#include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static doublecomplex c_b1 = {0.,0.}; static doublecomplex c_b2 = {1.,0.}; static integer c__1 = 1; static integer c__2 = 2; /* Subroutine */ int zhgeqz_(char *job, char *compq, char *compz, integer *n, integer *ilo, integer *ihi, doublecomplex *h__, integer *ldh, doublecomplex *t, integer *ldt, doublecomplex *alpha, doublecomplex * beta, doublecomplex *q, integer *ldq, doublecomplex *z__, integer * ldz, doublecomplex *work, integer *lwork, doublereal *rwork, integer * info) { /* System generated locals */ integer h_dim1, h_offset, q_dim1, q_offset, t_dim1, t_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5, i__6; doublereal d__1, d__2, d__3, d__4, d__5, d__6; doublecomplex z__1, z__2, z__3, z__4, z__5, z__6; /* Builtin functions */ double z_abs(doublecomplex *); void d_cnjg(doublecomplex *, doublecomplex *); double d_imag(doublecomplex *); void z_div(doublecomplex *, doublecomplex *, doublecomplex *), pow_zi( doublecomplex *, doublecomplex *, integer *), z_sqrt( doublecomplex *, doublecomplex *); /* Local variables */ doublereal c__; integer j; doublecomplex s, t1; integer jc, in; doublecomplex u12; integer jr; doublecomplex ad11, ad12, ad21, ad22; integer jch; logical ilq, ilz; doublereal ulp; doublecomplex abi22; doublereal absb, atol, btol, temp; extern /* Subroutine */ int zrot_(integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, doublecomplex *); doublereal temp2; extern logical lsame_(char *, char *); doublecomplex ctemp; integer iiter, ilast, jiter; doublereal anorm, bnorm; integer maxit; doublecomplex shift; extern /* Subroutine */ int zscal_(integer *, doublecomplex *, doublecomplex *, integer *); doublereal tempr; doublecomplex ctemp2, ctemp3; logical ilazr2; doublereal ascale, bscale; extern doublereal dlamch_(char *); doublecomplex signbc; doublereal safmin; extern /* Subroutine */ int xerbla_(char *, integer *); doublecomplex eshift; logical ilschr; integer icompq, ilastm; doublecomplex rtdisc; integer ischur; extern doublereal zlanhs_(char *, integer *, doublecomplex *, integer *, doublereal *); logical ilazro; integer icompz, ifirst; extern /* Subroutine */ int zlartg_(doublecomplex *, doublecomplex *, doublereal *, doublecomplex *, doublecomplex *); integer ifrstm; extern /* Subroutine */ int zlaset_(char *, integer *, integer *, doublecomplex *, doublecomplex *, doublecomplex *, integer *); integer istart; logical lquery; /* -- LAPACK routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* 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 (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. */ /* 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 .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Statement Functions .. */ /* .. */ /* .. Statement Function definitions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode JOB, COMPQ, COMPZ */ /* Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1; h__ -= h_offset; t_dim1 = *ldt; t_offset = 1 + t_dim1; t -= t_offset; --alpha; --beta; q_dim1 = *ldq; q_offset = 1 + q_dim1; q -= q_offset; z_dim1 = *ldz; z_offset = 1 + z_dim1; z__ -= z_offset; --work; --rwork; /* Function Body */ if (lsame_(job, "E")) { ilschr = FALSE_; ischur = 1; } else if (lsame_(job, "S")) { ilschr = TRUE_; ischur = 2; } else { ischur = 0; } if (lsame_(compq, "N")) { ilq = FALSE_; icompq = 1; } else if (lsame_(compq, "V")) { ilq = TRUE_; icompq = 2; } else if (lsame_(compq, "I")) { ilq = TRUE_; icompq = 3; } else { icompq = 0; } if (lsame_(compz, "N")) { ilz = FALSE_; icompz = 1; } else if (lsame_(compz, "V")) { ilz = TRUE_; icompz = 2; } else if (lsame_(compz, "I")) { ilz = TRUE_; icompz = 3; } else { icompz = 0; } /* Check Argument Values */ *info = 0; i__1 = max(1,*n); work[1].r = (doublereal) i__1, work[1].i = 0.; lquery = *lwork == -1; if (ischur == 0) { *info = -1; } else if (icompq == 0) { *info = -2; } else if (icompz == 0) { *info = -3; } else if (*n < 0) { *info = -4; } else if (*ilo < 1) { *info = -5; } else if (*ihi > *n || *ihi < *ilo - 1) { *info = -6; } else if (*ldh < *n) { *info = -8; } else if (*ldt < *n) { *info = -10; } else if (*ldq < 1 || ilq && *ldq < *n) { *info = -14; } else if (*ldz < 1 || ilz && *ldz < *n) { *info = -16; } else if (*lwork < max(1,*n) && ! lquery) { *info = -18; } if (*info != 0) { i__1 = -(*info); xerbla_("ZHGEQZ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ /* WORK( 1 ) = CMPLX( 1 ) */ if (*n <= 0) { work[1].r = 1., work[1].i = 0.; return 0; } /* Initialize Q and Z */ if (icompq == 3) { zlaset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq); } if (icompz == 3) { zlaset_("Full", n, n, &c_b1, &c_b2, &z__[z_offset], ldz); } /* Machine Constants */ in = *ihi + 1 - *ilo; safmin = dlamch_("S"); ulp = dlamch_("E") * dlamch_("B"); anorm = zlanhs_("F", &in, &h__[*ilo + *ilo * h_dim1], ldh, &rwork[1]); bnorm = zlanhs_("F", &in, &t[*ilo + *ilo * t_dim1], ldt, &rwork[1]); /* Computing MAX */ d__1 = safmin, d__2 = ulp * anorm; atol = max(d__1,d__2); /* Computing MAX */ d__1 = safmin, d__2 = ulp * bnorm; btol = max(d__1,d__2); ascale = 1. / max(safmin,anorm); bscale = 1. / max(safmin,bnorm); /* Set Eigenvalues IHI+1:N */ i__1 = *n; for (j = *ihi + 1; j <= i__1; ++j) { absb = z_abs(&t[j + j * t_dim1]); if (absb > safmin) { i__2 = j + j * t_dim1; z__2.r = t[i__2].r / absb, z__2.i = t[i__2].i / absb; d_cnjg(&z__1, &z__2); signbc.r = z__1.r, signbc.i = z__1.i; i__2 = j + j * t_dim1; t[i__2].r = absb, t[i__2].i = 0.; if (ilschr) { i__2 = j - 1; zscal_(&i__2, &signbc, &t[j * t_dim1 + 1], &c__1); zscal_(&j, &signbc, &h__[j * h_dim1 + 1], &c__1); } else { i__2 = j + j * h_dim1; i__3 = j + j * h_dim1; z__1.r = h__[i__3].r * signbc.r - h__[i__3].i * signbc.i, z__1.i = h__[i__3].r * signbc.i + h__[i__3].i * signbc.r; h__[i__2].r = z__1.r, h__[i__2].i = z__1.i; } if (ilz) { zscal_(n, &signbc, &z__[j * z_dim1 + 1], &c__1); } } else { i__2 = j + j * t_dim1; t[i__2].r = 0., t[i__2].i = 0.; } i__2 = j; i__3 = j + j * h_dim1; alpha[i__2].r = h__[i__3].r, alpha[i__2].i = h__[i__3].i; i__2 = j; i__3 = j + j * t_dim1; beta[i__2].r = t[i__3].r, beta[i__2].i = t[i__3].i; /* L10: */ } /* If IHI < ILO, skip QZ steps */ if (*ihi < *ilo) { goto L190; } /* 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) { ifrstm = 1; ilastm = *n; } else { ifrstm = *ilo; ilastm = *ihi; } iiter = 0; eshift.r = 0., eshift.i = 0.; maxit = (*ihi - *ilo + 1) * 30; i__1 = maxit; for (jiter = 1; jiter <= i__1; ++jiter) { /* Check for too many iterations. */ if (jiter > maxit) { goto L180; } /* 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 == *ilo) { goto L60; } else { i__2 = ilast + (ilast - 1) * h_dim1; if ((d__1 = h__[i__2].r, abs(d__1)) + (d__2 = d_imag(&h__[ilast + (ilast - 1) * h_dim1]), abs(d__2)) <= atol) { i__2 = ilast + (ilast - 1) * h_dim1; h__[i__2].r = 0., h__[i__2].i = 0.; goto L60; } } if (z_abs(&t[ilast + ilast * t_dim1]) <= btol) { i__2 = ilast + ilast * t_dim1; t[i__2].r = 0., t[i__2].i = 0.; goto L50; } /* General case: j= i__2; --j) { /* Test 1: for H(j,j-1)=0 or j=ILO */ if (j == *ilo) { ilazro = TRUE_; } else { i__3 = j + (j - 1) * h_dim1; if ((d__1 = h__[i__3].r, abs(d__1)) + (d__2 = d_imag(&h__[j + (j - 1) * h_dim1]), abs(d__2)) <= atol) { i__3 = j + (j - 1) * h_dim1; h__[i__3].r = 0., h__[i__3].i = 0.; ilazro = TRUE_; } else { ilazro = FALSE_; } } /* Test 2: for T(j,j)=0 */ if (z_abs(&t[j + j * t_dim1]) < btol) { i__3 = j + j * t_dim1; t[i__3].r = 0., t[i__3].i = 0.; /* Test 1a: Check for 2 consecutive small subdiagonals in A */ ilazr2 = FALSE_; if (! ilazro) { i__3 = j + (j - 1) * h_dim1; i__4 = j + 1 + j * h_dim1; i__5 = j + j * h_dim1; if (((d__1 = h__[i__3].r, abs(d__1)) + (d__2 = d_imag(& h__[j + (j - 1) * h_dim1]), abs(d__2))) * (ascale * ((d__3 = h__[i__4].r, abs(d__3)) + (d__4 = d_imag(&h__[j + 1 + j * h_dim1]), abs(d__4)))) <= ((d__5 = h__[i__5].r, abs(d__5)) + (d__6 = d_imag( &h__[j + j * h_dim1]), abs(d__6))) * (ascale * atol)) { ilazr2 = TRUE_; } } /* 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 || ilazr2) { i__3 = ilast - 1; for (jch = j; jch <= i__3; ++jch) { i__4 = jch + jch * h_dim1; ctemp.r = h__[i__4].r, ctemp.i = h__[i__4].i; zlartg_(&ctemp, &h__[jch + 1 + jch * h_dim1], &c__, & s, &h__[jch + jch * h_dim1]); i__4 = jch + 1 + jch * h_dim1; h__[i__4].r = 0., h__[i__4].i = 0.; i__4 = ilastm - jch; zrot_(&i__4, &h__[jch + (jch + 1) * h_dim1], ldh, & h__[jch + 1 + (jch + 1) * h_dim1], ldh, &c__, &s); i__4 = ilastm - jch; zrot_(&i__4, &t[jch + (jch + 1) * t_dim1], ldt, &t[ jch + 1 + (jch + 1) * t_dim1], ldt, &c__, &s); if (ilq) { d_cnjg(&z__1, &s); zrot_(n, &q[jch * q_dim1 + 1], &c__1, &q[(jch + 1) * q_dim1 + 1], &c__1, &c__, &z__1); } if (ilazr2) { i__4 = jch + (jch - 1) * h_dim1; i__5 = jch + (jch - 1) * h_dim1; z__1.r = c__ * h__[i__5].r, z__1.i = c__ * h__[ i__5].i; h__[i__4].r = z__1.r, h__[i__4].i = z__1.i; } ilazr2 = FALSE_; i__4 = jch + 1 + (jch + 1) * t_dim1; if ((d__1 = t[i__4].r, abs(d__1)) + (d__2 = d_imag(&t[ jch + 1 + (jch + 1) * t_dim1]), abs(d__2)) >= btol) { if (jch + 1 >= ilast) { goto L60; } else { ifirst = jch + 1; goto L70; } } i__4 = jch + 1 + (jch + 1) * t_dim1; t[i__4].r = 0., t[i__4].i = 0.; /* L20: */ } goto L50; } else { /* Only test 2 passed -- chase the zero to T(ILAST,ILAST) */ /* Then process as in the case T(ILAST,ILAST)=0 */ i__3 = ilast - 1; for (jch = j; jch <= i__3; ++jch) { i__4 = jch + (jch + 1) * t_dim1; ctemp.r = t[i__4].r, ctemp.i = t[i__4].i; zlartg_(&ctemp, &t[jch + 1 + (jch + 1) * t_dim1], & c__, &s, &t[jch + (jch + 1) * t_dim1]); i__4 = jch + 1 + (jch + 1) * t_dim1; t[i__4].r = 0., t[i__4].i = 0.; if (jch < ilastm - 1) { i__4 = ilastm - jch - 1; zrot_(&i__4, &t[jch + (jch + 2) * t_dim1], ldt, & t[jch + 1 + (jch + 2) * t_dim1], ldt, & c__, &s); } i__4 = ilastm - jch + 2; zrot_(&i__4, &h__[jch + (jch - 1) * h_dim1], ldh, & h__[jch + 1 + (jch - 1) * h_dim1], ldh, &c__, &s); if (ilq) { d_cnjg(&z__1, &s); zrot_(n, &q[jch * q_dim1 + 1], &c__1, &q[(jch + 1) * q_dim1 + 1], &c__1, &c__, &z__1); } i__4 = jch + 1 + jch * h_dim1; ctemp.r = h__[i__4].r, ctemp.i = h__[i__4].i; zlartg_(&ctemp, &h__[jch + 1 + (jch - 1) * h_dim1], & c__, &s, &h__[jch + 1 + jch * h_dim1]); i__4 = jch + 1 + (jch - 1) * h_dim1; h__[i__4].r = 0., h__[i__4].i = 0.; i__4 = jch + 1 - ifrstm; zrot_(&i__4, &h__[ifrstm + jch * h_dim1], &c__1, &h__[ ifrstm + (jch - 1) * h_dim1], &c__1, &c__, &s) ; i__4 = jch - ifrstm; zrot_(&i__4, &t[ifrstm + jch * t_dim1], &c__1, &t[ ifrstm + (jch - 1) * t_dim1], &c__1, &c__, &s) ; if (ilz) { zrot_(n, &z__[jch * z_dim1 + 1], &c__1, &z__[(jch - 1) * z_dim1 + 1], &c__1, &c__, &s); } /* L30: */ } goto L50; } } else if (ilazro) { /* Only test 1 passed -- work on J:ILAST */ ifirst = j; goto L70; } /* Neither test passed -- try next J */ /* L40: */ } /* (Drop-through is "impossible") */ *info = (*n << 1) + 1; goto L210; /* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a */ /* 1x1 block. */ L50: i__2 = ilast + ilast * h_dim1; ctemp.r = h__[i__2].r, ctemp.i = h__[i__2].i; zlartg_(&ctemp, &h__[ilast + (ilast - 1) * h_dim1], &c__, &s, &h__[ ilast + ilast * h_dim1]); i__2 = ilast + (ilast - 1) * h_dim1; h__[i__2].r = 0., h__[i__2].i = 0.; i__2 = ilast - ifrstm; zrot_(&i__2, &h__[ifrstm + ilast * h_dim1], &c__1, &h__[ifrstm + ( ilast - 1) * h_dim1], &c__1, &c__, &s); i__2 = ilast - ifrstm; zrot_(&i__2, &t[ifrstm + ilast * t_dim1], &c__1, &t[ifrstm + (ilast - 1) * t_dim1], &c__1, &c__, &s); if (ilz) { zrot_(n, &z__[ilast * z_dim1 + 1], &c__1, &z__[(ilast - 1) * z_dim1 + 1], &c__1, &c__, &s); } /* H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA */ L60: absb = z_abs(&t[ilast + ilast * t_dim1]); if (absb > safmin) { i__2 = ilast + ilast * t_dim1; z__2.r = t[i__2].r / absb, z__2.i = t[i__2].i / absb; d_cnjg(&z__1, &z__2); signbc.r = z__1.r, signbc.i = z__1.i; i__2 = ilast + ilast * t_dim1; t[i__2].r = absb, t[i__2].i = 0.; if (ilschr) { i__2 = ilast - ifrstm; zscal_(&i__2, &signbc, &t[ifrstm + ilast * t_dim1], &c__1); i__2 = ilast + 1 - ifrstm; zscal_(&i__2, &signbc, &h__[ifrstm + ilast * h_dim1], &c__1); } else { i__2 = ilast + ilast * h_dim1; i__3 = ilast + ilast * h_dim1; z__1.r = h__[i__3].r * signbc.r - h__[i__3].i * signbc.i, z__1.i = h__[i__3].r * signbc.i + h__[i__3].i * signbc.r; h__[i__2].r = z__1.r, h__[i__2].i = z__1.i; } if (ilz) { zscal_(n, &signbc, &z__[ilast * z_dim1 + 1], &c__1); } } else { i__2 = ilast + ilast * t_dim1; t[i__2].r = 0., t[i__2].i = 0.; } i__2 = ilast; i__3 = ilast + ilast * h_dim1; alpha[i__2].r = h__[i__3].r, alpha[i__2].i = h__[i__3].i; i__2 = ilast; i__3 = ilast + ilast * t_dim1; beta[i__2].r = t[i__3].r, beta[i__2].i = t[i__3].i; /* Go to next block -- exit if finished. */ --ilast; if (ilast < *ilo) { goto L190; } /* Reset counters */ iiter = 0; eshift.r = 0., eshift.i = 0.; if (! ilschr) { ilastm = ilast; if (ifrstm > ilast) { ifrstm = *ilo; } } goto L160; /* QZ step */ /* This iteration only involves rows/columns IFIRST:ILAST. We */ /* assume IFIRST < ILAST, and that the diagonal of B is non-zero. */ L70: ++iiter; if (! ilschr) { ifrstm = ifirst; } /* Compute the Shift. */ /* 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 != iiter) { /* The Wilkinson shift (AEP p.512), i.e., the eigenvalue of */ /* the bottom-right 2x2 block of A inv(B) which is nearest to */ /* the bottom-right element. */ /* We factor B as U*D, where U has unit diagonals, and */ /* compute (A*inv(D))*inv(U). */ i__2 = ilast - 1 + ilast * t_dim1; z__2.r = bscale * t[i__2].r, z__2.i = bscale * t[i__2].i; i__3 = ilast + ilast * t_dim1; z__3.r = bscale * t[i__3].r, z__3.i = bscale * t[i__3].i; z_div(&z__1, &z__2, &z__3); u12.r = z__1.r, u12.i = z__1.i; i__2 = ilast - 1 + (ilast - 1) * h_dim1; z__2.r = ascale * h__[i__2].r, z__2.i = ascale * h__[i__2].i; i__3 = ilast - 1 + (ilast - 1) * t_dim1; z__3.r = bscale * t[i__3].r, z__3.i = bscale * t[i__3].i; z_div(&z__1, &z__2, &z__3); ad11.r = z__1.r, ad11.i = z__1.i; i__2 = ilast + (ilast - 1) * h_dim1; z__2.r = ascale * h__[i__2].r, z__2.i = ascale * h__[i__2].i; i__3 = ilast - 1 + (ilast - 1) * t_dim1; z__3.r = bscale * t[i__3].r, z__3.i = bscale * t[i__3].i; z_div(&z__1, &z__2, &z__3); ad21.r = z__1.r, ad21.i = z__1.i; i__2 = ilast - 1 + ilast * h_dim1; z__2.r = ascale * h__[i__2].r, z__2.i = ascale * h__[i__2].i; i__3 = ilast + ilast * t_dim1; z__3.r = bscale * t[i__3].r, z__3.i = bscale * t[i__3].i; z_div(&z__1, &z__2, &z__3); ad12.r = z__1.r, ad12.i = z__1.i; i__2 = ilast + ilast * h_dim1; z__2.r = ascale * h__[i__2].r, z__2.i = ascale * h__[i__2].i; i__3 = ilast + ilast * t_dim1; z__3.r = bscale * t[i__3].r, z__3.i = bscale * t[i__3].i; z_div(&z__1, &z__2, &z__3); ad22.r = z__1.r, ad22.i = z__1.i; z__2.r = u12.r * ad21.r - u12.i * ad21.i, z__2.i = u12.r * ad21.i + u12.i * ad21.r; z__1.r = ad22.r - z__2.r, z__1.i = ad22.i - z__2.i; abi22.r = z__1.r, abi22.i = z__1.i; z__2.r = ad11.r + abi22.r, z__2.i = ad11.i + abi22.i; z__1.r = z__2.r * .5, z__1.i = z__2.i * .5; t1.r = z__1.r, t1.i = z__1.i; pow_zi(&z__4, &t1, &c__2); z__5.r = ad12.r * ad21.r - ad12.i * ad21.i, z__5.i = ad12.r * ad21.i + ad12.i * ad21.r; z__3.r = z__4.r + z__5.r, z__3.i = z__4.i + z__5.i; z__6.r = ad11.r * ad22.r - ad11.i * ad22.i, z__6.i = ad11.r * ad22.i + ad11.i * ad22.r; z__2.r = z__3.r - z__6.r, z__2.i = z__3.i - z__6.i; z_sqrt(&z__1, &z__2); rtdisc.r = z__1.r, rtdisc.i = z__1.i; z__1.r = t1.r - abi22.r, z__1.i = t1.i - abi22.i; z__2.r = t1.r - abi22.r, z__2.i = t1.i - abi22.i; temp = z__1.r * rtdisc.r + d_imag(&z__2) * d_imag(&rtdisc); if (temp <= 0.) { z__1.r = t1.r + rtdisc.r, z__1.i = t1.i + rtdisc.i; shift.r = z__1.r, shift.i = z__1.i; } else { z__1.r = t1.r - rtdisc.r, z__1.i = t1.i - rtdisc.i; shift.r = z__1.r, shift.i = z__1.i; } } else { /* Exceptional shift. Chosen for no particularly good reason. */ i__2 = ilast - 1 + ilast * h_dim1; z__4.r = ascale * h__[i__2].r, z__4.i = ascale * h__[i__2].i; i__3 = ilast - 1 + (ilast - 1) * t_dim1; z__5.r = bscale * t[i__3].r, z__5.i = bscale * t[i__3].i; z_div(&z__3, &z__4, &z__5); d_cnjg(&z__2, &z__3); z__1.r = eshift.r + z__2.r, z__1.i = eshift.i + z__2.i; eshift.r = z__1.r, eshift.i = z__1.i; shift.r = eshift.r, shift.i = eshift.i; } /* Now check for two consecutive small subdiagonals. */ i__2 = ifirst + 1; for (j = ilast - 1; j >= i__2; --j) { istart = j; i__3 = j + j * h_dim1; z__2.r = ascale * h__[i__3].r, z__2.i = ascale * h__[i__3].i; i__4 = j + j * t_dim1; z__4.r = bscale * t[i__4].r, z__4.i = bscale * t[i__4].i; z__3.r = shift.r * z__4.r - shift.i * z__4.i, z__3.i = shift.r * z__4.i + shift.i * z__4.r; z__1.r = z__2.r - z__3.r, z__1.i = z__2.i - z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; temp = (d__1 = ctemp.r, abs(d__1)) + (d__2 = d_imag(&ctemp), abs( d__2)); i__3 = j + 1 + j * h_dim1; temp2 = ascale * ((d__1 = h__[i__3].r, abs(d__1)) + (d__2 = d_imag(&h__[j + 1 + j * h_dim1]), abs(d__2))); tempr = max(temp,temp2); if (tempr < 1. && tempr != 0.) { temp /= tempr; temp2 /= tempr; } i__3 = j + (j - 1) * h_dim1; if (((d__1 = h__[i__3].r, abs(d__1)) + (d__2 = d_imag(&h__[j + (j - 1) * h_dim1]), abs(d__2))) * temp2 <= temp * atol) { goto L90; } /* L80: */ } istart = ifirst; i__2 = ifirst + ifirst * h_dim1; z__2.r = ascale * h__[i__2].r, z__2.i = ascale * h__[i__2].i; i__3 = ifirst + ifirst * t_dim1; z__4.r = bscale * t[i__3].r, z__4.i = bscale * t[i__3].i; z__3.r = shift.r * z__4.r - shift.i * z__4.i, z__3.i = shift.r * z__4.i + shift.i * z__4.r; z__1.r = z__2.r - z__3.r, z__1.i = z__2.i - z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; L90: /* Do an implicit-shift QZ sweep. */ /* Initial Q */ i__2 = istart + 1 + istart * h_dim1; z__1.r = ascale * h__[i__2].r, z__1.i = ascale * h__[i__2].i; ctemp2.r = z__1.r, ctemp2.i = z__1.i; zlartg_(&ctemp, &ctemp2, &c__, &s, &ctemp3); /* Sweep */ i__2 = ilast - 1; for (j = istart; j <= i__2; ++j) { if (j > istart) { i__3 = j + (j - 1) * h_dim1; ctemp.r = h__[i__3].r, ctemp.i = h__[i__3].i; zlartg_(&ctemp, &h__[j + 1 + (j - 1) * h_dim1], &c__, &s, & h__[j + (j - 1) * h_dim1]); i__3 = j + 1 + (j - 1) * h_dim1; h__[i__3].r = 0., h__[i__3].i = 0.; } i__3 = ilastm; for (jc = j; jc <= i__3; ++jc) { i__4 = j + jc * h_dim1; z__2.r = c__ * h__[i__4].r, z__2.i = c__ * h__[i__4].i; i__5 = j + 1 + jc * h_dim1; z__3.r = s.r * h__[i__5].r - s.i * h__[i__5].i, z__3.i = s.r * h__[i__5].i + s.i * h__[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; i__4 = j + 1 + jc * h_dim1; d_cnjg(&z__4, &s); z__3.r = -z__4.r, z__3.i = -z__4.i; i__5 = j + jc * h_dim1; z__2.r = z__3.r * h__[i__5].r - z__3.i * h__[i__5].i, z__2.i = z__3.r * h__[i__5].i + z__3.i * h__[i__5].r; i__6 = j + 1 + jc * h_dim1; z__5.r = c__ * h__[i__6].r, z__5.i = c__ * h__[i__6].i; z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i; h__[i__4].r = z__1.r, h__[i__4].i = z__1.i; i__4 = j + jc * h_dim1; h__[i__4].r = ctemp.r, h__[i__4].i = ctemp.i; i__4 = j + jc * t_dim1; z__2.r = c__ * t[i__4].r, z__2.i = c__ * t[i__4].i; i__5 = j + 1 + jc * t_dim1; z__3.r = s.r * t[i__5].r - s.i * t[i__5].i, z__3.i = s.r * t[ i__5].i + s.i * t[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp2.r = z__1.r, ctemp2.i = z__1.i; i__4 = j + 1 + jc * t_dim1; d_cnjg(&z__4, &s); z__3.r = -z__4.r, z__3.i = -z__4.i; i__5 = j + jc * t_dim1; z__2.r = z__3.r * t[i__5].r - z__3.i * t[i__5].i, z__2.i = z__3.r * t[i__5].i + z__3.i * t[i__5].r; i__6 = j + 1 + jc * t_dim1; z__5.r = c__ * t[i__6].r, z__5.i = c__ * t[i__6].i; z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i; t[i__4].r = z__1.r, t[i__4].i = z__1.i; i__4 = j + jc * t_dim1; t[i__4].r = ctemp2.r, t[i__4].i = ctemp2.i; /* L100: */ } if (ilq) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { i__4 = jr + j * q_dim1; z__2.r = c__ * q[i__4].r, z__2.i = c__ * q[i__4].i; d_cnjg(&z__4, &s); i__5 = jr + (j + 1) * q_dim1; z__3.r = z__4.r * q[i__5].r - z__4.i * q[i__5].i, z__3.i = z__4.r * q[i__5].i + z__4.i * q[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; i__4 = jr + (j + 1) * q_dim1; z__3.r = -s.r, z__3.i = -s.i; i__5 = jr + j * q_dim1; z__2.r = z__3.r * q[i__5].r - z__3.i * q[i__5].i, z__2.i = z__3.r * q[i__5].i + z__3.i * q[i__5].r; i__6 = jr + (j + 1) * q_dim1; z__4.r = c__ * q[i__6].r, z__4.i = c__ * q[i__6].i; z__1.r = z__2.r + z__4.r, z__1.i = z__2.i + z__4.i; q[i__4].r = z__1.r, q[i__4].i = z__1.i; i__4 = jr + j * q_dim1; q[i__4].r = ctemp.r, q[i__4].i = ctemp.i; /* L110: */ } } i__3 = j + 1 + (j + 1) * t_dim1; ctemp.r = t[i__3].r, ctemp.i = t[i__3].i; zlartg_(&ctemp, &t[j + 1 + j * t_dim1], &c__, &s, &t[j + 1 + (j + 1) * t_dim1]); i__3 = j + 1 + j * t_dim1; t[i__3].r = 0., t[i__3].i = 0.; /* Computing MIN */ i__4 = j + 2; i__3 = min(i__4,ilast); for (jr = ifrstm; jr <= i__3; ++jr) { i__4 = jr + (j + 1) * h_dim1; z__2.r = c__ * h__[i__4].r, z__2.i = c__ * h__[i__4].i; i__5 = jr + j * h_dim1; z__3.r = s.r * h__[i__5].r - s.i * h__[i__5].i, z__3.i = s.r * h__[i__5].i + s.i * h__[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; i__4 = jr + j * h_dim1; d_cnjg(&z__4, &s); z__3.r = -z__4.r, z__3.i = -z__4.i; i__5 = jr + (j + 1) * h_dim1; z__2.r = z__3.r * h__[i__5].r - z__3.i * h__[i__5].i, z__2.i = z__3.r * h__[i__5].i + z__3.i * h__[i__5].r; i__6 = jr + j * h_dim1; z__5.r = c__ * h__[i__6].r, z__5.i = c__ * h__[i__6].i; z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i; h__[i__4].r = z__1.r, h__[i__4].i = z__1.i; i__4 = jr + (j + 1) * h_dim1; h__[i__4].r = ctemp.r, h__[i__4].i = ctemp.i; /* L120: */ } i__3 = j; for (jr = ifrstm; jr <= i__3; ++jr) { i__4 = jr + (j + 1) * t_dim1; z__2.r = c__ * t[i__4].r, z__2.i = c__ * t[i__4].i; i__5 = jr + j * t_dim1; z__3.r = s.r * t[i__5].r - s.i * t[i__5].i, z__3.i = s.r * t[ i__5].i + s.i * t[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; i__4 = jr + j * t_dim1; d_cnjg(&z__4, &s); z__3.r = -z__4.r, z__3.i = -z__4.i; i__5 = jr + (j + 1) * t_dim1; z__2.r = z__3.r * t[i__5].r - z__3.i * t[i__5].i, z__2.i = z__3.r * t[i__5].i + z__3.i * t[i__5].r; i__6 = jr + j * t_dim1; z__5.r = c__ * t[i__6].r, z__5.i = c__ * t[i__6].i; z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i; t[i__4].r = z__1.r, t[i__4].i = z__1.i; i__4 = jr + (j + 1) * t_dim1; t[i__4].r = ctemp.r, t[i__4].i = ctemp.i; /* L130: */ } if (ilz) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { i__4 = jr + (j + 1) * z_dim1; z__2.r = c__ * z__[i__4].r, z__2.i = c__ * z__[i__4].i; i__5 = jr + j * z_dim1; z__3.r = s.r * z__[i__5].r - s.i * z__[i__5].i, z__3.i = s.r * z__[i__5].i + s.i * z__[i__5].r; z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i; ctemp.r = z__1.r, ctemp.i = z__1.i; i__4 = jr + j * z_dim1; d_cnjg(&z__4, &s); z__3.r = -z__4.r, z__3.i = -z__4.i; i__5 = jr + (j + 1) * z_dim1; z__2.r = z__3.r * z__[i__5].r - z__3.i * z__[i__5].i, z__2.i = z__3.r * z__[i__5].i + z__3.i * z__[i__5] .r; i__6 = jr + j * z_dim1; z__5.r = c__ * z__[i__6].r, z__5.i = c__ * z__[i__6].i; z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i; z__[i__4].r = z__1.r, z__[i__4].i = z__1.i; i__4 = jr + (j + 1) * z_dim1; z__[i__4].r = ctemp.r, z__[i__4].i = ctemp.i; /* L140: */ } } /* L150: */ } L160: /* L170: */ ; } /* Drop-through = non-convergence */ L180: *info = ilast; goto L210; /* Successful completion of all QZ steps */ L190: /* Set Eigenvalues 1:ILO-1 */ i__1 = *ilo - 1; for (j = 1; j <= i__1; ++j) { absb = z_abs(&t[j + j * t_dim1]); if (absb > safmin) { i__2 = j + j * t_dim1; z__2.r = t[i__2].r / absb, z__2.i = t[i__2].i / absb; d_cnjg(&z__1, &z__2); signbc.r = z__1.r, signbc.i = z__1.i; i__2 = j + j * t_dim1; t[i__2].r = absb, t[i__2].i = 0.; if (ilschr) { i__2 = j - 1; zscal_(&i__2, &signbc, &t[j * t_dim1 + 1], &c__1); zscal_(&j, &signbc, &h__[j * h_dim1 + 1], &c__1); } else { i__2 = j + j * h_dim1; i__3 = j + j * h_dim1; z__1.r = h__[i__3].r * signbc.r - h__[i__3].i * signbc.i, z__1.i = h__[i__3].r * signbc.i + h__[i__3].i * signbc.r; h__[i__2].r = z__1.r, h__[i__2].i = z__1.i; } if (ilz) { zscal_(n, &signbc, &z__[j * z_dim1 + 1], &c__1); } } else { i__2 = j + j * t_dim1; t[i__2].r = 0., t[i__2].i = 0.; } i__2 = j; i__3 = j + j * h_dim1; alpha[i__2].r = h__[i__3].r, alpha[i__2].i = h__[i__3].i; i__2 = j; i__3 = j + j * t_dim1; beta[i__2].r = t[i__3].r, beta[i__2].i = t[i__3].i; /* L200: */ } /* Normal Termination */ *info = 0; /* Exit (other than argument error) -- return optimal workspace size */ L210: z__1.r = (doublereal) (*n), z__1.i = 0.; work[1].r = z__1.r, work[1].i = z__1.i; return 0; /* End of ZHGEQZ */ } /* zhgeqz_ */