is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

2

Specification

Fortran Interface

Subroutine f08tbf (

itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, jfail, info)

Integer, Intent (In)	::	itype, n, il, iu, ldz
Integer, Intent (Inout)	::	jfail(*)
Integer, Intent (Out)	::	m, iwork(5*n), info
Real (Kind=nag_wp), Intent (In)	::	vl, vu, abstol
Real (Kind=nag_wp), Intent (Inout)	::	ap(), bp(), z(ldz,*)
Real (Kind=nag_wp), Intent (Out)	::	w(n), work(8*n)
Character (1), Intent (In)	::	jobz, range, uplo

C Header Interface

#include nagmk26.h

void

f08tbf_ ( const Integer *itype, const char *jobz, const char *range, const char *uplo, const Integer *n, double ap[], double bp[], const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], double z[], const Integer *ldz, double work[], Integer iwork[], Integer jfail[], Integer *info, const Charlen length_jobz, const Charlen length_range, const Charlen length_uplo)

The routine may be called by its LAPACK name dspgvx.

3

Description

f08tbf (dspgvx) first performs a Cholesky factorization of the matrix

B

B = U^{T} U

, when

uplo ='U'

B = L L^{T}

, when

uplo ='L'

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

C x = λ x,

which is solved for the desired eigenvalues and 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

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

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

5

Arguments

1: $itype$ – 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

2: $jobz$ – Character(1)Input

On entry: indicates whether eigenvectors are computed.

$jobz ='N'$: Only eigenvalues are computed.
$jobz ='V'$: Eigenvalues and eigenvectors are computed.

Constraint:

jobz ='N'

'V'

3: $range$ – Character(1)Input

On entry: if

range ='A'

, all eigenvalues will be found.

range ='V'

, all eigenvalues in the half-open interval

(vl, vu]

will be found.

range ='I'

, the ilth to iuth eigenvalues will be found.

Constraint:

range ='A'

'V'

'I'

4: $uplo$ – Character(1)Input

On entry: if

uplo ='U'

, the upper triangles of

A

and

B

are stored.

uplo ='L'

, the lower triangles of

A

and

B

are stored.

Constraint:

uplo ='U'

'L'

5: $n$ – IntegerInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

6: $ap (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension 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 columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

On exit: the contents of ap are destroyed.

7: $bp (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension 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 columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $B$ must be stored with element $B_{i j}$ in $bp (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $B$ must be stored with element $B_{i j}$ in $bp (i + (2 n - j) (j - 1) / 2)$ 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: $vl$ – Real (Kind=nag_wp)Input

9: $vu$ – Real (Kind=nag_wp)Input

On entry: if

range ='V'

, the lower and upper bounds of the interval to be searched for eigenvalues.

range ='A'

'I'

, vl and vu are not referenced.

Constraint: if

range ='V'

vl < vu

10: $il$ – IntegerInput

11: $iu$ – IntegerInput

On entry: if

range ='I'

, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

range ='A'

'V'

, il and iu are not referenced.

Constraints:

if $range ='I'$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range ='I'$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .

12: $abstol$ – Real (Kind=nag_wp)Input

On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval

[a, b]

of width less than or equal to

abstol + ε \max (|a|, |b|),

where

ε

is the machine precision. If abstol is less than or equal to zero, then

ε {‖T‖}_{1}

will be used in its place, where

T

is the tridiagonal matrix obtained by reducing

C

to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold

2 \times x02amf ()

, not zero. If this routine returns with

info = 1 to n

, indicating that some eigenvectors did not converge, try setting abstol to

2 \times x02amf ()

. See Demmel and Kahan (1990).

13: $m$ – IntegerOutput

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range ='A'

m = n

range ='I'

m = iu - il + 1

14: $w (n)$ – Real (Kind=nag_wp) arrayOutput

On exit: the first m elements contain the selected eigenvalues in ascending order.

15: $z (ldz *)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array z must be at least

\max (1, m)

jobz ='V'

, and at least

1

otherwise.

On exit: if

jobz ='V'

, then

if info=0, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with wi. The eigenvectors are normalized as follows:
- if $itype = 1$ or $2$ , $Z^{T} B Z = I$ ;
- if $itype = 3$ , $Z^{T} B^{- 1} Z = I$ ;
if an eigenvector fails to converge ( $info = 1 to n$ ), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

jobz ='N'

, z is not referenced.

Note: you must ensure that at least

\max (1, m)

columns are supplied in the array z; if

range ='V'

, the exact value of m is not known in advance and an upper bound of at least n must be used.

16: $ldz$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08tbf (dspgvx) is called.

Constraints:

if $jobz ='V'$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

17: $work (8 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

18: $iwork (5 \times n)$ – Integer arrayWorkspace

19: $jfail (*)$ – Integer arrayOutput

Note: the dimension of the array jfail must be at least

\max (1, n)

On exit: if

jobz ='V'

, then

if $info = 0$ , the first m elements of jfail are zero;
if $info = 1 to n$ , jfail contains the indices of the eigenvectors that failed to converge.

jobz ='N'

, jfail is not referenced.

20: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info = 1 to n$: If $info = i$ , f08gbf (dspevx) failed to converge; $i$ eigenvectors failed to converge. Their indices are stored in array jfail.

$info > n$: f07gdf (dpptrf) returned an error code; i.e., if $info = n + i$ , for $1 \leq i \leq n$ , then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

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.

8

Parallelism and Performance

f08tbf (dspgvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08tbf (dspgvx) 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 routine. 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 routine is f08tpf (zhpgvx).

10

Example

This example finds the eigenvalues in the half-open interval

(- 1.0, 1.0]

, and corresponding eigenvectors, of the generalized symmetric eigenproblem

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}) .

The example program for f08tcf (dspgvd) illustrates solving a generalized symmetric eigenproblem of the form

A B z = λ z

NAG Library Routine Document

f08tbf (dspgvx)

▸▿ 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