*> \brief \b ZLARGV generates a vector of plane rotations with real cosines and complex sines. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLARGV + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLARGV( N, X, INCX, Y, INCY, C, INCC ) * * .. Scalar Arguments .. * INTEGER INCC, INCX, INCY, N * .. * .. Array Arguments .. * DOUBLE PRECISION C( * ) * COMPLEX*16 X( * ), Y( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZLARGV generates a vector of complex plane rotations with real *> cosines, determined by elements of the complex vectors x and y. *> For i = 1,2,...,n *> *> ( c(i) s(i) ) ( x(i) ) = ( r(i) ) *> ( -conjg(s(i)) c(i) ) ( y(i) ) = ( 0 ) *> *> where c(i)**2 + ABS(s(i))**2 = 1 *> *> The following conventions are used (these are the same as in ZLARTG, *> but differ from the BLAS1 routine ZROTG): *> If y(i)=0, then c(i)=1 and s(i)=0. *> If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real. *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The number of plane rotations to be generated. *> \endverbatim *> *> \param[in,out] X *> \verbatim *> X is COMPLEX*16 array, dimension (1+(N-1)*INCX) *> On entry, the vector x. *> On exit, x(i) is overwritten by r(i), for i = 1,...,n. *> \endverbatim *> *> \param[in] INCX *> \verbatim *> INCX is INTEGER *> The increment between elements of X. INCX > 0. *> \endverbatim *> *> \param[in,out] Y *> \verbatim *> Y is COMPLEX*16 array, dimension (1+(N-1)*INCY) *> On entry, the vector y. *> On exit, the sines of the plane rotations. *> \endverbatim *> *> \param[in] INCY *> \verbatim *> INCY is INTEGER *> The increment between elements of Y. INCY > 0. *> \endverbatim *> *> \param[out] C *> \verbatim *> C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC) *> The cosines of the plane rotations. *> \endverbatim *> *> \param[in] INCC *> \verbatim *> INCC is INTEGER *> The increment between elements of C. INCC > 0. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date September 2012 * *> \ingroup complex16OTHERauxiliary * *> \par Further Details: * ===================== *> *> \verbatim *> *> 6-6-96 - 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 ZLARGV( N, X, INCX, Y, INCY, C, INCC ) * * -- LAPACK auxiliary routine (version 3.4.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * September 2012 * * .. Scalar Arguments .. INTEGER INCC, INCX, INCY, N * .. * .. Array Arguments .. DOUBLE PRECISION C( * ) COMPLEX*16 X( * ), Y( * ) * .. * * ===================================================================== * * .. 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, IC, IX, IY, J DOUBLE PRECISION CS, D, DI, DR, EPS, F2, F2S, G2, G2S, SAFMIN, $ SAFMN2, SAFMX2, SCALE COMPLEX*16 F, FF, FS, G, GS, R, SN * .. * .. External Functions .. DOUBLE PRECISION DLAMCH, DLAPY2 EXTERNAL DLAMCH, DLAPY2 * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, INT, LOG, $ MAX, SQRT * .. * .. Statement Functions .. DOUBLE PRECISION ABS1, ABSSQ * .. * .. Save statement .. * SAVE FIRST, SAFMX2, SAFMIN, SAFMN2 * .. * .. Data statements .. * DATA FIRST / .TRUE. / * .. * .. Statement Function definitions .. ABS1( FF ) = MAX( ABS( DBLE( FF ) ), ABS( DIMAG( FF ) ) ) ABSSQ( FF ) = DBLE( FF )**2 + DIMAG( FF )**2 * .. * .. Executable Statements .. * * IF( FIRST ) THEN * FIRST = .FALSE. SAFMIN = DLAMCH( 'S' ) EPS = DLAMCH( 'E' ) SAFMN2 = DLAMCH( 'B' )**INT( LOG( SAFMIN / EPS ) / $ LOG( DLAMCH( 'B' ) ) / TWO ) SAFMX2 = ONE / SAFMN2 * END IF IX = 1 IY = 1 IC = 1 DO 60 I = 1, N F = X( IX ) G = Y( IY ) * * Use identical algorithm as in ZLARTG * 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 ) THEN CS = ONE SN = CZERO R = F GO TO 50 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 ) GO TO 50 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 J = 1, COUNT R = R*SAFMX2 30 CONTINUE ELSE DO 40 J = 1, -COUNT R = R*SAFMN2 40 CONTINUE END IF END IF END IF 50 CONTINUE C( IC ) = CS Y( IY ) = SN X( IX ) = R IC = IC + INCC IY = IY + INCY IX = IX + INCX 60 CONTINUE RETURN * * End of ZLARGV * END