*> \brief \b ZLARTG generates a plane rotation with real cosine and complex sine. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLARTG + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLARTG( F, G, CS, SN, R ) * * .. Scalar Arguments .. * DOUBLE PRECISION CS * COMPLEX*16 F, G, R, SN * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLARTG generates a plane rotation so that *> *> [ CS SN ] [ F ] [ R ] *> [ __ ] . [ ] = [ ] where CS**2 + |SN|**2 = 1. *> [ -SN CS ] [ G ] [ 0 ] *> *> This is a faster version of the BLAS1 routine ZROTG, except for *> the following differences: *> F and G are unchanged on return. *> If G=0, then CS=1 and SN=0. *> If F=0, then CS=0 and SN is chosen so that R is real. *> \endverbatim * * Arguments: * ========== * *> \param[in] F *> \verbatim *> F is COMPLEX*16 *> The first component of vector to be rotated. *> \endverbatim *> *> \param[in] G *> \verbatim *> G is COMPLEX*16 *> The second component of vector to be rotated. *> \endverbatim *> *> \param[out] CS *> \verbatim *> CS is DOUBLE PRECISION *> The cosine of the rotation. *> \endverbatim *> *> \param[out] SN *> \verbatim *> SN is COMPLEX*16 *> The sine of the rotation. *> \endverbatim *> *> \param[out] R *> \verbatim *> R is COMPLEX*16 *> The nonzero component of the rotated vector. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup complex16OTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> 3-5-96 - Modified with a new algorithm by W. Kahan and J. Demmel *> *> This version has a few statements commented out for thread safety *> (machine parameters are computed on each entry). 10 feb 03, SJH. *> \endverbatim *> * ===================================================================== SUBROUTINE ZLARTG( F, G, CS, SN, R ) * * -- 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 .. DOUBLE PRECISION CS COMPLEX*16 F, G, R, SN * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION TWO, ONE, ZERO PARAMETER ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 ) COMPLEX*16 CZERO PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. * LOGICAL FIRST INTEGER COUNT, I DOUBLE PRECISION D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN, \$ SAFMN2, SAFMX2, SCALE COMPLEX*16 FF, FS, GS * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 LOGICAL DISNAN EXTERNAL DLAMCH, DLAPY2, DISNAN * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, LOG, \$ MAX, SQRT * .. * .. Statement Functions .. DOUBLE PRECISION ABS1, ABSSQ * .. * .. Statement Function definitions .. ABS1( FF ) = MAX( ABS( DBLE( FF ) ), ABS( DIMAG( FF ) ) ) ABSSQ( FF ) = DBLE( FF )**2 + DIMAG( FF )**2 * .. * .. Executable Statements .. * SAFMIN = DLAMCH( 'S' ) EPS = DLAMCH( 'E' ) SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) / \$ LOG( DLAMCH( 'B' ) ) / TWO ) SAFMX2 = ONE / SAFMN2 SCALE = MAX( ABS1( F ), ABS1( G ) ) FS = F GS = G COUNT = 0 IF( SCALE.GE.SAFMX2 ) THEN 10 CONTINUE COUNT = COUNT + 1 FS = FS*SAFMN2 GS = GS*SAFMN2 SCALE = SCALE*SAFMN2 IF( SCALE.GE.SAFMX2 ) \$ GO TO 10 ELSE IF( SCALE.LE.SAFMN2 ) THEN IF( G.EQ.CZERO.OR.DISNAN( ABS( G ) ) ) THEN CS = ONE SN = CZERO R = F RETURN END IF 20 CONTINUE COUNT = COUNT - 1 FS = FS*SAFMX2 GS = GS*SAFMX2 SCALE = SCALE*SAFMX2 IF( SCALE.LE.SAFMN2 ) \$ GO TO 20 END IF F2 = ABSSQ( FS ) G2 = ABSSQ( GS ) IF( F2.LE.MAX( G2, ONE )*SAFMIN ) THEN * * This is a rare case: F is very small. * IF( F.EQ.CZERO ) THEN CS = ZERO R = DLAPY2( DBLE( G ), DIMAG( G ) ) * Do complex/real division explicitly with two real divisions D = DLAPY2( DBLE( GS ), DIMAG( GS ) ) SN = DCMPLX( DBLE( GS ) / D, -DIMAG( GS ) / D ) RETURN END IF F2S = DLAPY2( DBLE( FS ), DIMAG( FS ) ) * G2 and G2S are accurate * G2 is at least SAFMIN, and G2S is at least SAFMN2 G2S = SQRT( G2 ) * Error in CS from underflow in F2S is at most * UNFL / SAFMN2 .lt. sqrt(UNFL*EPS) .lt. EPS * If MAX(G2,ONE)=G2, then F2 .lt. G2*SAFMIN, * and so CS .lt. sqrt(SAFMIN) * If MAX(G2,ONE)=ONE, then F2 .lt. SAFMIN * and so CS .lt. sqrt(SAFMIN)/SAFMN2 = sqrt(EPS) * Therefore, CS = F2S/G2S / sqrt( 1 + (F2S/G2S)**2 ) = F2S/G2S CS = F2S / G2S * Make sure abs(FF) = 1 * Do complex/real division explicitly with 2 real divisions IF( ABS1( F ).GT.ONE ) THEN D = DLAPY2( DBLE( F ), DIMAG( F ) ) FF = DCMPLX( DBLE( F ) / D, DIMAG( F ) / D ) ELSE DR = SAFMX2*DBLE( F ) DI = SAFMX2*DIMAG( F ) D = DLAPY2( DR, DI ) FF = DCMPLX( DR / D, DI / D ) END IF SN = FF*DCMPLX( DBLE( GS ) / G2S, -DIMAG( GS ) / G2S ) R = CS*F + SN*G ELSE * * This is the most common case. * Neither F2 nor F2/G2 are less than SAFMIN * F2S cannot overflow, and it is accurate * F2S = SQRT( ONE+G2 / F2 ) * Do the F2S(real)*FS(complex) multiply with two real multiplies R = DCMPLX( F2S*DBLE( FS ), F2S*DIMAG( FS ) ) CS = ONE / F2S D = F2 + G2 * Do complex/real division explicitly with two real divisions SN = DCMPLX( DBLE( R ) / D, DIMAG( R ) / D ) SN = SN*DCONJG( GS ) IF( COUNT.NE.0 ) THEN IF( COUNT.GT.0 ) THEN DO 30 I = 1, COUNT R = R*SAFMX2 30 CONTINUE ELSE DO 40 I = 1, -COUNT R = R*SAFMN2 40 CONTINUE END IF END IF END IF RETURN * * End of ZLARTG * END