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Cloned library LAPACK-3.11.0 with extra build files for internal package management.
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LAPACK-3.11.0/SRC/dlaqz0.f

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*> \brief \b DLAQZ0
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLAQZ0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqz0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqz0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqz0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
* $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
* $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
* $ INFO )
* IMPLICIT NONE
*
* Arguments
* CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
* INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
* $ REC
*
* INTEGER, INTENT( OUT ) :: INFO
*
* DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
* $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
* $ ALPHAI( * ), BETA( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLAQZ0 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 DGGHRD.
*>
*> 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 DGGHRD 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.
*>
*> Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
*> Algorithm with Aggressive Early Deflation", SIAM J. Numer.
*> Anal., 29(2006), pp. 199--227.
*>
*> Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
*> multipole rational QZ method with aggressive early deflation"
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTS
*> \verbatim
*> WANTS is CHARACTER*1
*> = 'E': Compute eigenvalues only;
*> = 'S': Compute eigenvalues and the Schur form.
*> \endverbatim
*>
*> \param[in] WANTQ
*> \verbatim
*> WANTQ 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 (A,B) is returned;
*> = 'V': Q must contain an orthogonal matrix Q1 on entry and
*> the product Q1*Q is returned.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is 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 (A,B) is returned;
*> = 'V': Z must contain an orthogonal matrix Z1 on entry and
*> the product Z1*Z is returned.
*> \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 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] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the N-by-N upper Hessenberg matrix A.
*> On exit, if JOB = 'S', A contains the upper quasi-triangular
*> matrix S from the generalized Schur factorization.
*> If JOB = 'E', the diagonal blocks of A match those of S, but
*> the rest of A is unspecified.
*> \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 DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the N-by-N upper triangular matrix B.
*> On exit, if JOB = 'S', B 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 A(j+1,j) is
*> non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
*> B(j+1,j+1) > 0.
*> If JOB = 'E', the diagonal blocks of B match those of P, but
*> the rest of B is unspecified.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue
*> of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION 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).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION 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.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is DOUBLE PRECISION array, dimension (LDQ, N)
*> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
*> the reduction of (A,B) to generalized Hessenberg form.
*> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
*> vectors of (A,B), and if COMPQ = 'V', the orthogonal 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 DOUBLE PRECISION 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'.
*> \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 DOUBLE PRECISION 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[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
*> = 1,...,N: the QZ iteration did not converge. (A,B) is not
*> in Schur form, but ALPHAR(i), ALPHAI(i), and
*> BETA(i), i=INFO+1,...,N should be correct.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Thijs Steel, KU Leuven
*
*> \date May 2020
*
*> \ingroup doubleGEcomputational
*>
* =====================================================================
RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
$ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
$ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
$ INFO )
IMPLICIT NONE
* Arguments
CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
$ REC
INTEGER, INTENT( OUT ) :: INFO
DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
$ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
$ ALPHAI( * ), BETA( * ), WORK( * )
* Parameters
DOUBLE PRECISION :: ZERO, ONE, HALF
PARAMETER( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
* Local scalars
DOUBLE PRECISION :: SMLNUM, ULP, ESHIFT, SAFMIN, SAFMAX, C1, S1,
$ TEMP, SWAP, BNORM, BTOL
INTEGER :: ISTART, ISTOP, IITER, MAXIT, ISTART2, K, LD, NSHIFTS,
$ NBLOCK, NW, NMIN, NIBBLE, N_UNDEFLATED, N_DEFLATED,
$ NS, SWEEP_INFO, SHIFTPOS, LWORKREQ, K2, ISTARTM,
$ ISTOPM, IWANTS, IWANTQ, IWANTZ, NORM_INFO, AED_INFO,
$ NWR, NBR, NSR, ITEMP1, ITEMP2, RCOST, I
LOGICAL :: ILSCHUR, ILQ, ILZ
CHARACTER :: JBCMPZ*3
* External Functions
EXTERNAL :: XERBLA, DHGEQZ, DLASET, DLAQZ3, DLAQZ4,
$ DLARTG, DROT
DOUBLE PRECISION, EXTERNAL :: DLAMCH, DLANHS
LOGICAL, EXTERNAL :: LSAME
INTEGER, EXTERNAL :: ILAENV
*
* Decode wantS,wantQ,wantZ
*
IF( LSAME( WANTS, 'E' ) ) THEN
ILSCHUR = .FALSE.
IWANTS = 1
ELSE IF( LSAME( WANTS, 'S' ) ) THEN
ILSCHUR = .TRUE.
IWANTS = 2
ELSE
IWANTS = 0
END IF
IF( LSAME( WANTQ, 'N' ) ) THEN
ILQ = .FALSE.
IWANTQ = 1
ELSE IF( LSAME( WANTQ, 'V' ) ) THEN
ILQ = .TRUE.
IWANTQ = 2
ELSE IF( LSAME( WANTQ, 'I' ) ) THEN
ILQ = .TRUE.
IWANTQ = 3
ELSE
IWANTQ = 0
END IF
IF( LSAME( WANTZ, 'N' ) ) THEN
ILZ = .FALSE.
IWANTZ = 1
ELSE IF( LSAME( WANTZ, 'V' ) ) THEN
ILZ = .TRUE.
IWANTZ = 2
ELSE IF( LSAME( WANTZ, 'I' ) ) THEN
ILZ = .TRUE.
IWANTZ = 3
ELSE
IWANTZ = 0
END IF
*
* Check Argument Values
*
INFO = 0
IF( IWANTS.EQ.0 ) THEN
INFO = -1
ELSE IF( IWANTQ.EQ.0 ) THEN
INFO = -2
ELSE IF( IWANTZ.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 = -15
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -17
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAQZ0', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = DBLE( 1 )
RETURN
END IF
*
* Get the parameters
*
JBCMPZ( 1:1 ) = WANTS
JBCMPZ( 2:2 ) = WANTQ
JBCMPZ( 3:3 ) = WANTZ
NMIN = ILAENV( 12, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = ILAENV( 13, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
NWR = MAX( 2, NWR )
NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
NIBBLE = ILAENV( 14, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = ILAENV( 15, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
RCOST = ILAENV( 17, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
ITEMP1 = INT( NSR/SQRT( 1+2*NSR/( DBLE( RCOST )/100*N ) ) )
ITEMP1 = ( ( ITEMP1-1 )/4 )*4+4
NBR = NSR+ITEMP1
IF( N .LT. NMIN .OR. REC .GE. 2 ) THEN
CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
RETURN
END IF
*
* Find out required workspace
*
* Workspace query to dlaqz3
NW = MAX( NWR, NMIN )
CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB,
$ Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED, ALPHAR,
$ ALPHAI, BETA, WORK, NW, WORK, NW, WORK, -1, REC,
$ AED_INFO )
ITEMP1 = INT( WORK( 1 ) )
* Workspace query to dlaqz4
CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSR, NBR, ALPHAR,
$ ALPHAI, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK,
$ NBR, WORK, NBR, WORK, -1, SWEEP_INFO )
ITEMP2 = INT( WORK( 1 ) )
LWORKREQ = MAX( ITEMP1+2*NW**2, ITEMP2+2*NBR**2 )
IF ( LWORK .EQ.-1 ) THEN
WORK( 1 ) = DBLE( LWORKREQ )
RETURN
ELSE IF ( LWORK .LT. LWORKREQ ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DLAQZ0', INFO )
RETURN
END IF
*
* Initialize Q and Z
*
IF( IWANTQ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Q, LDQ )
IF( IWANTZ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Z, LDZ )
* Get machine constants
SAFMIN = DLAMCH( 'SAFE MINIMUM' )
SAFMAX = ONE/SAFMIN
ULP = DLAMCH( 'PRECISION' )
SMLNUM = SAFMIN*( DBLE( N )/ULP )
BNORM = DLANHS( 'F', IHI-ILO+1, B( ILO, ILO ), LDB, WORK )
BTOL = MAX( SAFMIN, ULP*BNORM )
ISTART = ILO
ISTOP = IHI
MAXIT = 3*( IHI-ILO+1 )
LD = 0
DO IITER = 1, MAXIT
IF( IITER .GE. MAXIT ) THEN
INFO = ISTOP+1
GOTO 80
END IF
IF ( ISTART+1 .GE. ISTOP ) THEN
ISTOP = ISTART
EXIT
END IF
* Check deflations at the end
IF ( ABS( A( ISTOP-1, ISTOP-2 ) ) .LE. MAX( SMLNUM,
$ ULP*( ABS( A( ISTOP-1, ISTOP-1 ) )+ABS( A( ISTOP-2,
$ ISTOP-2 ) ) ) ) ) THEN
A( ISTOP-1, ISTOP-2 ) = ZERO
ISTOP = ISTOP-2
LD = 0
ESHIFT = ZERO
ELSE IF ( ABS( A( ISTOP, ISTOP-1 ) ) .LE. MAX( SMLNUM,
$ ULP*( ABS( A( ISTOP, ISTOP ) )+ABS( A( ISTOP-1,
$ ISTOP-1 ) ) ) ) ) THEN
A( ISTOP, ISTOP-1 ) = ZERO
ISTOP = ISTOP-1
LD = 0
ESHIFT = ZERO
END IF
* Check deflations at the start
IF ( ABS( A( ISTART+2, ISTART+1 ) ) .LE. MAX( SMLNUM,
$ ULP*( ABS( A( ISTART+1, ISTART+1 ) )+ABS( A( ISTART+2,
$ ISTART+2 ) ) ) ) ) THEN
A( ISTART+2, ISTART+1 ) = ZERO
ISTART = ISTART+2
LD = 0
ESHIFT = ZERO
ELSE IF ( ABS( A( ISTART+1, ISTART ) ) .LE. MAX( SMLNUM,
$ ULP*( ABS( A( ISTART, ISTART ) )+ABS( A( ISTART+1,
$ ISTART+1 ) ) ) ) ) THEN
A( ISTART+1, ISTART ) = ZERO
ISTART = ISTART+1
LD = 0
ESHIFT = ZERO
END IF
IF ( ISTART+1 .GE. ISTOP ) THEN
EXIT
END IF
* Check interior deflations
ISTART2 = ISTART
DO K = ISTOP, ISTART+1, -1
IF ( ABS( A( K, K-1 ) ) .LE. MAX( SMLNUM, ULP*( ABS( A( K,
$ K ) )+ABS( A( K-1, K-1 ) ) ) ) ) THEN
A( K, K-1 ) = ZERO
ISTART2 = K
EXIT
END IF
END DO
* Get range to apply rotations to
IF ( ILSCHUR ) THEN
ISTARTM = 1
ISTOPM = N
ELSE
ISTARTM = ISTART2
ISTOPM = ISTOP
END IF
* Check infinite eigenvalues, this is done without blocking so might
* slow down the method when many infinite eigenvalues are present
K = ISTOP
DO WHILE ( K.GE.ISTART2 )
IF( ABS( B( K, K ) ) .LT. BTOL ) THEN
* A diagonal element of B is negligible, move it
* to the top and deflate it
DO K2 = K, ISTART2+1, -1
CALL DLARTG( B( K2-1, K2 ), B( K2-1, K2-1 ), C1, S1,
$ TEMP )
B( K2-1, K2 ) = TEMP
B( K2-1, K2-1 ) = ZERO
CALL DROT( K2-2-ISTARTM+1, B( ISTARTM, K2 ), 1,
$ B( ISTARTM, K2-1 ), 1, C1, S1 )
CALL DROT( MIN( K2+1, ISTOP )-ISTARTM+1, A( ISTARTM,
$ K2 ), 1, A( ISTARTM, K2-1 ), 1, C1, S1 )
IF ( ILZ ) THEN
CALL DROT( N, Z( 1, K2 ), 1, Z( 1, K2-1 ), 1, C1,
$ S1 )
END IF
IF( K2.LT.ISTOP ) THEN
CALL DLARTG( A( K2, K2-1 ), A( K2+1, K2-1 ), C1,
$ S1, TEMP )
A( K2, K2-1 ) = TEMP
A( K2+1, K2-1 ) = ZERO
CALL DROT( ISTOPM-K2+1, A( K2, K2 ), LDA, A( K2+1,
$ K2 ), LDA, C1, S1 )
CALL DROT( ISTOPM-K2+1, B( K2, K2 ), LDB, B( K2+1,
$ K2 ), LDB, C1, S1 )
IF( ILQ ) THEN
CALL DROT( N, Q( 1, K2 ), 1, Q( 1, K2+1 ), 1,
$ C1, S1 )
END IF
END IF
END DO
IF( ISTART2.LT.ISTOP )THEN
CALL DLARTG( A( ISTART2, ISTART2 ), A( ISTART2+1,
$ ISTART2 ), C1, S1, TEMP )
A( ISTART2, ISTART2 ) = TEMP
A( ISTART2+1, ISTART2 ) = ZERO
CALL DROT( ISTOPM-( ISTART2+1 )+1, A( ISTART2,
$ ISTART2+1 ), LDA, A( ISTART2+1,
$ ISTART2+1 ), LDA, C1, S1 )
CALL DROT( ISTOPM-( ISTART2+1 )+1, B( ISTART2,
$ ISTART2+1 ), LDB, B( ISTART2+1,
$ ISTART2+1 ), LDB, C1, S1 )
IF( ILQ ) THEN
CALL DROT( N, Q( 1, ISTART2 ), 1, Q( 1,
$ ISTART2+1 ), 1, C1, S1 )
END IF
END IF
ISTART2 = ISTART2+1
END IF
K = K-1
END DO
* istart2 now points to the top of the bottom right
* unreduced Hessenberg block
IF ( ISTART2 .GE. ISTOP ) THEN
ISTOP = ISTART2-1
LD = 0
ESHIFT = ZERO
CYCLE
END IF
NW = NWR
NSHIFTS = NSR
NBLOCK = NBR
IF ( ISTOP-ISTART2+1 .LT. NMIN ) THEN
* Setting nw to the size of the subblock will make AED deflate
* all the eigenvalues. This is slightly more efficient than just
* using DHGEQZ because the off diagonal part gets updated via BLAS.
IF ( ISTOP-ISTART+1 .LT. NMIN ) THEN
NW = ISTOP-ISTART+1
ISTART2 = ISTART
ELSE
NW = ISTOP-ISTART2+1
END IF
END IF
*
* Time for AED
*
CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NW, A, LDA,
$ B, LDB, Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED,
$ ALPHAR, ALPHAI, BETA, WORK, NW, WORK( NW**2+1 ),
$ NW, WORK( 2*NW**2+1 ), LWORK-2*NW**2, REC,
$ AED_INFO )
IF ( N_DEFLATED > 0 ) THEN
ISTOP = ISTOP-N_DEFLATED
LD = 0
ESHIFT = ZERO
END IF
IF ( 100*N_DEFLATED > NIBBLE*( N_DEFLATED+N_UNDEFLATED ) .OR.
$ ISTOP-ISTART2+1 .LT. NMIN ) THEN
* AED has uncovered many eigenvalues. Skip a QZ sweep and run
* AED again.
CYCLE
END IF
LD = LD+1
NS = MIN( NSHIFTS, ISTOP-ISTART2 )
NS = MIN( NS, N_UNDEFLATED )
SHIFTPOS = ISTOP-N_UNDEFLATED+1
*
* Shuffle shifts to put double shifts in front
* This ensures that we don't split up a double shift
*
DO I = SHIFTPOS, SHIFTPOS+N_UNDEFLATED-1, 2
IF( ALPHAI( I ).NE.-ALPHAI( I+1 ) ) THEN
*
SWAP = ALPHAR( I )
ALPHAR( I ) = ALPHAR( I+1 )
ALPHAR( I+1 ) = ALPHAR( I+2 )
ALPHAR( I+2 ) = SWAP
SWAP = ALPHAI( I )
ALPHAI( I ) = ALPHAI( I+1 )
ALPHAI( I+1 ) = ALPHAI( I+2 )
ALPHAI( I+2 ) = SWAP
SWAP = BETA( I )
BETA( I ) = BETA( I+1 )
BETA( I+1 ) = BETA( I+2 )
BETA( I+2 ) = SWAP
END IF
END DO
IF ( MOD( LD, 6 ) .EQ. 0 ) THEN
*
* Exceptional shift. Chosen for no particularly good reason.
*
IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ISTOP,
$ ISTOP-1 ) ).LT.ABS( A( ISTOP-1, ISTOP-1 ) ) ) THEN
ESHIFT = A( ISTOP, ISTOP-1 )/B( ISTOP-1, ISTOP-1 )
ELSE
ESHIFT = ESHIFT+ONE/( SAFMIN*DBLE( MAXIT ) )
END IF
ALPHAR( SHIFTPOS ) = ONE
ALPHAR( SHIFTPOS+1 ) = ZERO
ALPHAI( SHIFTPOS ) = ZERO
ALPHAI( SHIFTPOS+1 ) = ZERO
BETA( SHIFTPOS ) = ESHIFT
BETA( SHIFTPOS+1 ) = ESHIFT
NS = 2
END IF
*
* Time for a QZ sweep
*
CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NS, NBLOCK,
$ ALPHAR( SHIFTPOS ), ALPHAI( SHIFTPOS ),
$ BETA( SHIFTPOS ), A, LDA, B, LDB, Q, LDQ, Z, LDZ,
$ WORK, NBLOCK, WORK( NBLOCK**2+1 ), NBLOCK,
$ WORK( 2*NBLOCK**2+1 ), LWORK-2*NBLOCK**2,
$ SWEEP_INFO )
END DO
*
* Call DHGEQZ to normalize the eigenvalue blocks and set the eigenvalues
* If all the eigenvalues have been found, DHGEQZ will not do any iterations
* and only normalize the blocks. In case of a rare convergence failure,
* the single shift might perform better.
*
80 CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
$ NORM_INFO )
INFO = NORM_INFO
END SUBROUTINE