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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 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2009 -- * * .. 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. * 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 * 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 ), LDT, 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 |