--- rpl/lapack/lapack/zhgeqz.f 2012/08/22 09:48:33 1.11
+++ 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 )