NAME

ZHBGST - reduce a complex Hermitian-definite banded generalized eigenproblem A∗x = lambda∗B∗x to standard form C∗y = lambda∗y,

SYNOPSIS

SUBROUTINE ZHBGST(
VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, RWORK, INFO )

CHARACTER UPLO, VECT

INTEGER INFO, KA, KB, LDAB, LDBB, LDX, N

DOUBLE PRECISION RWORK( ∗ )

COMPLEX∗16 AB( LDAB, ∗ ), BB( LDBB, ∗ ), WORK( ∗ ), X( LDX, ∗ )

PURPOSE

ZHBGST reduces a complex Hermitian-definite banded generalized eigenproblem  A∗x = lambda∗B∗x  to standard form  C∗y = lambda∗y, such that C has the same bandwidth as A. 
 
B must have been previously factorized as S∗∗H∗S by ZPBSTF, using a split Cholesky factorization. A is overwritten by C = X∗∗H∗A∗X, where X = S∗∗(-1)∗Q and Q is a unitary matrix chosen to preserve the bandwidth of A.
 

ARGUMENTS

VECT    (input) CHARACTER∗1
= ’N’:  do not form the transformation matrix X;
= ’V’:  form X.

UPLO    (input) CHARACTER∗1
= ’U’:  Upper triangle of A is stored;
= ’L’:  Lower triangle of A is stored.

N       (input) INTEGER
The order of the matrices A and B.  N >= 0.

KA      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’, or the number of subdiagonals if UPLO = ’L’.  KA >= 0.

KB      (input) INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’, or the number of subdiagonals if UPLO = ’L’.  KA >= KB >= 0.

AB      (input/output) COMPLEX∗16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the Hermitian band matrix A, stored in the first ka+1 rows of the array.  The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = ’U’, AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = ’L’, AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
 
On exit, the transformed matrix X∗∗H∗A∗X, stored in the same format as A.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KA+1.

BB      (input) COMPLEX∗16 array, dimension (LDBB,N)
The banded factor S from the split Cholesky factorization of B, as returned by ZPBSTF, stored in the first kb+1 rows of the array.

LDBB    (input) INTEGER
The leading dimension of the array BB.  LDBB >= KB+1.

X       (output) COMPLEX∗16 array, dimension (LDX,N)
If VECT = ’V’, the n-by-n matrix X. If VECT = ’N’, the array X is not referenced.

LDX     (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N) if VECT = ’V’; LDX >= 1 otherwise.

WORK    (workspace) COMPLEX∗16 array, dimension (N)

RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.

  —  LAPACK version 2.0  —  08 October 1994