*> \brief \b SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLARFGP + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLARFGP( N, ALPHA, X, INCX, TAU ) * * .. Scalar Arguments .. * INTEGER INCX, N * REAL ALPHA, TAU * .. * .. Array Arguments .. * REAL X( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLARFGP generates a real elementary reflector H of order n, such *> that *> *> H * ( alpha ) = ( beta ), H**T * H = I. *> ( x ) ( 0 ) *> *> where alpha and beta are scalars, beta is non-negative, and x is *> an (n-1)-element real vector. H is represented in the form *> *> H = I - tau * ( 1 ) * ( 1 v**T ) , *> ( v ) *> *> where tau is a real scalar and v is a real (n-1)-element *> vector. *> *> If the elements of x are all zero, then tau = 0 and H is taken to be *> the unit matrix. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the elementary reflector. *> \endverbatim *> *> \param[in,out] ALPHA *> \verbatim *> ALPHA is REAL *> On entry, the value alpha. *> On exit, it is overwritten with the value beta. *> \endverbatim *> *> \param[in,out] X *> \verbatim *> X is REAL array, dimension *> (1+(N-2)*abs(INCX)) *> On entry, the vector x. *> On exit, it is overwritten with the vector v. *> \endverbatim *> *> \param[in] INCX *> \verbatim *> INCX is INTEGER *> The increment between elements of X. INCX > 0. *> \endverbatim *> *> \param[out] TAU *> \verbatim *> TAU is REAL *> The value tau. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup realOTHERauxiliary * * ===================================================================== SUBROUTINE SLARFGP( N, ALPHA, X, INCX, TAU ) * * -- LAPACK auxiliary 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 .. INTEGER INCX, N REAL ALPHA, TAU * .. * .. Array Arguments .. REAL X( * ) * .. * * ===================================================================== * * .. Parameters .. REAL TWO, ONE, ZERO PARAMETER ( TWO = 2.0E+0, ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. INTEGER J, KNT REAL BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM * .. * .. External Functions .. REAL SLAMCH, SLAPY2, SNRM2 EXTERNAL SLAMCH, SLAPY2, SNRM2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, SIGN * .. * .. External Subroutines .. EXTERNAL SSCAL * .. * .. Executable Statements .. * IF( N.LE.0 ) THEN TAU = ZERO RETURN END IF * XNORM = SNRM2( N-1, X, INCX ) * IF( XNORM.EQ.ZERO ) THEN * * H = [+/-1, 0; I], sign chosen so ALPHA >= 0. * IF( ALPHA.GE.ZERO ) THEN * When TAU.eq.ZERO, the vector is special-cased to be * all zeros in the application routines. We do not need * to clear it. TAU = ZERO ELSE * However, the application routines rely on explicit * zero checks when TAU.ne.ZERO, and we must clear X. TAU = TWO DO J = 1, N-1 X( 1 + (J-1)*INCX ) = 0 END DO ALPHA = -ALPHA END IF ELSE * * general case * BETA = SIGN( SLAPY2( ALPHA, XNORM ), ALPHA ) SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'E' ) KNT = 0 IF( ABS( BETA ).LT.SMLNUM ) THEN * * XNORM, BETA may be inaccurate; scale X and recompute them * BIGNUM = ONE / SMLNUM 10 CONTINUE KNT = KNT + 1 CALL SSCAL( N-1, BIGNUM, X, INCX ) BETA = BETA*BIGNUM ALPHA = ALPHA*BIGNUM IF( ABS( BETA ).LT.SMLNUM ) \$ GO TO 10 * * New BETA is at most 1, at least SMLNUM * XNORM = SNRM2( N-1, X, INCX ) BETA = SIGN( SLAPY2( ALPHA, XNORM ), ALPHA ) END IF SAVEALPHA = ALPHA ALPHA = ALPHA + BETA IF( BETA.LT.ZERO ) THEN BETA = -BETA TAU = -ALPHA / BETA ELSE ALPHA = XNORM * (XNORM/ALPHA) TAU = ALPHA / BETA ALPHA = -ALPHA END IF * IF ( ABS(TAU).LE.SMLNUM ) THEN * * In the case where the computed TAU ends up being a denormalized number, * it loses relative accuracy. This is a BIG problem. Solution: flush TAU * to ZERO. This explains the next IF statement. * * (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.) * (Thanks Pat. Thanks MathWorks.) * IF( SAVEALPHA.GE.ZERO ) THEN TAU = ZERO ELSE TAU = TWO DO J = 1, N-1 X( 1 + (J-1)*INCX ) = 0 END DO BETA = -SAVEALPHA END IF * ELSE * * This is the general case. * CALL SSCAL( N-1, ONE / ALPHA, X, INCX ) * END IF * * If BETA is subnormal, it may lose relative accuracy * DO 20 J = 1, KNT BETA = BETA*SMLNUM 20 CONTINUE ALPHA = BETA END IF * RETURN * * End of SLARFGP * END