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365 lines
9.7 KiB
365 lines
9.7 KiB
2 years ago
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*> \brief \b SLAIC1 applies one step of incremental condition estimation.
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SLAIC1 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaic1.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaic1.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaic1.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
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*
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* .. Scalar Arguments ..
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* INTEGER J, JOB
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* REAL C, GAMMA, S, SEST, SESTPR
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* ..
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* .. Array Arguments ..
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* REAL W( J ), X( J )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SLAIC1 applies one step of incremental condition estimation in
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*> its simplest version:
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*>
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*> Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
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*> lower triangular matrix L, such that
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*> twonorm(L*x) = sest
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*> Then SLAIC1 computes sestpr, s, c such that
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*> the vector
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*> [ s*x ]
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*> xhat = [ c ]
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*> is an approximate singular vector of
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*> [ L 0 ]
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*> Lhat = [ w**T gamma ]
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*> in the sense that
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*> twonorm(Lhat*xhat) = sestpr.
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*>
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*> Depending on JOB, an estimate for the largest or smallest singular
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*> value is computed.
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*>
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*> Note that [s c]**T and sestpr**2 is an eigenpair of the system
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*>
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*> diag(sest*sest, 0) + [alpha gamma] * [ alpha ]
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*> [ gamma ]
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*>
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*> where alpha = x**T*w.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOB
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*> \verbatim
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*> JOB is INTEGER
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*> = 1: an estimate for the largest singular value is computed.
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*> = 2: an estimate for the smallest singular value is computed.
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*> \endverbatim
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*>
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*> \param[in] J
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*> \verbatim
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*> J is INTEGER
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*> Length of X and W
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*> \endverbatim
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*>
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*> \param[in] X
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*> \verbatim
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*> X is REAL array, dimension (J)
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*> The j-vector x.
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*> \endverbatim
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*>
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*> \param[in] SEST
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*> \verbatim
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*> SEST is REAL
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*> Estimated singular value of j by j matrix L
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*> \endverbatim
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*>
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*> \param[in] W
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*> \verbatim
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*> W is REAL array, dimension (J)
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*> The j-vector w.
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*> \endverbatim
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*>
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*> \param[in] GAMMA
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*> \verbatim
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*> GAMMA is REAL
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*> The diagonal element gamma.
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*> \endverbatim
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*>
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*> \param[out] SESTPR
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*> \verbatim
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*> SESTPR is REAL
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*> Estimated singular value of (j+1) by (j+1) matrix Lhat.
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*> \endverbatim
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*>
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*> \param[out] S
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*> \verbatim
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*> S is REAL
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*> Sine needed in forming xhat.
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*> \endverbatim
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*>
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*> \param[out] C
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*> \verbatim
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*> C is REAL
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*> Cosine needed in forming xhat.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup realOTHERauxiliary
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*
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* =====================================================================
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SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
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*
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* -- LAPACK auxiliary routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER J, JOB
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REAL C, GAMMA, S, SEST, SESTPR
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* ..
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* .. Array Arguments ..
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REAL W( J ), X( J )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE, TWO
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 )
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REAL HALF, FOUR
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PARAMETER ( HALF = 0.5E0, FOUR = 4.0E0 )
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* ..
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* .. Local Scalars ..
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REAL ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS,
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$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, SIGN, SQRT
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* ..
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* .. External Functions ..
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REAL SDOT, SLAMCH
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EXTERNAL SDOT, SLAMCH
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* ..
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* .. Executable Statements ..
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*
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EPS = SLAMCH( 'Epsilon' )
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ALPHA = SDOT( J, X, 1, W, 1 )
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*
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ABSALP = ABS( ALPHA )
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ABSGAM = ABS( GAMMA )
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ABSEST = ABS( SEST )
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*
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IF( JOB.EQ.1 ) THEN
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*
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* Estimating largest singular value
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*
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* special cases
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*
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IF( SEST.EQ.ZERO ) THEN
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S1 = MAX( ABSGAM, ABSALP )
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IF( S1.EQ.ZERO ) THEN
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S = ZERO
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C = ONE
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SESTPR = ZERO
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ELSE
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S = ALPHA / S1
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C = GAMMA / S1
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TMP = SQRT( S*S+C*C )
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S = S / TMP
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C = C / TMP
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SESTPR = S1*TMP
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END IF
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RETURN
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ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
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S = ONE
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C = ZERO
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TMP = MAX( ABSEST, ABSALP )
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S1 = ABSEST / TMP
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S2 = ABSALP / TMP
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SESTPR = TMP*SQRT( S1*S1+S2*S2 )
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RETURN
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ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
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S1 = ABSGAM
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S2 = ABSEST
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IF( S1.LE.S2 ) THEN
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S = ONE
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C = ZERO
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SESTPR = S2
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ELSE
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S = ZERO
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C = ONE
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SESTPR = S1
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END IF
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RETURN
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ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
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S1 = ABSGAM
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S2 = ABSALP
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IF( S1.LE.S2 ) THEN
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TMP = S1 / S2
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S = SQRT( ONE+TMP*TMP )
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SESTPR = S2*S
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C = ( GAMMA / S2 ) / S
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S = SIGN( ONE, ALPHA ) / S
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ELSE
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TMP = S2 / S1
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C = SQRT( ONE+TMP*TMP )
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SESTPR = S1*C
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S = ( ALPHA / S1 ) / C
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C = SIGN( ONE, GAMMA ) / C
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END IF
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RETURN
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ELSE
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*
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* normal case
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*
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ZETA1 = ALPHA / ABSEST
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ZETA2 = GAMMA / ABSEST
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*
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B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF
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C = ZETA1*ZETA1
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IF( B.GT.ZERO ) THEN
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T = C / ( B+SQRT( B*B+C ) )
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ELSE
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T = SQRT( B*B+C ) - B
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END IF
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*
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SINE = -ZETA1 / T
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COSINE = -ZETA2 / ( ONE+T )
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TMP = SQRT( SINE*SINE+COSINE*COSINE )
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S = SINE / TMP
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C = COSINE / TMP
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SESTPR = SQRT( T+ONE )*ABSEST
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RETURN
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END IF
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*
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ELSE IF( JOB.EQ.2 ) THEN
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*
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* Estimating smallest singular value
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*
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* special cases
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*
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IF( SEST.EQ.ZERO ) THEN
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SESTPR = ZERO
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IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN
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SINE = ONE
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COSINE = ZERO
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ELSE
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SINE = -GAMMA
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COSINE = ALPHA
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END IF
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S1 = MAX( ABS( SINE ), ABS( COSINE ) )
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S = SINE / S1
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C = COSINE / S1
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TMP = SQRT( S*S+C*C )
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S = S / TMP
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C = C / TMP
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RETURN
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ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
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S = ZERO
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C = ONE
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SESTPR = ABSGAM
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RETURN
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ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
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S1 = ABSGAM
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S2 = ABSEST
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IF( S1.LE.S2 ) THEN
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S = ZERO
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C = ONE
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SESTPR = S1
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ELSE
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S = ONE
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C = ZERO
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SESTPR = S2
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END IF
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RETURN
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ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
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S1 = ABSGAM
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S2 = ABSALP
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IF( S1.LE.S2 ) THEN
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TMP = S1 / S2
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C = SQRT( ONE+TMP*TMP )
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SESTPR = ABSEST*( TMP / C )
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S = -( GAMMA / S2 ) / C
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C = SIGN( ONE, ALPHA ) / C
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ELSE
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TMP = S2 / S1
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S = SQRT( ONE+TMP*TMP )
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SESTPR = ABSEST / S
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C = ( ALPHA / S1 ) / S
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S = -SIGN( ONE, GAMMA ) / S
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END IF
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RETURN
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ELSE
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*
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* normal case
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*
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ZETA1 = ALPHA / ABSEST
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ZETA2 = GAMMA / ABSEST
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*
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NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ),
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$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 )
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*
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* See if root is closer to zero or to ONE
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*
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TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 )
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IF( TEST.GE.ZERO ) THEN
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*
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* root is close to zero, compute directly
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*
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B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF
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C = ZETA2*ZETA2
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T = C / ( B+SQRT( ABS( B*B-C ) ) )
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SINE = ZETA1 / ( ONE-T )
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COSINE = -ZETA2 / T
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SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST
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ELSE
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*
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* root is closer to ONE, shift by that amount
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*
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B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF
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C = ZETA1*ZETA1
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IF( B.GE.ZERO ) THEN
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T = -C / ( B+SQRT( B*B+C ) )
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ELSE
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T = B - SQRT( B*B+C )
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END IF
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SINE = -ZETA1 / T
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COSINE = -ZETA2 / ( ONE+T )
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SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST
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END IF
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TMP = SQRT( SINE*SINE+COSINE*COSINE )
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S = SINE / TMP
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C = COSINE / TMP
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RETURN
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*
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END IF
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END IF
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RETURN
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*
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* End of SLAIC1
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*
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END
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