*> \brief \b DLAIC1 applies one step of incremental condition estimation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
* .. Scalar Arguments ..
* INTEGER J, JOB
* DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR
* ..
* .. Array Arguments ..
* DOUBLE PRECISION W( J ), X( J )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAIC1 applies one step of incremental condition estimation in
*> its simplest version:
*>
*> Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
*> lower triangular matrix L, such that
*> twonorm(L*x) = sest
*> Then DLAIC1 computes sestpr, s, c such that
*> the vector
*> [ s*x ]
*> xhat = [ c ]
*> is an approximate singular vector of
*> [ L 0 ]
*> Lhat = [ w**T gamma ]
*> in the sense that
*> twonorm(Lhat*xhat) = sestpr.
*>
*> Depending on JOB, an estimate for the largest or smallest singular
*> value is computed.
*>
*> Note that [s c]**T and sestpr**2 is an eigenpair of the system
*>
*> diag(sest*sest, 0) + [alpha gamma] * [ alpha ]
*> [ gamma ]
*>
*> where alpha = x**T*w.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is INTEGER
*> = 1: an estimate for the largest singular value is computed.
*> = 2: an estimate for the smallest singular value is computed.
*> \endverbatim
*>
*> \param[in] J
*> \verbatim
*> J is INTEGER
*> Length of X and W
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*> X is DOUBLE PRECISION array, dimension (J)
*> The j-vector x.
*> \endverbatim
*>
*> \param[in] SEST
*> \verbatim
*> SEST is DOUBLE PRECISION
*> Estimated singular value of j by j matrix L
*> \endverbatim
*>
*> \param[in] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (J)
*> The j-vector w.
*> \endverbatim
*>
*> \param[in] GAMMA
*> \verbatim
*> GAMMA is DOUBLE PRECISION
*> The diagonal element gamma.
*> \endverbatim
*>
*> \param[out] SESTPR
*> \verbatim
*> SESTPR is DOUBLE PRECISION
*> Estimated singular value of (j+1) by (j+1) matrix Lhat.
*> \endverbatim
*>
*> \param[out] S
*> \verbatim
*> S is DOUBLE PRECISION
*> Sine needed in forming xhat.
*> \endverbatim
*>
*> \param[out] C
*> \verbatim
*> C is DOUBLE PRECISION
*> Cosine needed in forming xhat.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERauxiliary
*
* =====================================================================
SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
*
* -- LAPACK auxiliary 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 J, JOB
DOUBLE PRECISION C, GAMMA, S, SEST, SESTPR
* ..
* .. Array Arguments ..
DOUBLE PRECISION W( J ), X( J )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
DOUBLE PRECISION HALF, FOUR
PARAMETER ( HALF = 0.5D0, FOUR = 4.0D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION ABSALP, ABSEST, ABSGAM, ALPHA, B, COSINE, EPS,
$ NORMA, S1, S2, SINE, T, TEST, TMP, ZETA1, ZETA2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN, SQRT
* ..
* .. External Functions ..
DOUBLE PRECISION DDOT, DLAMCH
EXTERNAL DDOT, DLAMCH
* ..
* .. Executable Statements ..
*
EPS = DLAMCH( 'Epsilon' )
ALPHA = DDOT( J, X, 1, W, 1 )
*
ABSALP = ABS( ALPHA )
ABSGAM = ABS( GAMMA )
ABSEST = ABS( SEST )
*
IF( JOB.EQ.1 ) THEN
*
* Estimating largest singular value
*
* special cases
*
IF( SEST.EQ.ZERO ) THEN
S1 = MAX( ABSGAM, ABSALP )
IF( S1.EQ.ZERO ) THEN
S = ZERO
C = ONE
SESTPR = ZERO
ELSE
S = ALPHA / S1
C = GAMMA / S1
TMP = SQRT( S*S+C*C )
S = S / TMP
C = C / TMP
SESTPR = S1*TMP
END IF
RETURN
ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
S = ONE
C = ZERO
TMP = MAX( ABSEST, ABSALP )
S1 = ABSEST / TMP
S2 = ABSALP / TMP
SESTPR = TMP*SQRT( S1*S1+S2*S2 )
RETURN
ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
S1 = ABSGAM
S2 = ABSEST
IF( S1.LE.S2 ) THEN
S = ONE
C = ZERO
SESTPR = S2
ELSE
S = ZERO
C = ONE
SESTPR = S1
END IF
RETURN
ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
S1 = ABSGAM
S2 = ABSALP
IF( S1.LE.S2 ) THEN
TMP = S1 / S2
S = SQRT( ONE+TMP*TMP )
SESTPR = S2*S
C = ( GAMMA / S2 ) / S
S = SIGN( ONE, ALPHA ) / S
ELSE
TMP = S2 / S1
C = SQRT( ONE+TMP*TMP )
SESTPR = S1*C
S = ( ALPHA / S1 ) / C
C = SIGN( ONE, GAMMA ) / C
END IF
RETURN
ELSE
*
* normal case
*
ZETA1 = ALPHA / ABSEST
ZETA2 = GAMMA / ABSEST
*
B = ( ONE-ZETA1*ZETA1-ZETA2*ZETA2 )*HALF
C = ZETA1*ZETA1
IF( B.GT.ZERO ) THEN
T = C / ( B+SQRT( B*B+C ) )
ELSE
T = SQRT( B*B+C ) - B
END IF
*
SINE = -ZETA1 / T
COSINE = -ZETA2 / ( ONE+T )
TMP = SQRT( SINE*SINE+COSINE*COSINE )
S = SINE / TMP
C = COSINE / TMP
SESTPR = SQRT( T+ONE )*ABSEST
RETURN
END IF
*
ELSE IF( JOB.EQ.2 ) THEN
*
* Estimating smallest singular value
*
* special cases
*
IF( SEST.EQ.ZERO ) THEN
SESTPR = ZERO
IF( MAX( ABSGAM, ABSALP ).EQ.ZERO ) THEN
SINE = ONE
COSINE = ZERO
ELSE
SINE = -GAMMA
COSINE = ALPHA
END IF
S1 = MAX( ABS( SINE ), ABS( COSINE ) )
S = SINE / S1
C = COSINE / S1
TMP = SQRT( S*S+C*C )
S = S / TMP
C = C / TMP
RETURN
ELSE IF( ABSGAM.LE.EPS*ABSEST ) THEN
S = ZERO
C = ONE
SESTPR = ABSGAM
RETURN
ELSE IF( ABSALP.LE.EPS*ABSEST ) THEN
S1 = ABSGAM
S2 = ABSEST
IF( S1.LE.S2 ) THEN
S = ZERO
C = ONE
SESTPR = S1
ELSE
S = ONE
C = ZERO
SESTPR = S2
END IF
RETURN
ELSE IF( ABSEST.LE.EPS*ABSALP .OR. ABSEST.LE.EPS*ABSGAM ) THEN
S1 = ABSGAM
S2 = ABSALP
IF( S1.LE.S2 ) THEN
TMP = S1 / S2
C = SQRT( ONE+TMP*TMP )
SESTPR = ABSEST*( TMP / C )
S = -( GAMMA / S2 ) / C
C = SIGN( ONE, ALPHA ) / C
ELSE
TMP = S2 / S1
S = SQRT( ONE+TMP*TMP )
SESTPR = ABSEST / S
C = ( ALPHA / S1 ) / S
S = -SIGN( ONE, GAMMA ) / S
END IF
RETURN
ELSE
*
* normal case
*
ZETA1 = ALPHA / ABSEST
ZETA2 = GAMMA / ABSEST
*
NORMA = MAX( ONE+ZETA1*ZETA1+ABS( ZETA1*ZETA2 ),
$ ABS( ZETA1*ZETA2 )+ZETA2*ZETA2 )
*
* See if root is closer to zero or to ONE
*
TEST = ONE + TWO*( ZETA1-ZETA2 )*( ZETA1+ZETA2 )
IF( TEST.GE.ZERO ) THEN
*
* root is close to zero, compute directly
*
B = ( ZETA1*ZETA1+ZETA2*ZETA2+ONE )*HALF
C = ZETA2*ZETA2
T = C / ( B+SQRT( ABS( B*B-C ) ) )
SINE = ZETA1 / ( ONE-T )
COSINE = -ZETA2 / T
SESTPR = SQRT( T+FOUR*EPS*EPS*NORMA )*ABSEST
ELSE
*
* root is closer to ONE, shift by that amount
*
B = ( ZETA2*ZETA2+ZETA1*ZETA1-ONE )*HALF
C = ZETA1*ZETA1
IF( B.GE.ZERO ) THEN
T = -C / ( B+SQRT( B*B+C ) )
ELSE
T = B - SQRT( B*B+C )
END IF
SINE = -ZETA1 / T
COSINE = -ZETA2 / ( ONE+T )
SESTPR = SQRT( ONE+T+FOUR*EPS*EPS*NORMA )*ABSEST
END IF
TMP = SQRT( SINE*SINE+COSINE*COSINE )
S = SINE / TMP
C = COSINE / TMP
RETURN
*
END IF
END IF
RETURN
*
* End of DLAIC1
*
END