NAG Library Function Document

nag_dspgst (f08tec)

+− 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

1 Purpose

nag_dspgst (f08tec) reduces a real symmetric-definite generalized eigenproblem

A z = λ B z

A B z = λ z

B A z = λ z

to the standard form

C y = λ y

, where

A

is a real symmetric matrix and

B

has been factorized by nag_dpptrf (f07gdc), using packed storage.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_dspgst (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, double ap[], const double bp[], NagError *fail)

3 Description

To reduce the real symmetric-definite generalized eigenproblem

A z = λ B z

A B z = λ z

B A z = λ z

to the standard form

C y = λ y

using packed storage, nag_dspgst (f08tec) must be preceded by a call to nag_dpptrf (f07gdc) which computes the Cholesky factorization of

B

;

B

must be positive definite.

The different problem types are specified by the argument comp_type, as indicated in the table below. The table shows how

C

is computed by the function, and also how the eigenvectors

z

of the original problem can be recovered from the eigenvectors of the standard form.

comp_type	Problem	uplo	$B$	$C$	$z$
$Nag_Compute_1$	$A z = λ B z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U^{- T} A U^{- 1}$ $L^{- 1} A L^{- T}$	$U^{- 1} y$ $L^{- T} y$
$Nag_Compute_2$	$A B z = λ z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{- 1} y$ $L^{- T} y$
$Nag_Compute_3$	$B A z = λ z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{T} y$ $L y$

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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: comp_type – Nag_ComputeTypeInput

On entry: indicates how the standard form is computed.

$comp_type = Nag_Compute_1$

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

$comp_type = Nag_Compute_2$ or $Nag_Compute_3$

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

Constraint:

comp_type = Nag_Compute_1

Nag_Compute_2

Nag_Compute_3

3: uplo – Nag_UploTypeInput

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

Constraint:

uplo = Nag_Upper

Nag_Lower

4: n – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: ap[ $\dim$ ] – doubleInput/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

n

symmetric 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 doubleInput

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 nag_dpptrf (f07gdc).

7: 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, $n = ⟨value⟩$ .
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.

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 nag_dspgst (f08tec) 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

nag_dspgst (f08tec) is not threaded by NAG in any implementation.

nag_dspgst (f08tec) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately

n^{3}

The complex analogue of this function is nag_zhpgst (f08tsc).

10 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} 0.24 & 0.39 & 0.42 & - 0.16 \\ 0.39 & - 0.11 & 0.79 & 0.63 \\ 0.42 & 0.79 & - 0.25 & 0.48 \\ - 0.16 & 0.63 & 0.48 & - 0.03 \end{array}) and B = (\begin{array}{r} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.09 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.09 & 0.34 & 1.18 \end{array}),

using packed storage. Here

B

is symmetric positive definite and must first be factorized by nag_dpptrf (f07gdc). The program calls nag_dspgst (f08tec) to reduce the problem to the standard form

C y = λ y

; then nag_dsptrd (f08gec) to reduce

C

to tridiagonal form, and nag_dsterf (f08jfc) to compute the eigenvalues.

NAG Library Function Documentnag_dspgst (f08tec)

+− 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_dspgst (f08tec)