^{- T}

or equivalently

A^{- T}

;

A^{- H}

denotes

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

or equivalently

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

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 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 2.3.1.3 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

Nag_ColMajor

2: $uplo$ – 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

Nag_Lower

3: $trans$ – Nag_TransTypeInput

On entry: specifies the operation to be performed.

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

Constraint:

trans = Nag_NoTrans

Nag_Trans

Nag_ConjTrans

4: $diag$ – 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

Nag_UnitDiag

5: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

6: $alpha$ – ComplexInput

On entry: the scalar

α

7: $ap [\dim]$ – const ComplexInput

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

\max (1, n \times (n + 1) / 2)

On entry: the

n

n

triangular matrix

A

, packed by rows or columns.

The storage of elements

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 $ap [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$A_{i j}$ is stored in $ap [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$A_{i j}$ is stored in $ap [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

diag = Nag_UnitDiag

, the diagonal elements of

AP

are assumed to be

1

, and are not referenced; the same storage scheme is used whether

diag = Nag_NonUnitDiag

diag = Nag_UnitDiag

8: $x [\dim]$ – ComplexInput/Output

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

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

On entry: the vector

x

On exit: the solution vector

x

9: $incx$ – IntegerInput

On entry: the increment in the subscripts of x between successive elements of

x

Constraint:

incx \neq 0

10: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 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_INT: On entry, $incx = 〈value〉$ .
Constraint: $incx \neq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INTERNAL_ERROR: An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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_ztpsv (f16slc) is not threaded in any implementation.

9 Further Comments

None.

10 Example

Solves complex triangular system of linear equations,

A x = y

, where

A

is a complex triangular

4

4

matrix, stored in packed storage format, given by

A = (\begin{array}{r} 4.78 + 4.56 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array}),

and

y = (\begin{array}{r} - 14.78 - 32.36 i \\ 2.98 - 2.14 i \\ - 20.96 + 17.06 i \\ 9.54 + 9.91 i \end{array}) .

NAG Library Function Documentnag_ztpsv (f16slc)

▸▿ 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 Library Function Document

nag_ztpsv (f16slc)