void	nag_dspgvd (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_UploType uplo, Integer n, double ap[], double bp[], double w[], double z[], Integer pdz, NagError *fail)

3 Description

nag_dspgvd (f08tcc) first performs a Cholesky factorization of the matrix

B

B = U^{T} U

, when

uplo = Nag_Upper

B = L L^{T}

, when

uplo = Nag_Lower

. The generalized problem is then reduced to a standard symmetric eigenvalue problem

C x = λ x,

which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.

For the problem

A z = λ B z

, the eigenvectors are normalized so that the matrix of eigenvectors,

z

, satisfies

Z^{T} A Z = Λ and Z^{T} B Z = I,

where

Λ

is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem

A B z = λ z

we correspondingly have

Z^{- 1} A Z^{- T} = Λ and Z^{T} B Z = I,

and for

B A z = λ z

we have

Z^{T} A Z = Λ and Z^{T} B^{- 1} Z = I .

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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 2.3.1.3 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: $itype$ – IntegerInput

On entry: specifies the problem type to be solved.

$itype = 1$: $A z = λ B z$ .
$itype = 2$: $A B z = λ z$ .
$itype = 3$: $B A z = λ z$ .

Constraint:

itype = 1

2

3

3: $job$ – Nag_JobTypeInput

On entry: indicates whether eigenvectors are computed.

$job = Nag_EigVals$: Only eigenvalues are computed.
$job = Nag_DoBoth$: Eigenvalues and eigenvectors are computed.

Constraint:

job = Nag_EigVals

Nag_DoBoth

4: $uplo$ – Nag_UploTypeInput

On entry: if

uplo = Nag_Upper

, the upper triangles of

A

and

B

are stored.

uplo = Nag_Lower

, the lower triangles of

A

and

B

are stored.

Constraint:

uplo = Nag_Upper

Nag_Lower

5: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

6: $ap [\dim]$ – 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 contents of ap are destroyed.

7: $bp [\dim]$ – doubleInput/Output

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

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

On entry: the upper or lower triangle of the

n

n

symmetric matrix

B

, packed by rows or columns.

The storage of elements

B_{i j}

depends on the order and uplo arguments as follows:

if $order = Nag_ColMajor$ and $uplo = Nag_Upper$ ,
$B_{i j}$ is stored in $bp [(j - 1) \times j / 2 + i - 1]$ , for $i \leq j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,
$B_{i j}$ is stored in $bp [(2 n - j) \times (j - 1) / 2 + i - 1]$ , for $i \geq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,
$B_{i j}$ is stored in $bp [(2 n - i) \times (i - 1) / 2 + j - 1]$ , for $i \leq j$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,
$B_{i j}$ is stored in $bp [(i - 1) \times i / 2 + j - 1]$ , for $i \geq j$ .

On exit: the triangular factor

U

L

from the Cholesky factorization

B = U^{T} U

B = L L^{T}

, in the same storage format as

B

8: $w [n]$ – doubleOutput

On exit: the eigenvalues in ascending order.

9: $z [\dim]$ – doubleOutput

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

$\max (1, pdz \times n)$ when $job = Nag_DoBoth$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Z

is stored in

$z [(j - 1) \times pdz + i - 1]$ when $order = Nag_ColMajor$ ;
$z [(i - 1) \times pdz + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

job = Nag_DoBoth

, z contains the matrix

Z

of eigenvectors. The eigenvectors are normalized as follows:

if $itype = 1$ or $2$ , $Z^{T} B Z = I$ ;
if $itype = 3$ , $Z^{T} B^{- 1} Z = I$ .

job = Nag_EigVals

, z is not referenced.

10: $pdz$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array z.

Constraints:

if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

11: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 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_CONVERGENCE: The algorithm failed to converge; $〈value〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
NE_ENUM_INT_2: On entry, $job = 〈value〉$ , $n = 〈value〉$ and $pdz = 〈value〉$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .

On entry, $job = 〈value〉$ , $pdz = 〈value〉$ , $n = 〈value〉$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_INT: On entry, $itype = 〈value〉$ .
Constraint: $itype = 1$ , $2$ or $3$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

On entry, $pdz = 〈value〉$ .
Constraint: $pdz > 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAT_NOT_POS_DEF: If $fail . errnum = n + 〈value〉$ , for $1 \leq 〈value〉 \leq n$ , then the leading minor of order $〈value〉$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.
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

B

is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of

B

differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of

B

would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

The example program below illustrates the computation of approximate error bounds.

8 Parallelism and Performance

nag_dspgvd (f08tcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dspgvd (f08tcc) 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 proportional to

n^{3}

The complex analogue of this function is nag_zhpgvd (f08tqc).

10 Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem

A B z = λ 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}),

together with an estimate of the condition number of

B

, and approximate error bounds for the computed eigenvalues and eigenvectors.

The example program for nag_dspgv (f08tac) illustrates solving a generalized symmetric eigenproblem of the form

A z = λ B z

NAG Library Function Documentnag_dspgvd (f08tcc)

▸▿ 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_dspgvd (f08tcc)