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*> \brief \b DLARTGS generates a plane rotation designed to introduce a bulge in implicit QR iteration for the bidiagonal SVD problem.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARTGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlartgs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlartgs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlartgs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION CS, SIGMA, SN, X, Y
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARTGS generates a plane rotation designed to introduce a bulge in
*> Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
*> problem. X and Y are the top-row entries, and SIGMA is the shift.
*> The computed CS and SN define a plane rotation satisfying
*>
*> [ CS SN ] . [ X^2 - SIGMA ] = [ R ],
*> [ -SN CS ] [ X * Y ] [ 0 ]
*>
*> with R nonnegative. If X^2 - SIGMA and X * Y are 0, then the
*> rotation is by PI/2.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION
*> The (1,1) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] Y
*> \verbatim
*> Y is DOUBLE PRECISION
*> The (1,2) entry of an upper bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> The shift.
*> \endverbatim
*>
*> \param[out] CS
*> \verbatim
*> CS is DOUBLE PRECISION
*> The cosine of the rotation.
*> \endverbatim
*>
*> \param[out] SN
*> \verbatim
*> SN is DOUBLE PRECISION
*> The sine of the rotation.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2017
*
*> \ingroup auxOTHERcomputational
*
* =====================================================================
SUBROUTINE DLARTGS( X, Y, SIGMA, CS, SN )
*
* -- LAPACK computational routine (version 3.8.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2017
*
* .. Scalar Arguments ..
DOUBLE PRECISION CS, SIGMA, SN, X, Y
* ..
*
* ===================================================================
*
* .. Parameters ..
DOUBLE PRECISION NEGONE, ONE, ZERO
PARAMETER ( NEGONE = -1.0D0, ONE = 1.0D0, ZERO = 0.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION R, S, THRESH, W, Z
* ..
* .. External Subroutines ..
EXTERNAL DLARTGP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* .. Executable Statements ..
*
THRESH = DLAMCH('E')
*
* Compute the first column of B**T*B - SIGMA^2*I, up to a scale
* factor.
*
IF( (SIGMA .EQ. ZERO .AND. ABS(X) .LT. THRESH) .OR.
$ (ABS(X) .EQ. SIGMA .AND. Y .EQ. ZERO) ) THEN
Z = ZERO
W = ZERO
ELSE IF( SIGMA .EQ. ZERO ) THEN
IF( X .GE. ZERO ) THEN
Z = X
W = Y
ELSE
Z = -X
W = -Y
END IF
ELSE IF( ABS(X) .LT. THRESH ) THEN
Z = -SIGMA*SIGMA
W = ZERO
ELSE
IF( X .GE. ZERO ) THEN
S = ONE
ELSE
S = NEGONE
END IF
Z = S * (ABS(X)-SIGMA) * (S+SIGMA/X)
W = S * Y
END IF
*
* Generate the rotation.
* CALL DLARTGP( Z, W, CS, SN, R ) might seem more natural;
* reordering the arguments ensures that if Z = 0 then the rotation
* is by PI/2.
*
CALL DLARTGP( W, Z, SN, CS, R )
*
RETURN
*
* End DLARTGS
*
END
|
