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+ SUBROUTINE DHGEQZ( 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 ..
+ DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ),
+ $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
+ $ WORK( * ), Z( LDZ, * )
+* ..
+*
+* Purpose
+* =======
+*
+* DHGEQZ 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.
+*
+* 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
+* The real parts of each scalar alpha defining an eigenvalue
+* of GNEP.
+*
+* ALPHAI (output) 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).
+*
+* BETA (output) 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.
+*
+* Q (input/output) DOUBLE PRECISION 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) 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'.
+*
+* LDZ (input) INTEGER
+* The leading dimension of the array Z. LDZ >= 1.
+* If COMPZ='V' or 'I', then LDZ >= N.
+*
+* WORK (workspace/output) DOUBLE PRECISION 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 )
+ DOUBLE PRECISION HALF, ZERO, ONE, SAFETY
+ PARAMETER ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
+ $ SAFETY = 1.0D+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
+ DOUBLE PRECISION 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 ..
+ DOUBLE PRECISION V( 3 )
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ DOUBLE PRECISION DLAMCH, DLANHS, DLAPY2, DLAPY3
+ EXTERNAL LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3
+* ..
+* .. External Subroutines ..
+ EXTERNAL DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT,
+ $ XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, DBLE, MAX, MIN, 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( 'DHGEQZ', -INFO )
+ RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
+ END IF
+*
+* Quick return if possible
+*
+ IF( N.LE.0 ) THEN
+ WORK( 1 ) = DBLE( 1 )
+ RETURN
+ END IF
+*
+* Initialize Q and Z
+*
+ IF( ICOMPQ.EQ.3 )
+ $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
+ IF( ICOMPZ.EQ.3 )
+ $ CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
+*
+* Machine Constants
+*
+ IN = IHI + 1 - ILO
+ SAFMIN = DLAMCH( 'S' )
+ SAFMAX = ONE / SAFMIN
+ ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
+ ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
+ BNORM = DLANHS( '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
+*
+* Special case: j=ILAST
+*
+ GO TO 80
+ ELSE
+ IF( ABS( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
+ H( ILAST, ILAST-1 ) = ZERO
+ GO TO 80
+ END IF
+ END IF
+*
+ IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
+ T( ILAST, ILAST ) = ZERO
+ GO TO 70
+ END IF
+*
+* General case: j<ILAST
+*
+ DO 60 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( ABS( H( J, J-1 ) ).LE.ATOL ) THEN
+ H( J, J-1 ) = ZERO
+ 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 ) = ZERO
+*
+* Test 1a: Check for 2 consecutive small subdiagonals in A
+*
+ ILAZR2 = .FALSE.
+ IF( .NOT.ILAZRO ) THEN
+ TEMP = ABS( H( J, J-1 ) )
+ TEMP2 = ABS( H( J, J ) )
+ TEMPR = MAX( TEMP, TEMP2 )
+ IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
+ TEMP = TEMP / TEMPR
+ TEMP2 = TEMP2 / TEMPR
+ END IF
+ IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
+ $ ( 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 40 JCH = J, ILAST - 1
+ TEMP = H( JCH, JCH )
+ CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
+ $ H( JCH, JCH ) )
+ H( JCH+1, JCH ) = ZERO
+ CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
+ $ H( JCH+1, JCH+1 ), LDH, C, S )
+ CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
+ $ T( JCH+1, JCH+1 ), LDT, C, S )
+ IF( ILQ )
+ $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
+ $ C, S )
+ IF( ILAZR2 )
+ $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
+ ILAZR2 = .FALSE.
+ IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
+ IF( JCH+1.GE.ILAST ) THEN
+ GO TO 80
+ ELSE
+ IFIRST = JCH + 1
+ GO TO 110
+ END IF
+ END IF
+ T( JCH+1, JCH+1 ) = ZERO
+ 40 CONTINUE
+ GO TO 70
+ ELSE
+*
+* Only test 2 passed -- chase the zero to T(ILAST,ILAST)
+* Then process as in the case T(ILAST,ILAST)=0
+*
+ DO 50 JCH = J, ILAST - 1
+ TEMP = T( JCH, JCH+1 )
+ CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
+ $ T( JCH, JCH+1 ) )
+ T( JCH+1, JCH+1 ) = ZERO
+ IF( JCH.LT.ILASTM-1 )
+ $ CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
+ $ T( JCH+1, JCH+2 ), LDT, C, S )
+ CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
+ $ H( JCH+1, JCH-1 ), LDH, C, S )
+ IF( ILQ )
+ $ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
+ $ C, S )
+ TEMP = H( JCH+1, JCH )
+ CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
+ $ H( JCH+1, JCH ) )
+ H( JCH+1, JCH-1 ) = ZERO
+ CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
+ $ H( IFRSTM, JCH-1 ), 1, C, S )
+ CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
+ $ T( IFRSTM, JCH-1 ), 1, C, S )
+ IF( ILZ )
+ $ CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
+ $ C, S )
+ 50 CONTINUE
+ GO TO 70
+ END IF
+ ELSE IF( ILAZRO ) THEN
+*
+* Only test 1 passed -- work on J:ILAST
+*
+ IFIRST = J
+ GO TO 110
+ END IF
+*
+* Neither test passed -- try next J
+*
+ 60 CONTINUE
+*
+* (Drop-through is "impossible")
+*
+ INFO = N + 1
+ GO TO 420
+*
+* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
+* 1x1 block.
+*
+ 70 CONTINUE
+ TEMP = H( ILAST, ILAST )
+ CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
+ $ H( ILAST, ILAST ) )
+ H( ILAST, ILAST-1 ) = ZERO
+ CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
+ $ H( IFRSTM, ILAST-1 ), 1, C, S )
+ CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
+ $ T( IFRSTM, ILAST-1 ), 1, C, S )
+ IF( ILZ )
+ $ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
+*
+* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
+* and BETA
+*
+ 80 CONTINUE
+ IF( T( ILAST, ILAST ).LT.ZERO ) THEN
+ IF( ILSCHR ) THEN
+ DO 90 J = IFRSTM, ILAST
+ H( J, ILAST ) = -H( J, ILAST )
+ T( J, ILAST ) = -T( J, ILAST )
+ 90 CONTINUE
+ ELSE
+ H( ILAST, ILAST ) = -H( ILAST, ILAST )
+ T( ILAST, ILAST ) = -T( ILAST, ILAST )
+ END IF
+ IF( ILZ ) THEN
+ DO 100 J = 1, N
+ Z( J, ILAST ) = -Z( J, ILAST )
+ 100 CONTINUE
+ END IF
+ END IF
+ ALPHAR( ILAST ) = H( ILAST, ILAST )
+ ALPHAI( ILAST ) = ZERO
+ BETA( ILAST ) = T( ILAST, ILAST )
+*
+* Go to next block -- exit if finished.
+*
+ ILAST = ILAST - 1
+ IF( ILAST.LT.ILO )
+ $ GO TO 380
+*
+* Reset counters
+*
+ IITER = 0
+ ESHIFT = ZERO
+ IF( .NOT.ILSCHR ) THEN
+ ILASTM = ILAST
+ IF( IFRSTM.GT.ILAST )
+ $ IFRSTM = ILO
+ END IF
+ GO TO 350
+*
+* QZ step
+*
+* This iteration only involves rows/columns IFIRST:ILAST. We
+* assume IFIRST < ILAST, and that the diagonal of B is non-zero.
+*
+ 110 CONTINUE
+ IITER = IITER + 1
+ IF( .NOT.ILSCHR ) THEN
+ IFRSTM = IFIRST
+ END IF
+*
+* Compute single shifts.
+*
+* 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.EQ.IITER ) THEN
+*
+* Exceptional shift. Chosen for no particularly good reason.
+* (Single shift only.)
+*
+ IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST-1, ILAST ) ).LT.
+ $ ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
+ ESHIFT = ESHIFT + H( ILAST-1, ILAST ) /
+ $ T( ILAST-1, ILAST-1 )
+ ELSE
+ ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
+ END IF
+ S1 = ONE
+ WR = ESHIFT
+*
+ ELSE
+*
+* Shifts based on the generalized eigenvalues of the
+* bottom-right 2x2 block of A and B. The first eigenvalue
+* returned by DLAG2 is the Wilkinson shift (AEP p.512),
+*
+ CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
+ $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
+ $ S2, WR, WR2, WI )
+*
+ TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
+ IF( WI.NE.ZERO )
+ $ GO TO 200
+ END IF
+*
+* Fiddle with shift to avoid overflow
+*
+ TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
+ IF( S1.GT.TEMP ) THEN
+ SCALE = TEMP / S1
+ ELSE
+ SCALE = ONE
+ END IF
+*
+ TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
+ IF( ABS( WR ).GT.TEMP )
+ $ SCALE = MIN( SCALE, TEMP / ABS( WR ) )
+ S1 = SCALE*S1
+ WR = SCALE*WR
+*
+* Now check for two consecutive small subdiagonals.
+*
+ DO 120 J = ILAST - 1, IFIRST + 1, -1
+ ISTART = J
+ TEMP = ABS( S1*H( J, J-1 ) )
+ TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
+ TEMPR = MAX( TEMP, TEMP2 )
+ IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
+ TEMP = TEMP / TEMPR
+ TEMP2 = TEMP2 / TEMPR
+ END IF
+ IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
+ $ TEMP2 )GO TO 130
+ 120 CONTINUE
+*
+ ISTART = IFIRST
+ 130 CONTINUE
+*
+* Do an implicit single-shift QZ sweep.
+*
+* Initial Q
+*
+ TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
+ TEMP2 = S1*H( ISTART+1, ISTART )
+ CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
+*
+* Sweep
+*
+ DO 190 J = ISTART, ILAST - 1
+ IF( J.GT.ISTART ) THEN
+ TEMP = H( J, J-1 )
+ CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
+ H( J+1, J-1 ) = ZERO
+ END IF
+*
+ DO 140 JC = J, ILASTM
+ TEMP = C*H( J, JC ) + S*H( J+1, JC )
+ H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
+ H( J, JC ) = TEMP
+ TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
+ T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
+ T( J, JC ) = TEMP2
+ 140 CONTINUE
+ IF( ILQ ) THEN
+ DO 150 JR = 1, N
+ TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
+ Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
+ Q( JR, J ) = TEMP
+ 150 CONTINUE
+ END IF
+*
+ TEMP = T( J+1, J+1 )
+ CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
+ T( J+1, J ) = ZERO
+*
+ DO 160 JR = IFRSTM, MIN( J+2, ILAST )
+ TEMP = C*H( JR, J+1 ) + S*H( JR, J )
+ H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
+ H( JR, J+1 ) = TEMP
+ 160 CONTINUE
+ DO 170 JR = IFRSTM, J
+ TEMP = C*T( JR, J+1 ) + S*T( JR, J )
+ T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
+ T( JR, J+1 ) = TEMP
+ 170 CONTINUE
+ IF( ILZ ) THEN
+ DO 180 JR = 1, N
+ TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
+ Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
+ Z( JR, J+1 ) = TEMP
+ 180 CONTINUE
+ END IF
+ 190 CONTINUE
+*
+ GO TO 350
+*
+* Use Francis double-shift
+*
+* Note: the Francis double-shift should work with real shifts,
+* but only if the block is at least 3x3.
+* This code may break if this point is reached with
+* a 2x2 block with real eigenvalues.
+*
+ 200 CONTINUE
+ IF( IFIRST+1.EQ.ILAST ) THEN
+*
+* Special case -- 2x2 block with complex eigenvectors
+*
+* Step 1: Standardize, that is, rotate so that
+*
+* ( B11 0 )
+* B = ( ) with B11 non-negative.
+* ( 0 B22 )
+*
+ CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
+ $ T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
+*
+ IF( B11.LT.ZERO ) THEN
+ CR = -CR
+ SR = -SR
+ B11 = -B11
+ B22 = -B22
+ END IF
+*
+ CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
+ $ H( ILAST, ILAST-1 ), LDH, CL, SL )
+ CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
+ $ H( IFRSTM, ILAST ), 1, CR, SR )
+*
+ IF( ILAST.LT.ILASTM )
+ $ CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
+ $ T( ILAST, ILAST+1 ), LDH, CL, SL )
+ IF( IFRSTM.LT.ILAST-1 )
+ $ CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
+ $ T( IFRSTM, ILAST ), 1, CR, SR )
+*
+ IF( ILQ )
+ $ CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
+ $ SL )
+ IF( ILZ )
+ $ CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR,
+ $ SR )
+*
+ T( ILAST-1, ILAST-1 ) = B11
+ T( ILAST-1, ILAST ) = ZERO
+ T( ILAST, ILAST-1 ) = ZERO
+ T( ILAST, ILAST ) = B22
+*
+* If B22 is negative, negate column ILAST
+*
+ IF( B22.LT.ZERO ) THEN
+ DO 210 J = IFRSTM, ILAST
+ H( J, ILAST ) = -H( J, ILAST )
+ T( J, ILAST ) = -T( J, ILAST )
+ 210 CONTINUE
+*
+ IF( ILZ ) THEN
+ DO 220 J = 1, N
+ Z( J, ILAST ) = -Z( J, ILAST )
+ 220 CONTINUE
+ END IF
+ END IF
+*
+* Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
+*
+* Recompute shift
+*
+ CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
+ $ T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
+ $ TEMP, WR, TEMP2, WI )
+*
+* If standardization has perturbed the shift onto real line,
+* do another (real single-shift) QR step.
+*
+ IF( WI.EQ.ZERO )
+ $ GO TO 350
+ S1INV = ONE / S1
+*
+* Do EISPACK (QZVAL) computation of alpha and beta
+*
+ A11 = H( ILAST-1, ILAST-1 )
+ A21 = H( ILAST, ILAST-1 )
+ A12 = H( ILAST-1, ILAST )
+ A22 = H( ILAST, ILAST )
+*
+* Compute complex Givens rotation on right
+* (Assume some element of C = (sA - wB) > unfl )
+* __
+* (sA - wB) ( CZ -SZ )
+* ( SZ CZ )
+*
+ C11R = S1*A11 - WR*B11
+ C11I = -WI*B11
+ C12 = S1*A12
+ C21 = S1*A21
+ C22R = S1*A22 - WR*B22
+ C22I = -WI*B22
+*
+ IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
+ $ ABS( C22R )+ABS( C22I ) ) THEN
+ T1 = DLAPY3( C12, C11R, C11I )
+ CZ = C12 / T1
+ SZR = -C11R / T1
+ SZI = -C11I / T1
+ ELSE
+ CZ = DLAPY2( C22R, C22I )
+ IF( CZ.LE.SAFMIN ) THEN
+ CZ = ZERO
+ SZR = ONE
+ SZI = ZERO
+ ELSE
+ TEMPR = C22R / CZ
+ TEMPI = C22I / CZ
+ T1 = DLAPY2( CZ, C21 )
+ CZ = CZ / T1
+ SZR = -C21*TEMPR / T1
+ SZI = C21*TEMPI / T1
+ END IF
+ END IF
+*
+* Compute Givens rotation on left
+*
+* ( CQ SQ )
+* ( __ ) A or B
+* ( -SQ CQ )
+*
+ AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
+ BN = ABS( B11 ) + ABS( B22 )
+ WABS = ABS( WR ) + ABS( WI )
+ IF( S1*AN.GT.WABS*BN ) THEN
+ CQ = CZ*B11
+ SQR = SZR*B22
+ SQI = -SZI*B22
+ ELSE
+ A1R = CZ*A11 + SZR*A12
+ A1I = SZI*A12
+ A2R = CZ*A21 + SZR*A22
+ A2I = SZI*A22
+ CQ = DLAPY2( A1R, A1I )
+ IF( CQ.LE.SAFMIN ) THEN
+ CQ = ZERO
+ SQR = ONE
+ SQI = ZERO
+ ELSE
+ TEMPR = A1R / CQ
+ TEMPI = A1I / CQ
+ SQR = TEMPR*A2R + TEMPI*A2I
+ SQI = TEMPI*A2R - TEMPR*A2I
+ END IF
+ END IF
+ T1 = DLAPY3( CQ, SQR, SQI )
+ CQ = CQ / T1
+ SQR = SQR / T1
+ SQI = SQI / T1
+*
+* Compute diagonal elements of QBZ
+*
+ TEMPR = SQR*SZR - SQI*SZI
+ TEMPI = SQR*SZI + SQI*SZR
+ B1R = CQ*CZ*B11 + TEMPR*B22
+ B1I = TEMPI*B22
+ B1A = DLAPY2( B1R, B1I )
+ B2R = CQ*CZ*B22 + TEMPR*B11
+ B2I = -TEMPI*B11
+ B2A = DLAPY2( B2R, B2I )
+*
+* Normalize so beta > 0, and Im( alpha1 ) > 0
+*
+ BETA( ILAST-1 ) = B1A
+ BETA( ILAST ) = B2A
+ ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
+ ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
+ ALPHAR( ILAST ) = ( WR*B2A )*S1INV
+ ALPHAI( ILAST ) = -( WI*B2A )*S1INV
+*
+* Step 3: Go to next block -- exit if finished.
+*
+ ILAST = IFIRST - 1
+ IF( ILAST.LT.ILO )
+ $ GO TO 380
+*
+* Reset counters
+*
+ IITER = 0
+ ESHIFT = ZERO
+ IF( .NOT.ILSCHR ) THEN
+ ILASTM = ILAST
+ IF( IFRSTM.GT.ILAST )
+ $ IFRSTM = ILO
+ END IF
+ GO TO 350
+ ELSE
+*
+* Usual case: 3x3 or larger block, using Francis implicit
+* double-shift
+*
+* 2
+* Eigenvalue equation is w - c w + d = 0,
+*
+* -1 2 -1
+* so compute 1st column of (A B ) - c A B + d
+* using the formula in QZIT (from EISPACK)
+*
+* We assume that the block is at least 3x3
+*
+ 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 ) )
+ U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
+ AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
+ $ ( BSCALE*T( IFIRST, IFIRST ) )
+ AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
+ $ ( BSCALE*T( IFIRST, IFIRST ) )
+ AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
+ $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
+ AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
+ $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
+ AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
+ $ ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
+ U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
+*
+ V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
+ $ AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
+ V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
+ $ ( AD22-AD11L )+AD21*U12 )*AD21L
+ V( 3 ) = AD32L*AD21L
+*
+ ISTART = IFIRST
+*
+ CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
+ V( 1 ) = ONE
+*
+* Sweep
+*
+ DO 290 J = ISTART, ILAST - 2
+*
+* All but last elements: use 3x3 Householder transforms.
+*
+* Zero (j-1)st column of A
+*
+ IF( J.GT.ISTART ) THEN
+ V( 1 ) = H( J, J-1 )
+ V( 2 ) = H( J+1, J-1 )
+ V( 3 ) = H( J+2, J-1 )
+*
+ CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
+ V( 1 ) = ONE
+ H( J+1, J-1 ) = ZERO
+ H( J+2, J-1 ) = ZERO
+ END IF
+*
+ DO 230 JC = J, ILASTM
+ TEMP = TAU*( H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
+ $ H( J+2, JC ) )
+ H( J, JC ) = H( J, JC ) - TEMP
+ H( J+1, JC ) = H( J+1, JC ) - TEMP*V( 2 )
+ H( J+2, JC ) = H( J+2, JC ) - TEMP*V( 3 )
+ TEMP2 = TAU*( T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
+ $ T( J+2, JC ) )
+ T( J, JC ) = T( J, JC ) - TEMP2
+ T( J+1, JC ) = T( J+1, JC ) - TEMP2*V( 2 )
+ T( J+2, JC ) = T( J+2, JC ) - TEMP2*V( 3 )
+ 230 CONTINUE
+ IF( ILQ ) THEN
+ DO 240 JR = 1, N
+ TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
+ $ Q( JR, J+2 ) )
+ Q( JR, J ) = Q( JR, J ) - TEMP
+ Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 )
+ Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 )
+ 240 CONTINUE
+ END IF
+*
+* Zero j-th column of B (see DLAGBC for details)
+*
+* Swap rows to pivot
+*
+ ILPIVT = .FALSE.
+ TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
+ TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
+ IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
+ SCALE = ZERO
+ U1 = ONE
+ U2 = ZERO
+ GO TO 250
+ ELSE IF( TEMP.GE.TEMP2 ) THEN
+ W11 = T( J+1, J+1 )
+ W21 = T( J+2, J+1 )
+ W12 = T( J+1, J+2 )
+ W22 = T( J+2, J+2 )
+ U1 = T( J+1, J )
+ U2 = T( J+2, J )
+ ELSE
+ W21 = T( J+1, J+1 )
+ W11 = T( J+2, J+1 )
+ W22 = T( J+1, J+2 )
+ W12 = T( J+2, J+2 )
+ U2 = T( J+1, J )
+ U1 = T( J+2, J )
+ END IF
+*
+* Swap columns if nec.
+*
+ IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
+ ILPIVT = .TRUE.
+ TEMP = W12
+ TEMP2 = W22
+ W12 = W11
+ W22 = W21
+ W11 = TEMP
+ W21 = TEMP2
+ END IF
+*
+* LU-factor
+*
+ TEMP = W21 / W11
+ U2 = U2 - TEMP*U1
+ W22 = W22 - TEMP*W12
+ W21 = ZERO
+*
+* Compute SCALE
+*
+ SCALE = ONE
+ IF( ABS( W22 ).LT.SAFMIN ) THEN
+ SCALE = ZERO
+ U2 = ONE
+ U1 = -W12 / W11
+ GO TO 250
+ END IF
+ IF( ABS( W22 ).LT.ABS( U2 ) )
+ $ SCALE = ABS( W22 / U2 )
+ IF( ABS( W11 ).LT.ABS( U1 ) )
+ $ SCALE = MIN( SCALE, ABS( W11 / U1 ) )
+*
+* Solve
+*
+ U2 = ( SCALE*U2 ) / W22
+ U1 = ( SCALE*U1-W12*U2 ) / W11
+*
+ 250 CONTINUE
+ IF( ILPIVT ) THEN
+ TEMP = U2
+ U2 = U1
+ U1 = TEMP
+ END IF
+*
+* Compute Householder Vector
+*
+ T1 = SQRT( SCALE**2+U1**2+U2**2 )
+ TAU = ONE + SCALE / T1
+ VS = -ONE / ( SCALE+T1 )
+ V( 1 ) = ONE
+ V( 2 ) = VS*U1
+ V( 3 ) = VS*U2
+*
+* Apply transformations from the right.
+*
+ DO 260 JR = IFRSTM, MIN( J+3, ILAST )
+ TEMP = TAU*( H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
+ $ H( JR, J+2 ) )
+ H( JR, J ) = H( JR, J ) - TEMP
+ H( JR, J+1 ) = H( JR, J+1 ) - TEMP*V( 2 )
+ H( JR, J+2 ) = H( JR, J+2 ) - TEMP*V( 3 )
+ 260 CONTINUE
+ DO 270 JR = IFRSTM, J + 2
+ TEMP = TAU*( T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
+ $ T( JR, J+2 ) )
+ T( JR, J ) = T( JR, J ) - TEMP
+ T( JR, J+1 ) = T( JR, J+1 ) - TEMP*V( 2 )
+ T( JR, J+2 ) = T( JR, J+2 ) - TEMP*V( 3 )
+ 270 CONTINUE
+ IF( ILZ ) THEN
+ DO 280 JR = 1, N
+ TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
+ $ Z( JR, J+2 ) )
+ Z( JR, J ) = Z( JR, J ) - TEMP
+ Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 )
+ Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 )
+ 280 CONTINUE
+ END IF
+ T( J+1, J ) = ZERO
+ T( J+2, J ) = ZERO
+ 290 CONTINUE
+*
+* Last elements: Use Givens rotations
+*
+* Rotations from the left
+*
+ J = ILAST - 1
+ TEMP = H( J, J-1 )
+ CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
+ H( J+1, J-1 ) = ZERO
+*
+ DO 300 JC = J, ILASTM
+ TEMP = C*H( J, JC ) + S*H( J+1, JC )
+ H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
+ H( J, JC ) = TEMP
+ TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
+ T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
+ T( J, JC ) = TEMP2
+ 300 CONTINUE
+ IF( ILQ ) THEN
+ DO 310 JR = 1, N
+ TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
+ Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
+ Q( JR, J ) = TEMP
+ 310 CONTINUE
+ END IF
+*
+* Rotations from the right.
+*
+ TEMP = T( J+1, J+1 )
+ CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
+ T( J+1, J ) = ZERO
+*
+ DO 320 JR = IFRSTM, ILAST
+ TEMP = C*H( JR, J+1 ) + S*H( JR, J )
+ H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
+ H( JR, J+1 ) = TEMP
+ 320 CONTINUE
+ DO 330 JR = IFRSTM, ILAST - 1
+ TEMP = C*T( JR, J+1 ) + S*T( JR, J )
+ T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
+ T( JR, J+1 ) = TEMP
+ 330 CONTINUE
+ IF( ILZ ) THEN
+ DO 340 JR = 1, N
+ TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
+ Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
+ Z( JR, J+1 ) = TEMP
+ 340 CONTINUE
+ END IF
+*
+* End of Double-Shift code
+*
+ END IF
+*
+ GO TO 350
+*
+* End of iteration loop
+*
+ 350 CONTINUE
+ 360 CONTINUE
+*
+* Drop-through = non-convergence
+*
+ INFO = ILAST
+ GO TO 420
+*
+* Successful completion of all QZ steps
+*
+ 380 CONTINUE
+*
+* Set Eigenvalues 1:ILO-1
+*
+ DO 410 J = 1, ILO - 1
+ IF( T( J, J ).LT.ZERO ) THEN
+ IF( ILSCHR ) THEN
+ DO 390 JR = 1, J
+ H( JR, J ) = -H( JR, J )
+ T( JR, J ) = -T( JR, J )
+ 390 CONTINUE
+ ELSE
+ H( J, J ) = -H( J, J )
+ T( J, J ) = -T( J, J )
+ END IF
+ IF( ILZ ) THEN
+ DO 400 JR = 1, N
+ Z( JR, J ) = -Z( JR, J )
+ 400 CONTINUE
+ END IF
+ END IF
+ ALPHAR( J ) = H( J, J )
+ ALPHAI( J ) = ZERO
+ BETA( J ) = T( J, J )
+ 410 CONTINUE
+*
+* Normal Termination
+*
+ INFO = 0
+*
+* Exit (other than argument error) -- return optimal workspace size
+*
+ 420 CONTINUE
+ WORK( 1 ) = DBLE( N )
+ RETURN
+*
+* End of DHGEQZ
+*
+ END
generated by cgit (git 2.34.1) at 2024-12-20 11:38:56 +0000