void	nag_dsygst (Nag_OrderType order, Nag_ComputeType comp_type, Nag_UploType uplo, Integer n, double a[], Integer pda, const double b[], Integer pdb, 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

, nag_dsygst (f08sec) must be preceded by a call to nag_dpotrf (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_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 3.3.1.3 in How to Use the NAG Library and its Documentation 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 $uplo = Nag_Upper$ , $C = U^{- T} A U^{- 1}$ 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_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 $B = U^{T} U$ 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: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $a [\dim]$ – doubleInput/Output

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

\max (1, pda \times n)

On entry: the

n

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: $pda$ – IntegerInput

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 doubleInput

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 nag_dpotrf (f07fdc).

8: $pdb$ – IntegerInput

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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 nag_dsygst (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

nag_dsygst (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 nag_zhegst (f08ssc).

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}) .

Here

B

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

C y = λ y

; then nag_dsytrd (f08fec) to reduce

C

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

NAG Library Function Document

nag_dsygst (f08sec)

▸▿ 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

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