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*> \brief \b DSYGS2 reduces a symmetric definite generalized eigenproblem to standard form, using the factorization results obtained from spotrf (unblocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DSYGS2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsygs2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsygs2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsygs2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSYGS2 reduces a real symmetric-definite generalized eigenproblem
*> to standard form.
*>
*> 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 DPOTRF.
*> \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
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored, and how B has been factorized.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n by n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n by n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*>
*> On exit, if INFO = 0, the transformed matrix, stored in the
*> same format as A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB,N)
*> The triangular factor from the Cholesky factorization of B,
*> as returned by DPOTRF.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \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.
*
*> \date December 2016
*
*> \ingroup doubleSYcomputational
*
* =====================================================================
SUBROUTINE DSYGS2( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, ITYPE, LDA, LDB, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, HALF
PARAMETER ( ONE = 1.0D0, HALF = 0.5D0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER K
DOUBLE PRECISION AKK, BKK, CT
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DSCAL, DSYR2, DTRMV, DTRSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. 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
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYGS2', -INFO )
RETURN
END IF
*
IF( ITYPE.EQ.1 ) THEN
IF( UPPER ) THEN
*
* Compute inv(U**T)*A*inv(U)
*
DO 10 K = 1, N
*
* Update the upper triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K, K+1 ), LDA )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DSYR2( UPLO, N-K, -ONE, A( K, K+1 ), LDA,
$ B( K, K+1 ), LDB, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K, K+1 ), LDB, A( K, K+1 ),
$ LDA )
CALL DTRSV( UPLO, 'Transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K, K+1 ), LDA )
END IF
10 CONTINUE
ELSE
*
* Compute inv(L)*A*inv(L**T)
*
DO 20 K = 1, N
*
* Update the lower triangle of A(k:n,k:n)
*
AKK = A( K, K )
BKK = B( K, K )
AKK = AKK / BKK**2
A( K, K ) = AKK
IF( K.LT.N ) THEN
CALL DSCAL( N-K, ONE / BKK, A( K+1, K ), 1 )
CT = -HALF*AKK
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
CALL DSYR2( UPLO, N-K, -ONE, A( K+1, K ), 1,
$ B( K+1, K ), 1, A( K+1, K+1 ), LDA )
CALL DAXPY( N-K, CT, B( K+1, K ), 1, A( K+1, K ), 1 )
CALL DTRSV( UPLO, 'No transpose', 'Non-unit', N-K,
$ B( K+1, K+1 ), LDB, A( K+1, K ), 1 )
END IF
20 CONTINUE
END IF
ELSE
IF( UPPER ) THEN
*
* Compute U*A*U**T
*
DO 30 K = 1, N
*
* Update the upper triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'No transpose', 'Non-unit', K-1, B,
$ LDB, A( 1, K ), 1 )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSYR2( UPLO, K-1, ONE, A( 1, K ), 1, B( 1, K ), 1,
$ A, LDA )
CALL DAXPY( K-1, CT, B( 1, K ), 1, A( 1, K ), 1 )
CALL DSCAL( K-1, BKK, A( 1, K ), 1 )
A( K, K ) = AKK*BKK**2
30 CONTINUE
ELSE
*
* Compute L**T *A*L
*
DO 40 K = 1, N
*
* Update the lower triangle of A(1:k,1:k)
*
AKK = A( K, K )
BKK = B( K, K )
CALL DTRMV( UPLO, 'Transpose', 'Non-unit', K-1, B, LDB,
$ A( K, 1 ), LDA )
CT = HALF*AKK
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSYR2( UPLO, K-1, ONE, A( K, 1 ), LDA, B( K, 1 ),
$ LDB, A, LDA )
CALL DAXPY( K-1, CT, B( K, 1 ), LDB, A( K, 1 ), LDA )
CALL DSCAL( K-1, BKK, A( K, 1 ), LDA )
A( K, K ) = AKK*BKK**2
40 CONTINUE
END IF
END IF
RETURN
*
* End of DSYGS2
*
END
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