*> \brief \b CLARFGP 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 CLARFGP + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CLARFGP( N, ALPHA, X, INCX, TAU ) * * .. Scalar Arguments .. * INTEGER INCX, N * COMPLEX ALPHA, TAU * .. * .. Array Arguments .. * COMPLEX X( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CLARFGP generates a complex elementary reflector H of order n, such *> that *> *> H**H * ( alpha ) = ( beta ), H**H * H = I. *> ( x ) ( 0 ) *> *> where alpha and beta are scalars, beta is real and non-negative, and *> x is an (n-1)-element complex vector. H is represented in the form *> *> H = I - tau * ( 1 ) * ( 1 v**H ) , *> ( v ) *> *> where tau is a complex scalar and v is a complex (n-1)-element *> vector. Note that H is not hermitian. *> *> If the elements of x are all zero and alpha is real, 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 COMPLEX *> On entry, the value alpha. *> On exit, it is overwritten with the value beta. *> \endverbatim *> *> \param[in,out] X *> \verbatim *> X is COMPLEX 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 COMPLEX *> The value tau. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup complexOTHERauxiliary * * ===================================================================== SUBROUTINE CLARFGP( N, ALPHA, X, INCX, TAU ) * * -- 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 INCX, N COMPLEX ALPHA, TAU * .. * .. Array Arguments .. COMPLEX 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 ALPHI, ALPHR, BETA, BIGNUM, SMLNUM, XNORM COMPLEX SAVEALPHA * .. * .. External Functions .. REAL SCNRM2, SLAMCH, SLAPY3, SLAPY2 COMPLEX CLADIV EXTERNAL SCNRM2, SLAMCH, SLAPY3, SLAPY2, CLADIV * .. * .. Intrinsic Functions .. INTRINSIC ABS, AIMAG, CMPLX, REAL, SIGN * .. * .. External Subroutines .. EXTERNAL CSCAL, CSSCAL * .. * .. Executable Statements .. * IF( N.LE.0 ) THEN TAU = ZERO RETURN END IF * XNORM = SCNRM2( N-1, X, INCX ) ALPHR = REAL( ALPHA ) ALPHI = AIMAG( ALPHA ) * IF( XNORM.EQ.ZERO ) THEN * * H = [1-alpha/abs(alpha) 0; 0 I], sign chosen so ALPHA >= 0. * IF( ALPHI.EQ.ZERO ) THEN IF( ALPHR.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 ) = ZERO END DO ALPHA = -ALPHA END IF ELSE * Only "reflecting" the diagonal entry to be real and non-negative. XNORM = SLAPY2( ALPHR, ALPHI ) TAU = CMPLX( ONE - ALPHR / XNORM, -ALPHI / XNORM ) DO J = 1, N-1 X( 1 + (J-1)*INCX ) = ZERO END DO ALPHA = XNORM END IF ELSE * * general case * BETA = SIGN( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR ) SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'E' ) BIGNUM = ONE / SMLNUM * KNT = 0 IF( ABS( BETA ).LT.SMLNUM ) THEN * * XNORM, BETA may be inaccurate; scale X and recompute them * 10 CONTINUE KNT = KNT + 1 CALL CSSCAL( N-1, BIGNUM, X, INCX ) BETA = BETA*BIGNUM ALPHI = ALPHI*BIGNUM ALPHR = ALPHR*BIGNUM IF( (ABS( BETA ).LT.SMLNUM) .AND. (KNT .LT. 20) ) \$ GO TO 10 * * New BETA is at most 1, at least SMLNUM * XNORM = SCNRM2( N-1, X, INCX ) ALPHA = CMPLX( ALPHR, ALPHI ) BETA = SIGN( SLAPY3( ALPHR, ALPHI, XNORM ), ALPHR ) END IF SAVEALPHA = ALPHA ALPHA = ALPHA + BETA IF( BETA.LT.ZERO ) THEN BETA = -BETA TAU = -ALPHA / BETA ELSE ALPHR = ALPHI * (ALPHI/REAL( ALPHA )) ALPHR = ALPHR + XNORM * (XNORM/REAL( ALPHA )) TAU = CMPLX( ALPHR/BETA, -ALPHI/BETA ) ALPHA = CMPLX( -ALPHR, ALPHI ) END IF ALPHA = CLADIV( CMPLX( ONE ), ALPHA ) * 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 (or TWO or whatever makes a nonnegative real number for BETA). * * (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.) * (Thanks Pat. Thanks MathWorks.) * ALPHR = REAL( SAVEALPHA ) ALPHI = AIMAG( SAVEALPHA ) IF( ALPHI.EQ.ZERO ) THEN IF( ALPHR.GE.ZERO ) THEN TAU = ZERO ELSE TAU = TWO DO J = 1, N-1 X( 1 + (J-1)*INCX ) = ZERO END DO BETA = REAL( -SAVEALPHA ) END IF ELSE XNORM = SLAPY2( ALPHR, ALPHI ) TAU = CMPLX( ONE - ALPHR / XNORM, -ALPHI / XNORM ) DO J = 1, N-1 X( 1 + (J-1)*INCX ) = ZERO END DO BETA = XNORM END IF * ELSE * * This is the general case. * CALL CSCAL( N-1, 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 CLARFGP * END