f08unc computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form

A z = λ B z,

where

A

and

B

are Hermitian and banded, and

B

is also positive definite.

2 Specification

#include <nag.h>

void	f08unc (Nag_OrderType order, Nag_JobType job, Nag_UploType uplo, Integer n, Integer ka, Integer kb, Complex ab[], Integer pdab, Complex bb[], Integer pdbb, double w[], Complex z[], Integer pdz, NagError *fail)

The function may be called by the names: f08unc, nag_lapackeig_zhbgv or nag_zhbgv.

3 Description

The generalized Hermitian-definite band problem

A z = λ B z

is first reduced to a standard band Hermitian problem

C x = λ x,

where

C

is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.

The eigenvectors are normalized so that the matrix of eigenvectors,

Z

, satisfies

Z^{H} A Z = Λ and Z^{H} B Z = I,

where

Λ

is the diagonal matrix whose diagonal elements are the eigenvalues.

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

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44

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

Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

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: $job$ – 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

3: $uplo$ – 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

4: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

5: $ka$ – Integer Input

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{a}

, of the matrix

A

uplo = Nag_Lower

, the number of subdiagonals,

k_{a}

, of the matrix

A

Constraint:

ka \geq 0

6: $kb$ – Integer Input

On entry: if

uplo = Nag_Upper

, the number of superdiagonals,

k_{b}

, of the matrix

B

uplo = Nag_Lower

, the number of subdiagonals,

k_{b}

, of the matrix

B

Constraint:

ka \geq kb \geq 0

7: $ab [\dim]$ – Complex Input/Output

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

\max (1, pdab \times n)

On entry: the upper or lower triangle of the

n \times n

Hermitian band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

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 $ab [k_{a} + i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{a}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ab [i - j + (j - 1) \times pdab]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{a})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $A_{i j}$ is stored in $ab [j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{a})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $A_{i j}$ is stored in $ab [k_{a} + j - i + (i - 1) \times pdab]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{a}), \dots, i$ .

On exit: the contents of ab are overwritten.

8: $pdab$ – Integer Input

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

A

in the array ab.

Constraint:

pdab \geq ka + 1

9: $bb [\dim]$ – Complex Input/Output

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

\max (1, pdbb \times n)

On entry: the upper or lower triangle of the

n \times n

Hermitian band matrix

B

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of

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 $bb [k_{b} + i - j + (j - 1) \times pdbb]$ , for $j = 1, \dots, n$ and $i = \max (1, j - k_{b}), \dots, j$ ;
if $order = Nag_ColMajor$ and $uplo = Nag_Lower$ ,: $B_{i j}$ is stored in $bb [i - j + (j - 1) \times pdbb]$ , for $j = 1, \dots, n$ and $i = j, \dots, \min (n, j + k_{b})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Upper$ ,: $B_{i j}$ is stored in $bb [j - i + (i - 1) \times pdbb]$ , for $i = 1, \dots, n$ and $j = i, \dots, \min (n, i + k_{b})$ ;
if $order = Nag_RowMajor$ and $uplo = Nag_Lower$ ,: $B_{i j}$ is stored in $bb [k_{b} + j - i + (i - 1) \times pdbb]$ , for $i = 1, \dots, n$ and $j = \max (1, i - k_{b}), \dots, i$ .

On exit: the factor

S

from the split Cholesky factorization

B = S^{H} S

, as returned by f08utc.

10: $pdbb$ – Integer Input

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

B

in the array bb.

Constraint:

pdbb \geq kb + 1

11: $w [n]$ – double Output

On exit: the eigenvalues in ascending order.

12: $z [\dim]$ – Complex 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

, z contains the matrix

Z

of eigenvectors, with the

i

th column of

Z

holding the eigenvector associated with

w [i - 1]

. The eigenvectors are normalized so that

Z^{H} B Z = I

job = Nag_EigVals

, z is not referenced.

13: $pdz$ – 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 $pdz \geq 1$ .

14: $fail$ – 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; $⟨ value ⟩$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
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_INT: On entry, $ka = ⟨ value ⟩$ .
Constraint: $ka \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

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

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

On entry, $pdz = ⟨ value ⟩$ .
Constraint: $pdz > 0$ .
NE_INT_2: On entry, $ka = ⟨ value ⟩$ and $kb = ⟨ value ⟩$ .
Constraint: $ka \geq kb \geq 0$ .

On entry, $pdab = ⟨ value ⟩$ and $ka = ⟨ value ⟩$ .
Constraint: $pdab \geq ka + 1$ .

On entry, $pdbb = ⟨ value ⟩$ and $kb = ⟨ value ⟩$ .
Constraint: $pdbb \geq kb + 1$ .
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 $1 \leq ⟨ value ⟩ \leq n$ , f08utc returned $fail . errnum = ⟨ value ⟩$ : $B$ 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

f08unc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08unc 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}

job = Nag_DoBoth

and, assuming that

n ≫ k_{a}

, is approximately proportional to

n^{2} k_{a}

otherwise.

The real analogue of this function is f08uac.

10 Example

This example finds all the eigenvalues of the generalized band Hermitian eigenproblem

A z = λ B z

, where

A = (\begin{array}{r} - 1.13 & 1.94 - 2.10 i & - 1.40 + 0.25 i & 0 \\ 1.94 + 2.10 i & - 1.91 & - 0.82 - 0.89 i & - 0.67 + 0.34 i \\ - 1.40 - 0.25 i & - 0.82 + 0.89 i & - 1.87 & - 1.10 - 0.16 i \\ 0 & - 0.67 - 0.34 i & - 1.10 + 0.16 i & 0.50 \end{array})

and

B = (\begin{array}{r} 9.89 & 1.08 - 1.73 i & 0 & 0 \\ 1.08 + 1.73 i & 1.69 & - 0.04 + 0.29 i & 0 \\ 0 & - 0.04 - 0.29 i & 2.65 & - 0.33 + 2.24 i \\ 0 & 0 & - 0.33 - 2.24 i & 2.17 \end{array}) .

f08un: FL CL CPP AD

NAG CL Interfacef08unc (zhbgv)

▸▿ 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
f08unc (zhbgv)