void f08sec (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, double a[], Integer pda, const double b[], Integer pdb, NagError *fail)

The function may be called by the names: f08sec, nag_lapackeig_dsygst or nag_dsygst.

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

, f08sec must be preceded by a call to f07fdc 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.

			$order = Nag_ColMajor$			$order = Nag_RowMajor$
comp_type	Problem	uplo	$B$	$C$	$z$	$B$	$C$	$z$
$1$	$A z = λ B z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U^{- T} A U^{- 1}$ $L^{- 1} A L^{- T}$	$U^{- 1} y$ $L^{- T} y$	$U U^{T}$ $L^{T} L$	$U^{- 1} A U^{- T}$ $L^{- T} A L^{- 1}$	$U^{- T} y$ $L^{- 1} y$
$2$	$A B z = λ z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{- 1} y$ $L^{- T} y$	$U U^{T}$ $L^{T} L$	$U^{T} A U$ $L A L^{T}$	$U^{- T} y$ $L^{- 1} y$
$3$	$B A z = λ z$	$Nag_Upper$ $Nag_Lower$	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{T} y$ $L y$	$U U^{T}$ $L^{T} L$	$U^{T} A U$ $L A L^{T}$	$U y$ $L^{T} y$

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_OrderType Input

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 $uplo = Nag_Upper$ , $C = U^{- T} A U^{- 1}$ when $order = Nag_ColMajor$ and $C = U^{- 1} A U^{- T}$ when $order = Nag_RowMajor$ ;
if $uplo = Nag_Lower$ , $C = L^{- 1} A L^{- T}$ when $order = Nag_ColMajor$ and $C = L^{- T} A L^{- 1}$ when $order = Nag_RowMajor$ .

$comp_type = Nag_Compute_2$ or $Nag_Compute_3$

if $uplo = Nag_Upper$ , $C = U A U^{T}$ when $order = Nag_ColMajor$ and $C = U^{T} A U$ when $order = Nag_RowMajor$ ;
if $uplo = Nag_Lower$ , $C = L^{T} A L$ when $order = Nag_ColMajor$ and $C = L A L^{T}$ when $order = Nag_RowMajor$ .

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

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: $a [\dim]$ – double Input/Output

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

\max (1, pda \times n)

On entry: the

n \times n

symmetric matrix

A

order = Nag_ColMajor

A_{i j}

is stored in

a [(j - 1) \times pda + i - 1]

order = Nag_RowMajor

A_{i j}

is stored in

a [(i - 1) \times pda + j - 1]

uplo = Nag_Upper

, the upper triangular part of

A

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

A

must be stored and the elements of the array above the diagonal are not referenced.

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

C

as specified by comp_type and uplo.

6: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array a.

Constraint:

pda \geq \max (1, n)

7: $b [\dim]$ – const double Input

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

\max (1, pdb \times n)

On entry: the Cholesky factor of

B

as specified by uplo and returned by f07fdc.

8: $pdb$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

B

in the array b.

Constraint:

pdb \geq \max (1, n)

9: $fail$ – 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: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdb = ⟨ value ⟩$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .

On entry, $pdb = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdb \geq \max (1, n)$ .
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 f08sec 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.

f08sec 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 floating-point operations is approximately

n^{3}

The complex analogue of this function is f08ssc.

10 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{matrix} 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{matrix}) and B = (\begin{matrix} 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{matrix}) .

Here

B

is symmetric positive definite and must first be factorized by f07fdc. The program calls f08sec to reduce the problem to the standard form

C y = λ y

; then f08fec to reduce

C

to tridiagonal form, and f08jfc to compute the eigenvalues.

f08se: FL CL CPP AD PY MB

NAG CL Interfacef08sec (dsygst)

▸▿ 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
f08sec (dsygst)