chgeqz.f

计算矩阵的经典开源库．全世界都在用它．相信你也不能例外．

      SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
     $                   ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
     $                   RWORK, INFO )
*
*  -- LAPACK routine (instrumented to count operations, version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      REAL               RWORK( * )
      COMPLEX            A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
*     ..
*
*     ----------------------- Begin Timing Code ------------------------
*     Common block to return operation count and iteration count
*     ITCNT is initialized to 0, OPS is only incremented
*     OPST is used to accumulate small contributions to OPS
*     to avoid roundoff error
*     .. Common blocks ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      REAL               ITCNT, OPS
*     ..
*     ------------------------ End Timing Code -------------------------
*
*
*  Purpose
*  =======
*
*  CHGEQZ implements a single-shift version of the QZ
*  method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i)
*  of the equation
*
*       det( A - w(i) B ) = 0
*
*  If JOB='S', then the pair (A,B) is simultaneously
*  reduced to Schur form (i.e., A and B are both upper triangular) by
*  applying one unitary tranformation (usually called Q) on the left and
*  another (usually called Z) on the right.  The diagonal elements of
*  A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
*
*  If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary
*  transformations used to reduce (A,B) are accumulated into the arrays
*  Q and Z s.t.:
*
*       Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
*       Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
*
*  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
*       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
*       pp. 241--256.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          = 'E': compute only ALPHA and BETA.  A and B will not
*                 necessarily be put into generalized Schur form.
*          = 'S': put A and B into generalized Schur form, as well
*                 as computing ALPHA and BETA.
*
*  COMPQ   (input) CHARACTER*1
*          = 'N': do not modify Q.
*          = 'V': multiply the array Q on the right by the conjugate
*                 transpose of the unitary tranformation that is
*                 applied to the left side of A and B to reduce them
*                 to Schur form.
*          = 'I': like COMPQ='V', except that Q will be initialized to
*                 the identity first.
*
*  COMPZ   (input) CHARACTER*1
*          = 'N': do not modify Z.
*          = 'V': multiply the array Z on the right by the unitary
*                 tranformation that is applied to the right side of
*                 A and B to reduce them to Schur form.
*          = 'I': like COMPZ='V', except that Z will be initialized to
*                 the identity first.
*
*  N       (input) INTEGER
*          The order of the matrices A, B, Q, and Z.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) INTEGER
*          It is assumed that A is already upper triangular in rows and
*          columns 1:ILO-1 and IHI+1:N.
*          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
*
*  A       (input/output) COMPLEX array, dimension (LDA, N)
*          On entry, the N-by-N upper Hessenberg matrix A.  Elements
*          below the subdiagonal must be zero.
*          If JOB='S', then on exit A and B will have been
*             simultaneously reduced to upper triangular form.
*          If JOB='E', then on exit A will have been destroyed.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max( 1, N ).
*
*  B       (input/output) COMPLEX array, dimension (LDB, N)
*          On entry, the N-by-N upper triangular matrix B.  Elements
*          below the diagonal must be zero.
*          If JOB='S', then on exit A and B will have been
*             simultaneously reduced to upper triangular form.
*          If JOB='E', then on exit B will have been destroyed.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max( 1, N ).
*
*  ALPHA   (output) COMPLEX array, dimension (N)
*          The diagonal elements of A when the pair (A,B) has been
*          reduced to Schur form.  ALPHA(i)/BETA(i) i=1,...,N
*          are the generalized eigenvalues.
*
*  BETA    (output) COMPLEX array, dimension (N)
*          The diagonal elements of B when the pair (A,B) has been
*          reduced to Schur form.  ALPHA(i)/BETA(i) i=1,...,N
*          are the generalized eigenvalues.  A and B are normalized
*          so that BETA(1),...,BETA(N) are non-negative real numbers.
*
*  Q       (input/output) COMPLEX array, dimension (LDQ, N)
*          If COMPQ='N', then Q will not be referenced.
*          If COMPQ='V' or 'I', then the conjugate transpose of the
*             unitary transformations which are applied to A and B on
*             the left will be applied to the array Q on the right.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= 1.
*          If COMPQ='V' or 'I', then LDQ >= N.
*
*  Z       (input/output) COMPLEX array, dimension (LDZ, N)
*          If COMPZ='N', then Z will not be referenced.
*          If COMPZ='V' or 'I', then the unitary transformations which
*             are applied to A and B on the right will be applied to the
*             array Z on the right.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1.
*          If COMPZ='V' or 'I', then LDZ >= N.
*
*  WORK    (workspace/output) COMPLEX array, dimension (LWORK)
*          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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.
*
*  RWORK   (workspace) REAL array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          = 1,...,N: the QZ iteration did not converge.  (A,B) 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.  (A,B) is not
*                     in Schur form, but ALPHA(i) and BETA(i),
*                     i=INFO-N+1,...,N should be correct.
*          > 2*N:     various "impossible" errors.
*
*  Further Details
*  ===============
*
*  We assume that complex ABS works as long as its value is less than
*  overflow.
*
*  =====================================================================
*
*     .. 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, NQ, NZ
      REAL               ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
     $                   C, OPST, SAFMIN, TEMP, TEMP2, TEMPR, ULP
      COMPLEX            ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
     $                   CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T,
     $                   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 ..
*     ----------------------- Begin Timing Code ------------------------
      ITCNT = ZERO
*     ------------------------ End Timing Code -------------------------
*
*     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
         NQ = 0
      ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 2
         NQ = N
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ILQ = .TRUE.
         ICOMPQ = 3
         NQ = N
      ELSE
         ICOMPQ = 0
      END IF
*
      IF( LSAME( COMPZ, 'N' ) ) THEN
         ILZ = .FALSE.
         ICOMPZ = 1
         NZ = 0
      ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 2
         NZ = N
      ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
         ILZ = .TRUE.
         ICOMPZ = 3
         NZ = N
      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( LDA.LT.N ) THEN
         INFO = -8
      ELSE IF( LDB.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
*
c     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, A( ILO, ILO ), LDA, RWORK )
      BNORM = CLANHS( 'F', IN, B( ILO, ILO ), LDB, RWORK )
      ATOL = MAX( SAFMIN, ULP*ANORM )
      BTOL = MAX( SAFMIN, ULP*BNORM )
      ASCALE = ONE / MAX( SAFMIN, ANORM )
      BSCALE = ONE / MAX( SAFMIN, BNORM )
*
*     ---------------------- Begin Timing Code -------------------------
*     Count ops for norms, etc.
      OPST = ZERO
      OPS = OPS + REAL( 4*N**2+12*N-5 )
*     ----------------------- End Timing Code --------------------------
*
*
*
*     Set Eigenvalues IHI+1:N
*
      DO 10 J = IHI + 1, N
         ABSB = ABS( B( J, J ) )
         IF( ABSB.GT.SAFMIN ) THEN
            SIGNBC = CONJG( B( J, J ) / ABSB )
            B( J, J ) = ABSB
            IF( ILSCHR ) THEN
               CALL CSCAL( J-1, SIGNBC, B( 1, J ), 1 )
               CALL CSCAL( J, SIGNBC, A( 1, J ), 1 )
*              ----------------- Begin Timing Code ---------------------
               OPST = OPST + REAL( 12*( J-1 ) )
*              ------------------ End Timing Code ----------------------
            ELSE
               A( J, J ) = A( J, J )*SIGNBC
            END IF
            IF( ILZ )
     $         CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 )
*           ------------------- Begin Timing Code ----------------------
            OPST = OPST + REAL( 6*NZ+13 )
*           -------------------- End Timing Code -----------------------
         ELSE
            B( J, J ) = CZERO
         END IF
         ALPHA( J ) = A( J, J )
         BETA( J ) = B( 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: A(j,j-1)=0  or  j=ILO
*           2: B(j,j)=0
*
*        Special case: j=ILAST
*
         IF( ILAST.EQ.ILO ) THEN
            GO TO 60
         ELSE
            IF( ABS1( A( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
               A( ILAST, ILAST-1 ) = CZERO
               GO TO 60
            END IF
         END IF
*
         IF( ABS( B( ILAST, ILAST ) ).LE.BTOL ) THEN
            B( ILAST, ILAST ) = CZERO
            GO TO 50