SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, \$ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, \$ RWORK, INFO ) * * -- LAPACK routine (instrumented to count operations, version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * May 3, 2001 * * .. Scalar Arguments .. CHARACTER COMPQ, COMPZ, JOB INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION RWORK( * ) COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ), \$ Q( LDQ, * ), T( LDT, * ), WORK( * ), \$ Z( LDZ, * ) * .. * * ----------------------- Begin Timing Code ------------------------ * Common block to return operation count and iteration count * ITCNT is initialized to 0, OPS is only incremented * OPST is used to accumulate small contributions to OPS * to avoid roundoff error * .. Common blocks .. COMMON / LATIME / OPS, ITCNT * .. * .. Scalars in Common .. DOUBLE PRECISION ITCNT, OPS * .. * ------------------------ End Timing Code ------------------------- * * * Purpose * ======= * * ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T), * where H is an upper Hessenberg matrix and T is upper triangular, * using the single-shift QZ method. * Matrix pairs of this type are produced by the reduction to * generalized upper Hessenberg form of a complex matrix pair (A,B): * * A = Q1*H*Z1**H, B = Q1*T*Z1**H, * * as computed by ZGGHRD. * * If JOB='S', then the Hessenberg-triangular pair (H,T) is * also reduced to generalized Schur form, * * H = Q*S*Z**H, T = Q*P*Z**H, * * where Q and Z are unitary matrices and S and P are upper triangular. * * Optionally, the unitary matrix Q from the generalized Schur * factorization may be postmultiplied into an input matrix Q1, and the * unitary matrix Z may be postmultiplied into an input matrix Z1. * If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced * the matrix pair (A,B) to generalized Hessenberg form, then the output * matrices Q1*Q and Z1*Z are the unitary factors from the generalized * Schur factorization of (A,B): * * A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. * * To avoid overflow, eigenvalues of the matrix pair (H,T) * (equivalently, of (A,B)) are computed as a pair of complex values * (alpha,beta). 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. * The values of alpha and beta for the i-th eigenvalue 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': Computer 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 a unitary matrix Q1 on entry and * the product Q1*Q is returned. * * COMPZ (input) CHARACTER*1 * = 'N': Right Schur vectors (Z) are not computed; * = 'I': Q is initialized to the unit matrix and the matrix Z * of right Schur vectors of (H,T) is returned; * = 'V': Z must contain a unitary 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) COMPLEX*16 array, dimension (LDH, N) * On entry, the N-by-N upper Hessenberg matrix H. * On exit, if JOB = 'S', H contains the upper triangular * matrix S from the generalized Schur factorization. * If JOB = 'E', the diagonal of H matches that 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) COMPLEX*16 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. * If JOB = 'E', the diagonal of T matches that of P, but * the rest of T is unspecified. * * LDT (input) INTEGER * The leading dimension of the array T. LDT >= max( 1, N ). * * ALPHA (output) COMPLEX*16 array, dimension (N) * The complex scalars alpha that define the eigenvalues of * GNEP. ALPHA(i) = S(i,i) in the generalized Schur * factorization. * * BETA (output) COMPLEX*16 array, dimension (N) * The real non-negative scalars beta that define the * eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized * Schur factorization. * * Together, the quantities alpha = ALPHA(j) and beta = BETA(j) * represent the j-th eigenvalue of the matrix pair (A,B), in * one of the forms lambda = alpha/beta or mu = beta/alpha. * Since either lambda or mu may overflow, they should not, * in general, be computed. * * Q (input/output) COMPLEX*16 array, dimension (LDQ, N) * On entry, if COMPZ = 'V', the unitary matrix Q1 used in the * reduction of (A,B) to generalized Hessenberg form. * On exit, if COMPZ = 'I', the unitary matrix of left Schur * vectors of (H,T), and if COMPZ = 'V', the unitary 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) COMPLEX*16 array, dimension (LDZ, N) * On entry, if COMPZ = 'V', the unitary matrix Z1 used in the * reduction of (A,B) to generalized Hessenberg form. * On exit, if COMPZ = 'I', the unitary matrix of right Schur * vectors of (H,T), and if COMPZ = 'V', the unitary 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) COMPLEX*16 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). * * 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. * * RWORK (workspace) DOUBLE PRECISION array, dimension (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. (H,T) is not * in Schur form, but ALPHA(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 ALPHA(i) and BETA(i), * i=INFO-N+1,...,N should be correct. * * Further Details * =============== * * We assume that complex ABS works as long as its value is less than * overflow. * * ===================================================================== * * .. Parameters .. COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ), \$ CONE = ( 1.0D+0, 0.0D+0 ) ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 0.5D+0 ) * .. * .. Local Scalars .. LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST, \$ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER, \$ JR, MAXIT, NQ, NZ DOUBLE PRECISION ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL, \$ C, OPST, SAFMIN, TEMP, TEMP2, TEMPR, ULP COMPLEX*16 ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2, \$ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1, \$ U12, X * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANHS EXTERNAL LSAME, DLAMCH, ZLANHS * .. * .. External Subroutines .. EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN, \$ SQRT * .. * .. Statement Functions .. DOUBLE PRECISION ABS1 * .. * .. Statement Function definitions .. ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) ) * .. * .. Executable Statements .. * ----------------------- Begin Timing Code ------------------------ ITCNT = ZERO * ------------------------ End Timing Code ------------------------- * * 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 NQ = 0 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN ILQ = .TRUE. ICOMPQ = 2 NQ = N ELSE IF( LSAME( COMPQ, 'I' ) ) THEN ILQ = .TRUE. ICOMPQ = 3 NQ = N ELSE ICOMPQ = 0 END IF * IF( LSAME( COMPZ, 'N' ) ) THEN ILZ = .FALSE. ICOMPZ = 1 NZ = 0 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN ILZ = .TRUE. ICOMPZ = 2 NZ = N ELSE IF( LSAME( COMPZ, 'I' ) ) THEN ILZ = .TRUE. ICOMPZ = 3 NZ = N 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 = -14 ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN INFO = -16 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN INFO = -18 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHGEQZ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * * WORK( 1 ) = CMPLX( 1 ) IF( N.LE.0 ) THEN WORK( 1 ) = DCMPLX( 1 ) RETURN END IF * * Initialize Q and Z * IF( ICOMPQ.EQ.3 ) \$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) IF( ICOMPZ.EQ.3 ) \$ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ ) * * Machine Constants * IN = IHI + 1 - ILO SAFMIN = DLAMCH( 'S' ) ULP = DLAMCH( 'E' )*DLAMCH( 'B' ) ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK ) BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK ) ATOL = MAX( SAFMIN, ULP*ANORM ) BTOL = MAX( SAFMIN, ULP*BNORM ) ASCALE = ONE / MAX( SAFMIN, ANORM ) BSCALE = ONE / MAX( SAFMIN, BNORM ) * * ---------------------- Begin Timing Code ------------------------- * Count ops for norms, etc. OPST = ZERO OPS = OPS + DBLE( 4*N**2+12*N-5 ) * ----------------------- End Timing Code -------------------------- * * * * Set Eigenvalues IHI+1:N * DO 10 J = IHI + 1, N ABSB = ABS( T( J, J ) ) IF( ABSB.GT.SAFMIN ) THEN SIGNBC = DCONJG( T( J, J ) / ABSB ) T( J, J ) = ABSB IF( ILSCHR ) THEN CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 ) CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 ) * ----------------- Begin Timing Code --------------------- OPST = OPST + DBLE( 12*( J-1 ) ) * ------------------ End Timing Code ---------------------- ELSE H( J, J ) = H( J, J )*SIGNBC END IF IF( ILZ ) \$ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 ) * ------------------- Begin Timing Code ---------------------- OPST = OPST + DBLE( 6*NZ+13 ) * -------------------- End Timing Code ----------------------- ELSE T( J, J ) = CZERO END IF ALPHA( J ) = H( J, J ) BETA( J ) = T( J, J ) 10 CONTINUE * * If IHI < ILO, skip QZ steps * IF( IHI.LT.ILO ) \$ GO TO 190 * * 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 = CZERO MAXIT = 30*( IHI-ILO+1 ) * DO 170 JITER = 1, MAXIT * * Check for too many iterations. * IF( JITER.GT.MAXIT ) \$ GO TO 180 * * Split the matrix if possible. * * Two tests: * 1: H(j,j-1)=0 or j=ILO * 2: T(j,j)=0 * * Special case: j=ILAST * IF( ILAST.EQ.ILO ) THEN GO TO 60 ELSE IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN H( ILAST, ILAST-1 ) = CZERO GO TO 60 END IF END IF * IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN T( ILAST, ILAST ) = CZERO GO TO 50 END IF * * General case: j