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

#include <nag.h>

void	f08tbc (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_RangeType range, Nag_UploType uplo, Integer n, double ap[], double bp[], double vl, double vu, Integer il, Integer iu, double abstol, Integer m, double w[], double z[], Integer pdz, Integer jfail[], NagError fail)

The function may be called by the names: f08tbc, nag_lapackeig_dspgvx or nag_dspgvx.

3 Description

f08tbc 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 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 https://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: $order$ – 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: $itype$ – Integer Input

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_JobType Input

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: $range$ – Nag_RangeType Input

On entry: if

range = Nag_AllValues

, all eigenvalues will be found.

range = Nag_Interval

, all eigenvalues in the half-open interval

(vl, vu]

will be found.

range = Nag_Indices

, the ilth to iuth eigenvalues will be found.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

5: $uplo$ – Nag_UploType Input

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

6: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

7: $ap [\dim]$ – double Input/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 \times 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.

8: $bp [\dim]$ – double Input/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 \times 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

9: $vl$ – double Input

10: $vu$ – double Input

On entry: if

range = Nag_Interval

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

range = Nag_AllValues

Nag_Indices

, vl and vu are not referenced.

Constraint: if

range = Nag_Interval

vl < vu

11: $il$ – Integer Input

12: $iu$ – Integer Input

On entry: if

range = Nag_Indices

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

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

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

13: $abstol$ – double 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 nag_real_safe_small_number ()

, not zero. If this function returns with

fail . code =

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

2 \times nag_real_safe_small_number ()

. See Demmel and Kahan (1990).

14: $m$ – Integer * Output

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range = Nag_AllValues

m = n

range = Nag_Indices

m = iu - il + 1

15: $w [n]$ – double Output

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

16: $z [\dim]$ – double Output

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

, then

if $fail . code =$ NE_NOERROR, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$ th column of $Z$ holding the eigenvector associated with $w [i - 1]$ . 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 ( $fail . code =$ NE_CONVERGENCE), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

job = Nag_EigVals

, z is not referenced.

17: $pdz$ – Integer Input

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 $pdz \geq 1$ .

18: $jfail [\dim]$ – Integer Output

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

\max (1, n)

On exit: if

job = Nag_DoBoth

, then

if $fail . code =$ NE_NOERROR, the first m elements of jfail are zero;
if $fail . code =$ NE_CONVERGENCE, the first $fail . errnum$ elements of jfail contains the indices of the eigenvectors that failed to converge.

job = Nag_EigVals

, jfail is not referenced.

19: $fail$ – 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_CONVERGENCE: The algorithm failed to converge; $⟨ value ⟩$ eigenvectors did not converge. Their indices are stored in array jfail.
NE_ENUM_INT_2: On entry, $job = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoBoth$ , $pdz \geq \max (1, n)$ ;
otherwise $pdz \geq 1$ .
NE_ENUM_INT_3: On entry, $range = ⟨ value ⟩$ , $il = ⟨ value ⟩$ , $iu = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .
NE_ENUM_REAL_2: On entry, $range = ⟨ value ⟩$ , $vl = ⟨ value ⟩$ and $vu = ⟨ value ⟩$ .
Constraint: if $range = Nag_Interval$ , $vl < vu$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08tbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08tbc 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 f08tpc.

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 f08tcc illustrates solving a generalized symmetric eigenproblem of the form

A B z = λ z

f08tb: FL CL CPP AD PY MB

NAG CL Interfacef08tbc (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

NAG CL Interface
f08tbc (dspgvx)