NAG Library Function Document

nag_zhegv (f08snc)

+− 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

nag_zhegv (f08snc) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form

A z = λ B z, A B z = λ z or B A z = λ z,

where

A

and

B

are Hermitian and

B

is also positive definite.

2 Specification

#include <nag.h>

#include <nagf08.h>

void	nag_zhegv (Nag_OrderType order, Integer itype, Nag_JobType job, Nag_UploType uplo, Integer n, Complex a[], Integer pda, Complex b[], Integer pdb, double w[], NagError *fail)

3 Description

nag_zhegv (f08snc) first performs a Cholesky factorization of the matrix

B

B = U^{H} U

, when

uplo = Nag_Upper

B = L L^{H}

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

and for

B A z = λ z

we have

Z^{H} A Z = Λ and Z^{H} 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 3.2.1.3 in the Essential Introduction 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: a[ $\dim$ ] – ComplexInput/Output

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

\max (1, pda \times n)

On entry: the

n

n

Hermitian 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: if

job = Nag_DoBoth

, a contains the matrix

Z

of eigenvectors. The eigenvectors are normalized as follows:

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

job = Nag_EigVals

, the upper triangle (if

uplo = Nag_Upper

) or the lower triangle (if

uplo = Nag_Lower

) of a, including the diagonal, is overwritten.

7: pda – IntegerInput

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

Constraint:

pda \geq \max (1, n)

8: b[ $\dim$ ] – ComplexInput/Output

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

\max (1, pdb \times n)

On entry: the

n

n

Hermitian positive definite matrix

B

order = Nag_ColMajor

B_{i j}

is stored in

b [(j - 1) \times pdb + i - 1]

order = Nag_RowMajor

B_{i j}

is stored in

b [(i - 1) \times pdb + j - 1]

uplo = Nag_Upper

, the upper triangular part of

B

must be stored and the elements of the array below the diagonal are not referenced.

uplo = Nag_Lower

, the lower triangular part of

B

must be stored and the elements of the array above the diagonal are not referenced.

On exit: if

fail . code =

NE_NOERROR or NE_CONVERGENCE, the part of b containing the matrix is overwritten by the triangular factor

U

L

from the Cholesky factorization

B = U^{H} U

B = L L^{H}

9: pdb – IntegerInput

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

Constraint:

pdb \geq \max (1, n)

10: w[n] – doubleOutput

On exit: the eigenvalues in ascending order.

11: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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_INT: On entry, $itype = ⟨value⟩$ .
Constraint: $itype = 1$ , $2$ or $3$ .
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.
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.

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_zhegv (f08snc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zhegv (f08snc) 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 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 real analogue of this function is nag_dsygv (f08sac).

10 Example

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

A z = λ B z

, where

A = (\begin{array}{r} - 7.36 & 0.77 - 0.43 i & - 0.64 - 0.92 i & 3.01 - 6.97 i \\ 0.77 + 0.43 i & 3.49 & 2.19 + 4.45 i & 1.90 + 3.73 i \\ - 0.64 + 0.92 i & 2.19 - 4.45 i & 0.12 & 2.88 - 3.17 i \\ 3.01 + 6.97 i & 1.90 - 3.73 i & 2.88 + 3.17 i & - 2.54 \end{array})

and

B = (\begin{array}{r} 3.23 & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 \end{array}),

together with and estimate of the condition number of

B

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

The example program for nag_zhegvd (f08sqc) illustrates solving a generalized Hermitian eigenproblem of the form

A B z = λ z

NAG Library Function Documentnag_zhegv (f08snc)

+− 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_zhegv (f08snc)