NAG Library Function Document

nag_ztbmv (f16sgc)

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

3 Description

nag_ztbmv (f16sgc) performs one of the matrix-vector operations

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

where

A

is an

n

n

complex triangular band matrix with

k

subdiagonals or superdiagonals,

x

is an

n

-element complex vector and

α

is a complex scalar.

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 3.2.1.3 in the Essential Introduction 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.

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

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: k – IntegerInput

On entry:

k

, the number of subdiagonals or superdiagonals of the matrix

A

Constraint:

k \geq 0

7: alpha – ComplexInput

On entry: the scalar

α

8: ab[ $\dim$ ] – const ComplexInput

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

\max (1, pdab \times n)

On entry: the

n

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 – IntegerInput

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[ $\dim$ ] – ComplexInput/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 – IntegerInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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: $pdab \geq k + 1$ .

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

Not applicable.

9 Further Comments

No test for singularity or near-singularity of

A

is included in nag_ztbmv (f16sgc). Such tests must be performed before calling this function.

10 Example

This example computes the matrix-vector product

y = α A x

where

A = (\begin{array}{r} 1.0 + 1.0 i & 0.0 + 0.0 i & 0.0 + 0.0 i & 0.0 + 0.0 i \\ 2.0 + 1.0 i & 2.0 + 2.0 i & 0.0 + 0.0 i & 0.0 + 0.0 i \\ 0.0 + 0.0 i & 3.0 + 2.0 i & 3.0 + 3.0 i & 0.0 + 0.0 i \\ 0.0 + 0.0 i & 0.0 + 0.0 i & 4.0 + 3.0 i & 4.0 + 4.0 i \end{array}),

x = (\begin{array}{r} 1.0 + 1.0 i \\ - 2.0 + 2.0 i \\ 3.0 - 2.0 i \\ - 1.0 + 1.0 i \end{array})

and

α = 1.0 + 0.0 i .

NAG Library Function Documentnag_ztbmv (f16sgc)

+− 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_ztbmv (f16sgc)