#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(_Fcomplex x, integer n) { _Fcomplex 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 = _FCmulcc (pow,x); if(u >>= 1) x = _FCmulcc (x,x); else break; } } return pow; } #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 CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using th e double-shift/single-shift QR algorithm. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download CLAHQR + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE CLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, */ /* IHIZ, Z, LDZ, INFO ) */ /* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, N */ /* LOGICAL WANTT, WANTZ */ /* COMPLEX H( LDH, * ), W( * ), Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > CLAHQR is an auxiliary routine called by CHSEQR to update the */ /* > eigenvalues and Schur decomposition already computed by CHSEQR, by */ /* > dealing with the Hessenberg submatrix in rows and columns ILO to */ /* > IHI. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] WANTT */ /* > \verbatim */ /* > WANTT is LOGICAL */ /* > = .TRUE. : the full Schur form T is required; */ /* > = .FALSE.: only eigenvalues are required. */ /* > \endverbatim */ /* > */ /* > \param[in] WANTZ */ /* > \verbatim */ /* > WANTZ is LOGICAL */ /* > = .TRUE. : the matrix of Schur vectors Z is required; */ /* > = .FALSE.: Schur vectors are not required. */ /* > \endverbatim */ /* > */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The order of the matrix H. N >= 0. */ /* > \endverbatim */ /* > */ /* > \param[in] ILO */ /* > \verbatim */ /* > ILO is INTEGER */ /* > \endverbatim */ /* > */ /* > \param[in] IHI */ /* > \verbatim */ /* > IHI is INTEGER */ /* > It is assumed that H is already upper triangular in rows and */ /* > columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). */ /* > CLAHQR works primarily with the Hessenberg submatrix in rows */ /* > and columns ILO to IHI, but applies transformations to all of */ /* > H if WANTT is .TRUE.. */ /* > 1 <= ILO <= f2cmax(1,IHI); IHI <= N. */ /* > \endverbatim */ /* > */ /* > \param[in,out] H */ /* > \verbatim */ /* > H is COMPLEX array, dimension (LDH,N) */ /* > On entry, the upper Hessenberg matrix H. */ /* > On exit, if INFO is zero and if WANTT is .TRUE., then H */ /* > is upper triangular in rows and columns ILO:IHI. If INFO */ /* > is zero and if WANTT is .FALSE., then the contents of H */ /* > are unspecified on exit. The output state of H in case */ /* > INF is positive is below under the description of INFO. */ /* > \endverbatim */ /* > */ /* > \param[in] LDH */ /* > \verbatim */ /* > LDH is INTEGER */ /* > The leading dimension of the array H. LDH >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] W */ /* > \verbatim */ /* > W is COMPLEX array, dimension (N) */ /* > The computed eigenvalues ILO to IHI are stored in the */ /* > corresponding elements of W. If WANTT is .TRUE., the */ /* > eigenvalues are stored in the same order as on the diagonal */ /* > of the Schur form returned in H, with W(i) = H(i,i). */ /* > \endverbatim */ /* > */ /* > \param[in] ILOZ */ /* > \verbatim */ /* > ILOZ is INTEGER */ /* > \endverbatim */ /* > */ /* > \param[in] IHIZ */ /* > \verbatim */ /* > IHIZ is INTEGER */ /* > Specify the rows of Z to which transformations must be */ /* > applied if WANTZ is .TRUE.. */ /* > 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. */ /* > \endverbatim */ /* > */ /* > \param[in,out] Z */ /* > \verbatim */ /* > Z is COMPLEX array, dimension (LDZ,N) */ /* > If WANTZ is .TRUE., on entry Z must contain the current */ /* > matrix Z of transformations accumulated by CHSEQR, and on */ /* > exit Z has been updated; transformations are applied only to */ /* > the submatrix Z(ILOZ:IHIZ,ILO:IHI). */ /* > If WANTZ is .FALSE., Z is not referenced. */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. LDZ >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > > 0: if INFO = i, CLAHQR failed to compute all the */ /* > eigenvalues ILO to IHI in a total of 30 iterations */ /* > per eigenvalue; elements i+1:ihi of W contain */ /* > those eigenvalues which have been successfully */ /* > computed. */ /* > */ /* > If INFO > 0 and WANTT is .FALSE., then on exit, */ /* > the remaining unconverged eigenvalues are the */ /* > eigenvalues of the upper Hessenberg matrix */ /* > rows and columns ILO through INFO of the final, */ /* > output value of H. */ /* > */ /* > If INFO > 0 and WANTT is .TRUE., then on exit */ /* > (*) (initial value of H)*U = U*(final value of H) */ /* > where U is an orthogonal matrix. The final */ /* > value of H is upper Hessenberg and triangular in */ /* > rows and columns INFO+1 through IHI. */ /* > */ /* > If INFO > 0 and WANTZ is .TRUE., then on exit */ /* > (final value of Z) = (initial value of Z)*U */ /* > where U is the orthogonal matrix in (*) */ /* > (regardless of the value of WANTT.) */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup complexOTHERauxiliary */ /* > \par Contributors: */ /* ================== */ /* > */ /* > \verbatim */ /* > */ /* > 02-96 Based on modifications by */ /* > David Day, Sandia National Laboratory, USA */ /* > */ /* > 12-04 Further modifications by */ /* > Ralph Byers, University of Kansas, USA */ /* > This is a modified version of CLAHQR from LAPACK version 3.0. */ /* > It is (1) more robust against overflow and underflow and */ /* > (2) adopts the more conservative Ahues & Tisseur stopping */ /* > criterion (LAWN 122, 1997). */ /* > \endverbatim */ /* > */ /* ===================================================================== */ /* Subroutine */ int clahqr_(logical *wantt, logical *wantz, integer *n, integer *ilo, integer *ihi, complex *h__, integer *ldh, complex *w, integer *iloz, integer *ihiz, complex *z__, integer *ldz, integer * info) { /* System generated locals */ integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2, r__3, r__4, r__5, r__6; complex q__1, q__2, q__3, q__4, q__5, q__6, q__7; /* Local variables */ complex temp; integer i__, j, k, l, m; real s; complex t, u, v[2], x, y; extern /* Subroutine */ int cscal_(integer *, complex *, complex *, integer *), ccopy_(integer *, complex *, integer *, complex *, integer *); integer itmax; real rtemp; integer i1, i2; complex t1; real t2; complex v2; real aa, ab, ba, bb, h10; complex h11; real h21; complex h22, sc; integer nh; extern /* Subroutine */ int slabad_(real *, real *), clarfg_(integer *, complex *, complex *, integer *, complex *); extern /* Complex */ VOID cladiv_(complex *, complex *, complex *); extern real slamch_(char *); integer nz; real sx, safmin, safmax, smlnum; integer jhi; complex h11s; integer jlo, its; real ulp; complex sum; real tst; /* -- LAPACK auxiliary 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..-- */ /* December 2016 */ /* ========================================================= */ /* Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1 * 1; h__ -= h_offset; --w; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; /* Function Body */ *info = 0; /* Quick return if possible */ if (*n == 0) { return 0; } if (*ilo == *ihi) { i__1 = *ilo; i__2 = *ilo + *ilo * h_dim1; w[i__1].r = h__[i__2].r, w[i__1].i = h__[i__2].i; return 0; } /* ==== clear out the trash ==== */ i__1 = *ihi - 3; for (j = *ilo; j <= i__1; ++j) { i__2 = j + 2 + j * h_dim1; h__[i__2].r = 0.f, h__[i__2].i = 0.f; i__2 = j + 3 + j * h_dim1; h__[i__2].r = 0.f, h__[i__2].i = 0.f; /* L10: */ } if (*ilo <= *ihi - 2) { i__1 = *ihi + (*ihi - 2) * h_dim1; h__[i__1].r = 0.f, h__[i__1].i = 0.f; } /* ==== ensure that subdiagonal entries are real ==== */ if (*wantt) { jlo = 1; jhi = *n; } else { jlo = *ilo; jhi = *ihi; } i__1 = *ihi; for (i__ = *ilo + 1; i__ <= i__1; ++i__) { if (r_imag(&h__[i__ + (i__ - 1) * h_dim1]) != 0.f) { /* ==== The following redundant normalization */ /* . avoids problems with both gradual and */ /* . sudden underflow in ABS(H(I,I-1)) ==== */ i__2 = i__ + (i__ - 1) * h_dim1; i__3 = i__ + (i__ - 1) * h_dim1; r__3 = (r__1 = h__[i__3].r, abs(r__1)) + (r__2 = r_imag(&h__[i__ + (i__ - 1) * h_dim1]), abs(r__2)); q__1.r = h__[i__2].r / r__3, q__1.i = h__[i__2].i / r__3; sc.r = q__1.r, sc.i = q__1.i; r_cnjg(&q__2, &sc); r__1 = c_abs(&sc); q__1.r = q__2.r / r__1, q__1.i = q__2.i / r__1; sc.r = q__1.r, sc.i = q__1.i; i__2 = i__ + (i__ - 1) * h_dim1; r__1 = c_abs(&h__[i__ + (i__ - 1) * h_dim1]); h__[i__2].r = r__1, h__[i__2].i = 0.f; i__2 = jhi - i__ + 1; cscal_(&i__2, &sc, &h__[i__ + i__ * h_dim1], ldh); /* Computing MIN */ i__3 = jhi, i__4 = i__ + 1; i__2 = f2cmin(i__3,i__4) - jlo + 1; r_cnjg(&q__1, &sc); cscal_(&i__2, &q__1, &h__[jlo + i__ * h_dim1], &c__1); if (*wantz) { i__2 = *ihiz - *iloz + 1; r_cnjg(&q__1, &sc); cscal_(&i__2, &q__1, &z__[*iloz + i__ * z_dim1], &c__1); } } /* L20: */ } nh = *ihi - *ilo + 1; nz = *ihiz - *iloz + 1; /* Set machine-dependent constants for the stopping criterion. */ safmin = slamch_("SAFE MINIMUM"); safmax = 1.f / safmin; slabad_(&safmin, &safmax); ulp = slamch_("PRECISION"); smlnum = safmin * ((real) nh / ulp); /* I1 and I2 are the indices of the first row and last column of H */ /* to which transformations must be applied. If eigenvalues only are */ /* being computed, I1 and I2 are set inside the main loop. */ if (*wantt) { i1 = 1; i2 = *n; } /* ITMAX is the total number of QR iterations allowed. */ itmax = f2cmax(10,nh) * 30; /* The main loop begins here. I is the loop index and decreases from */ /* IHI to ILO in steps of 1. Each iteration of the loop works */ /* with the active submatrix in rows and columns L to I. */ /* Eigenvalues I+1 to IHI have already converged. Either L = ILO, or */ /* H(L,L-1) is negligible so that the matrix splits. */ i__ = *ihi; L30: if (i__ < *ilo) { goto L150; } /* Perform QR iterations on rows and columns ILO to I until a */ /* submatrix of order 1 splits off at the bottom because a */ /* subdiagonal element has become negligible. */ l = *ilo; i__1 = itmax; for (its = 0; its <= i__1; ++its) { /* Look for a single small subdiagonal element. */ i__2 = l + 1; for (k = i__; k >= i__2; --k) { i__3 = k + (k - 1) * h_dim1; if ((r__1 = h__[i__3].r, abs(r__1)) + (r__2 = r_imag(&h__[k + (k - 1) * h_dim1]), abs(r__2)) <= smlnum) { goto L50; } i__3 = k - 1 + (k - 1) * h_dim1; i__4 = k + k * h_dim1; tst = (r__1 = h__[i__3].r, abs(r__1)) + (r__2 = r_imag(&h__[k - 1 + (k - 1) * h_dim1]), abs(r__2)) + ((r__3 = h__[i__4].r, abs(r__3)) + (r__4 = r_imag(&h__[k + k * h_dim1]), abs( r__4))); if (tst == 0.f) { if (k - 2 >= *ilo) { i__3 = k - 1 + (k - 2) * h_dim1; tst += (r__1 = h__[i__3].r, abs(r__1)); } if (k + 1 <= *ihi) { i__3 = k + 1 + k * h_dim1; tst += (r__1 = h__[i__3].r, abs(r__1)); } } /* ==== The following is a conservative small subdiagonal */ /* . deflation criterion due to Ahues & Tisseur (LAWN 122, */ /* . 1997). It has better mathematical foundation and */ /* . improves accuracy in some examples. ==== */ i__3 = k + (k - 1) * h_dim1; if ((r__1 = h__[i__3].r, abs(r__1)) <= ulp * tst) { /* Computing MAX */ i__3 = k + (k - 1) * h_dim1; i__4 = k - 1 + k * h_dim1; r__5 = (r__1 = h__[i__3].r, abs(r__1)) + (r__2 = r_imag(&h__[ k + (k - 1) * h_dim1]), abs(r__2)), r__6 = (r__3 = h__[i__4].r, abs(r__3)) + (r__4 = r_imag(&h__[k - 1 + k * h_dim1]), abs(r__4)); ab = f2cmax(r__5,r__6); /* Computing MIN */ i__3 = k + (k - 1) * h_dim1; i__4 = k - 1 + k * h_dim1; r__5 = (r__1 = h__[i__3].r, abs(r__1)) + (r__2 = r_imag(&h__[ k + (k - 1) * h_dim1]), abs(r__2)), r__6 = (r__3 = h__[i__4].r, abs(r__3)) + (r__4 = r_imag(&h__[k - 1 + k * h_dim1]), abs(r__4)); ba = f2cmin(r__5,r__6); i__3 = k - 1 + (k - 1) * h_dim1; i__4 = k + k * h_dim1; q__2.r = h__[i__3].r - h__[i__4].r, q__2.i = h__[i__3].i - h__[i__4].i; q__1.r = q__2.r, q__1.i = q__2.i; /* Computing MAX */ i__5 = k + k * h_dim1; r__5 = (r__1 = h__[i__5].r, abs(r__1)) + (r__2 = r_imag(&h__[ k + k * h_dim1]), abs(r__2)), r__6 = (r__3 = q__1.r, abs(r__3)) + (r__4 = r_imag(&q__1), abs(r__4)); aa = f2cmax(r__5,r__6); i__3 = k - 1 + (k - 1) * h_dim1; i__4 = k + k * h_dim1; q__2.r = h__[i__3].r - h__[i__4].r, q__2.i = h__[i__3].i - h__[i__4].i; q__1.r = q__2.r, q__1.i = q__2.i; /* Computing MIN */ i__5 = k + k * h_dim1; r__5 = (r__1 = h__[i__5].r, abs(r__1)) + (r__2 = r_imag(&h__[ k + k * h_dim1]), abs(r__2)), r__6 = (r__3 = q__1.r, abs(r__3)) + (r__4 = r_imag(&q__1), abs(r__4)); bb = f2cmin(r__5,r__6); s = aa + ab; /* Computing MAX */ r__1 = smlnum, r__2 = ulp * (bb * (aa / s)); if (ba * (ab / s) <= f2cmax(r__1,r__2)) { goto L50; } } /* L40: */ } L50: l = k; if (l > *ilo) { /* H(L,L-1) is negligible */ i__2 = l + (l - 1) * h_dim1; h__[i__2].r = 0.f, h__[i__2].i = 0.f; } /* Exit from loop if a submatrix of order 1 has split off. */ if (l >= i__) { goto L140; } /* Now the active submatrix is in rows and columns L to I. If */ /* eigenvalues only are being computed, only the active submatrix */ /* need be transformed. */ if (! (*wantt)) { i1 = l; i2 = i__; } if (its == 10) { /* Exceptional shift. */ i__2 = l + 1 + l * h_dim1; s = (r__1 = h__[i__2].r, abs(r__1)) * .75f; i__2 = l + l * h_dim1; q__1.r = s + h__[i__2].r, q__1.i = h__[i__2].i; t.r = q__1.r, t.i = q__1.i; } else if (its == 20) { /* Exceptional shift. */ i__2 = i__ + (i__ - 1) * h_dim1; s = (r__1 = h__[i__2].r, abs(r__1)) * .75f; i__2 = i__ + i__ * h_dim1; q__1.r = s + h__[i__2].r, q__1.i = h__[i__2].i; t.r = q__1.r, t.i = q__1.i; } else { /* Wilkinson's shift. */ i__2 = i__ + i__ * h_dim1; t.r = h__[i__2].r, t.i = h__[i__2].i; c_sqrt(&q__2, &h__[i__ - 1 + i__ * h_dim1]); c_sqrt(&q__3, &h__[i__ + (i__ - 1) * h_dim1]); q__1.r = q__2.r * q__3.r - q__2.i * q__3.i, q__1.i = q__2.r * q__3.i + q__2.i * q__3.r; u.r = q__1.r, u.i = q__1.i; s = (r__1 = u.r, abs(r__1)) + (r__2 = r_imag(&u), abs(r__2)); if (s != 0.f) { i__2 = i__ - 1 + (i__ - 1) * h_dim1; q__2.r = h__[i__2].r - t.r, q__2.i = h__[i__2].i - t.i; q__1.r = q__2.r * .5f, q__1.i = q__2.i * .5f; x.r = q__1.r, x.i = q__1.i; sx = (r__1 = x.r, abs(r__1)) + (r__2 = r_imag(&x), abs(r__2)); /* Computing MAX */ r__3 = s, r__4 = (r__1 = x.r, abs(r__1)) + (r__2 = r_imag(&x), abs(r__2)); s = f2cmax(r__3,r__4); q__5.r = x.r / s, q__5.i = x.i / s; pow_ci(&q__4, &q__5, &c__2); q__7.r = u.r / s, q__7.i = u.i / s; pow_ci(&q__6, &q__7, &c__2); q__3.r = q__4.r + q__6.r, q__3.i = q__4.i + q__6.i; c_sqrt(&q__2, &q__3); q__1.r = s * q__2.r, q__1.i = s * q__2.i; y.r = q__1.r, y.i = q__1.i; if (sx > 0.f) { q__1.r = x.r / sx, q__1.i = x.i / sx; q__2.r = x.r / sx, q__2.i = x.i / sx; if (q__1.r * y.r + r_imag(&q__2) * r_imag(&y) < 0.f) { q__3.r = -y.r, q__3.i = -y.i; y.r = q__3.r, y.i = q__3.i; } } q__4.r = x.r + y.r, q__4.i = x.i + y.i; cladiv_(&q__3, &u, &q__4); q__2.r = u.r * q__3.r - u.i * q__3.i, q__2.i = u.r * q__3.i + u.i * q__3.r; q__1.r = t.r - q__2.r, q__1.i = t.i - q__2.i; t.r = q__1.r, t.i = q__1.i; } } /* Look for two consecutive small subdiagonal elements. */ i__2 = l + 1; for (m = i__ - 1; m >= i__2; --m) { /* Determine the effect of starting the single-shift QR */ /* iteration at row M, and see if this would make H(M,M-1) */ /* negligible. */ i__3 = m + m * h_dim1; h11.r = h__[i__3].r, h11.i = h__[i__3].i; i__3 = m + 1 + (m + 1) * h_dim1; h22.r = h__[i__3].r, h22.i = h__[i__3].i; q__1.r = h11.r - t.r, q__1.i = h11.i - t.i; h11s.r = q__1.r, h11s.i = q__1.i; i__3 = m + 1 + m * h_dim1; h21 = h__[i__3].r; s = (r__1 = h11s.r, abs(r__1)) + (r__2 = r_imag(&h11s), abs(r__2)) + abs(h21); q__1.r = h11s.r / s, q__1.i = h11s.i / s; h11s.r = q__1.r, h11s.i = q__1.i; h21 /= s; v[0].r = h11s.r, v[0].i = h11s.i; v[1].r = h21, v[1].i = 0.f; i__3 = m + (m - 1) * h_dim1; h10 = h__[i__3].r; if (abs(h10) * abs(h21) <= ulp * (((r__1 = h11s.r, abs(r__1)) + ( r__2 = r_imag(&h11s), abs(r__2))) * ((r__3 = h11.r, abs( r__3)) + (r__4 = r_imag(&h11), abs(r__4)) + ((r__5 = h22.r, abs(r__5)) + (r__6 = r_imag(&h22), abs(r__6)))))) { goto L70; } /* L60: */ } i__2 = l + l * h_dim1; h11.r = h__[i__2].r, h11.i = h__[i__2].i; i__2 = l + 1 + (l + 1) * h_dim1; h22.r = h__[i__2].r, h22.i = h__[i__2].i; q__1.r = h11.r - t.r, q__1.i = h11.i - t.i; h11s.r = q__1.r, h11s.i = q__1.i; i__2 = l + 1 + l * h_dim1; h21 = h__[i__2].r; s = (r__1 = h11s.r, abs(r__1)) + (r__2 = r_imag(&h11s), abs(r__2)) + abs(h21); q__1.r = h11s.r / s, q__1.i = h11s.i / s; h11s.r = q__1.r, h11s.i = q__1.i; h21 /= s; v[0].r = h11s.r, v[0].i = h11s.i; v[1].r = h21, v[1].i = 0.f; L70: /* Single-shift QR step */ i__2 = i__ - 1; for (k = m; k <= i__2; ++k) { /* The first iteration of this loop determines a reflection G */ /* from the vector V and applies it from left and right to H, */ /* thus creating a nonzero bulge below the subdiagonal. */ /* Each subsequent iteration determines a reflection G to */ /* restore the Hessenberg form in the (K-1)th column, and thus */ /* chases the bulge one step toward the bottom of the active */ /* submatrix. */ /* V(2) is always real before the call to CLARFG, and hence */ /* after the call T2 ( = T1*V(2) ) is also real. */ if (k > m) { ccopy_(&c__2, &h__[k + (k - 1) * h_dim1], &c__1, v, &c__1); } clarfg_(&c__2, v, &v[1], &c__1, &t1); if (k > m) { i__3 = k + (k - 1) * h_dim1; h__[i__3].r = v[0].r, h__[i__3].i = v[0].i; i__3 = k + 1 + (k - 1) * h_dim1; h__[i__3].r = 0.f, h__[i__3].i = 0.f; } v2.r = v[1].r, v2.i = v[1].i; q__1.r = t1.r * v2.r - t1.i * v2.i, q__1.i = t1.r * v2.i + t1.i * v2.r; t2 = q__1.r; /* Apply G from the left to transform the rows of the matrix */ /* in columns K to I2. */ i__3 = i2; for (j = k; j <= i__3; ++j) { r_cnjg(&q__3, &t1); i__4 = k + j * h_dim1; q__2.r = q__3.r * h__[i__4].r - q__3.i * h__[i__4].i, q__2.i = q__3.r * h__[i__4].i + q__3.i * h__[i__4].r; i__5 = k + 1 + j * h_dim1; q__4.r = t2 * h__[i__5].r, q__4.i = t2 * h__[i__5].i; q__1.r = q__2.r + q__4.r, q__1.i = q__2.i + q__4.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = k + j * h_dim1; i__5 = k + j * h_dim1; q__1.r = h__[i__5].r - sum.r, q__1.i = h__[i__5].i - sum.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; i__4 = k + 1 + j * h_dim1; i__5 = k + 1 + j * h_dim1; q__2.r = sum.r * v2.r - sum.i * v2.i, q__2.i = sum.r * v2.i + sum.i * v2.r; q__1.r = h__[i__5].r - q__2.r, q__1.i = h__[i__5].i - q__2.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; /* L80: */ } /* Apply G from the right to transform the columns of the */ /* matrix in rows I1 to f2cmin(K+2,I). */ /* Computing MIN */ i__4 = k + 2; i__3 = f2cmin(i__4,i__); for (j = i1; j <= i__3; ++j) { i__4 = j + k * h_dim1; q__2.r = t1.r * h__[i__4].r - t1.i * h__[i__4].i, q__2.i = t1.r * h__[i__4].i + t1.i * h__[i__4].r; i__5 = j + (k + 1) * h_dim1; q__3.r = t2 * h__[i__5].r, q__3.i = t2 * h__[i__5].i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = j + k * h_dim1; i__5 = j + k * h_dim1; q__1.r = h__[i__5].r - sum.r, q__1.i = h__[i__5].i - sum.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; i__4 = j + (k + 1) * h_dim1; i__5 = j + (k + 1) * h_dim1; r_cnjg(&q__3, &v2); q__2.r = sum.r * q__3.r - sum.i * q__3.i, q__2.i = sum.r * q__3.i + sum.i * q__3.r; q__1.r = h__[i__5].r - q__2.r, q__1.i = h__[i__5].i - q__2.i; h__[i__4].r = q__1.r, h__[i__4].i = q__1.i; /* L90: */ } if (*wantz) { /* Accumulate transformations in the matrix Z */ i__3 = *ihiz; for (j = *iloz; j <= i__3; ++j) { i__4 = j + k * z_dim1; q__2.r = t1.r * z__[i__4].r - t1.i * z__[i__4].i, q__2.i = t1.r * z__[i__4].i + t1.i * z__[i__4].r; i__5 = j + (k + 1) * z_dim1; q__3.r = t2 * z__[i__5].r, q__3.i = t2 * z__[i__5].i; q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i; sum.r = q__1.r, sum.i = q__1.i; i__4 = j + k * z_dim1; i__5 = j + k * z_dim1; q__1.r = z__[i__5].r - sum.r, q__1.i = z__[i__5].i - sum.i; z__[i__4].r = q__1.r, z__[i__4].i = q__1.i; i__4 = j + (k + 1) * z_dim1; i__5 = j + (k + 1) * z_dim1; r_cnjg(&q__3, &v2); q__2.r = sum.r * q__3.r - sum.i * q__3.i, q__2.i = sum.r * q__3.i + sum.i * q__3.r; q__1.r = z__[i__5].r - q__2.r, q__1.i = z__[i__5].i - q__2.i; z__[i__4].r = q__1.r, z__[i__4].i = q__1.i; /* L100: */ } } if (k == m && m > l) { /* If the QR step was started at row M > L because two */ /* consecutive small subdiagonals were found, then extra */ /* scaling must be performed to ensure that H(M,M-1) remains */ /* real. */ q__1.r = 1.f - t1.r, q__1.i = 0.f - t1.i; temp.r = q__1.r, temp.i = q__1.i; r__1 = c_abs(&temp); q__1.r = temp.r / r__1, q__1.i = temp.i / r__1; temp.r = q__1.r, temp.i = q__1.i; i__3 = m + 1 + m * h_dim1; i__4 = m + 1 + m * h_dim1; r_cnjg(&q__2, &temp); q__1.r = h__[i__4].r * q__2.r - h__[i__4].i * q__2.i, q__1.i = h__[i__4].r * q__2.i + h__[i__4].i * q__2.r; h__[i__3].r = q__1.r, h__[i__3].i = q__1.i; if (m + 2 <= i__) { i__3 = m + 2 + (m + 1) * h_dim1; i__4 = m + 2 + (m + 1) * h_dim1; q__1.r = h__[i__4].r * temp.r - h__[i__4].i * temp.i, q__1.i = h__[i__4].r * temp.i + h__[i__4].i * temp.r; h__[i__3].r = q__1.r, h__[i__3].i = q__1.i; } i__3 = i__; for (j = m; j <= i__3; ++j) { if (j != m + 1) { if (i2 > j) { i__4 = i2 - j; cscal_(&i__4, &temp, &h__[j + (j + 1) * h_dim1], ldh); } i__4 = j - i1; r_cnjg(&q__1, &temp); cscal_(&i__4, &q__1, &h__[i1 + j * h_dim1], &c__1); if (*wantz) { r_cnjg(&q__1, &temp); cscal_(&nz, &q__1, &z__[*iloz + j * z_dim1], & c__1); } } /* L110: */ } } /* L120: */ } /* Ensure that H(I,I-1) is real. */ i__2 = i__ + (i__ - 1) * h_dim1; temp.r = h__[i__2].r, temp.i = h__[i__2].i; if (r_imag(&temp) != 0.f) { rtemp = c_abs(&temp); i__2 = i__ + (i__ - 1) * h_dim1; h__[i__2].r = rtemp, h__[i__2].i = 0.f; q__1.r = temp.r / rtemp, q__1.i = temp.i / rtemp; temp.r = q__1.r, temp.i = q__1.i; if (i2 > i__) { i__2 = i2 - i__; r_cnjg(&q__1, &temp); cscal_(&i__2, &q__1, &h__[i__ + (i__ + 1) * h_dim1], ldh); } i__2 = i__ - i1; cscal_(&i__2, &temp, &h__[i1 + i__ * h_dim1], &c__1); if (*wantz) { cscal_(&nz, &temp, &z__[*iloz + i__ * z_dim1], &c__1); } } /* L130: */ } /* Failure to converge in remaining number of iterations */ *info = i__; return 0; L140: /* H(I,I-1) is negligible: one eigenvalue has converged. */ i__1 = i__; i__2 = i__ + i__ * h_dim1; w[i__1].r = h__[i__2].r, w[i__1].i = h__[i__2].i; /* return to start of the main loop with new value of I. */ i__ = l - 1; goto L30; L150: return 0; /* End of CLAHQR */ } /* clahqr_ */