SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
$ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
$ LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ, JOB
INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
* ..
* .. Array Arguments ..
REAL ALPHAI( * ), ALPHAR( * ), BETA( * ),
$ H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
$ WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SHGEQZ computes the eigenvalues of a real matrix pair (H,T),
* where H is an upper Hessenberg matrix and T is upper triangular,
* using the double-shift QZ method.
* Matrix pairs of this type are produced by the reduction to
* generalized upper Hessenberg form of a real matrix pair (A,B):
*
* A = Q1*H*Z1**T, B = Q1*T*Z1**T,
*
* as computed by SGGHRD.
*
* If JOB='S', then the Hessenberg-triangular pair (H,T) is
* also reduced to generalized Schur form,
*
* H = Q*S*Z**T, T = Q*P*Z**T,
*
* where Q and Z are orthogonal matrices, P is an upper triangular
* matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
* diagonal blocks.
*
* The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
* (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
* eigenvalues.
*
* Additionally, the 2-by-2 upper triangular diagonal blocks of P
* corresponding to 2-by-2 blocks of S are reduced to positive diagonal
* form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
* P(j,j) > 0, and P(j+1,j+1) > 0.
*
* Optionally, the orthogonal matrix Q from the generalized Schur
* factorization may be postmultiplied into an input matrix Q1, and the
* orthogonal matrix Z may be postmultiplied into an input matrix Z1.
* If Q1 and Z1 are the orthogonal matrices from SGGHRD that reduced
* the matrix pair (A,B) to generalized upper Hessenberg form, then the
* output matrices Q1*Q and Z1*Z are the orthogonal factors from the
* generalized Schur factorization of (A,B):
*
* A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
*
* To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
* of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
* complex and beta real.
* If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
* generalized nonsymmetric eigenvalue problem (GNEP)
* A*x = lambda*B*x
* and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
* alternate form of the GNEP
* mu*A*y = B*y.
* Real eigenvalues can be read directly from the generalized Schur
* form:
* alpha = S(i,i), beta = P(i,i).
*
* Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
* Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
* pp. 241--256.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* = 'E': Compute eigenvalues only;
* = 'S': Compute eigenvalues and the Schur form.
*
* COMPQ (input) CHARACTER*1
* = 'N': Left Schur vectors (Q) are not computed;
* = 'I': Q is initialized to the unit matrix and the matrix Q
* of left Schur vectors of (H,T) is returned;
* = 'V': Q must contain an orthogonal matrix Q1 on entry and
* the product Q1*Q is returned.
*
* COMPZ (input) CHARACTER*1
* = 'N': Right Schur vectors (Z) are not computed;
* = 'I': Z is initialized to the unit matrix and the matrix Z
* of right Schur vectors of (H,T) is returned;
* = 'V': Z must contain an orthogonal matrix Z1 on entry and
* the product Z1*Z is returned.
*
* N (input) INTEGER
* The order of the matrices H, T, Q, and Z. N >= 0.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* ILO and IHI mark the rows and columns of H which are in
* Hessenberg form. It is assumed that A is already upper
* triangular in rows and columns 1:ILO-1 and IHI+1:N.
* If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
*
* H (input/output) REAL array, dimension (LDH, N)
* On entry, the N-by-N upper Hessenberg matrix H.
* On exit, if JOB = 'S', H contains the upper quasi-triangular
* matrix S from the generalized Schur factorization;
* 2-by-2 diagonal blocks (corresponding to complex conjugate
* pairs of eigenvalues) are returned in standard form, with
* H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
* If JOB = 'E', the diagonal blocks of H match those of S, but
* the rest of H is unspecified.
*
* LDH (input) INTEGER
* The leading dimension of the array H. LDH >= max( 1, N ).
*
* T (input/output) REAL array, dimension (LDT, N)
* On entry, the N-by-N upper triangular matrix T.
* On exit, if JOB = 'S', T contains the upper triangular
* matrix P from the generalized Schur factorization;
* 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
* are reduced to positive diagonal form, i.e., if H(j+1,j) is
* non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
* T(j+1,j+1) > 0.
* If JOB = 'E', the diagonal blocks of T match those of P, but
* the rest of T is unspecified.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= max( 1, N ).
*
* ALPHAR (output) REAL array, dimension (N)
* The real parts of each scalar alpha defining an eigenvalue
* of GNEP.
*
* ALPHAI (output) REAL array, dimension (N)
* The imaginary parts of each scalar alpha defining an
* eigenvalue of GNEP.
* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
* positive, then the j-th and (j+1)-st eigenvalues are a
* complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
*
* BETA (output) REAL array, dimension (N)
* The scalars beta that define the eigenvalues of GNEP.
* Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
* beta = BETA(j) represent the j-th eigenvalue of the matrix
* pair (A,B), in one of the forms lambda = alpha/beta or
* mu = beta/alpha. Since either lambda or mu may overflow,
* they should not, in general, be computed.
*
* Q (input/output) REAL array, dimension (LDQ, N)
* On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
* the reduction of (A,B) to generalized Hessenberg form.
* On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
* vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
* of left Schur vectors of (A,B).
* Not referenced if COMPZ = 'N'.
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1.
* If COMPQ='V' or 'I', then LDQ >= N.
*
* Z (input/output) REAL array, dimension (LDZ, N)
* On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
* the reduction of (A,B) to generalized Hessenberg form.
* On exit, if COMPZ = 'I', the orthogonal matrix of
* right Schur vectors of (H,T), and if COMPZ = 'V', the
* orthogonal matrix of right Schur vectors of (A,B).
* Not referenced if COMPZ = 'N'.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1.
* If COMPZ='V' or 'I', then LDZ >= N.
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,N).
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* = 1,...,N: the QZ iteration did not converge. (H,T) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO+1,...,N should be correct.
* = N+1,...,2*N: the shift calculation failed. (H,T) is not
* in Schur form, but ALPHAR(i), ALPHAI(i), and
* BETA(i), i=INFO-N+1,...,N should be correct.
*
* Further Details
* ===============
*
* Iteration counters:
*
* JITER -- counts iterations.
* IITER -- counts iterations run since ILAST was last
* changed. This is therefore reset only when a 1-by-1 or
* 2-by-2 block deflates off the bottom.
*
* =====================================================================
*
* .. Parameters ..
* $ SAFETY = 1.0E+0 )
REAL HALF, ZERO, ONE, SAFETY
PARAMETER ( HALF = 0.5E+0, ZERO = 0.0E+0, ONE = 1.0E+0,
$ SAFETY = 1.0E+2 )
* ..
* .. Local Scalars ..
LOGICAL ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
$ LQUERY
INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
$ JR, MAXIT
REAL A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
$ AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
$ AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
$ B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
$ BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
$ CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
$ SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
$ TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
$ U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
$ WR2
* ..
* .. Local Arrays ..
REAL V( 3 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANHS, SLAPY2, SLAPY3
EXTERNAL LSAME, SLAMCH, SLANHS, SLAPY2, SLAPY3
* ..
* .. External Subroutines ..
EXTERNAL SLAG2, SLARFG, SLARTG, SLASET, SLASV2, SROT,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL, SQRT
* ..
* .. Executable Statements ..
*
* Decode JOB, COMPQ, COMPZ
*
IF( LSAME( JOB, 'E' ) ) THEN
ILSCHR = .FALSE.
ISCHUR = 1
ELSE IF( LSAME( JOB, 'S' ) ) THEN
ILSCHR = .TRUE.
ISCHUR = 2
ELSE
ISCHUR = 0
END IF
*
IF( LSAME( COMPQ, 'N' ) ) THEN
ILQ = .FALSE.
ICOMPQ = 1
ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 2
ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
ILQ = .TRUE.
ICOMPQ = 3
ELSE
ICOMPQ = 0
END IF
*
IF( LSAME( COMPZ, 'N' ) ) THEN
ILZ = .FALSE.
ICOMPZ = 1
ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 2
ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
ILZ = .TRUE.
ICOMPZ = 3
ELSE
ICOMPZ = 0
END IF
*
* Check Argument Values
*
INFO = 0
WORK( 1 ) = MAX( 1, N )
LQUERY = ( LWORK.EQ.-1 )
IF( ISCHUR.EQ.0 ) THEN
INFO = -1
ELSE IF( ICOMPQ.EQ.0 ) THEN
INFO = -2
ELSE IF( ICOMPZ.EQ.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( ILO.LT.1 ) THEN
INFO = -5
ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
INFO = -6
ELSE IF( LDH.LT.N ) THEN
INFO = -8
ELSE IF( LDT.LT.N ) THEN
INFO = -10
ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
INFO = -15
ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
INFO = -17
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -19
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SHGEQZ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.LE.0 ) THEN
WORK( 1 ) = REAL( 1 )
RETURN
END IF
*
* Initialize Q and Z
*
IF( ICOMPQ.EQ.3 )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
IF( ICOMPZ.EQ.3 )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
*
* Machine Constants
*
IN = IHI + 1 - ILO
SAFMIN = SLAMCH( 'S' )
SAFMAX = ONE / SAFMIN
ULP = SLAMCH( 'E' )*SLAMCH( 'B' )
ANORM = SLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
BNORM = SLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
ATOL = MAX( SAFMIN, ULP*ANORM )
BTOL = MAX( SAFMIN, ULP*BNORM )
ASCALE = ONE / MAX( SAFMIN, ANORM )
BSCALE = ONE / MAX( SAFMIN, BNORM )
*
* Set Eigenvalues IHI+1:N
*
DO 30 J = IHI + 1, N
IF( T( J, J ).LT.ZERO ) THEN
IF( ILSCHR ) THEN
DO 10 JR = 1, J
H( JR, J ) = -H( JR, J )
T( JR, J ) = -T( JR, J )
10 CONTINUE
ELSE
H( J, J ) = -H( J, J )
T( J, J ) = -T( J, J )
END IF
IF( ILZ ) THEN
DO 20 JR = 1, N
Z( JR, J ) = -Z( JR, J )
20 CONTINUE
END IF
END IF
ALPHAR( J ) = H( J, J )
ALPHAI( J ) = ZERO
BETA( J ) = T( J, J )
30 CONTINUE
*
* If IHI < ILO, skip QZ steps
*
IF( IHI.LT.ILO )
$ GO TO 380
*
* MAIN QZ ITERATION LOOP
*
* Initialize dynamic indices
*
* Eigenvalues ILAST+1:N have been found.
* Column operations modify rows IFRSTM:whatever.
* Row operations modify columns whatever:ILASTM.
*
* If only eigenvalues are being computed, then
* IFRSTM is the row of the last splitting row above row ILAST;
* this is always at least ILO.
* IITER counts iterations since the last eigenvalue was found,
* to tell when to use an extraordinary shift.
* MAXIT is the maximum number of QZ sweeps allowed.
*
ILAST = IHI
IF( ILSCHR ) THEN
IFRSTM = 1
ILASTM = N
ELSE
IFRSTM = ILO
ILASTM = IHI
END IF
IITER = 0
ESHIFT = ZERO
MAXIT = 30*( IHI-ILO+1 )
*
DO 360 JITER = 1, MAXIT
*
* Split the matrix if possible.
*
* Two tests:
* 1: H(j,j-1)=0 or j=ILO
* 2: T(j,j)=0
*
IF( ILAST.EQ.ILO ) THEN
*
* 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 SLARTG( TEMP, H( JCH+1, JCH ), C, S,
$ H( JCH, JCH ) )
H( JCH+1, JCH ) = ZERO
CALL SROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
$ H( JCH+1, JCH+1 ), LDH, C, S )
CALL SROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
$ T( JCH+1, JCH+1 ), LDT, C, S )
IF( ILQ )
$ CALL SROT( 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 SLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
$ T( JCH, JCH+1 ) )
T( JCH+1, JCH+1 ) = ZERO
IF( JCH.LT.ILASTM-1 )
$ CALL SROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
$ T( JCH+1, JCH+2 ), LDT, C, S )
CALL SROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
$ H( JCH+1, JCH-1 ), LDH, C, S )
IF( ILQ )
$ CALL SROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
$ C, S )
TEMP = H( JCH+1, JCH )
CALL SLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
$ H( JCH+1, JCH ) )
H( JCH+1, JCH-1 ) = ZERO
CALL SROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
$ H( IFRSTM, JCH-1 ), 1, C, S )
CALL SROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
$ T( IFRSTM, JCH-1 ), 1, C, S )
IF( ILZ )
$ CALL SROT( 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 SLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
$ H( ILAST, ILAST ) )
H( ILAST, ILAST-1 ) = ZERO
CALL SROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
$ H( IFRSTM, ILAST-1 ), 1, C, S )
CALL SROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
$ T( IFRSTM, ILAST-1 ), 1, C, S )
IF( ILZ )
$ CALL SROT( 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( ( REAL( 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*REAL( 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 SLAG2 is the Wilkinson shift (AEP p.512),
*
CALL SLAG2( 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 SLARTG( TEMP, TEMP2, C, S, TEMPR )
*
* Sweep
*
DO 190 J = ISTART, ILAST - 1
IF( J.GT.ISTART ) THEN
TEMP = H( J, J-1 )
CALL SLARTG( 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 SLARTG( 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 SLASV2( 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 SROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
$ H( ILAST, ILAST-1 ), LDH, CL, SL )
CALL SROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
$ H( IFRSTM, ILAST ), 1, CR, SR )
*
IF( ILAST.LT.ILASTM )
$ CALL SROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
$ T( ILAST, ILAST+1 ), LDH, CL, SL )
IF( IFRSTM.LT.ILAST-1 )
$ CALL SROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
$ T( IFRSTM, ILAST ), 1, CR, SR )
*
IF( ILQ )
$ CALL SROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
$ SL )
IF( ILZ )
$ CALL SROT( 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 SLAG2( 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 = SLAPY3( C12, C11R, C11I )
CZ = C12 / T1
SZR = -C11R / T1
SZI = -C11I / T1
ELSE
CZ = SLAPY2( C22R, C22I )
IF( CZ.LE.SAFMIN ) THEN
CZ = ZERO
SZR = ONE
SZI = ZERO
ELSE
TEMPR = C22R / CZ
TEMPI = C22I / CZ
T1 = SLAPY2( 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 = SLAPY2( 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 = SLAPY3( 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 = SLAPY2( B1R, B1I )
B2R = CQ*CZ*B22 + TEMPR*B11
B2I = -TEMPI*B11
B2A = SLAPY2( 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 SLARFG( 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 SLARFG( 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 SLAGBC 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 SLARTG( 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 SLARTG( 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 ) = REAL( N )
RETURN
*
* End of SHGEQZ
*
END