F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentF08SSF (ZHEGST)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

1  Purpose

F08SSF (ZHEGST) reduces a complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, where $A$ is a complex Hermitian matrix and $B$ has been factorized by F07FRF (ZPOTRF).

2  Specification

 SUBROUTINE F08SSF ( ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
 INTEGER ITYPE, N, LDA, LDB, INFO COMPLEX (KIND=nag_wp) A(LDA,*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zhegst.

3  Description

To reduce the complex Hermitian-definite generalized eigenproblem $Az=\lambda Bz$, $ABz=\lambda z$ or $BAz=\lambda z$ to the standard form $Cy=\lambda y$, F08SSF (ZHEGST) must be preceded by a call to F07FRF (ZPOTRF) which computes the Cholesky factorization of $B$; $B$ must be positive definite.
The different problem types are specified by the parameter ITYPE, as indicated in the table below. The table shows how $C$ is computed by the routine, and also how the eigenvectors $z$ of the original problem can be recovered from the eigenvectors of the standard form.
 ITYPE Problem UPLO $B$ $C$ $z$ $1$ $Az=\lambda Bz$ 'U' 'L' ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ ${U}^{-\mathrm{H}}A{U}^{-1}$  ${L}^{-1}A{L}^{-\mathrm{H}}$ ${U}^{-1}y$  ${L}^{-\mathrm{H}}y$ $2$ $ABz=\lambda z$ 'U' 'L' ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ $UA{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}AL$ ${U}^{-1}y$  ${L}^{-\mathrm{H}}y$ $3$ $BAz=\lambda z$ 'U' 'L' ${U}^{\mathrm{H}}U$  $L{L}^{\mathrm{H}}$ $UA{U}^{\mathrm{H}}$  ${L}^{\mathrm{H}}AL$ ${U}^{\mathrm{H}}y$  $Ly$

4  References

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

5  Parameters

1:     ITYPE – INTEGERInput
On entry: indicates how the standard form is computed.
${\mathbf{ITYPE}}=1$
• if ${\mathbf{UPLO}}=\text{'U'}$, $C={U}^{-\mathrm{H}}A{U}^{-1}$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $C={L}^{-1}A{L}^{-\mathrm{H}}$.
${\mathbf{ITYPE}}=2$ or $3$
• if ${\mathbf{UPLO}}=\text{'U'}$, $C=UA{U}^{\mathrm{H}}$;
• if ${\mathbf{UPLO}}=\text{'L'}$, $C={L}^{\mathrm{H}}AL$.
Constraint: ${\mathbf{ITYPE}}=1$, $2$ or $3$.
2:     UPLO – CHARACTER(1)Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored and how $B$ has been factorized.
${\mathbf{UPLO}}=\text{'U'}$
The upper triangular part of $A$ is stored and $B={U}^{\mathrm{H}}U$.
${\mathbf{UPLO}}=\text{'L'}$
The lower triangular part of $A$ is stored and $B=L{L}^{\mathrm{H}}$.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     A(LDA,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ Hermitian matrix $A$.
• If ${\mathbf{UPLO}}=\text{'U'}$, the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of A is overwritten by the corresponding upper or lower triangle of $C$ as specified by ITYPE and UPLO.
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08SSF (ZHEGST) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
6:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the Cholesky factor of $B$ as specified by UPLO and returned by F07FRF (ZPOTRF).
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08SSF (ZHEGST) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

7  Accuracy

Forming the reduced matrix $C$ is a stable procedure. However it involves implicit multiplication by ${B}^{-1}$ (if ${\mathbf{ITYPE}}=1$) or $B$ (if ${\mathbf{ITYPE}}=2$ or $3$). When F08SSF (ZHEGST) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if $B$ is ill-conditioned with respect to inversion. See the document for F08SNF (ZHEGV) for further details.

The total number of real floating point operations is approximately $4{n}^{3}$.
The real analogue of this routine is F08SEF (DSYGST).

9  Example

This example computes all the eigenvalues of $Az=\lambda Bz$, where
 $A = -7.36+0.00i 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49+0.00i 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12+0.00i 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54+0.00i$
and
 $B = 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .$
Here $B$ is Hermitian positive definite and must first be factorized by F07FRF (ZPOTRF). The program calls F08SSF (ZHEGST) to reduce the problem to the standard form $Cy=\lambda y$; then F08FSF (ZHETRD) to reduce $C$ to tridiagonal form, and F08JFF (DSTERF) to compute the eigenvalues.

9.1  Program Text

Program Text (f08ssfe.f90)

9.2  Program Data

Program Data (f08ssfe.d)

9.3  Program Results

Program Results (f08ssfe.r)