*> \brief \b SORBDB6
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sorbdb6.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
* LDQ2, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INCX1, INCX2, INFO, LDQ1, LDQ2, LWORK, M1, M2,
* $ N
* ..
* .. Array Arguments ..
* REAL Q1(LDQ1,*), Q2(LDQ2,*), WORK(*), X1(*), X2(*)
* ..
*
*
*> \par Purpose:
* =============
*>
*>\verbatim
*>
*> SORBDB6 orthogonalizes the column vector
*> X = [ X1 ]
*> [ X2 ]
*> with respect to the columns of
*> Q = [ Q1 ] .
*> [ Q2 ]
*> The columns of Q must be orthonormal.
*>
*> If the projection is zero according to Kahan's "twice is enough"
*> criterion, then the zero vector is returned.
*>
*>\endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M1
*> \verbatim
*> M1 is INTEGER
*> The dimension of X1 and the number of rows in Q1. 0 <= M1.
*> \endverbatim
*>
*> \param[in] M2
*> \verbatim
*> M2 is INTEGER
*> The dimension of X2 and the number of rows in Q2. 0 <= M2.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in Q1 and Q2. 0 <= N.
*> \endverbatim
*>
*> \param[in,out] X1
*> \verbatim
*> X1 is REAL array, dimension (M1)
*> On entry, the top part of the vector to be orthogonalized.
*> On exit, the top part of the projected vector.
*> \endverbatim
*>
*> \param[in] INCX1
*> \verbatim
*> INCX1 is INTEGER
*> Increment for entries of X1.
*> \endverbatim
*>
*> \param[in,out] X2
*> \verbatim
*> X2 is REAL array, dimension (M2)
*> On entry, the bottom part of the vector to be
*> orthogonalized. On exit, the bottom part of the projected
*> vector.
*> \endverbatim
*>
*> \param[in] INCX2
*> \verbatim
*> INCX2 is INTEGER
*> Increment for entries of X2.
*> \endverbatim
*>
*> \param[in] Q1
*> \verbatim
*> Q1 is REAL array, dimension (LDQ1, N)
*> The top part of the orthonormal basis matrix.
*> \endverbatim
*>
*> \param[in] LDQ1
*> \verbatim
*> LDQ1 is INTEGER
*> The leading dimension of Q1. LDQ1 >= M1.
*> \endverbatim
*>
*> \param[in] Q2
*> \verbatim
*> Q2 is REAL array, dimension (LDQ2, N)
*> The bottom part of the orthonormal basis matrix.
*> \endverbatim
*>
*> \param[in] LDQ2
*> \verbatim
*> LDQ2 is INTEGER
*> The leading dimension of Q2. LDQ2 >= M2.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 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.
*
*> \ingroup realOTHERcomputational
*
* =====================================================================
SUBROUTINE SORBDB6( M1, M2, N, X1, INCX1, X2, INCX2, Q1, LDQ1, Q2,
$ LDQ2, WORK, LWORK, 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 ..
INTEGER INCX1, INCX2, INFO, LDQ1, LDQ2, LWORK, M1, M2,
$ N
* ..
* .. Array Arguments ..
REAL Q1(LDQ1,*), Q2(LDQ2,*), WORK(*), X1(*), X2(*)
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ALPHASQ, REALONE, REALZERO
PARAMETER ( ALPHASQ = 0.01E0, REALONE = 1.0E0,
$ REALZERO = 0.0E0 )
REAL NEGONE, ONE, ZERO
PARAMETER ( NEGONE = -1.0E0, ONE = 1.0E0, ZERO = 0.0E0 )
* ..
* .. Local Scalars ..
INTEGER I
REAL NORMSQ1, NORMSQ2, SCL1, SCL2, SSQ1, SSQ2
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, SLASSQ, XERBLA
* ..
* .. Intrinsic Function ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
IF( M1 .LT. 0 ) THEN
INFO = -1
ELSE IF( M2 .LT. 0 ) THEN
INFO = -2
ELSE IF( N .LT. 0 ) THEN
INFO = -3
ELSE IF( INCX1 .LT. 1 ) THEN
INFO = -5
ELSE IF( INCX2 .LT. 1 ) THEN
INFO = -7
ELSE IF( LDQ1 .LT. MAX( 1, M1 ) ) THEN
INFO = -9
ELSE IF( LDQ2 .LT. MAX( 1, M2 ) ) THEN
INFO = -11
ELSE IF( LWORK .LT. N ) THEN
INFO = -13
END IF
*
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'SORBDB6', -INFO )
RETURN
END IF
*
* First, project X onto the orthogonal complement of Q's column
* space
*
SCL1 = REALZERO
SSQ1 = REALONE
CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
SCL2 = REALZERO
SSQ2 = REALONE
CALL SLASSQ( M2, X2, INCX2, SCL2, SSQ2 )
NORMSQ1 = SCL1**2*SSQ1 + SCL2**2*SSQ2
*
IF( M1 .EQ. 0 ) THEN
DO I = 1, N
WORK(I) = ZERO
END DO
ELSE
CALL SGEMV( 'C', M1, N, ONE, Q1, LDQ1, X1, INCX1, ZERO, WORK,
$ 1 )
END IF
*
CALL SGEMV( 'C', M2, N, ONE, Q2, LDQ2, X2, INCX2, ONE, WORK, 1 )
*
CALL SGEMV( 'N', M1, N, NEGONE, Q1, LDQ1, WORK, 1, ONE, X1,
$ INCX1 )
CALL SGEMV( 'N', M2, N, NEGONE, Q2, LDQ2, WORK, 1, ONE, X2,
$ INCX2 )
*
SCL1 = REALZERO
SSQ1 = REALONE
CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
SCL2 = REALZERO
SSQ2 = REALONE
CALL SLASSQ( M2, X2, INCX2, SCL2, SSQ2 )
NORMSQ2 = SCL1**2*SSQ1 + SCL2**2*SSQ2
*
* If projection is sufficiently large in norm, then stop.
* If projection is zero, then stop.
* Otherwise, project again.
*
IF( NORMSQ2 .GE. ALPHASQ*NORMSQ1 ) THEN
RETURN
END IF
*
IF( NORMSQ2 .EQ. ZERO ) THEN
RETURN
END IF
*
NORMSQ1 = NORMSQ2
*
DO I = 1, N
WORK(I) = ZERO
END DO
*
IF( M1 .EQ. 0 ) THEN
DO I = 1, N
WORK(I) = ZERO
END DO
ELSE
CALL SGEMV( 'C', M1, N, ONE, Q1, LDQ1, X1, INCX1, ZERO, WORK,
$ 1 )
END IF
*
CALL SGEMV( 'C', M2, N, ONE, Q2, LDQ2, X2, INCX2, ONE, WORK, 1 )
*
CALL SGEMV( 'N', M1, N, NEGONE, Q1, LDQ1, WORK, 1, ONE, X1,
$ INCX1 )
CALL SGEMV( 'N', M2, N, NEGONE, Q2, LDQ2, WORK, 1, ONE, X2,
$ INCX2 )
*
SCL1 = REALZERO
SSQ1 = REALONE
CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
SCL2 = REALZERO
SSQ2 = REALONE
CALL SLASSQ( M1, X1, INCX1, SCL1, SSQ1 )
NORMSQ2 = SCL1**2*SSQ1 + SCL2**2*SSQ2
*
* If second projection is sufficiently large in norm, then do
* nothing more. Alternatively, if it shrunk significantly, then
* truncate it to zero.
*
IF( NORMSQ2 .LT. ALPHASQ*NORMSQ1 ) THEN
DO I = 1, M1
X1(I) = ZERO
END DO
DO I = 1, M2
X2(I) = ZERO
END DO
END IF
*
RETURN
*
* End of SORBDB6
*
END