zlaqz2.f

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*> \brief \b ZLAQZ2
*
*  =========== DOCUMENTATION ===========
*
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*
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*      SUBROUTINE ZLAQZ2( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B,
*     $    LDB, Q, LDQ, Z, LDZ, NS, ND, ALPHA, BETA, QC, LDQC, ZC, LDZC,
*     $    WORK, LWORK, RWORK, REC, INFO )
*      IMPLICIT NONE
*
*      Arguments
*      LOGICAL, INTENT( IN ) :: ILSCHUR, ILQ, ILZ
*      INTEGER, INTENT( IN ) :: N, ILO, IHI, NW, LDA, LDB, LDQ, LDZ,
*     $    LDQC, LDZC, LWORK, REC
*
*      COMPLEX*16, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ), Q( LDQ,
*     $    * ), Z( LDZ, * ), ALPHA( * ), BETA( * )
*      INTEGER, INTENT( OUT ) :: NS, ND, INFO
*      COMPLEX*16 :: QC( LDQC, * ), ZC( LDZC, * ), WORK( * )
*      DOUBLE PRECISION :: RWORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZLAQZ2 performs AED
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] ILSCHUR
*> \verbatim
*>          ILSCHUR is LOGICAL
*>              Determines whether or not to update the full Schur form
*> \endverbatim
*> \param[in] ILQ
*> \verbatim
*>          ILQ is LOGICAL
*>              Determines whether or not to update the matrix Q
*> \endverbatim
*>
*> \param[in] ILZ
*> \verbatim
*>          ILZ is LOGICAL
*>              Determines whether or not to update the matrix Z
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, 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 (A,B) which
*>          are to be normalized
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          The desired size of the deflation window.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension (LDA, N)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension (LDB, N)
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is COMPLEX*16 array, dimension (LDQ, N)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is COMPLEX*16 array, dimension (LDZ, N)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*>          NS is INTEGER
*>          The number of unconverged eigenvalues available to
*>          use as shifts.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*>          ND is INTEGER
*>          The number of converged eigenvalues found.
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is COMPLEX*16 array, dimension (N)
*>          Each scalar alpha defining an eigenvalue
*>          of GNEP.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is COMPLEX*16 array, dimension (N)
*>          The scalars beta that define the eigenvalues of GNEP.
*>          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] QC
*> \verbatim
*>          QC is COMPLEX*16 array, dimension (LDQC, NW)
*> \endverbatim
*>
*> \param[in] LDQC
*> \verbatim
*>          LDQC is INTEGER
*> \endverbatim
*>
*> \param[in,out] ZC
*> \verbatim
*>          ZC is COMPLEX*16 array, dimension (LDZC, NW)
*> \endverbatim
*>
*> \param[in] LDZC
*> \verbatim
*>          LDZ is INTEGER
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 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 DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[in] REC
*> \verbatim
*>          REC is INTEGER
*>             REC indicates the current recursion level. Should be set
*>             to 0 on first call.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0: successful exit
*>          < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Thijs Steel, KU Leuven
*
*> \date May 2020
*
*> \ingroup complex16GEcomputational
*>
*  =====================================================================
      RECURSIVE SUBROUTINE zlaqz2( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW,
     $                             A, LDA, B, LDB, Q, LDQ, Z, LDZ, NS,
     $                             ND, ALPHA, BETA, QC, LDQC, ZC, LDZC,
     $                             WORK, LWORK, RWORK, REC, INFO )
      IMPLICIT NONE
 
*     Arguments
      LOGICAL, INTENT( IN ) :: ilschur, ilq, ilz
      INTEGER, INTENT( IN ) :: n, ilo, ihi, nw, lda, ldb, ldq, ldz,
     $         ldqc, ldzc, lwork, rec
 
      COMPLEX*16, INTENT( INOUT ) :: A( lda, * ), b( ldb, * ), q( ldq,
     $   * ), z( ldz, * ), alpha( * ), beta( * )
      INTEGER, INTENT( OUT ) :: NS, ND, info
      COMPLEX*16 :: qc( ldqc, * ), ZC( ldzc, * ), work( * )
      DOUBLE PRECISION :: rwork( * )
 
*     Parameters
      COMPLEX*16         czero, cone
      PARAMETER          ( czero = ( 0.0d+0, 0.0d+0 ), cone = ( 1.0d+0,
     $                     0.0d+0 ) )
      DOUBLE PRECISION :: zero, one, half
      parameter( zero = 0.0d0, one = 1.0d0, half = 0.5d0 )
 
*     Local Scalars
      INTEGER :: jw, kwtop, kwbot, istopm, istartm, k, k2, ztgexc_info,
     $           ifst, ilst, lworkreq, qz_small_info
      DOUBLE PRECISION ::smlnum, ulp, safmin, safmax, c1, tempr
      COMPLEX*16 :: s, s1, temp
 
*     External Functions
      EXTERNAL :: xerbla, zlaqz0, zlaqz1, dlabad, zlacpy, zlaset, zgemm,
     $            ztgexc, zlartg, zrot
      DOUBLE PRECISION, EXTERNAL :: dlamch
 
      info = 0
 
*     Set up deflation window
      jw = min( nw, ihi-ilo+1 )
      kwtop = ihi-jw+1
      IF ( kwtop .EQ. ilo ) THEN
         s = czero
      ELSE
         s = a( kwtop, kwtop-1 )
      END IF
 
*     Determine required workspace
      ifst = 1
      ilst = jw
      CALL zlaqz0( 'S', 'V', 'V', jw, 1, jw, a( kwtop, kwtop ), lda,
     $             b( kwtop, kwtop ), ldb, alpha, beta, qc, ldqc, zc,
     $             ldzc, work, -1, rwork, rec+1, qz_small_info )
      lworkreq = int( work( 1 ) )+2*jw**2
      lworkreq = max( lworkreq, n*nw, 2*nw**2+n )
      IF ( lwork .EQ.-1 ) THEN
*        workspace query, quick return
         work( 1 ) = lworkreq
         RETURN
      ELSE IF ( lwork .LT. lworkreq ) THEN
         info = -26
      END IF
 
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZLAQZ2', -info )
         RETURN
      END IF
 
*     Get machine constants
      safmin = dlamch( 'SAFE MINIMUM' )
      safmax = one/safmin
      CALL dlabad( safmin, safmax )
      ulp = dlamch( 'PRECISION' )
      smlnum = safmin*( dble( n )/ulp )
 
      IF ( ihi .EQ. kwtop ) THEN
*        1 by 1 deflation window, just try a regular deflation
         alpha( kwtop ) = a( kwtop, kwtop )
         beta( kwtop ) = b( kwtop, kwtop )
         ns = 1
         nd = 0
         IF ( abs( s ) .LE. max( smlnum, ulp*abs( a( kwtop,
     $      kwtop ) ) ) ) THEN
            ns = 0
            nd = 1
            IF ( kwtop .GT. ilo ) THEN
               a( kwtop, kwtop-1 ) = czero
            END IF
         END IF
      END IF
 
 
*     Store window in case of convergence failure
      CALL zlacpy( 'ALL', jw, jw, a( kwtop, kwtop ), lda, work, jw )
      CALL zlacpy( 'ALL', jw, jw, b( kwtop, kwtop ), ldb, work( jw**2+
     $             1 ), jw )
 
*     Transform window to real schur form
      CALL zlaset( 'full', JW, JW, CZERO, CONE, QC, LDQC )
      CALL ZLASET( 'full', JW, JW, CZERO, CONE, ZC, LDZC )
      CALL ZLAQZ0( 's', 'v', 'v', JW, 1, JW, A( KWTOP, KWTOP ), LDA,
     $             B( KWTOP, KWTOP ), LDB, ALPHA, BETA, QC, LDQC, ZC,
     $             LDZC, WORK( 2*JW**2+1 ), LWORK-2*JW**2, RWORK,
     $             REC+1, QZ_SMALL_INFO )
 
.NE.      IF( QZ_SMALL_INFO  0 ) THEN
*        Convergence failure, restore the window and exit
         ND = 0
         NS = JW-QZ_SMALL_INFO
         CALL ZLACPY( 'all', JW, JW, WORK, JW, A( KWTOP, KWTOP ), LDA )
         CALL ZLACPY( 'all', JW, JW, WORK( JW**2+1 ), JW, B( KWTOP,
     $                KWTOP ), LDB )
         RETURN
      END IF
 
*     Deflation detection loop
.EQ..OR..EQ.      IF ( KWTOP  ILO  S  CZERO ) THEN
         KWBOT = KWTOP-1
      ELSE
         KWBOT = IHI
         K = 1
         K2 = 1
.LE.         DO WHILE ( K  JW )
*              Try to deflate eigenvalue
               TEMPR = ABS( A( KWBOT, KWBOT ) )
.EQ.               IF( TEMPR  ZERO ) THEN
                  TEMPR = ABS( S )
               END IF
.LE.               IF ( ( ABS( S*QC( 1, KWBOT-KWTOP+1 ) ) )  MAX( ULP*
     $            TEMPR, SMLNUM ) ) THEN
*                 Deflatable
                  KWBOT = KWBOT-1
               ELSE
*                 Not deflatable, move out of the way
                  IFST = KWBOT-KWTOP+1
                  ILST = K2
                  CALL ZTGEXC( .TRUE., .TRUE., JW, A( KWTOP, KWTOP ),
     $                         LDA, B( KWTOP, KWTOP ), LDB, QC, LDQC,
     $                         ZC, LDZC, IFST, ILST, ZTGEXC_INFO )
                  K2 = K2+1
               END IF
 
               K = K+1
         END DO
      END IF
 
*     Store eigenvalues
      ND = IHI-KWBOT
      NS = JW-ND
      K = KWTOP
.LE.      DO WHILE ( K  IHI )
         ALPHA( K ) = A( K, K )
         BETA( K ) = B( K, K )
         K = K+1
      END DO
 
.NE..AND..NE.      IF ( KWTOP  ILO  S  CZERO ) THEN
*        Reflect spike back, this will create optimally packed bulges
         A( KWTOP:KWBOT, KWTOP-1 ) = A( KWTOP, KWTOP-1 ) *DCONJG( QC( 1,
     $      1:JW-ND ) )
         DO K = KWBOT-1, KWTOP, -1
            CALL ZLARTG( A( K, KWTOP-1 ), A( K+1, KWTOP-1 ), C1, S1,
     $                   TEMP )
            A( K, KWTOP-1 ) = TEMP
            A( K+1, KWTOP-1 ) = CZERO
            K2 = MAX( KWTOP, K-1 )
            CALL ZROT( IHI-K2+1, A( K, K2 ), LDA, A( K+1, K2 ), LDA, C1,
     $                 S1 )
            CALL ZROT( IHI-( K-1 )+1, B( K, K-1 ), LDB, B( K+1, K-1 ),
     $                 LDB, C1, S1 )
            CALL ZROT( JW, QC( 1, K-KWTOP+1 ), 1, QC( 1, K+1-KWTOP+1 ),
     $                 1, C1, DCONJG( S1 ) )
         END DO
 
*        Chase bulges down
         ISTARTM = KWTOP
         ISTOPM = IHI
         K = KWBOT-1
.GE.         DO WHILE ( K  KWTOP )
 
*           Move bulge down and remove it
            DO K2 = K, KWBOT-1
               CALL ZLAQZ1( .TRUE., .TRUE., K2, KWTOP, KWTOP+JW-1,
     $                      KWBOT, A, LDA, B, LDB, JW, KWTOP, QC, LDQC,
     $                      JW, KWTOP, ZC, LDZC )
            END DO
 
            K = K-1
         END DO
 
      END IF
 
*     Apply Qc and Zc to rest of the matrix
      IF ( ILSCHUR ) THEN
         ISTARTM = 1
         ISTOPM = N
      ELSE
         ISTARTM = ILO
         ISTOPM = IHI
      END IF
 
      IF ( ISTOPM-IHI > 0 ) THEN
         CALL ZGEMM( 'c', 'n', JW, ISTOPM-IHI, JW, CONE, QC, LDQC,
     $               A( KWTOP, IHI+1 ), LDA, CZERO, WORK, JW )
         CALL ZLACPY( 'all', JW, ISTOPM-IHI, WORK, JW, A( KWTOP,
     $                IHI+1 ), LDA )
         CALL ZGEMM( 'c', 'n', JW, ISTOPM-IHI, JW, CONE, QC, LDQC,
     $               B( KWTOP, IHI+1 ), LDB, CZERO, WORK, JW )
         CALL ZLACPY( 'all', JW, ISTOPM-IHI, WORK, JW, B( KWTOP,
     $                IHI+1 ), LDB )
      END IF
      IF ( ILQ ) THEN
         CALL ZGEMM( 'n', 'n', N, JW, JW, CONE, Q( 1, KWTOP ), LDQ, QC,
     $               LDQC, CZERO, WORK, N )
         CALL ZLACPY( 'all', N, JW, WORK, N, Q( 1, KWTOP ), LDQ )
      END IF
 
      IF ( KWTOP-1-ISTARTM+1 > 0 ) THEN
         CALL ZGEMM( 'n', 'n', KWTOP-ISTARTM, JW, JW, CONE, A( ISTARTM,
     $               KWTOP ), LDA, ZC, LDZC, CZERO, WORK,
     $               KWTOP-ISTARTM )
        CALL ZLACPY( 'all', KWTOP-ISTARTM, JW, WORK, KWTOP-ISTARTM,
     $               A( ISTARTM, KWTOP ), LDA )
         CALL ZGEMM( 'n', 'n', KWTOP-ISTARTM, JW, JW, CONE, B( ISTARTM,
     $               KWTOP ), LDB, ZC, LDZC, CZERO, WORK,
     $               KWTOP-ISTARTM )
        CALL ZLACPY( 'all', KWTOP-ISTARTM, JW, WORK, KWTOP-ISTARTM,
     $               B( ISTARTM, KWTOP ), LDB )
      END IF
      IF ( ILZ ) THEN
         CALL ZGEMM( 'n', 'n', N, JW, JW, CONE, Z( 1, KWTOP ), LDZ, ZC,
     $               LDZC, CZERO, WORK, N )
         CALL ZLACPY( 'all', N, JW, WORK, N, Z( 1, KWTOP ), LDZ )
      END IF
 
      END SUBROUTINE