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: $comp_type$ – Nag_ComputeType Input

On entry: indicates how the standard form is computed.

$comp_type = Nag_Compute_1$

if $uplo = Nag_Upper$ , $C = U^{- H} A U^{- 1}$ ;
if $uplo = Nag_Lower$ , $C = L^{- 1} A L^{- H}$ .

$comp_type = Nag_Compute_2$ or $Nag_Compute_3$

if $uplo = Nag_Upper$ , $C = U A U^{H}$ ;
if $uplo = Nag_Lower$ , $C = L^{H} A L$ .

Constraint:

comp_type = Nag_Compute_1

Nag_Compute_2

Nag_Compute_3

3: $uplo$ – Nag_UploType Input

On entry: indicates whether the upper or lower triangular part of

A

is stored and how

B

has been factorized.

$uplo = Nag_Upper$: The upper triangular part of $A$ is stored and $B = U^{H} U$ .
$uplo = Nag_Lower$: The lower triangular part of $A$ is stored and $B = L L^{H}$ .

Constraint:

uplo = Nag_Upper

Nag_Lower

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ap [\dim]$ – Complex Input/Output

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

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

On entry: the upper or lower triangle of the

n \times n

Hermitian 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 $i \geq j$ .

On exit: the upper or lower triangle of ap is overwritten by the corresponding upper or lower triangle of

C

as specified by comp_type and uplo, using the same packed storage format as described above.

6: $bp [\dim]$ – const Complex Input

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

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

On entry: the Cholesky factor of

B

as specified by uplo and returned by f07grc.

7: $fail$ – 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, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
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

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

if (

comp_type = Nag_Compute_1

) or

B

(if

comp_type = Nag_Compute_2

Nag_Compute_3

). When f08tsc is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if

B

is ill-conditioned with respect to inversion.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08tsc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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

The total number of real floating-point operations is approximately

4 n^{3}

The real analogue of this function is f08tec.

10 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} - 7.36 + 0.00 i & 0.77 - 0.43 i & - 0.64 - 0.92 i & 3.01 - 6.97 i \\ 0.77 + 0.43 i & 3.49 + 0.00 i & 2.19 + 4.45 i & 1.90 + 3.73 i \\ - 0.64 + 0.92 i & 2.19 - 4.45 i & 0.12 + 0.00 i & 2.88 - 3.17 i \\ 3.01 + 6.97 i & 1.90 - 3.73 i & 2.88 + 3.17 i & - 2.54 + 0.00 i \end{array})

and

B = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}),

using packed storage. Here

B

is Hermitian positive definite and must first be factorized by f07grc. The program calls f08tsc to reduce the problem to the standard form

C y = λ y

; then f08gsc to reduce

C

to tridiagonal form, and f08jfc to compute the eigenvalues.

f08ts: FL CL CPP AD PY MB

NAG CL Interfacef08tsc (zhpgst)

▸▿ 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
f08tsc (zhpgst)