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|
SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
*
* -- LAPACK routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( * ),
$ Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* DHGEQZ implements a single-/double-shift version of the QZ method for
* finding the generalized eigenvalues
*
* w(j)=(ALPHAR(j) + i*ALPHAI(j))/BETAR(j) of the equation
*
* det( A - w(i) B ) = 0
*
* In addition, the pair A,B may be reduced to generalized Schur form:
* B is upper triangular, and A is block upper triangular, where the
* diagonal blocks are either 1-by-1 or 2-by-2, the 2-by-2 blocks having
* complex generalized eigenvalues (see the description of the argument
* JOB.)
*
* If JOB='S', then the pair (A,B) is simultaneously reduced to Schur
* form by applying one orthogonal tranformation (usually called Q) on
* the left and another (usually called Z) on the right. The 2-by-2
* upper-triangular diagonal blocks of B corresponding to 2-by-2 blocks
* of A will be reduced to positive diagonal matrices. (I.e.,
* if A(j+1,j) is non-zero, then B(j+1,j)=B(j,j+1)=0 and B(j,j) and
* B(j+1,j+1) will be positive.)
*
* If JOB='E', then at each iteration, the same transformations
* are computed, but they are only applied to those parts of A and B
* which are needed to compute ALPHAR, ALPHAI, and BETAR.
*
* If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the orthogonal
* 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 ALPHAR, ALPHAI, 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 ALPHAR, ALPHAI, and BETA.
*
* COMPQ (input) CHARACTER*1
* = 'N': do not modify Q.
* = 'V': multiply the array Q on the right by the transpose of
* the orthogonal 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 orthogonal
* 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) DOUBLE PRECISION 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 generalized Schur form.
* If JOB='E', then on exit A will have been destroyed.
* The diagonal blocks will be correct, but the off-diagonal
* portion will be meaningless.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max( 1, N ).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
* On entry, the N-by-N upper triangular matrix B. Elements
* below the diagonal must be zero. 2-by-2 blocks in B
* corresponding to 2-by-2 blocks in A will be 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 and B(j,j) and B(j+1,j+1) will be
* positive.)
* If JOB='S', then on exit A and B will have been
* simultaneously reduced to Schur form.
* If JOB='E', then on exit B will have been destroyed.
* Elements corresponding to diagonal blocks of A will be
* correct, but the off-diagonal portion will be meaningless.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max( 1, N ).
*
* ALPHAR (output) DOUBLE PRECISION array, dimension (N)
* ALPHAR(1:N) will be set to real parts of the diagonal
* elements of A that would result from reducing A and B to
* Schur form and then further reducing them both to triangular
* form using unitary transformations s.t. the diagonal of B
* was non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=A(j,j).
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
*
* ALPHAI (output) DOUBLE PRECISION array, dimension (N)
* ALPHAI(1:N) will be set to imaginary parts of the diagonal
* elements of A that would result from reducing A and B to
* Schur form and then further reducing them both to triangular
* form using unitary transformations s.t. the diagonal of B
* was non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then ALPHAR(j)=0.
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
*
* BETA (output) DOUBLE PRECISION array, dimension (N)
* BETA(1:N) will be set to the (real) diagonal elements of B
* that would result from reducing A and B to Schur form and
* then further reducing them both to triangular form using
* unitary transformations s.t. the diagonal of B was
* non-negative real. Thus, if A(j,j) is in a 1-by-1 block
* (i.e., A(j+1,j)=A(j,j+1)=0), then BETA(j)=B(j,j).
* Note that the (real or complex) values
* (ALPHAR(j) + i*ALPHAI(j))/BETA(j), j=1,...,N, are the
* generalized eigenvalues of the matrix pencil A - wB.
* (Note that BETA(1:N) will always be non-negative, and no
* BETAI is necessary.)
*
* Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
* If COMPQ='N', then Q will not be referenced.
* If COMPQ='V' or 'I', then the transpose of the orthogonal
* 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) DOUBLE PRECISION array, dimension (LDZ, N)
* If COMPZ='N', then Z will not be referenced.
* If COMPZ='V' or 'I', then the orthogonal 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) DOUBLE PRECISION 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).
*
* 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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO-N+1,...,N should be correct.
* > 2*N: various "impossible" errors.
*
* 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
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, T,
$ 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
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 = -15
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -17
ELSE IF( LWORK.LT.MAX( 1, N ) ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DHGEQZ', -INFO )
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, A( ILO, ILO ), LDA, WORK )
BNORM = DLANHS( 'F', IN, B( ILO, ILO ), LDB, 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( B( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 10 JR = 1, J
A( JR, J ) = -A( JR, J )
B( JR, J ) = -B( JR, J )
10 CONTINUE
ELSE
A( J, J ) = -A( J, J )
B( J, J ) = -B( 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 ) = A( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = B( 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: A(j,j-1)=0 or j=ILO
* 2: B(j,j)=0
*
IF( ILAST.EQ.ILO ) THEN
*
* Special case: j=ILAST
*
GO TO 80
ELSE
IF( ABS( A( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
A( ILAST, ILAST-1 ) = ZERO
GO TO 80
END IF
END IF
*
IF( ABS( B( ILAST, ILAST ) ).LE.BTOL ) THEN
B( ILAST, ILAST ) = ZERO
GO TO 70
END IF
*
* General case: j<ILAST
*
DO 60 J = ILAST - 1, ILO, -1
*
* Test 1: for A(j,j-1)=0 or j=ILO
*
IF( J.EQ.ILO ) THEN
ILAZRO = .TRUE.
ELSE
IF( ABS( A( J, J-1 ) ).LE.ATOL ) THEN
A( J, J-1 ) = ZERO
ILAZRO = .TRUE.
ELSE
ILAZRO = .FALSE.
END IF
END IF
*
* Test 2: for B(j,j)=0
*
IF( ABS( B( J, J ) ).LT.BTOL ) THEN
B( J, J ) = ZERO
*
* Test 1a: Check for 2 consecutive small subdiagonals in A
*
ILAZR2 = .FALSE.
IF( .NOT.ILAZRO ) THEN
TEMP = ABS( A( J, J-1 ) )
TEMP2 = ABS( A( 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( A( 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 = A( JCH, JCH )
CALL DLARTG( TEMP, A( JCH+1, JCH ), C, S,
$ A( JCH, JCH ) )
A( JCH+1, JCH ) = ZERO
CALL DROT( ILASTM-JCH, A( JCH, JCH+1 ), LDA,
$ A( JCH+1, JCH+1 ), LDA, C, S )
CALL DROT( ILASTM-JCH, B( JCH, JCH+1 ), LDB,
$ B( JCH+1, JCH+1 ), LDB, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
$ C, S )
IF( ILAZR2 )
$ A( JCH, JCH-1 ) = A( JCH, JCH-1 )*C
ILAZR2 = .FALSE.
IF( ABS( B( 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
B( JCH+1, JCH+1 ) = ZERO
40 CONTINUE
GO TO 70
ELSE
*
* Only test 2 passed -- chase the zero to B(ILAST,ILAST)
* Then process as in the case B(ILAST,ILAST)=0
*
DO 50 JCH = J, ILAST - 1
TEMP = B( JCH, JCH+1 )
CALL DLARTG( TEMP, B( JCH+1, JCH+1 ), C, S,
$ B( JCH, JCH+1 ) )
B( JCH+1, JCH+1 ) = ZERO
IF( JCH.LT.ILASTM-1 )
$ CALL DROT( ILASTM-JCH-1, B( JCH, JCH+2 ), LDB,
$ B( JCH+1, JCH+2 ), LDB, C, S )
CALL DROT( ILASTM-JCH+2, A( JCH, JCH-1 ), LDA,
$ A( JCH+1, JCH-1 ), LDA, C, S )
IF( ILQ )
$ CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
$ C, S )
TEMP = A( JCH+1, JCH )
CALL DLARTG( TEMP, A( JCH+1, JCH-1 ), C, S,
$ A( JCH+1, JCH ) )
A( JCH+1, JCH-1 ) = ZERO
CALL DROT( JCH+1-IFRSTM, A( IFRSTM, JCH ), 1,
$ A( IFRSTM, JCH-1 ), 1, C, S )
CALL DROT( JCH-IFRSTM, B( IFRSTM, JCH ), 1,
$ B( 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
*
* B(ILAST,ILAST)=0 -- clear A(ILAST,ILAST-1) to split off a
* 1x1 block.
*
70 CONTINUE
TEMP = A( ILAST, ILAST )
CALL DLARTG( TEMP, A( ILAST, ILAST-1 ), C, S,
$ A( ILAST, ILAST ) )
A( ILAST, ILAST-1 ) = ZERO
CALL DROT( ILAST-IFRSTM, A( IFRSTM, ILAST ), 1,
$ A( IFRSTM, ILAST-1 ), 1, C, S )
CALL DROT( ILAST-IFRSTM, B( IFRSTM, ILAST ), 1,
$ B( IFRSTM, ILAST-1 ), 1, C, S )
IF( ILZ )
$ CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
*
* A(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
* and BETA
*
80 CONTINUE
IF( B( ILAST, ILAST ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 90 J = IFRSTM, ILAST
A( J, ILAST ) = -A( J, ILAST )
B( J, ILAST ) = -B( J, ILAST )
90 CONTINUE
ELSE
A( ILAST, ILAST ) = -A( ILAST, ILAST )
B( ILAST, ILAST ) = -B( 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 ) = A( ILAST, ILAST )
ALPHAI( ILAST ) = ZERO
BETA( ILAST ) = B( 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
* B(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( A( ILAST-1, ILAST ) ).LT.
$ ABS( B( ILAST-1, ILAST-1 ) ) ) THEN
ESHIFT = ESHIFT + A( ILAST-1, ILAST ) /
$ B( 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( A( ILAST-1, ILAST-1 ), LDA,
$ B( ILAST-1, ILAST-1 ), LDB, 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*A( J, J-1 ) )
TEMP2 = ABS( S1*A( J, J )-WR*B( 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*A( 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*A( ISTART, ISTART ) - WR*B( ISTART, ISTART )
TEMP2 = S1*A( ISTART+1, ISTART )
CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
*
* Sweep
*
DO 190 J = ISTART, ILAST - 1
IF( J.GT.ISTART ) THEN
TEMP = A( J, J-1 )
CALL DLARTG( TEMP, A( J+1, J-1 ), C, S, A( J, J-1 ) )
A( J+1, J-1 ) = ZERO
END IF
*
DO 140 JC = J, ILASTM
TEMP = C*A( J, JC ) + S*A( J+1, JC )
A( J+1, JC ) = -S*A( J, JC ) + C*A( J+1, JC )
A( J, JC ) = TEMP
TEMP2 = C*B( J, JC ) + S*B( J+1, JC )
B( J+1, JC ) = -S*B( J, JC ) + C*B( J+1, JC )
B( 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 = B( J+1, J+1 )
CALL DLARTG( TEMP, B( J+1, J ), C, S, B( J+1, J+1 ) )
B( J+1, J ) = ZERO
*
DO 160 JR = IFRSTM, MIN( J+2, ILAST )
TEMP = C*A( JR, J+1 ) + S*A( JR, J )
A( JR, J ) = -S*A( JR, J+1 ) + C*A( JR, J )
A( JR, J+1 ) = TEMP
160 CONTINUE
DO 170 JR = IFRSTM, J
TEMP = C*B( JR, J+1 ) + S*B( JR, J )
B( JR, J ) = -S*B( JR, J+1 ) + C*B( JR, J )
B( 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( B( ILAST-1, ILAST-1 ), B( ILAST-1, ILAST ),
$ B( 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, A( ILAST-1, ILAST-1 ), LDA,
$ A( ILAST, ILAST-1 ), LDA, CL, SL )
CALL DROT( ILAST+1-IFRSTM, A( IFRSTM, ILAST-1 ), 1,
$ A( IFRSTM, ILAST ), 1, CR, SR )
*
IF( ILAST.LT.ILASTM )
$ CALL DROT( ILASTM-ILAST, B( ILAST-1, ILAST+1 ), LDB,
$ B( ILAST, ILAST+1 ), LDA, CL, SL )
IF( IFRSTM.LT.ILAST-1 )
$ CALL DROT( IFIRST-IFRSTM, B( IFRSTM, ILAST-1 ), 1,
$ B( 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 )
*
B( ILAST-1, ILAST-1 ) = B11
B( ILAST-1, ILAST ) = ZERO
B( ILAST, ILAST-1 ) = ZERO
B( ILAST, ILAST ) = B22
*
* If B22 is negative, negate column ILAST
*
IF( B22.LT.ZERO ) THEN
DO 210 J = IFRSTM, ILAST
A( J, ILAST ) = -A( J, ILAST )
B( J, ILAST ) = -B( 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( A( ILAST-1, ILAST-1 ), LDA,
$ B( ILAST-1, ILAST-1 ), LDB, 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 = A( ILAST-1, ILAST-1 )
A21 = A( ILAST, ILAST-1 )
A12 = A( ILAST-1, ILAST )
A22 = A( 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
T = DLAPY3( C12, C11R, C11I )
CZ = C12 / T
SZR = -C11R / T
SZI = -C11I / T
ELSE
CZ = DLAPY2( C22R, C22I )
IF( CZ.LE.SAFMIN ) THEN
CZ = ZERO
SZR = ONE
SZI = ZERO
ELSE
TEMPR = C22R / CZ
TEMPI = C22I / CZ
T = DLAPY2( CZ, C21 )
CZ = CZ / T
SZR = -C21*TEMPR / T
SZI = C21*TEMPI / T
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
T = DLAPY3( CQ, SQR, SQI )
CQ = CQ / T
SQR = SQR / T
SQI = SQI / T
*
* 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*A( ILAST-1, ILAST-1 ) ) /
$ ( BSCALE*B( ILAST-1, ILAST-1 ) )
AD21 = ( ASCALE*A( ILAST, ILAST-1 ) ) /
$ ( BSCALE*B( ILAST-1, ILAST-1 ) )
AD12 = ( ASCALE*A( ILAST-1, ILAST ) ) /
$ ( BSCALE*B( ILAST, ILAST ) )
AD22 = ( ASCALE*A( ILAST, ILAST ) ) /
$ ( BSCALE*B( ILAST, ILAST ) )
U12 = B( ILAST-1, ILAST ) / B( ILAST, ILAST )
AD11L = ( ASCALE*A( IFIRST, IFIRST ) ) /
$ ( BSCALE*B( IFIRST, IFIRST ) )
AD21L = ( ASCALE*A( IFIRST+1, IFIRST ) ) /
$ ( BSCALE*B( IFIRST, IFIRST ) )
AD12L = ( ASCALE*A( IFIRST, IFIRST+1 ) ) /
$ ( BSCALE*B( IFIRST+1, IFIRST+1 ) )
AD22L = ( ASCALE*A( IFIRST+1, IFIRST+1 ) ) /
$ ( BSCALE*B( IFIRST+1, IFIRST+1 ) )
AD32L = ( ASCALE*A( IFIRST+2, IFIRST+1 ) ) /
$ ( BSCALE*B( IFIRST+1, IFIRST+1 ) )
U12L = B( IFIRST, IFIRST+1 ) / B( 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 ) = A( J, J-1 )
V( 2 ) = A( J+1, J-1 )
V( 3 ) = A( J+2, J-1 )
*
CALL DLARFG( 3, A( J, J-1 ), V( 2 ), 1, TAU )
V( 1 ) = ONE
A( J+1, J-1 ) = ZERO
A( J+2, J-1 ) = ZERO
END IF
*
DO 230 JC = J, ILASTM
TEMP = TAU*( A( J, JC )+V( 2 )*A( J+1, JC )+V( 3 )*
$ A( J+2, JC ) )
A( J, JC ) = A( J, JC ) - TEMP
A( J+1, JC ) = A( J+1, JC ) - TEMP*V( 2 )
A( J+2, JC ) = A( J+2, JC ) - TEMP*V( 3 )
TEMP2 = TAU*( B( J, JC )+V( 2 )*B( J+1, JC )+V( 3 )*
$ B( J+2, JC ) )
B( J, JC ) = B( J, JC ) - TEMP2
B( J+1, JC ) = B( J+1, JC ) - TEMP2*V( 2 )
B( J+2, JC ) = B( 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( B( J+1, J+1 ) ), ABS( B( J+1, J+2 ) ) )
TEMP2 = MAX( ABS( B( J+2, J+1 ) ), ABS( B( 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 = B( J+1, J+1 )
W21 = B( J+2, J+1 )
W12 = B( J+1, J+2 )
W22 = B( J+2, J+2 )
U1 = B( J+1, J )
U2 = B( J+2, J )
ELSE
W21 = B( J+1, J+1 )
W11 = B( J+2, J+1 )
W22 = B( J+1, J+2 )
W12 = B( J+2, J+2 )
U2 = B( J+1, J )
U1 = B( 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
*
T = SQRT( SCALE**2+U1**2+U2**2 )
TAU = ONE + SCALE / T
VS = -ONE / ( SCALE+T )
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*( A( JR, J )+V( 2 )*A( JR, J+1 )+V( 3 )*
$ A( JR, J+2 ) )
A( JR, J ) = A( JR, J ) - TEMP
A( JR, J+1 ) = A( JR, J+1 ) - TEMP*V( 2 )
A( JR, J+2 ) = A( JR, J+2 ) - TEMP*V( 3 )
260 CONTINUE
DO 270 JR = IFRSTM, J + 2
TEMP = TAU*( B( JR, J )+V( 2 )*B( JR, J+1 )+V( 3 )*
$ B( JR, J+2 ) )
B( JR, J ) = B( JR, J ) - TEMP
B( JR, J+1 ) = B( JR, J+1 ) - TEMP*V( 2 )
B( JR, J+2 ) = B( 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
B( J+1, J ) = ZERO
B( J+2, J ) = ZERO
290 CONTINUE
*
* Last elements: Use Givens rotations
*
* Rotations from the left
*
J = ILAST - 1
TEMP = A( J, J-1 )
CALL DLARTG( TEMP, A( J+1, J-1 ), C, S, A( J, J-1 ) )
A( J+1, J-1 ) = ZERO
*
DO 300 JC = J, ILASTM
TEMP = C*A( J, JC ) + S*A( J+1, JC )
A( J+1, JC ) = -S*A( J, JC ) + C*A( J+1, JC )
A( J, JC ) = TEMP
TEMP2 = C*B( J, JC ) + S*B( J+1, JC )
B( J+1, JC ) = -S*B( J, JC ) + C*B( J+1, JC )
B( 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 = B( J+1, J+1 )
CALL DLARTG( TEMP, B( J+1, J ), C, S, B( J+1, J+1 ) )
B( J+1, J ) = ZERO
*
DO 320 JR = IFRSTM, ILAST
TEMP = C*A( JR, J+1 ) + S*A( JR, J )
A( JR, J ) = -S*A( JR, J+1 ) + C*A( JR, J )
A( JR, J+1 ) = TEMP
320 CONTINUE
DO 330 JR = IFRSTM, ILAST - 1
TEMP = C*B( JR, J+1 ) + S*B( JR, J )
B( JR, J ) = -S*B( JR, J+1 ) + C*B( JR, J )
B( 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
*
370 CONTINUE
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( B( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 390 JR = 1, J
A( JR, J ) = -A( JR, J )
B( JR, J ) = -B( JR, J )
390 CONTINUE
ELSE
A( J, J ) = -A( J, J )
B( J, J ) = -B( 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 ) = A( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = B( 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
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