shgeqz.f

famous linear algebra library (LAPACK) ports to windows

      SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
     $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
     $                   LWORK, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
*     ..
*     .. Array Arguments ..
      REAL               ALPHAI( * ), ALPHAR( * ), BETA( * ),
     $                   H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
*  where H is an upper Hessenberg matrix and T is upper triangular,
*  using the double-shift QZ method.
*  Matrix pairs of this type are produced by the reduction to
*  generalized upper Hessenberg form of a real matrix pair (A,B):
*
*     A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
*
*  as computed by SGGHRD.
*
*  If JOB='S', then the Hessenberg-triangular pair (H,T) is
*  also reduced to generalized Schur form,
*  
*     H = Q*S*Z**T,  T = Q*P*Z**T,
*  
*  where Q and Z are orthogonal matrices, P is an upper triangular
*  matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
*  diagonal blocks.
*
*  The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
*  (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
*  eigenvalues.
*
*  Additionally, the 2-by-2 upper triangular diagonal blocks of P
*  corresponding to 2-by-2 blocks of S are reduced to positive diagonal
*  form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
*  P(j,j) > 0, and P(j+1,j+1) > 0.
*
*  Optionally, the orthogonal matrix Q from the generalized Schur
*  factorization may be postmultiplied into an input matrix Q1, and the
*  orthogonal matrix Z may be postmultiplied into an input matrix Z1.
*  If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
*  the matrix pair (A,B) to generalized upper Hessenberg form, then the
*  output matrices Q1*Q and Z1*Z are the orthogonal factors from the
*  generalized Schur factorization of (A,B):
*
*     A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
*  
*  To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
*  of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
*  complex and beta real.
*  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.
*  Real eigenvalues 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.
*
*  Arguments
*  =========
*
*  JOB     (input) CHARACTER*1
*          = 'E': Compute eigenvalues only;
*          = 'S': Compute eigenvalues and the Schur form. 
*
*  COMPQ   (input) 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 an orthogonal matrix Q1 on entry and
*                 the product Q1*Q is returned.
*
*  COMPZ   (input) CHARACTER*1
*          = 'N': Right Schur vectors (Z) are not computed;
*          = 'I': Z is initialized to the unit matrix and the matrix Z
*                 of right Schur vectors of (H,T) is returned;
*          = 'V': Z must contain an orthogonal matrix Z1 on entry and
*                 the product Z1*Z is returned.
*
*  N       (input) INTEGER
*          The order of the matrices H, T, Q, and Z.  N >= 0.
*
*  ILO     (input) INTEGER
*  IHI     (input) 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.
*
*  H       (input/output) REAL array, dimension (LDH, N)
*          On entry, the N-by-N upper Hessenberg matrix H.
*          On exit, if JOB = 'S', H contains the upper quasi-triangular
*          matrix S from the generalized Schur factorization;
*          2-by-2 diagonal blocks (corresponding to complex conjugate
*          pairs of eigenvalues) are returned in standard form, with
*          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
*          If JOB = 'E', the diagonal blocks of H match those of S, but
*          the rest of H is unspecified.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H.  LDH >= max( 1, N ).
*
*  T       (input/output) REAL 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;
*          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
*          are reduced to positive diagonal form, i.e., if H(j+1,j) is
*          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
*          T(j+1,j+1) > 0.
*          If JOB = 'E', the diagonal blocks of T match those of P, but
*          the rest of T is unspecified.
*
*  LDT     (input) INTEGER
*          The leading dimension of the array T.  LDT >= max( 1, N ).
*
*  ALPHAR  (output) REAL array, dimension (N)
*          The real parts of each scalar alpha defining an eigenvalue
*          of GNEP.
*
*  ALPHAI  (output) REAL array, dimension (N)
*          The imaginary parts of each scalar alpha defining an
*          eigenvalue of GNEP.
*          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*          positive, then the j-th and (j+1)-st eigenvalues are a
*          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
*
*  BETA    (output) REAL array, dimension (N)
*          The scalars beta that define the eigenvalues of GNEP.
*          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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.
*
*  Q       (input/output) REAL array, dimension (LDQ, N)
*          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
*          the reduction of (A,B) to generalized Hessenberg form.
*          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
*          vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
*          of left Schur vectors of (A,B).
*          Not referenced if COMPZ = 'N'.
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= 1.
*          If COMPQ='V' or 'I', then LDQ >= N.
*
*  Z       (input/output) REAL array, dimension (LDZ, N)
*          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
*          the reduction of (A,B) to generalized Hessenberg form.
*          On exit, if COMPZ = 'I', the orthogonal matrix of
*          right Schur vectors of (H,T), and if COMPZ = 'V', the
*          orthogonal matrix of right Schur vectors of (A,B).
*          Not referenced if COMPZ = 'N'.
*
*  LDZ     (input) INTEGER
*          The leading dimension of the array Z.  LDZ >= 1.
*          If COMPZ='V' or 'I', then LDZ >= N.
*
*  WORK    (workspace/output) REAL array, dimension (MAX(1,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.
*
*  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.  (H,T) is not
*                     in Schur form, but ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
*                     BETA(i), i=INFO-N+1,...,N should be correct.
*
*  Further Details
*  ===============
*
*  Iteration counters:
*
*  JITER  -- counts iterations.
*  IITER  -- counts iterations run since ILAST was last
*            changed.  This is therefore reset only when a 1-by-1 or
*            2-by-2 block deflates off the bottom.
*
*  =====================================================================
*
*     .. Parameters ..
*    $                     SAFETY = 1.0E+0 )
      REAL               HALF, ZERO, ONE, SAFETY
      PARAMETER          ( HALF = 0.5E+0, ZERO = 0.0E+0, ONE = 1.0E+0,
     $                   SAFETY = 1.0E+2 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
     $                   LQUERY
      INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
     $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
     $                   JR, MAXIT
      REAL               A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
     $                   AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
     $                   AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
     $                   B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
     $                   BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
     $                   CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
     $                   SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
     $                   TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
     $                   U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
     $                   WR2
*     ..
*     .. Local Arrays ..
      REAL               V( 3 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANHS, SLAPY2, SLAPY3
      EXTERNAL           LSAME, SLAMCH, SLANHS, SLAPY2, SLAPY3
*     ..
*     .. External Subroutines ..
      EXTERNAL           SLAG2, SLARFG, SLARTG, SLASET, SLASV2, SROT,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, MIN, REAL, SQRT
*     ..
*     .. 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 = -15
      ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
         INFO = -17
      ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
         INFO = -19
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SHGEQZ', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.LE.0 ) THEN
         WORK( 1 ) = REAL( 1 )
         RETURN
      END IF
*
*     Initialize Q and Z
*
      IF( ICOMPQ.EQ.3 )
     $   CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
      IF( ICOMPZ.EQ.3 )
     $   CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
*     Machine Constants
*
      IN = IHI + 1 - ILO
      SAFMIN = SLAMCH( 'S' )
      SAFMAX = ONE / SAFMIN
      ULP = SLAMCH( 'E' )*SLAMCH( 'B' )
      ANORM = SLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
      BNORM = SLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
      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 30 J = IHI + 1, N
         IF( T( J, J ).LT.ZERO ) THEN
            IF( ILSCHR ) THEN
               DO 10 JR = 1, J
                  H( JR, J ) = -H( JR, J )
                  T( JR, J ) = -T( JR, J )
   10          CONTINUE
            ELSE
               H( J, J ) = -H( J, J )
               T( J, J ) = -T( J, J )
            END IF
            IF( ILZ ) THEN
               DO 20 JR = 1, N
                  Z( JR, J ) = -Z( JR, J )
   20          CONTINUE
            END IF
         END IF
         ALPHAR( J ) = H( J, J )
         ALPHAI( J ) = ZERO
         BETA( J ) = T( J, J )
   30 CONTINUE
*
*     If IHI < ILO, skip QZ steps
*
      IF( IHI.LT.ILO )
     $   GO TO 380
*
*     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 = ZERO
      MAXIT = 30*( IHI-ILO+1 )
*
      DO 360 JITER = 1, MAXIT
*
*        Split the matrix if possible.
*
*        Two tests:
*           1: H(j,j-1)=0  or  j=ILO
*           2: T(j,j)=0
*
         IF( ILAST.EQ.ILO ) THEN