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chgeqz.f
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*> \brief \b CHGEQZ
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHGEQZ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chgeqz.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chgeqz.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chgeqz.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CHGEQZ( 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
* ..
* .. Array Arguments ..
* REAL RWORK( * )
* COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
* $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
*> where H is an upper Hessenberg matrix and T is upper triangular,
*> 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 CGGHRD.
*>
*> 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.
*> If Q1 and Z1 are the unitary matrices from CGGHRD that reduced
*> 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
*> eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
*> A*x = lambda*B*x
*> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
*> alternate form of the GNEP
*> mu*A*y = B*y.
*> The values of alpha and beta for the i-th eigenvalue can be read
*> directly from the generalized Schur form: alpha = S(i,i),
*> beta = P(i,i).
*>
*> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
*> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
*> pp. 241--256.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> = 'E': Compute eigenvalues only;
*> = 'S': Computer eigenvalues and the Schur form.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*> COMPQ is CHARACTER*1
*> = 'N': Left Schur vectors (Q) are not computed;
*> = 'I': Q is initialized to the unit matrix and the matrix Q
*> of left Schur vectors of (H,T) is returned;
*> = 'V': Q must contain a unitary matrix Q1 on entry and
*> the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> COMPZ is CHARACTER*1
*> = 'N': Right Schur vectors (Z) are not computed;
*> = 'I': Q is initialized to the unit matrix and the matrix Z
*> of right Schur vectors of (H,T) is returned;
*> = 'V': Z must contain a unitary matrix Z1 on entry and
*> the product Z1*Z is returned.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices H, T, Q, and Z. N >= 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI mark the rows and columns of H which are in
*> Hessenberg form. It is assumed that A is already upper
*> triangular in rows and columns 1:ILO-1 and IHI+1:N.
*> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> H is COMPLEX array, dimension (LDH, N)
*> On entry, the N-by-N upper Hessenberg matrix H.
*> On exit, if JOB = 'S', H contains the upper triangular
*> matrix S from the generalized Schur factorization.
*> If JOB = 'E', the diagonal of H matches that of S, but
*> the rest of H is unspecified.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] T
*> \verbatim
*> T is COMPLEX array, dimension (LDT, N)
*> On entry, the N-by-N upper triangular matrix T.
*> On exit, if JOB = 'S', T contains the upper triangular
*> matrix P from the generalized Schur factorization.
*> If JOB = 'E', the diagonal of T matches that of P, but
*> the rest of T is unspecified.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX array, dimension (N)
*> The complex scalars alpha that define the eigenvalues of
*> GNEP. ALPHA(i) = S(i,i) in the generalized Schur
*> factorization.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX array, dimension (N)
*> The real non-negative scalars beta that define the
*> eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
*> Schur factorization.
*>
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*> represent the j-th eigenvalue of the matrix pair (A,B), in
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
*> Since either lambda or mu may overflow, they should not,
*> in general, be computed.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is COMPLEX array, dimension (LDQ, N)
*> On entry, if COMPQ = 'V', the unitary matrix Q1 used in the
*> reduction of (A,B) to generalized Hessenberg form.
*> 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 COMPQ = 'N'.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1.
*> If COMPQ='V' or 'I', then LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is COMPLEX array, dimension (LDZ, N)
*> On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
*> reduction of (A,B) to generalized Hessenberg form.
*> On exit, if COMPZ = 'I', the unitary matrix of right Schur
*> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
*> right Schur vectors of (A,B).
*> Not referenced if COMPZ = 'N'.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1.
*> If COMPZ='V' or 'I', then LDZ >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (MAX(1,LWORK))
*> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= max(1,N).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> = 1,...,N: the QZ iteration did not converge. (H,T) is not
*> in Schur form, but ALPHA(i) and BETA(i),
*> i=INFO+1,...,N should be correct.
*> = N+1,...,2*N: the shift calculation failed. (H,T) is not
*> in Schur form, but ALPHA(i) and BETA(i),
*> i=INFO-N+1,...,N should be correct.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date April 2012
*
*> \ingroup complexGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> We assume that complex ABS works as long as its value is less than
*> overflow.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
$ RWORK, INFO )
*
* -- LAPACK computational 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..--
* April 2012
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ),
$ Q( LDQ, * ), T( LDT, * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
REAL HALF
PARAMETER ( HALF = 0.5E+0 )
* ..
* .. Local Scalars ..
LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
$ JR, MAXIT
REAL ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
$ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
COMPLEX ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
$ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
$ U12, X
* ..
* .. External Functions ..
LOGICAL LSAME
REAL CLANHS, SLAMCH
EXTERNAL LSAME, CLANHS, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CLARTG, CLASET, CROT, CSCAL, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL, SQRT
* ..
* .. Statement Functions ..
REAL ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode JOB, COMPQ, COMPZ
*
IF( LSAME( JOB, 'E' ) ) THEN
ILSCHR = .FALSE.
ISCHUR = 1
ELSE IF( LSAME( JOB, 'S' ) ) THEN
ILSCHR = .TRUE.
ISCHUR = 2
ELSE
ISCHUR = 0
END IF
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Check Argument Values
*
INFO = 0
WORK( 1 ) = MAX( 1, N )
LQUERY = ( LWORK.EQ.-1 )
IF( ISCHUR.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.EQ.0 ) THEN
INFO = -2
ELSE IF( ICOMPZ.EQ.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 ) THEN
INFO = -5
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -6
ELSE IF( LDH.LT.N ) THEN
INFO = -8
ELSE IF( LDT.LT.N ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
INFO = -14
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -16
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHGEQZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
* WORK( 1 ) = CMPLX( 1 )
IF( N.LE.0 ) THEN
WORK( 1 ) = CMPLX( 1 )
RETURN
END IF
*
* Initialize Q and Z
*
IF( ICOMPQ.EQ.3 )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
*
* Machine Constants
*
IN = IHI + 1 - ILO
SAFMIN = SLAMCH( 'S' )
ULP = SLAMCH( 'E' )*SLAMCH( 'B' )
ANORM = CLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
BNORM = CLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
ATOL = MAX( SAFMIN, ULP*ANORM )
BTOL = MAX( SAFMIN, ULP*BNORM )
ASCALE = ONE / MAX( SAFMIN, ANORM )
BSCALE = ONE / MAX( SAFMIN, BNORM )
*
*
* Set Eigenvalues IHI+1:N
*
DO 10 J = IHI + 1, N
ABSB = ABS( T( J, J ) )
IF( ABSB.GT.SAFMIN ) THEN
SIGNBC = CONJG( T( J, J ) / ABSB )
T( J, J ) = ABSB
IF( ILSCHR ) THEN
CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
ELSE
CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
END IF
IF( ILZ )
$ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
ELSE
T( J, J ) = CZERO
END IF
ALPHA( J ) = H( J, J )
BETA( J ) = T( J, J )
10 CONTINUE
*
* If IHI < ILO, skip QZ steps
*
IF( IHI.LT.ILO )
$ GO TO 190
*
* MAIN QZ ITERATION LOOP
*
* Initialize dynamic indices
*
* Eigenvalues ILAST+1:N have been found.
* Column operations modify rows IFRSTM:whatever
* Row operations modify columns whatever:ILASTM
*
* If only eigenvalues are being computed, then
* IFRSTM is the row of the last splitting row above row ILAST;
* this is always at least ILO.
* IITER counts iterations since the last eigenvalue was found,
* to tell when to use an extraordinary shift.
* MAXIT is the maximum number of QZ sweeps allowed.
*
ILAST = IHI
IF( ILSCHR ) THEN
IFRSTM = 1
ILASTM = N
ELSE
IFRSTM = ILO
ILASTM = IHI
END IF
IITER = 0
ESHIFT = CZERO
MAXIT = 30*( IHI-ILO+1 )
*
DO 170 JITER = 1, MAXIT
*
* Check for too many iterations.
*
IF( JITER.GT.MAXIT )
$ GO TO 180
*
* Split the matrix if possible.
*
* Two tests:
* 1: H(j,j-1)=0 or j=ILO
* 2: T(j,j)=0
*
* Special case: j=ILAST
*
IF( ILAST.EQ.ILO ) THEN
GO TO 60
ELSE
IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
H( ILAST, ILAST-1 ) = CZERO
GO TO 60
END IF
END IF
*
IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
T( ILAST, ILAST ) = CZERO
GO TO 50
END IF
*
* General case: j<ILAST
*
DO 40 J = ILAST - 1, ILO, -1
*
* Test 1: for H(j,j-1)=0 or j=ILO
*
IF( J.EQ.ILO ) THEN
ILAZRO = .TRUE.
ELSE
IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
H( J, J-1 ) = CZERO
ILAZRO = .TRUE.
ELSE
ILAZRO = .FALSE.
END IF
END IF
*
* Test 2: for T(j,j)=0
*
IF( ABS( T( J, J ) ).LT.BTOL ) THEN
T( J, J ) = CZERO
*
* Test 1a: Check for 2 consecutive small subdiagonals in A
*
ILAZR2 = .FALSE.
IF( .NOT.ILAZRO ) THEN
IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
$ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
$ ILAZR2 = .TRUE.
END IF
*
* If both tests pass (1 & 2), i.e., the leading diagonal
* element of B in the block is zero, split a 1x1 block off
* at the top. (I.e., at the J-th row/column) The leading
* diagonal element of the remainder can also be zero, so
* this may have to be done repeatedly.
*
IF( ILAZRO .OR. ILAZR2 ) THEN
DO 20 JCH = J, ILAST - 1
CTEMP = H( JCH, JCH )
CALL CLARTG( CTEMP, H( JCH+1, JCH ), C, S,
$ H( JCH, JCH ) )
H( JCH+1, JCH ) = CZERO
CALL CROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
$ H( JCH+1, JCH+1 ), LDH, C, S )
CALL CROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
$ T( JCH+1, JCH+1 ), LDT, C, S )
IF( ILQ )
$ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
$ C, CONJG( S ) )
IF( ILAZR2 )
$ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
ILAZR2 = .FALSE.
IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
IF( JCH+1.GE.ILAST ) THEN
GO TO 60
ELSE
IFIRST = JCH + 1
GO TO 70
END IF
END IF
T( JCH+1, JCH+1 ) = CZERO
20 CONTINUE
GO TO 50
ELSE
*
* Only test 2 passed -- chase the zero to T(ILAST,ILAST)
* Then process as in the case T(ILAST,ILAST)=0
*
DO 30 JCH = J, ILAST - 1
CTEMP = T( JCH, JCH+1 )
CALL CLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
$ T( JCH, JCH+1 ) )
T( JCH+1, JCH+1 ) = CZERO
IF( JCH.LT.ILASTM-1 )
$ CALL CROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
$ T( JCH+1, JCH+2 ), LDT, C, S )
CALL CROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
$ H( JCH+1, JCH-1 ), LDH, C, S )
IF( ILQ )
$ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
$ C, CONJG( S ) )
CTEMP = H( JCH+1, JCH )
CALL CLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
$ H( JCH+1, JCH ) )
H( JCH+1, JCH-1 ) = CZERO
CALL CROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
$ H( IFRSTM, JCH-1 ), 1, C, S )
CALL CROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
$ T( IFRSTM, JCH-1 ), 1, C, S )
IF( ILZ )
$ CALL CROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
$ C, S )
30 CONTINUE
GO TO 50
END IF
ELSE IF( ILAZRO ) THEN
*
* Only test 1 passed -- work on J:ILAST
*
IFIRST = J
GO TO 70
END IF
*
* Neither test passed -- try next J
*
40 CONTINUE
*
* (Drop-through is "impossible")
*
INFO = 2*N + 1
GO TO 210
*
* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
* 1x1 block.
*
50 CONTINUE
CTEMP = H( ILAST, ILAST )
CALL CLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
$ H( ILAST, ILAST ) )
H( ILAST, ILAST-1 ) = CZERO
CALL CROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
$ H( IFRSTM, ILAST-1 ), 1, C, S )
CALL CROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
$ T( IFRSTM, ILAST-1 ), 1, C, S )
IF( ILZ )
$ CALL CROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
*
* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
*
60 CONTINUE
ABSB = ABS( T( ILAST, ILAST ) )
IF( ABSB.GT.SAFMIN ) THEN
SIGNBC = CONJG( T( ILAST, ILAST ) / ABSB )
T( ILAST, ILAST ) = ABSB
IF( ILSCHR ) THEN
CALL CSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
CALL CSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
$ 1 )
ELSE
CALL CSCAL( 1, SIGNBC, H( ILAST, ILAST ), 1 )
END IF
IF( ILZ )
$ CALL CSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
ELSE
T( ILAST, ILAST ) = CZERO
END IF
ALPHA( ILAST ) = H( ILAST, ILAST )
BETA( ILAST ) = T( ILAST, ILAST )
*
* Go to next block -- exit if finished.
*
ILAST = ILAST - 1
IF( ILAST.LT.ILO )
$ GO TO 190
*
* Reset counters
*
IITER = 0
ESHIFT = CZERO
IF( .NOT.ILSCHR ) THEN
ILASTM = ILAST
IF( IFRSTM.GT.ILAST )
$ IFRSTM = ILO
END IF
GO TO 160
*
* QZ step
*
* This iteration only involves rows/columns IFIRST:ILAST. We
* assume IFIRST < ILAST, and that the diagonal of B is non-zero.
*
70 CONTINUE
IITER = IITER + 1
IF( .NOT.ILSCHR ) THEN
IFRSTM = IFIRST
END IF
*
* Compute the Shift.
*
* At this point, IFIRST < ILAST, and the diagonal elements of
* T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
* magnitude)
*
IF( ( IITER / 10 )*10.NE.IITER ) THEN
*
* The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
* the bottom-right 2x2 block of A inv(B) which is nearest to
* the bottom-right element.
*
* We factor B as U*D, where U has unit diagonals, and
* compute (A*inv(D))*inv(U).
*
U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
$ ( BSCALE*T( ILAST, ILAST ) )
AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
$ ( BSCALE*T( ILAST-1, ILAST-1 ) )
AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
$ ( BSCALE*T( ILAST, ILAST ) )
AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
$ ( BSCALE*T( ILAST, ILAST ) )
ABI22 = AD22 - U12*AD21
*
T1 = HALF*( AD11+ABI22 )
RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
TEMP = REAL( T1-ABI22 )*REAL( RTDISC ) +
$ AIMAG( T1-ABI22 )*AIMAG( RTDISC )
IF( TEMP.LE.ZERO ) THEN
SHIFT = T1 + RTDISC
ELSE
SHIFT = T1 - RTDISC
END IF
ELSE
*
* Exceptional shift. Chosen for no particularly good reason.
*
ESHIFT = ESHIFT + (ASCALE*H(ILAST,ILAST-1))/
$ (BSCALE*T(ILAST-1,ILAST-1))
SHIFT = ESHIFT
END IF
*
* Now check for two consecutive small subdiagonals.
*
DO 80 J = ILAST - 1, IFIRST + 1, -1
ISTART = J
CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
TEMP = ABS1( CTEMP )
TEMP2 = ASCALE*ABS1( H( J+1, J ) )
TEMPR = MAX( TEMP, TEMP2 )
IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
TEMP = TEMP / TEMPR
TEMP2 = TEMP2 / TEMPR
END IF
IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
$ GO TO 90
80 CONTINUE
*
ISTART = IFIRST
CTEMP = ASCALE*H( IFIRST, IFIRST ) -
$ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
90 CONTINUE
*
* Do an implicit-shift QZ sweep.
*
* Initial Q
*
CTEMP2 = ASCALE*H( ISTART+1, ISTART )
CALL CLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
*
* Sweep
*
DO 150 J = ISTART, ILAST - 1
IF( J.GT.ISTART ) THEN
CTEMP = H( J, J-1 )
CALL CLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
H( J+1, J-1 ) = CZERO
END IF
*
DO 100 JC = J, ILASTM
CTEMP = C*H( J, JC ) + S*H( J+1, JC )
H( J+1, JC ) = -CONJG( S )*H( J, JC ) + C*H( J+1, JC )
H( J, JC ) = CTEMP
CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
T( J+1, JC ) = -CONJG( S )*T( J, JC ) + C*T( J+1, JC )
T( J, JC ) = CTEMP2
100 CONTINUE
IF( ILQ ) THEN
DO 110 JR = 1, N
CTEMP = C*Q( JR, J ) + CONJG( S )*Q( JR, J+1 )
Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
Q( JR, J ) = CTEMP
110 CONTINUE
END IF
*
CTEMP = T( J+1, J+1 )
CALL CLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
T( J+1, J ) = CZERO
*
DO 120 JR = IFRSTM, MIN( J+2, ILAST )
CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
H( JR, J ) = -CONJG( S )*H( JR, J+1 ) + C*H( JR, J )
H( JR, J+1 ) = CTEMP
120 CONTINUE
DO 130 JR = IFRSTM, J
CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
T( JR, J ) = -CONJG( S )*T( JR, J+1 ) + C*T( JR, J )
T( JR, J+1 ) = CTEMP
130 CONTINUE
IF( ILZ ) THEN
DO 140 JR = 1, N
CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
Z( JR, J ) = -CONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
Z( JR, J+1 ) = CTEMP
140 CONTINUE
END IF
150 CONTINUE
*
160 CONTINUE
*
170 CONTINUE
*
* Drop-through = non-convergence
*
180 CONTINUE
INFO = ILAST
GO TO 210
*
* Successful completion of all QZ steps
*
190 CONTINUE
*
* Set Eigenvalues 1:ILO-1
*
DO 200 J = 1, ILO - 1
ABSB = ABS( T( J, J ) )
IF( ABSB.GT.SAFMIN ) THEN
SIGNBC = CONJG( T( J, J ) / ABSB )
T( J, J ) = ABSB
IF( ILSCHR ) THEN
CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 )
CALL CSCAL( J, SIGNBC, H( 1, J ), 1 )
ELSE
CALL CSCAL( 1, SIGNBC, H( J, J ), 1 )
END IF
IF( ILZ )
$ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
ELSE
T( J, J ) = CZERO
END IF
ALPHA( J ) = H( J, J )
BETA( J ) = T( J, J )
200 CONTINUE
*
* Normal Termination
*
INFO = 0
*
* Exit (other than argument error) -- return optimal workspace size
*
210 CONTINUE
WORK( 1 ) = CMPLX( N )
RETURN
*
* End of CHGEQZ
*
END