*> \brief \b DSPGST
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*
* Definition:
* ===========
*
* SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AP( * ), BP( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSPGST reduces a real symmetric-definite generalized eigenproblem
*> to standard form, using packed storage.
*>
*> If ITYPE = 1, the problem is A*x = lambda*B*x,
*> and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*>
*> If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*> B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*>
*> B must have been previously factorized as U**T*U or L*L**T by DPPTRF.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] ITYPE
*> \verbatim
*> ITYPE is INTEGER
*> = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*> = 2 or 3: compute U*A*U**T or L**T*A*L.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored and B is factored as
*> U**T*U;
*> = 'L': Lower triangle of A is stored and B is factored as
*> L*L**T.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the upper or lower triangle of the symmetric matrix
*> A, packed columnwise in a linear array. The j-th column of A
*> is stored in the array AP as follows:
*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] BP
*> \verbatim
*> BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The triangular factor from the Cholesky factorization of B,
*> stored in the same format as A, as returned by DPPTRF.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpgst
*
* =====================================================================
SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AP( * ), BP( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, J1, J1J1, JJ, K, K1, K1K1, KK
DOUBLE PRECISION AJJ, AKK, BJJ, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSPMV, DSPR2, DTPMV,
$ DTPSV,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
INFO = -1
ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSPGST', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
* J1 and JJ are the indices of A(1,j) and A(j,j)
*
JJ = 0
DO 10 J = 1, N
J1 = JJ + 1
JJ = JJ + J
*
* Compute the j-th column of the upper triangle of A
*
BJJ = BP( JJ )
CALL DTPSV( UPLO, 'Transpose', 'Nonunit', J, BP,
$ AP( J1 ), 1 )
CALL DSPMV( UPLO, J-1, -ONE, AP, BP( J1 ), 1, ONE,
$ AP( J1 ), 1 )
CALL DSCAL( J-1, ONE / BJJ, AP( J1 ), 1 )
AP( JJ ) = ( AP( JJ )-DDOT( J-1, AP( J1 ), 1,
$ BP( J1 ),
$ 1 ) ) / BJJ
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
* KK and K1K1 are the indices of A(k,k) and A(k+1,k+1)
*
KK = 1
DO 20 K = 1, N
K1K1 = KK + N - K + 1
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = AP( KK )
BKK = BP( KK )
AKK = AKK / BKK**2
AP( KK ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, AP( KK+1 ), 1 )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL DSPR2( UPLO, N-K, -ONE, AP( KK+1 ), 1,
$ BP( KK+1 ), 1, AP( K1K1 ) )
CALL DAXPY( N-K, CT, BP( KK+1 ), 1, AP( KK+1 ), 1 )
CALL DTPSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ BP( K1K1 ), AP( KK+1 ), 1 )
END IF
KK = K1K1
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
* K1 and KK are the indices of A(1,k) and A(k,k)
*
KK = 0
DO 30 K = 1, N
K1 = KK + 1
KK = KK + K
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = AP( KK )
BKK = BP( KK )
CALL DTPMV( UPLO, 'No transpose', 'Non-unit', K-1, BP,
$ AP( K1 ), 1 )
CT = HALF*AKK
CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL DSPR2( UPLO, K-1, ONE, AP( K1 ), 1, BP( K1 ), 1,
$ AP )
CALL DAXPY( K-1, CT, BP( K1 ), 1, AP( K1 ), 1 )
CALL DSCAL( K-1, BKK, AP( K1 ), 1 )
AP( KK ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
* JJ and J1J1 are the indices of A(j,j) and A(j+1,j+1)
*
JJ = 1
DO 40 J = 1, N
J1J1 = JJ + N - J + 1
*
* Compute the j-th column of the lower triangle of A
*
AJJ = AP( JJ )
BJJ = BP( JJ )
AP( JJ ) = AJJ*BJJ + DDOT( N-J, AP( JJ+1 ), 1,
$ BP( JJ+1 ), 1 )
CALL DSCAL( N-J, BJJ, AP( JJ+1 ), 1 )
CALL DSPMV( UPLO, N-J, ONE, AP( J1J1 ), BP( JJ+1 ), 1,
$ ONE, AP( JJ+1 ), 1 )
CALL DTPMV( UPLO, 'Transpose', 'Non-unit', N-J+1,
$ BP( JJ ), AP( JJ ), 1 )
JJ = J1J1
40 CONTINUE
END IF
END IF
RETURN
*
* End of DSPGST
*
END