#include #include #include #include #include #ifdef complex #undef complex #endif #ifdef I #undef I #endif #if defined(_WIN64) typedef long long BLASLONG; typedef unsigned long long BLASULONG; #else typedef long BLASLONG; typedef unsigned long BLASULONG; #endif #ifdef LAPACK_ILP64 typedef BLASLONG blasint; #if defined(_WIN64) #define blasabs(x) llabs(x) #else #define blasabs(x) labs(x) #endif #else typedef int blasint; #define blasabs(x) abs(x) #endif typedef blasint integer; typedef unsigned int uinteger; typedef char *address; typedef short int shortint; typedef float real; typedef double doublereal; typedef struct { real r, i; } complex; typedef struct { doublereal r, i; } doublecomplex; #ifdef _MSC_VER static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;} static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;} static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;} static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;} #else static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;} static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;} static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;} static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;} #endif #define pCf(z) (*_pCf(z)) #define pCd(z) (*_pCd(z)) typedef int logical; typedef short int shortlogical; typedef char logical1; typedef char integer1; #define TRUE_ (1) #define FALSE_ (0) /* Extern is for use with -E */ #ifndef Extern #define Extern extern #endif /* I/O stuff */ typedef int flag; typedef int ftnlen; typedef int ftnint; /*external read, write*/ typedef struct { flag cierr; ftnint ciunit; flag ciend; char *cifmt; ftnint cirec; } cilist; /*internal read, write*/ typedef struct { flag icierr; char *iciunit; flag iciend; char *icifmt; ftnint icirlen; ftnint icirnum; } icilist; /*open*/ typedef struct { flag oerr; ftnint ounit; char *ofnm; ftnlen ofnmlen; char *osta; char *oacc; char *ofm; ftnint orl; char *oblnk; } olist; /*close*/ typedef struct { flag cerr; ftnint cunit; char *csta; } cllist; /*rewind, backspace, endfile*/ typedef struct { flag aerr; ftnint aunit; } alist; /* inquire */ typedef struct { flag inerr; ftnint inunit; char *infile; ftnlen infilen; ftnint *inex; /*parameters in standard's order*/ ftnint *inopen; ftnint *innum; ftnint *innamed; char *inname; ftnlen innamlen; char *inacc; ftnlen inacclen; char *inseq; ftnlen inseqlen; char *indir; ftnlen indirlen; char *infmt; ftnlen infmtlen; char *inform; ftnint informlen; char *inunf; ftnlen inunflen; ftnint *inrecl; ftnint *innrec; char *inblank; ftnlen inblanklen; } inlist; #define VOID void union Multitype { /* for multiple entry points */ integer1 g; shortint h; integer i; /* longint j; */ real r; doublereal d; complex c; doublecomplex z; }; typedef union Multitype Multitype; struct Vardesc { /* for Namelist */ char *name; char *addr; ftnlen *dims; int type; }; typedef struct Vardesc Vardesc; struct Namelist { char *name; Vardesc **vars; int nvars; }; typedef struct Namelist Namelist; #define abs(x) ((x) >= 0 ? (x) : -(x)) #define dabs(x) (fabs(x)) #define f2cmin(a,b) ((a) <= (b) ? (a) : (b)) #define f2cmax(a,b) ((a) >= (b) ? (a) : (b)) #define dmin(a,b) (f2cmin(a,b)) #define dmax(a,b) (f2cmax(a,b)) #define bit_test(a,b) ((a) >> (b) & 1) #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b))) #define bit_set(a,b) ((a) | ((uinteger)1 << (b))) #define abort_() { sig_die("Fortran abort routine called", 1); } #define c_abs(z) (cabsf(Cf(z))) #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); } #ifdef _MSC_VER #define c_div(c, a, b) {Cf(c)._Val[0] = (Cf(a)._Val[0]/Cf(b)._Val[0]); Cf(c)._Val[1]=(Cf(a)._Val[1]/Cf(b)._Val[1]);} #define z_div(c, a, b) {Cd(c)._Val[0] = (Cd(a)._Val[0]/Cd(b)._Val[0]); Cd(c)._Val[1]=(Cd(a)._Val[1]/df(b)._Val[1]);} #else #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);} #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);} #endif #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));} #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));} #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));} //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));} #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));} #define d_abs(x) (fabs(*(x))) #define d_acos(x) (acos(*(x))) #define d_asin(x) (asin(*(x))) #define d_atan(x) (atan(*(x))) #define d_atn2(x, y) (atan2(*(x),*(y))) #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); } #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); } #define d_cos(x) (cos(*(x))) #define d_cosh(x) (cosh(*(x))) #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 ) #define d_exp(x) (exp(*(x))) #define d_imag(z) (cimag(Cd(z))) #define r_imag(z) (cimagf(Cf(z))) #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x))) #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) ) #define d_log(x) (log(*(x))) #define d_mod(x, y) (fmod(*(x), *(y))) #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x))) #define d_nint(x) u_nint(*(x)) #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a))) #define d_sign(a,b) u_sign(*(a),*(b)) #define r_sign(a,b) u_sign(*(a),*(b)) #define d_sin(x) (sin(*(x))) #define d_sinh(x) (sinh(*(x))) #define d_sqrt(x) (sqrt(*(x))) #define d_tan(x) (tan(*(x))) #define d_tanh(x) (tanh(*(x))) #define i_abs(x) abs(*(x)) #define i_dnnt(x) ((integer)u_nint(*(x))) #define i_len(s, n) (n) #define i_nint(x) ((integer)u_nint(*(x))) #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b))) #define pow_dd(ap, bp) ( pow(*(ap), *(bp))) #define pow_si(B,E) spow_ui(*(B),*(E)) #define pow_ri(B,E) spow_ui(*(B),*(E)) #define pow_di(B,E) dpow_ui(*(B),*(E)) #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));} #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));} #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));} #define s_cat(lpp, rpp, rnp, np, llp) { ftnlen i, nc, ll; char *f__rp, *lp; ll = (llp); lp = (lpp); for(i=0; i < (int)*(np); ++i) { nc = ll; if((rnp)[i] < nc) nc = (rnp)[i]; ll -= nc; f__rp = (rpp)[i]; while(--nc >= 0) *lp++ = *(f__rp)++; } while(--ll >= 0) *lp++ = ' '; } #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d)))) #define s_copy(A,B,C,D) { int __i,__m; for (__i=0, __m=f2cmin((C),(D)); __i<__m && (B)[__i] != 0; ++__i) (A)[__i] = (B)[__i]; } #define sig_die(s, kill) { exit(1); } #define s_stop(s, n) {exit(0);} static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n"; #define z_abs(z) (cabs(Cd(z))) #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));} #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));} #define myexit_() break; #define mycycle() continue; #define myceiling(w) {ceil(w)} #define myhuge(w) {HUGE_VAL} //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);} #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)} /* procedure parameter types for -A and -C++ */ #define F2C_proc_par_types 1 #ifdef __cplusplus typedef logical (*L_fp)(...); #else typedef logical (*L_fp)(); #endif static float spow_ui(float x, integer n) { float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static double dpow_ui(double x, integer n) { double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #ifdef _MSC_VER static _Fcomplex cpow_ui(complex x, integer n) { complex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i; for(u = n; ; ) { if(u & 01) pow.r *= x.r, pow.i *= x.i; if(u >>= 1) x.r *= x.r, x.i *= x.i; else break; } } _Fcomplex p={pow.r, pow.i}; return p; } #else static _Complex float cpow_ui(_Complex float x, integer n) { _Complex float pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif #ifdef _MSC_VER static _Dcomplex zpow_ui(_Dcomplex x, integer n) { _Dcomplex pow={1.0,0.0}; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1]; for(u = n; ; ) { if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1]; if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1]; else break; } } _Dcomplex p = {pow._Val[0], pow._Val[1]}; return p; } #else static _Complex double zpow_ui(_Complex double x, integer n) { _Complex double pow=1.0; unsigned long int u; if(n != 0) { if(n < 0) n = -n, x = 1/x; for(u = n; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } #endif static integer pow_ii(integer x, integer n) { integer pow; unsigned long int u; if (n <= 0) { if (n == 0 || x == 1) pow = 1; else if (x != -1) pow = x == 0 ? 1/x : 0; else n = -n; } if ((n > 0) || !(n == 0 || x == 1 || x != -1)) { u = n; for(pow = 1; ; ) { if(u & 01) pow *= x; if(u >>= 1) x *= x; else break; } } return pow; } static integer dmaxloc_(double *w, integer s, integer e, integer *n) { double m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static integer smaxloc_(float *w, integer s, integer e, integer *n) { float m; integer i, mi; for(m=w[s-1], mi=s, i=s+1; i<=e; i++) if (w[i-1]>m) mi=i ,m=w[i-1]; return mi-s+1; } static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) { integer n = *n_, incx = *incx_, incy = *incy_, i; #ifdef _MSC_VER _Fcomplex zdotc = {0.0, 0.0}; if (incx == 1 && incy == 1) { for (i=0;i \brief \b DHGEQZ */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DHGEQZ + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, */ /* ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, */ /* LWORK, INFO ) */ /* CHARACTER COMPQ, COMPZ, JOB */ /* INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N */ /* DOUBLE PRECISION ALPHAI( * ), ALPHAR( * ), BETA( * ), */ /* $ H( LDH, * ), Q( LDQ, * ), T( LDT, * ), */ /* $ WORK( * ), Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DHGEQZ 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 DGGHRD. */ /* > */ /* > 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 DGGHRD 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. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] JOB */ /* > \verbatim */ /* > JOB is CHARACTER*1 */ /* > = 'E': Compute eigenvalues only; */ /* > = 'S': Compute eigenvalues and the Schur form. */ /* > \endverbatim */ /* > */ /* > \param[in] COMPQ */ /* > \verbatim */ /* > COMPQ is 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. */ /* > \endverbatim */ /* > */ /* > \param[in] COMPZ */ /* > \verbatim */ /* > COMPZ is 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. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrices H, T, Q, and Z. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] ILO */ /* > \verbatim */ /* > ILO is INTEGER */ /* > \endverbatim */ /* > */ /* > \param[in] IHI */ /* > \verbatim */ /* > IHI is 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. */ /* > \endverbatim */ /* > */ /* > \param[in,out] H */ /* > \verbatim */ /* > H is DOUBLE PRECISION 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. */ /* > \endverbatim */ /* > */ /* > \param[in] LDH */ /* > \verbatim */ /* > LDH is INTEGER */ /* > The leading dimension of the array H. LDH >= f2cmax( 1, N ). */ /* > \endverbatim */ /* > */ /* > \param[in,out] T */ /* > \verbatim */ /* > T is DOUBLE PRECISION 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. */ /* > \endverbatim */ /* > */ /* > \param[in] LDT */ /* > \verbatim */ /* > LDT is INTEGER */ /* > The leading dimension of the array T. LDT >= f2cmax( 1, N ). */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAR */ /* > \verbatim */ /* > ALPHAR is DOUBLE PRECISION array, dimension (N) */ /* > The real parts of each scalar alpha defining an eigenvalue */ /* > of GNEP. */ /* > \endverbatim */ /* > */ /* > \param[out] ALPHAI */ /* > \verbatim */ /* > ALPHAI is DOUBLE PRECISION 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). */ /* > \endverbatim */ /* > */ /* > \param[out] BETA */ /* > \verbatim */ /* > BETA is DOUBLE PRECISION 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. */ /* > \endverbatim */ /* > */ /* > \param[in,out] Q */ /* > \verbatim */ /* > Q is DOUBLE PRECISION array, dimension (LDQ, N) */ /* > On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in */ /* > the reduction of (A,B) to generalized Hessenberg form. */ /* > On exit, if COMPQ = 'I', the orthogonal matrix of left Schur */ /* > vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix */ /* > of left Schur vectors of (A,B). */ /* > Not referenced if COMPQ = 'N'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDQ */ /* > \verbatim */ /* > LDQ is INTEGER */ /* > The leading dimension of the array Q. LDQ >= 1. */ /* > If COMPQ='V' or 'I', then LDQ >= N. */ /* > \endverbatim */ /* > */ /* > \param[in,out] Z */ /* > \verbatim */ /* > Z is DOUBLE PRECISION 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'. */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. LDZ >= 1. */ /* > If COMPZ='V' or 'I', then LDZ >= N. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */ /* > On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > The dimension of the array WORK. LWORK >= f2cmax(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. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is 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. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date June 2016 */ /* > \ingroup doubleGEcomputational */ /* > \par Further Details: */ /* ===================== */ /* > */ /* > \verbatim */ /* > */ /* > 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. */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int dhgeqz_(char *job, char *compq, char *compz, integer *n, integer *ilo, integer *ihi, doublereal *h__, integer *ldh, doublereal *t, integer *ldt, doublereal *alphar, doublereal *alphai, doublereal * beta, doublereal *q, integer *ldq, doublereal *z__, integer *ldz, doublereal *work, integer *lwork, 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; doublereal d__1, d__2, d__3, d__4; /* Local variables */ doublereal ad11l, ad12l, ad21l, ad22l, ad32l, wabs, atol, btol, temp; extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *), dlag2_( doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); doublereal temp2, s1inv, c__; integer j; doublereal s, v[3], scale; extern logical lsame_(char *, char *); integer iiter, ilast, jiter; doublereal anorm, bnorm; integer maxit; doublereal tempi, tempr, s1, s2, t1, u1, u2; extern doublereal dlapy2_(doublereal *, doublereal *), dlapy3_(doublereal *, doublereal *, doublereal *); extern /* Subroutine */ int dlasv2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); logical ilazr2; doublereal a11, a12, a21, a22, b11, b22, c12, c21; integer jc; doublereal an, bn, cl, cq, cr; integer in; doublereal ascale, bscale, u12, w11; integer jr; doublereal cz, sl, w12, w21, w22, wi; extern doublereal dlamch_(char *); doublereal sr; extern /* Subroutine */ int dlarfg_(integer *, doublereal *, doublereal *, integer *, doublereal *); doublereal vs, wr; extern doublereal dlanhs_(char *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *); doublereal safmin; extern /* Subroutine */ int dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); doublereal safmax; extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen); doublereal eshift; logical ilschr; doublereal b1a, b2a; integer icompq, ilastm; doublereal a1i; integer ischur; doublereal a2i, b1i; logical ilazro; integer icompz, ifirst; doublereal b2i; integer ifrstm; doublereal a1r; integer istart; logical ilpivt; doublereal a2r, b1r, b2r; logical lquery; doublereal wr2, ad11, ad12, ad21, ad22, c11i, c22i; integer jch; doublereal c11r, c22r; logical ilq; doublereal u12l, tau, sqi; logical ilz; doublereal ulp, sqr, szi, szr; /* -- LAPACK computational routine (version 3.7.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* June 2016 */ /* ===================================================================== */ /* $ SAFETY = 1.0E+0 ) */ /* Decode JOB, COMPQ, COMPZ */ /* Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1 * 1; h__ -= h_offset; t_dim1 = *ldt; t_offset = 1 + t_dim1 * 1; t -= t_offset; --alphar; --alphai; --beta; q_dim1 = *ldq; q_offset = 1 + q_dim1 * 1; q -= q_offset; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --work; /* 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; work[1] = (doublereal) f2cmax(1,*n); 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 = -15; } else if (*ldz < 1 || ilz && *ldz < *n) { *info = -17; } else if (*lwork < f2cmax(1,*n) && ! lquery) { *info = -19; } if (*info != 0) { i__1 = -(*info); xerbla_("DHGEQZ", &i__1, (ftnlen)6); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n <= 0) { work[1] = 1.; return 0; } /* Initialize Q and Z */ if (icompq == 3) { dlaset_("Full", n, n, &c_b12, &c_b13, &q[q_offset], ldq); } if (icompz == 3) { dlaset_("Full", n, n, &c_b12, &c_b13, &z__[z_offset], ldz); } /* Machine Constants */ in = *ihi + 1 - *ilo; safmin = dlamch_("S"); safmax = 1. / safmin; ulp = dlamch_("E") * dlamch_("B"); anorm = dlanhs_("F", &in, &h__[*ilo + *ilo * h_dim1], ldh, &work[1]); bnorm = dlanhs_("F", &in, &t[*ilo + *ilo * t_dim1], ldt, &work[1]); /* Computing MAX */ d__1 = safmin, d__2 = ulp * anorm; atol = f2cmax(d__1,d__2); /* Computing MAX */ d__1 = safmin, d__2 = ulp * bnorm; btol = f2cmax(d__1,d__2); ascale = 1. / f2cmax(safmin,anorm); bscale = 1. / f2cmax(safmin,bnorm); /* Set Eigenvalues IHI+1:N */ i__1 = *n; for (j = *ihi + 1; j <= i__1; ++j) { if (t[j + j * t_dim1] < 0.) { if (ilschr) { i__2 = j; for (jr = 1; jr <= i__2; ++jr) { h__[jr + j * h_dim1] = -h__[jr + j * h_dim1]; t[jr + j * t_dim1] = -t[jr + j * t_dim1]; /* L10: */ } } else { h__[j + j * h_dim1] = -h__[j + j * h_dim1]; t[j + j * t_dim1] = -t[j + j * t_dim1]; } if (ilz) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { z__[jr + j * z_dim1] = -z__[jr + j * z_dim1]; /* L20: */ } } } alphar[j] = h__[j + j * h_dim1]; alphai[j] = 0.; beta[j] = t[j + j * t_dim1]; /* L30: */ } /* If IHI < ILO, skip QZ steps */ if (*ihi < *ilo) { goto L380; } /* 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 = 0.; maxit = (*ihi - *ilo + 1) * 30; i__1 = maxit; for (jiter = 1; jiter <= i__1; ++jiter) { /* Split the matrix if possible. */ /* Two tests: */ /* 1: H(j,j-1)=0 or j=ILO */ /* 2: T(j,j)=0 */ if (ilast == *ilo) { /* Special case: j=ILAST */ goto L80; } else { if ((d__1 = h__[ilast + (ilast - 1) * h_dim1], abs(d__1)) <= atol) { h__[ilast + (ilast - 1) * h_dim1] = 0.; goto L80; } } if ((d__1 = t[ilast + ilast * t_dim1], abs(d__1)) <= btol) { t[ilast + ilast * t_dim1] = 0.; goto L70; } /* General case: j= i__2; --j) { /* Test 1: for H(j,j-1)=0 or j=ILO */ if (j == *ilo) { ilazro = TRUE_; } else { if ((d__1 = h__[j + (j - 1) * h_dim1], abs(d__1)) <= atol) { h__[j + (j - 1) * h_dim1] = 0.; ilazro = TRUE_; } else { ilazro = FALSE_; } } /* Test 2: for T(j,j)=0 */ if ((d__1 = t[j + j * t_dim1], abs(d__1)) < btol) { t[j + j * t_dim1] = 0.; /* Test 1a: Check for 2 consecutive small subdiagonals in A */ ilazr2 = FALSE_; if (! ilazro) { temp = (d__1 = h__[j + (j - 1) * h_dim1], abs(d__1)); temp2 = (d__1 = h__[j + j * h_dim1], abs(d__1)); tempr = f2cmax(temp,temp2); if (tempr < 1. && tempr != 0.) { temp /= tempr; temp2 /= tempr; } if (temp * (ascale * (d__1 = h__[j + 1 + j * h_dim1], abs( d__1))) <= temp2 * (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) { temp = h__[jch + jch * h_dim1]; dlartg_(&temp, &h__[jch + 1 + jch * h_dim1], &c__, &s, &h__[jch + jch * h_dim1]); h__[jch + 1 + jch * h_dim1] = 0.; i__4 = ilastm - jch; drot_(&i__4, &h__[jch + (jch + 1) * h_dim1], ldh, & h__[jch + 1 + (jch + 1) * h_dim1], ldh, &c__, &s); i__4 = ilastm - jch; drot_(&i__4, &t[jch + (jch + 1) * t_dim1], ldt, &t[ jch + 1 + (jch + 1) * t_dim1], ldt, &c__, &s); if (ilq) { drot_(n, &q[jch * q_dim1 + 1], &c__1, &q[(jch + 1) * q_dim1 + 1], &c__1, &c__, &s); } if (ilazr2) { h__[jch + (jch - 1) * h_dim1] *= c__; } ilazr2 = FALSE_; if ((d__1 = t[jch + 1 + (jch + 1) * t_dim1], abs(d__1) ) >= btol) { if (jch + 1 >= ilast) { goto L80; } else { ifirst = jch + 1; goto L110; } } t[jch + 1 + (jch + 1) * t_dim1] = 0.; /* L40: */ } goto L70; } 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) { temp = t[jch + (jch + 1) * t_dim1]; dlartg_(&temp, &t[jch + 1 + (jch + 1) * t_dim1], &c__, &s, &t[jch + (jch + 1) * t_dim1]); t[jch + 1 + (jch + 1) * t_dim1] = 0.; if (jch < ilastm - 1) { i__4 = ilastm - jch - 1; drot_(&i__4, &t[jch + (jch + 2) * t_dim1], ldt, & t[jch + 1 + (jch + 2) * t_dim1], ldt, & c__, &s); } i__4 = ilastm - jch + 2; drot_(&i__4, &h__[jch + (jch - 1) * h_dim1], ldh, & h__[jch + 1 + (jch - 1) * h_dim1], ldh, &c__, &s); if (ilq) { drot_(n, &q[jch * q_dim1 + 1], &c__1, &q[(jch + 1) * q_dim1 + 1], &c__1, &c__, &s); } temp = h__[jch + 1 + jch * h_dim1]; dlartg_(&temp, &h__[jch + 1 + (jch - 1) * h_dim1], & c__, &s, &h__[jch + 1 + jch * h_dim1]); h__[jch + 1 + (jch - 1) * h_dim1] = 0.; i__4 = jch + 1 - ifrstm; drot_(&i__4, &h__[ifrstm + jch * h_dim1], &c__1, &h__[ ifrstm + (jch - 1) * h_dim1], &c__1, &c__, &s) ; i__4 = jch - ifrstm; drot_(&i__4, &t[ifrstm + jch * t_dim1], &c__1, &t[ ifrstm + (jch - 1) * t_dim1], &c__1, &c__, &s) ; if (ilz) { drot_(n, &z__[jch * z_dim1 + 1], &c__1, &z__[(jch - 1) * z_dim1 + 1], &c__1, &c__, &s); } /* L50: */ } goto L70; } } else if (ilazro) { /* Only test 1 passed -- work on J:ILAST */ ifirst = j; goto L110; } /* Neither test passed -- try next J */ /* L60: */ } /* (Drop-through is "impossible") */ *info = *n + 1; goto L420; /* T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a */ /* 1x1 block. */ L70: temp = h__[ilast + ilast * h_dim1]; dlartg_(&temp, &h__[ilast + (ilast - 1) * h_dim1], &c__, &s, &h__[ ilast + ilast * h_dim1]); h__[ilast + (ilast - 1) * h_dim1] = 0.; i__2 = ilast - ifrstm; drot_(&i__2, &h__[ifrstm + ilast * h_dim1], &c__1, &h__[ifrstm + ( ilast - 1) * h_dim1], &c__1, &c__, &s); i__2 = ilast - ifrstm; drot_(&i__2, &t[ifrstm + ilast * t_dim1], &c__1, &t[ifrstm + (ilast - 1) * t_dim1], &c__1, &c__, &s); if (ilz) { drot_(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 ALPHAR, ALPHAI, */ /* and BETA */ L80: if (t[ilast + ilast * t_dim1] < 0.) { if (ilschr) { i__2 = ilast; for (j = ifrstm; j <= i__2; ++j) { h__[j + ilast * h_dim1] = -h__[j + ilast * h_dim1]; t[j + ilast * t_dim1] = -t[j + ilast * t_dim1]; /* L90: */ } } else { h__[ilast + ilast * h_dim1] = -h__[ilast + ilast * h_dim1]; t[ilast + ilast * t_dim1] = -t[ilast + ilast * t_dim1]; } if (ilz) { i__2 = *n; for (j = 1; j <= i__2; ++j) { z__[j + ilast * z_dim1] = -z__[j + ilast * z_dim1]; /* L100: */ } } } alphar[ilast] = h__[ilast + ilast * h_dim1]; alphai[ilast] = 0.; beta[ilast] = t[ilast + ilast * t_dim1]; /* Go to next block -- exit if finished. */ --ilast; if (ilast < *ilo) { goto L380; } /* Reset counters */ iiter = 0; eshift = 0.; if (! ilschr) { ilastm = ilast; if (ifrstm > ilast) { ifrstm = *ilo; } } goto L350; /* QZ step */ /* This iteration only involves rows/columns IFIRST:ILAST. We */ /* assume IFIRST < ILAST, and that the diagonal of B is non-zero. */ L110: ++iiter; if (! ilschr) { ifrstm = ifirst; } /* 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 == iiter) { /* Exceptional shift. Chosen for no particularly good reason. */ /* (Single shift only.) */ if ((doublereal) maxit * safmin * (d__1 = h__[ilast + (ilast - 1) * h_dim1], abs(d__1)) < (d__2 = t[ilast - 1 + (ilast - 1) * t_dim1], abs(d__2))) { eshift = h__[ilast + (ilast - 1) * h_dim1] / t[ilast - 1 + ( ilast - 1) * t_dim1]; } else { eshift += 1. / (safmin * (doublereal) maxit); } s1 = 1.; wr = eshift; } else { /* Shifts based on the generalized eigenvalues of the */ /* bottom-right 2x2 block of A and B. The first eigenvalue */ /* returned by DLAG2 is the Wilkinson shift (AEP p.512), */ d__1 = safmin * 100.; dlag2_(&h__[ilast - 1 + (ilast - 1) * h_dim1], ldh, &t[ilast - 1 + (ilast - 1) * t_dim1], ldt, &d__1, &s1, &s2, &wr, &wr2, &wi); if ((d__1 = wr / s1 * t[ilast + ilast * t_dim1] - h__[ilast + ilast * h_dim1], abs(d__1)) > (d__2 = wr2 / s2 * t[ilast + ilast * t_dim1] - h__[ilast + ilast * h_dim1], abs(d__2) )) { temp = wr; wr = wr2; wr2 = temp; temp = s1; s1 = s2; s2 = temp; } /* Computing MAX */ /* Computing MAX */ d__3 = 1., d__4 = abs(wr), d__3 = f2cmax(d__3,d__4), d__4 = abs(wi); d__1 = s1, d__2 = safmin * f2cmax(d__3,d__4); temp = f2cmax(d__1,d__2); if (wi != 0.) { goto L200; } } /* Fiddle with shift to avoid overflow */ temp = f2cmin(ascale,1.) * (safmax * .5); if (s1 > temp) { scale = temp / s1; } else { scale = 1.; } temp = f2cmin(bscale,1.) * (safmax * .5); if (abs(wr) > temp) { /* Computing MIN */ d__1 = scale, d__2 = temp / abs(wr); scale = f2cmin(d__1,d__2); } s1 = scale * s1; wr = scale * wr; /* Now check for two consecutive small subdiagonals. */ i__2 = ifirst + 1; for (j = ilast - 1; j >= i__2; --j) { istart = j; temp = (d__1 = s1 * h__[j + (j - 1) * h_dim1], abs(d__1)); temp2 = (d__1 = s1 * h__[j + j * h_dim1] - wr * t[j + j * t_dim1], abs(d__1)); tempr = f2cmax(temp,temp2); if (tempr < 1. && tempr != 0.) { temp /= tempr; temp2 /= tempr; } if ((d__1 = ascale * h__[j + 1 + j * h_dim1] * temp, abs(d__1)) <= ascale * atol * temp2) { goto L130; } /* L120: */ } istart = ifirst; L130: /* Do an implicit single-shift QZ sweep. */ /* Initial Q */ temp = s1 * h__[istart + istart * h_dim1] - wr * t[istart + istart * t_dim1]; temp2 = s1 * h__[istart + 1 + istart * h_dim1]; dlartg_(&temp, &temp2, &c__, &s, &tempr); /* Sweep */ i__2 = ilast - 1; for (j = istart; j <= i__2; ++j) { if (j > istart) { temp = h__[j + (j - 1) * h_dim1]; dlartg_(&temp, &h__[j + 1 + (j - 1) * h_dim1], &c__, &s, &h__[ j + (j - 1) * h_dim1]); h__[j + 1 + (j - 1) * h_dim1] = 0.; } i__3 = ilastm; for (jc = j; jc <= i__3; ++jc) { temp = c__ * h__[j + jc * h_dim1] + s * h__[j + 1 + jc * h_dim1]; h__[j + 1 + jc * h_dim1] = -s * h__[j + jc * h_dim1] + c__ * h__[j + 1 + jc * h_dim1]; h__[j + jc * h_dim1] = temp; temp2 = c__ * t[j + jc * t_dim1] + s * t[j + 1 + jc * t_dim1]; t[j + 1 + jc * t_dim1] = -s * t[j + jc * t_dim1] + c__ * t[j + 1 + jc * t_dim1]; t[j + jc * t_dim1] = temp2; /* L140: */ } if (ilq) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { temp = c__ * q[jr + j * q_dim1] + s * q[jr + (j + 1) * q_dim1]; q[jr + (j + 1) * q_dim1] = -s * q[jr + j * q_dim1] + c__ * q[jr + (j + 1) * q_dim1]; q[jr + j * q_dim1] = temp; /* L150: */ } } temp = t[j + 1 + (j + 1) * t_dim1]; dlartg_(&temp, &t[j + 1 + j * t_dim1], &c__, &s, &t[j + 1 + (j + 1) * t_dim1]); t[j + 1 + j * t_dim1] = 0.; /* Computing MIN */ i__4 = j + 2; i__3 = f2cmin(i__4,ilast); for (jr = ifrstm; jr <= i__3; ++jr) { temp = c__ * h__[jr + (j + 1) * h_dim1] + s * h__[jr + j * h_dim1]; h__[jr + j * h_dim1] = -s * h__[jr + (j + 1) * h_dim1] + c__ * h__[jr + j * h_dim1]; h__[jr + (j + 1) * h_dim1] = temp; /* L160: */ } i__3 = j; for (jr = ifrstm; jr <= i__3; ++jr) { temp = c__ * t[jr + (j + 1) * t_dim1] + s * t[jr + j * t_dim1] ; t[jr + j * t_dim1] = -s * t[jr + (j + 1) * t_dim1] + c__ * t[ jr + j * t_dim1]; t[jr + (j + 1) * t_dim1] = temp; /* L170: */ } if (ilz) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { temp = c__ * z__[jr + (j + 1) * z_dim1] + s * z__[jr + j * z_dim1]; z__[jr + j * z_dim1] = -s * z__[jr + (j + 1) * z_dim1] + c__ * z__[jr + j * z_dim1]; z__[jr + (j + 1) * z_dim1] = temp; /* L180: */ } } /* L190: */ } goto L350; /* 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. */ L200: if (ifirst + 1 == ilast) { /* Special case -- 2x2 block with complex eigenvectors */ /* Step 1: Standardize, that is, rotate so that */ /* ( B11 0 ) */ /* B = ( ) with B11 non-negative. */ /* ( 0 B22 ) */ dlasv2_(&t[ilast - 1 + (ilast - 1) * t_dim1], &t[ilast - 1 + ilast * t_dim1], &t[ilast + ilast * t_dim1], &b22, &b11, & sr, &cr, &sl, &cl); if (b11 < 0.) { cr = -cr; sr = -sr; b11 = -b11; b22 = -b22; } i__2 = ilastm + 1 - ifirst; drot_(&i__2, &h__[ilast - 1 + (ilast - 1) * h_dim1], ldh, &h__[ ilast + (ilast - 1) * h_dim1], ldh, &cl, &sl); i__2 = ilast + 1 - ifrstm; drot_(&i__2, &h__[ifrstm + (ilast - 1) * h_dim1], &c__1, &h__[ ifrstm + ilast * h_dim1], &c__1, &cr, &sr); if (ilast < ilastm) { i__2 = ilastm - ilast; drot_(&i__2, &t[ilast - 1 + (ilast + 1) * t_dim1], ldt, &t[ ilast + (ilast + 1) * t_dim1], ldt, &cl, &sl); } if (ifrstm < ilast - 1) { i__2 = ifirst - ifrstm; drot_(&i__2, &t[ifrstm + (ilast - 1) * t_dim1], &c__1, &t[ ifrstm + ilast * t_dim1], &c__1, &cr, &sr); } if (ilq) { drot_(n, &q[(ilast - 1) * q_dim1 + 1], &c__1, &q[ilast * q_dim1 + 1], &c__1, &cl, &sl); } if (ilz) { drot_(n, &z__[(ilast - 1) * z_dim1 + 1], &c__1, &z__[ilast * z_dim1 + 1], &c__1, &cr, &sr); } t[ilast - 1 + (ilast - 1) * t_dim1] = b11; t[ilast - 1 + ilast * t_dim1] = 0.; t[ilast + (ilast - 1) * t_dim1] = 0.; t[ilast + ilast * t_dim1] = b22; /* If B22 is negative, negate column ILAST */ if (b22 < 0.) { i__2 = ilast; for (j = ifrstm; j <= i__2; ++j) { h__[j + ilast * h_dim1] = -h__[j + ilast * h_dim1]; t[j + ilast * t_dim1] = -t[j + ilast * t_dim1]; /* L210: */ } if (ilz) { i__2 = *n; for (j = 1; j <= i__2; ++j) { z__[j + ilast * z_dim1] = -z__[j + ilast * z_dim1]; /* L220: */ } } b22 = -b22; } /* Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.) */ /* Recompute shift */ d__1 = safmin * 100.; dlag2_(&h__[ilast - 1 + (ilast - 1) * h_dim1], ldh, &t[ilast - 1 + (ilast - 1) * t_dim1], ldt, &d__1, &s1, &temp, &wr, & temp2, &wi); /* If standardization has perturbed the shift onto real line, */ /* do another (real single-shift) QR step. */ if (wi == 0.) { goto L350; } s1inv = 1. / s1; /* Do EISPACK (QZVAL) computation of alpha and beta */ a11 = h__[ilast - 1 + (ilast - 1) * h_dim1]; a21 = h__[ilast + (ilast - 1) * h_dim1]; a12 = h__[ilast - 1 + ilast * h_dim1]; a22 = h__[ilast + ilast * h_dim1]; /* 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) > abs(c21) + abs(c22r) + abs( c22i)) { t1 = dlapy3_(&c12, &c11r, &c11i); cz = c12 / t1; szr = -c11r / t1; szi = -c11i / t1; } else { cz = dlapy2_(&c22r, &c22i); if (cz <= safmin) { cz = 0.; szr = 1.; szi = 0.; } else { tempr = c22r / cz; tempi = c22i / cz; t1 = dlapy2_(&cz, &c21); cz /= t1; szr = -c21 * tempr / t1; szi = c21 * tempi / t1; } } /* 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 > wabs * bn) { cq = cz * b11; sqr = szr * b22; sqi = -szi * b22; } else { a1r = cz * a11 + szr * a12; a1i = szi * a12; a2r = cz * a21 + szr * a22; a2i = szi * a22; cq = dlapy2_(&a1r, &a1i); if (cq <= safmin) { cq = 0.; sqr = 1.; sqi = 0.; } else { tempr = a1r / cq; tempi = a1i / cq; sqr = tempr * a2r + tempi * a2i; sqi = tempi * a2r - tempr * a2i; } } t1 = dlapy3_(&cq, &sqr, &sqi); cq /= t1; sqr /= t1; 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 = dlapy2_(&b1r, &b1i); b2r = cq * cz * b22 + tempr * b11; b2i = -tempi * b11; b2a = dlapy2_(&b2r, &b2i); /* Normalize so beta > 0, and Im( alpha1 ) > 0 */ beta[ilast - 1] = b1a; beta[ilast] = b2a; alphar[ilast - 1] = wr * b1a * s1inv; alphai[ilast - 1] = wi * b1a * s1inv; alphar[ilast] = wr * b2a * s1inv; alphai[ilast] = -(wi * b2a) * s1inv; /* Step 3: Go to next block -- exit if finished. */ ilast = ifirst - 1; if (ilast < *ilo) { goto L380; } /* Reset counters */ iiter = 0; eshift = 0.; if (! ilschr) { ilastm = ilast; if (ifrstm > ilast) { ifrstm = *ilo; } } goto L350; } 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) * h_dim1] / (bscale * t[ilast - 1 + (ilast - 1) * t_dim1]); ad21 = ascale * h__[ilast + (ilast - 1) * h_dim1] / (bscale * t[ ilast - 1 + (ilast - 1) * t_dim1]); ad12 = ascale * h__[ilast - 1 + ilast * h_dim1] / (bscale * t[ ilast + ilast * t_dim1]); ad22 = ascale * h__[ilast + ilast * h_dim1] / (bscale * t[ilast + ilast * t_dim1]); u12 = t[ilast - 1 + ilast * t_dim1] / t[ilast + ilast * t_dim1]; ad11l = ascale * h__[ifirst + ifirst * h_dim1] / (bscale * t[ ifirst + ifirst * t_dim1]); ad21l = ascale * h__[ifirst + 1 + ifirst * h_dim1] / (bscale * t[ ifirst + ifirst * t_dim1]); ad12l = ascale * h__[ifirst + (ifirst + 1) * h_dim1] / (bscale * t[ifirst + 1 + (ifirst + 1) * t_dim1]); ad22l = ascale * h__[ifirst + 1 + (ifirst + 1) * h_dim1] / ( bscale * t[ifirst + 1 + (ifirst + 1) * t_dim1]); ad32l = ascale * h__[ifirst + 2 + (ifirst + 1) * h_dim1] / ( bscale * t[ifirst + 1 + (ifirst + 1) * t_dim1]); u12l = t[ifirst + (ifirst + 1) * t_dim1] / t[ifirst + 1 + (ifirst + 1) * t_dim1]; v[0] = (ad11 - ad11l) * (ad22 - ad11l) - ad12 * ad21 + ad21 * u12 * ad11l + (ad12l - ad11l * u12l) * ad21l; v[1] = (ad22l - ad11l - ad21l * u12l - (ad11 - ad11l) - (ad22 - ad11l) + ad21 * u12) * ad21l; v[2] = ad32l * ad21l; istart = ifirst; dlarfg_(&c__3, v, &v[1], &c__1, &tau); v[0] = 1.; /* Sweep */ i__2 = ilast - 2; for (j = istart; j <= i__2; ++j) { /* All but last elements: use 3x3 Householder transforms. */ /* Zero (j-1)st column of A */ if (j > istart) { v[0] = h__[j + (j - 1) * h_dim1]; v[1] = h__[j + 1 + (j - 1) * h_dim1]; v[2] = h__[j + 2 + (j - 1) * h_dim1]; dlarfg_(&c__3, &h__[j + (j - 1) * h_dim1], &v[1], &c__1, & tau); v[0] = 1.; h__[j + 1 + (j - 1) * h_dim1] = 0.; h__[j + 2 + (j - 1) * h_dim1] = 0.; } i__3 = ilastm; for (jc = j; jc <= i__3; ++jc) { temp = tau * (h__[j + jc * h_dim1] + v[1] * h__[j + 1 + jc * h_dim1] + v[2] * h__[j + 2 + jc * h_dim1]); h__[j + jc * h_dim1] -= temp; h__[j + 1 + jc * h_dim1] -= temp * v[1]; h__[j + 2 + jc * h_dim1] -= temp * v[2]; temp2 = tau * (t[j + jc * t_dim1] + v[1] * t[j + 1 + jc * t_dim1] + v[2] * t[j + 2 + jc * t_dim1]); t[j + jc * t_dim1] -= temp2; t[j + 1 + jc * t_dim1] -= temp2 * v[1]; t[j + 2 + jc * t_dim1] -= temp2 * v[2]; /* L230: */ } if (ilq) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { temp = tau * (q[jr + j * q_dim1] + v[1] * q[jr + (j + 1) * q_dim1] + v[2] * q[jr + (j + 2) * q_dim1] ); q[jr + j * q_dim1] -= temp; q[jr + (j + 1) * q_dim1] -= temp * v[1]; q[jr + (j + 2) * q_dim1] -= temp * v[2]; /* L240: */ } } /* Zero j-th column of B (see DLAGBC for details) */ /* Swap rows to pivot */ ilpivt = FALSE_; /* Computing MAX */ d__3 = (d__1 = t[j + 1 + (j + 1) * t_dim1], abs(d__1)), d__4 = (d__2 = t[j + 1 + (j + 2) * t_dim1], abs(d__2)); temp = f2cmax(d__3,d__4); /* Computing MAX */ d__3 = (d__1 = t[j + 2 + (j + 1) * t_dim1], abs(d__1)), d__4 = (d__2 = t[j + 2 + (j + 2) * t_dim1], abs(d__2)); temp2 = f2cmax(d__3,d__4); if (f2cmax(temp,temp2) < safmin) { scale = 0.; u1 = 1.; u2 = 0.; goto L250; } else if (temp >= temp2) { w11 = t[j + 1 + (j + 1) * t_dim1]; w21 = t[j + 2 + (j + 1) * t_dim1]; w12 = t[j + 1 + (j + 2) * t_dim1]; w22 = t[j + 2 + (j + 2) * t_dim1]; u1 = t[j + 1 + j * t_dim1]; u2 = t[j + 2 + j * t_dim1]; } else { w21 = t[j + 1 + (j + 1) * t_dim1]; w11 = t[j + 2 + (j + 1) * t_dim1]; w22 = t[j + 1 + (j + 2) * t_dim1]; w12 = t[j + 2 + (j + 2) * t_dim1]; u2 = t[j + 1 + j * t_dim1]; u1 = t[j + 2 + j * t_dim1]; } /* Swap columns if nec. */ if (abs(w12) > abs(w11)) { ilpivt = TRUE_; temp = w12; temp2 = w22; w12 = w11; w22 = w21; w11 = temp; w21 = temp2; } /* LU-factor */ temp = w21 / w11; u2 -= temp * u1; w22 -= temp * w12; w21 = 0.; /* Compute SCALE */ scale = 1.; if (abs(w22) < safmin) { scale = 0.; u2 = 1.; u1 = -w12 / w11; goto L250; } if (abs(w22) < abs(u2)) { scale = (d__1 = w22 / u2, abs(d__1)); } if (abs(w11) < abs(u1)) { /* Computing MIN */ d__2 = scale, d__3 = (d__1 = w11 / u1, abs(d__1)); scale = f2cmin(d__2,d__3); } /* Solve */ u2 = scale * u2 / w22; u1 = (scale * u1 - w12 * u2) / w11; L250: if (ilpivt) { temp = u2; u2 = u1; u1 = temp; } /* Compute Householder Vector */ /* Computing 2nd power */ d__1 = scale; /* Computing 2nd power */ d__2 = u1; /* Computing 2nd power */ d__3 = u2; t1 = sqrt(d__1 * d__1 + d__2 * d__2 + d__3 * d__3); tau = scale / t1 + 1.; vs = -1. / (scale + t1); v[0] = 1.; v[1] = vs * u1; v[2] = vs * u2; /* Apply transformations from the right. */ /* Computing MIN */ i__4 = j + 3; i__3 = f2cmin(i__4,ilast); for (jr = ifrstm; jr <= i__3; ++jr) { temp = tau * (h__[jr + j * h_dim1] + v[1] * h__[jr + (j + 1) * h_dim1] + v[2] * h__[jr + (j + 2) * h_dim1]); h__[jr + j * h_dim1] -= temp; h__[jr + (j + 1) * h_dim1] -= temp * v[1]; h__[jr + (j + 2) * h_dim1] -= temp * v[2]; /* L260: */ } i__3 = j + 2; for (jr = ifrstm; jr <= i__3; ++jr) { temp = tau * (t[jr + j * t_dim1] + v[1] * t[jr + (j + 1) * t_dim1] + v[2] * t[jr + (j + 2) * t_dim1]); t[jr + j * t_dim1] -= temp; t[jr + (j + 1) * t_dim1] -= temp * v[1]; t[jr + (j + 2) * t_dim1] -= temp * v[2]; /* L270: */ } if (ilz) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { temp = tau * (z__[jr + j * z_dim1] + v[1] * z__[jr + ( j + 1) * z_dim1] + v[2] * z__[jr + (j + 2) * z_dim1]); z__[jr + j * z_dim1] -= temp; z__[jr + (j + 1) * z_dim1] -= temp * v[1]; z__[jr + (j + 2) * z_dim1] -= temp * v[2]; /* L280: */ } } t[j + 1 + j * t_dim1] = 0.; t[j + 2 + j * t_dim1] = 0.; /* L290: */ } /* Last elements: Use Givens rotations */ /* Rotations from the left */ j = ilast - 1; temp = h__[j + (j - 1) * h_dim1]; dlartg_(&temp, &h__[j + 1 + (j - 1) * h_dim1], &c__, &s, &h__[j + (j - 1) * h_dim1]); h__[j + 1 + (j - 1) * h_dim1] = 0.; i__2 = ilastm; for (jc = j; jc <= i__2; ++jc) { temp = c__ * h__[j + jc * h_dim1] + s * h__[j + 1 + jc * h_dim1]; h__[j + 1 + jc * h_dim1] = -s * h__[j + jc * h_dim1] + c__ * h__[j + 1 + jc * h_dim1]; h__[j + jc * h_dim1] = temp; temp2 = c__ * t[j + jc * t_dim1] + s * t[j + 1 + jc * t_dim1]; t[j + 1 + jc * t_dim1] = -s * t[j + jc * t_dim1] + c__ * t[j + 1 + jc * t_dim1]; t[j + jc * t_dim1] = temp2; /* L300: */ } if (ilq) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { temp = c__ * q[jr + j * q_dim1] + s * q[jr + (j + 1) * q_dim1]; q[jr + (j + 1) * q_dim1] = -s * q[jr + j * q_dim1] + c__ * q[jr + (j + 1) * q_dim1]; q[jr + j * q_dim1] = temp; /* L310: */ } } /* Rotations from the right. */ temp = t[j + 1 + (j + 1) * t_dim1]; dlartg_(&temp, &t[j + 1 + j * t_dim1], &c__, &s, &t[j + 1 + (j + 1) * t_dim1]); t[j + 1 + j * t_dim1] = 0.; i__2 = ilast; for (jr = ifrstm; jr <= i__2; ++jr) { temp = c__ * h__[jr + (j + 1) * h_dim1] + s * h__[jr + j * h_dim1]; h__[jr + j * h_dim1] = -s * h__[jr + (j + 1) * h_dim1] + c__ * h__[jr + j * h_dim1]; h__[jr + (j + 1) * h_dim1] = temp; /* L320: */ } i__2 = ilast - 1; for (jr = ifrstm; jr <= i__2; ++jr) { temp = c__ * t[jr + (j + 1) * t_dim1] + s * t[jr + j * t_dim1] ; t[jr + j * t_dim1] = -s * t[jr + (j + 1) * t_dim1] + c__ * t[ jr + j * t_dim1]; t[jr + (j + 1) * t_dim1] = temp; /* L330: */ } if (ilz) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { temp = c__ * z__[jr + (j + 1) * z_dim1] + s * z__[jr + j * z_dim1]; z__[jr + j * z_dim1] = -s * z__[jr + (j + 1) * z_dim1] + c__ * z__[jr + j * z_dim1]; z__[jr + (j + 1) * z_dim1] = temp; /* L340: */ } } /* End of Double-Shift code */ } goto L350; /* End of iteration loop */ L350: /* L360: */ ; } /* Drop-through = non-convergence */ *info = ilast; goto L420; /* Successful completion of all QZ steps */ L380: /* Set Eigenvalues 1:ILO-1 */ i__1 = *ilo - 1; for (j = 1; j <= i__1; ++j) { if (t[j + j * t_dim1] < 0.) { if (ilschr) { i__2 = j; for (jr = 1; jr <= i__2; ++jr) { h__[jr + j * h_dim1] = -h__[jr + j * h_dim1]; t[jr + j * t_dim1] = -t[jr + j * t_dim1]; /* L390: */ } } else { h__[j + j * h_dim1] = -h__[j + j * h_dim1]; t[j + j * t_dim1] = -t[j + j * t_dim1]; } if (ilz) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { z__[jr + j * z_dim1] = -z__[jr + j * z_dim1]; /* L400: */ } } } alphar[j] = h__[j + j * h_dim1]; alphai[j] = 0.; beta[j] = t[j + j * t_dim1]; /* L410: */ } /* Normal Termination */ *info = 0; /* Exit (other than argument error) -- return optimal workspace size */ L420: work[1] = (doublereal) (*n); return 0; /* End of DHGEQZ */ } /* dhgeqz_ */