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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 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

or 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 Further Comments

The total number of floating point operations is approximately

n^{3}

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

9 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 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_dspgst (f08tec)