--- rpl/lapack/lapack/zhgeqz.f 2012/07/31 11:06:38 1.10 +++ rpl/lapack/lapack/zhgeqz.f 2023/08/07 08:39:25 1.20 @@ -2,18 +2,18 @@ * * =========== DOCUMENTATION =========== * -* Online html documentation available at -* http://www.netlib.org/lapack/explore-html/ +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ * *> \htmlonly -*> Download ZHGEQZ + dependencies -*> -*> [TGZ] -*> -*> [ZIP] -*> +*> Download ZHGEQZ + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> *> [TXT] -*> \endhtmlonly +*> \endhtmlonly * * Definition: * =========== @@ -21,7 +21,7 @@ * SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, * ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, * RWORK, INFO ) -* +* * .. Scalar Arguments .. * CHARACTER COMPQ, COMPZ, JOB * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N @@ -32,7 +32,7 @@ * $ Q( LDQ, * ), T( LDT, * ), WORK( * ), * $ Z( LDZ, * ) * .. -* +* * *> \par Purpose: * ============= @@ -44,18 +44,18 @@ *> using the single-shift QZ method. *> Matrix pairs of this type are produced by the reduction to *> generalized upper Hessenberg form of a complex matrix pair (A,B): -*> +*> *> A = Q1*H*Z1**H, B = Q1*T*Z1**H, -*> +*> *> as computed by ZGGHRD. -*> +*> *> If JOB='S', then the Hessenberg-triangular pair (H,T) is *> also reduced to generalized Schur form, -*> +*> *> H = Q*S*Z**H, T = Q*P*Z**H, -*> +*> *> where Q and Z are unitary matrices and S and P are upper triangular. -*> +*> *> Optionally, the unitary matrix Q from the generalized Schur *> factorization may be postmultiplied into an input matrix Q1, and the *> unitary matrix Z may be postmultiplied into an input matrix Z1. @@ -63,9 +63,9 @@ *> the matrix pair (A,B) to generalized Hessenberg form, then the output *> matrices Q1*Q and Z1*Z are the unitary factors from the generalized *> Schur factorization of (A,B): -*> +*> *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. -*> +*> *> To avoid overflow, eigenvalues of the matrix pair (H,T) *> (equivalently, of (A,B)) are computed as a pair of complex values *> (alpha,beta). If beta is nonzero, lambda = alpha / beta is an @@ -190,12 +190,12 @@ *> \param[in,out] Q *> \verbatim *> Q is COMPLEX*16 array, dimension (LDQ, N) -*> On entry, if COMPZ = 'V', the unitary matrix Q1 used in the +*> On entry, if COMPQ = 'V', the unitary matrix Q1 used in the *> reduction of (A,B) to generalized Hessenberg form. -*> On exit, if COMPZ = 'I', the unitary matrix of left Schur -*> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of +*> On exit, if COMPQ = 'I', the unitary matrix of left Schur +*> vectors of (H,T), and if COMPQ = 'V', the unitary matrix of *> left Schur vectors of (A,B). -*> Not referenced if COMPZ = 'N'. +*> Not referenced if COMPQ = 'N'. *> \endverbatim *> *> \param[in] LDQ @@ -261,12 +261,10 @@ * Authors: * ======== * -*> \author Univ. of Tennessee -*> \author Univ. of California Berkeley -*> \author Univ. of Colorado Denver -*> \author NAG Ltd. -* -*> \date April 2012 +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. * *> \ingroup complex16GEcomputational * @@ -284,10 +282,9 @@ $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, $ RWORK, INFO ) * -* -- LAPACK computational routine (version 3.4.1) -- +* -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* April 2012 * * .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, JOB @@ -319,13 +316,14 @@ DOUBLE PRECISION ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL, $ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP COMPLEX*16 ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2, - $ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1, - $ U12, X + $ CTEMP3, ESHIFT, S, SHIFT, SIGNBC, + $ U12, X, ABI12, Y * .. * .. External Functions .. + COMPLEX*16 ZLADIV LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANHS - EXTERNAL LSAME, DLAMCH, ZLANHS + EXTERNAL ZLADIV, LSAME, DLAMCH, ZLANHS * .. * .. External Subroutines .. EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL @@ -351,6 +349,7 @@ ILSCHR = .TRUE. ISCHUR = 2 ELSE + ILSCHR = .TRUE. ISCHUR = 0 END IF * @@ -364,6 +363,7 @@ ILQ = .TRUE. ICOMPQ = 3 ELSE + ILQ = .TRUE. ICOMPQ = 0 END IF * @@ -377,6 +377,7 @@ ILZ = .TRUE. ICOMPZ = 3 ELSE + ILZ = .TRUE. ICOMPZ = 0 END IF * @@ -454,7 +455,7 @@ CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) ELSE - H( J, J ) = H( J, J )*SIGNBC + CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 ) END IF IF( ILZ ) $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) @@ -515,7 +516,9 @@ IF( ILAST.EQ.ILO ) THEN GO TO 60 ELSE - IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN + IF( ABS1( H( ILAST, ILAST-1 ) ).LE.MAX( SAFMIN, ULP*( + $ ABS1( H( ILAST, ILAST ) ) + ABS1( H( ILAST-1, ILAST-1 ) + $ ) ) ) ) THEN H( ILAST, ILAST-1 ) = CZERO GO TO 60 END IF @@ -535,7 +538,9 @@ IF( J.EQ.ILO ) THEN ILAZRO = .TRUE. ELSE - IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN + IF( ABS1( H( J, J-1 ) ).LE.MAX( SAFMIN, ULP*( + $ ABS1( H( J, J ) ) + ABS1( H( J-1, J-1 ) ) + $ ) ) ) THEN H( J, J-1 ) = CZERO ILAZRO = .TRUE. ELSE @@ -666,7 +671,7 @@ CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ), $ 1 ) ELSE - H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC + CALL ZSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 ) END IF IF( ILZ ) $ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 ) @@ -730,22 +735,34 @@ AD22 = ( ASCALE*H( ILAST, ILAST ) ) / $ ( BSCALE*T( ILAST, ILAST ) ) ABI22 = AD22 - U12*AD21 + ABI12 = AD12 - U12*AD11 * - T1 = HALF*( AD11+ABI22 ) - RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 ) - TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) + - $ DIMAG( T1-ABI22 )*DIMAG( RTDISC ) - IF( TEMP.LE.ZERO ) THEN - SHIFT = T1 + RTDISC - ELSE - SHIFT = T1 - RTDISC + SHIFT = ABI22 + CTEMP = SQRT( ABI12 )*SQRT( AD21 ) + TEMP = ABS1( CTEMP ) + IF( CTEMP.NE.ZERO ) THEN + X = HALF*( AD11-SHIFT ) + TEMP2 = ABS1( X ) + TEMP = MAX( TEMP, ABS1( X ) ) + Y = TEMP*SQRT( ( X / TEMP )**2+( CTEMP / TEMP )**2 ) + IF( TEMP2.GT.ZERO ) THEN + IF( DBLE( X / TEMP2 )*DBLE( Y )+ + $ DIMAG( X / TEMP2 )*DIMAG( Y ).LT.ZERO )Y = -Y + END IF + SHIFT = SHIFT - CTEMP*ZLADIV( CTEMP, ( X+Y ) ) END IF ELSE * * Exceptional shift. Chosen for no particularly good reason. * - ESHIFT = ESHIFT + (ASCALE*H(ILAST,ILAST-1))/ - $ (BSCALE*T(ILAST-1,ILAST-1)) + IF( ( IITER / 20 )*20.EQ.IITER .AND. + $ BSCALE*ABS1(T( ILAST, ILAST )).GT.SAFMIN ) THEN + ESHIFT = ESHIFT + ( ASCALE*H( ILAST, + $ ILAST ) )/( BSCALE*T( ILAST, ILAST ) ) + ELSE + ESHIFT = ESHIFT + ( ASCALE*H( ILAST, + $ ILAST-1 ) )/( BSCALE*T( ILAST-1, ILAST-1 ) ) + END IF SHIFT = ESHIFT END IF * @@ -850,7 +867,7 @@ CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) ELSE - H( J, J ) = H( J, J )*SIGNBC + CALL ZSCAL( 1, SIGNBC, H( J, J ), 1 ) END IF IF( ILZ ) $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )