#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 DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Sc hur decomposition. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download DLAQR4 + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE DLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, */ /* ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO ) */ /* INTEGER IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N */ /* LOGICAL WANTT, WANTZ */ /* DOUBLE PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), */ /* $ Z( LDZ, * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > DLAQR4 implements one level of recursion for DLAQR0. */ /* > It is a complete implementation of the small bulge multi-shift */ /* > QR algorithm. It may be called by DLAQR0 and, for large enough */ /* > deflation window size, it may be called by DLAQR3. This */ /* > subroutine is identical to DLAQR0 except that it calls DLAQR2 */ /* > instead of DLAQR3. */ /* > */ /* > DLAQR4 computes the eigenvalues of a Hessenberg matrix H */ /* > and, optionally, the matrices T and Z from the Schur decomposition */ /* > H = Z T Z**T, where T is an upper quasi-triangular matrix (the */ /* > Schur form), and Z is the orthogonal matrix of Schur vectors. */ /* > */ /* > Optionally Z may be postmultiplied into an input orthogonal */ /* > matrix Q so that this routine can give the Schur factorization */ /* > of a matrix A which has been reduced to the Hessenberg form H */ /* > by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. */ /* > \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 1:ILO-1 and IHI+1:N and, if ILO > 1, */ /* > H(ILO,ILO-1) is zero. ILO and IHI are normally set by a */ /* > previous call to DGEBAL, and then passed to DGEHRD when the */ /* > matrix output by DGEBAL is reduced to Hessenberg form. */ /* > Otherwise, ILO and IHI should be set to 1 and N, */ /* > respectively. If N > 0, then 1 <= ILO <= IHI <= N. */ /* > If N = 0, then ILO = 1 and IHI = 0. */ /* > \endverbatim */ /* > */ /* > \param[in,out] H */ /* > \verbatim */ /* > H is DOUBLE PRECISION array, dimension (LDH,N) */ /* > On entry, the upper Hessenberg matrix H. */ /* > On exit, if INFO = 0 and WANTT is .TRUE., then H contains */ /* > the upper quasi-triangular matrix T from the Schur */ /* > decomposition (the Schur form); 2-by-2 diagonal blocks */ /* > (corresponding to complex conjugate pairs of eigenvalues) */ /* > are returned in standard form, with H(i,i) = H(i+1,i+1) */ /* > and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT is */ /* > .FALSE., then the contents of H are unspecified on exit. */ /* > (The output value of H when INFO > 0 is given under the */ /* > description of INFO below.) */ /* > */ /* > This subroutine may explicitly set H(i,j) = 0 for i > j and */ /* > j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. */ /* > \endverbatim */ /* > */ /* > \param[in] LDH */ /* > \verbatim */ /* > LDH is INTEGER */ /* > The leading dimension of the array H. LDH >= f2cmax(1,N). */ /* > \endverbatim */ /* > */ /* > \param[out] WR */ /* > \verbatim */ /* > WR is DOUBLE PRECISION array, dimension (IHI) */ /* > \endverbatim */ /* > */ /* > \param[out] WI */ /* > \verbatim */ /* > WI is DOUBLE PRECISION array, dimension (IHI) */ /* > The real and imaginary parts, respectively, of the computed */ /* > eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) */ /* > and WI(ILO:IHI). If two eigenvalues are computed as a */ /* > complex conjugate pair, they are stored in consecutive */ /* > elements of WR and WI, say the i-th and (i+1)th, with */ /* > WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., then */ /* > the eigenvalues are stored in the same order as on the */ /* > diagonal of the Schur form returned in H, with */ /* > WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal */ /* > block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and */ /* > WI(i+1) = -WI(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 DOUBLE PRECISION array, dimension (LDZ,IHI) */ /* > If WANTZ is .FALSE., then Z is not referenced. */ /* > If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is */ /* > replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the */ /* > orthogonal Schur factor of H(ILO:IHI,ILO:IHI). */ /* > (The output value of Z when INFO > 0 is given under */ /* > the description of INFO below.) */ /* > \endverbatim */ /* > */ /* > \param[in] LDZ */ /* > \verbatim */ /* > LDZ is INTEGER */ /* > The leading dimension of the array Z. if WANTZ is .TRUE. */ /* > then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is DOUBLE PRECISION array, dimension LWORK */ /* > On exit, if LWORK = -1, WORK(1) returns an estimate of */ /* > the optimal value for LWORK. */ /* > \endverbatim */ /* > */ /* > \param[in] LWORK */ /* > \verbatim */ /* > LWORK is INTEGER */ /* > The dimension of the array WORK. LWORK >= f2cmax(1,N) */ /* > is sufficient, but LWORK typically as large as 6*N may */ /* > be required for optimal performance. A workspace query */ /* > to determine the optimal workspace size is recommended. */ /* > */ /* > If LWORK = -1, then DLAQR4 does a workspace query. */ /* > In this case, DLAQR4 checks the input parameters and */ /* > estimates the optimal workspace size for the given */ /* > values of N, ILO and IHI. The estimate is returned */ /* > in WORK(1). No error message related to LWORK is */ /* > issued by XERBLA. Neither H nor Z are accessed. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > > 0: if INFO = i, DLAQR4 failed to compute all of */ /* > the eigenvalues. Elements 1:ilo-1 and i+1:n of WR */ /* > and WI contain those eigenvalues which have been */ /* > successfully computed. (Failures are rare.) */ /* > */ /* > If INFO > 0 and WANT is .FALSE., then on exit, */ /* > the remaining unconverged eigenvalues are the eigen- */ /* > values 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 a 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(ILO:IHI,ILOZ:IHIZ) */ /* > = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U */ /* > */ /* > where U is the orthogonal matrix in (*) (regard- */ /* > less of the value of WANTT.) */ /* > */ /* > If INFO > 0 and WANTZ is .FALSE., then Z is not */ /* > accessed. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup doubleOTHERauxiliary */ /* > \par Contributors: */ /* ================== */ /* > */ /* > Karen Braman and Ralph Byers, Department of Mathematics, */ /* > University of Kansas, USA */ /* > \par References: */ /* ================ */ /* > */ /* > K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */ /* > Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 */ /* > Performance, SIAM Journal of Matrix Analysis, volume 23, pages */ /* > 929--947, 2002. */ /* > \n */ /* > K. Braman, R. Byers and R. Mathias, The Multi-Shift QR */ /* > Algorithm Part II: Aggressive Early Deflation, SIAM Journal */ /* > of Matrix Analysis, volume 23, pages 948--973, 2002. */ /* > */ /* ===================================================================== */ /* Subroutine */ int dlaqr4_(logical *wantt, logical *wantz, integer *n, integer *ilo, integer *ihi, doublereal *h__, integer *ldh, doublereal *wr, doublereal *wi, integer *iloz, integer *ihiz, doublereal *z__, integer *ldz, doublereal *work, integer *lwork, integer *info) { /* System generated locals */ integer h_dim1, h_offset, z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5; doublereal d__1, d__2, d__3, d__4; /* Local variables */ integer ndec, ndfl, kbot, nmin; doublereal swap; integer ktop; doublereal zdum[1] /* was [1][1] */; integer kacc22, i__, k, itmax, nsmax, nwmax, kwtop; extern /* Subroutine */ int dlaqr2_(logical *, logical *, integer *, integer *, integer *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlanv2_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *), dlaqr5_( logical *, logical *, integer *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *); doublereal aa, bb, cc, dd; integer ld; doublereal cs; integer nh, nibble, it, ks, kt; doublereal sn; integer ku, kv, ls, ns; doublereal ss; integer nw; extern /* Subroutine */ int dlahqr_(logical *, logical *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); char jbcmpz[2]; integer nwupbd; logical sorted; integer lwkopt, inf, kdu, nho, nve, kwh, nsr, nwr, kwv; /* -- 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 */ /* ================================================================ */ /* ==== Matrices of order NTINY or smaller must be processed by */ /* . DLAHQR because of insufficient subdiagonal scratch space. */ /* . (This is a hard limit.) ==== */ /* ==== Exceptional deflation windows: try to cure rare */ /* . slow convergence by varying the size of the */ /* . deflation window after KEXNW iterations. ==== */ /* ==== Exceptional shifts: try to cure rare slow convergence */ /* . with ad-hoc exceptional shifts every KEXSH iterations. */ /* . ==== */ /* ==== The constants WILK1 and WILK2 are used to form the */ /* . exceptional shifts. ==== */ /* Parameter adjustments */ h_dim1 = *ldh; h_offset = 1 + h_dim1 * 1; h__ -= h_offset; --wr; --wi; z_dim1 = *ldz; z_offset = 1 + z_dim1 * 1; z__ -= z_offset; --work; /* Function Body */ *info = 0; /* ==== Quick return for N = 0: nothing to do. ==== */ if (*n == 0) { work[1] = 1.; return 0; } if (*n <= 15) { /* ==== Tiny matrices must use DLAHQR. ==== */ lwkopt = 1; if (*lwork != -1) { dlahqr_(wantt, wantz, n, ilo, ihi, &h__[h_offset], ldh, &wr[1], & wi[1], iloz, ihiz, &z__[z_offset], ldz, info); } } else { /* ==== Use small bulge multi-shift QR with aggressive early */ /* . deflation on larger-than-tiny matrices. ==== */ /* ==== Hope for the best. ==== */ *info = 0; /* ==== Set up job flags for ILAENV. ==== */ if (*wantt) { *(unsigned char *)jbcmpz = 'S'; } else { *(unsigned char *)jbcmpz = 'E'; } if (*wantz) { *(unsigned char *)&jbcmpz[1] = 'V'; } else { *(unsigned char *)&jbcmpz[1] = 'N'; } /* ==== NWR = recommended deflation window size. At this */ /* . point, N .GT. NTINY = 15, so there is enough */ /* . subdiagonal workspace for NWR.GE.2 as required. */ /* . (In fact, there is enough subdiagonal space for */ /* . NWR.GE.4.) ==== */ nwr = ilaenv_(&c__13, "DLAQR4", jbcmpz, n, ilo, ihi, lwork, (ftnlen)6, (ftnlen)2); nwr = f2cmax(2,nwr); /* Computing MIN */ i__1 = *ihi - *ilo + 1, i__2 = (*n - 1) / 3, i__1 = f2cmin(i__1,i__2); nwr = f2cmin(i__1,nwr); /* ==== NSR = recommended number of simultaneous shifts. */ /* . At this point N .GT. NTINY = 15, so there is at */ /* . enough subdiagonal workspace for NSR to be even */ /* . and greater than or equal to two as required. ==== */ nsr = ilaenv_(&c__15, "DLAQR4", jbcmpz, n, ilo, ihi, lwork, (ftnlen)6, (ftnlen)2); /* Computing MIN */ i__1 = nsr, i__2 = (*n - 3) / 6, i__1 = f2cmin(i__1,i__2), i__2 = *ihi - *ilo; nsr = f2cmin(i__1,i__2); /* Computing MAX */ i__1 = 2, i__2 = nsr - nsr % 2; nsr = f2cmax(i__1,i__2); /* ==== Estimate optimal workspace ==== */ /* ==== Workspace query call to DLAQR2 ==== */ i__1 = nwr + 1; dlaqr2_(wantt, wantz, n, ilo, ihi, &i__1, &h__[h_offset], ldh, iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1], &h__[ h_offset], ldh, n, &h__[h_offset], ldh, n, &h__[h_offset], ldh, &work[1], &c_n1); /* ==== Optimal workspace = MAX(DLAQR5, DLAQR2) ==== */ /* Computing MAX */ i__1 = nsr * 3 / 2, i__2 = (integer) work[1]; lwkopt = f2cmax(i__1,i__2); /* ==== Quick return in case of workspace query. ==== */ if (*lwork == -1) { work[1] = (doublereal) lwkopt; return 0; } /* ==== DLAHQR/DLAQR0 crossover point ==== */ nmin = ilaenv_(&c__12, "DLAQR4", jbcmpz, n, ilo, ihi, lwork, (ftnlen) 6, (ftnlen)2); nmin = f2cmax(15,nmin); /* ==== Nibble crossover point ==== */ nibble = ilaenv_(&c__14, "DLAQR4", jbcmpz, n, ilo, ihi, lwork, ( ftnlen)6, (ftnlen)2); nibble = f2cmax(0,nibble); /* ==== Accumulate reflections during ttswp? Use block */ /* . 2-by-2 structure during matrix-matrix multiply? ==== */ kacc22 = ilaenv_(&c__16, "DLAQR4", jbcmpz, n, ilo, ihi, lwork, ( ftnlen)6, (ftnlen)2); kacc22 = f2cmax(0,kacc22); kacc22 = f2cmin(2,kacc22); /* ==== NWMAX = the largest possible deflation window for */ /* . which there is sufficient workspace. ==== */ /* Computing MIN */ i__1 = (*n - 1) / 3, i__2 = *lwork / 2; nwmax = f2cmin(i__1,i__2); nw = nwmax; /* ==== NSMAX = the Largest number of simultaneous shifts */ /* . for which there is sufficient workspace. ==== */ /* Computing MIN */ i__1 = (*n - 3) / 6, i__2 = (*lwork << 1) / 3; nsmax = f2cmin(i__1,i__2); nsmax -= nsmax % 2; /* ==== NDFL: an iteration count restarted at deflation. ==== */ ndfl = 1; /* ==== ITMAX = iteration limit ==== */ /* Computing MAX */ i__1 = 10, i__2 = *ihi - *ilo + 1; itmax = 30 * f2cmax(i__1,i__2); /* ==== Last row and column in the active block ==== */ kbot = *ihi; /* ==== Main Loop ==== */ i__1 = itmax; for (it = 1; it <= i__1; ++it) { /* ==== Done when KBOT falls below ILO ==== */ if (kbot < *ilo) { goto L90; } /* ==== Locate active block ==== */ i__2 = *ilo + 1; for (k = kbot; k >= i__2; --k) { if (h__[k + (k - 1) * h_dim1] == 0.) { goto L20; } /* L10: */ } k = *ilo; L20: ktop = k; /* ==== Select deflation window size: */ /* . Typical Case: */ /* . If possible and advisable, nibble the entire */ /* . active block. If not, use size MIN(NWR,NWMAX) */ /* . or MIN(NWR+1,NWMAX) depending upon which has */ /* . the smaller corresponding subdiagonal entry */ /* . (a heuristic). */ /* . */ /* . Exceptional Case: */ /* . If there have been no deflations in KEXNW or */ /* . more iterations, then vary the deflation window */ /* . size. At first, because, larger windows are, */ /* . in general, more powerful than smaller ones, */ /* . rapidly increase the window to the maximum possible. */ /* . Then, gradually reduce the window size. ==== */ nh = kbot - ktop + 1; nwupbd = f2cmin(nh,nwmax); if (ndfl < 5) { nw = f2cmin(nwupbd,nwr); } else { /* Computing MIN */ i__2 = nwupbd, i__3 = nw << 1; nw = f2cmin(i__2,i__3); } if (nw < nwmax) { if (nw >= nh - 1) { nw = nh; } else { kwtop = kbot - nw + 1; if ((d__1 = h__[kwtop + (kwtop - 1) * h_dim1], abs(d__1)) > (d__2 = h__[kwtop - 1 + (kwtop - 2) * h_dim1], abs(d__2))) { ++nw; } } } if (ndfl < 5) { ndec = -1; } else if (ndec >= 0 || nw >= nwupbd) { ++ndec; if (nw - ndec < 2) { ndec = 0; } nw -= ndec; } /* ==== Aggressive early deflation: */ /* . split workspace under the subdiagonal into */ /* . - an nw-by-nw work array V in the lower */ /* . left-hand-corner, */ /* . - an NW-by-at-least-NW-but-more-is-better */ /* . (NW-by-NHO) horizontal work array along */ /* . the bottom edge, */ /* . - an at-least-NW-but-more-is-better (NHV-by-NW) */ /* . vertical work array along the left-hand-edge. */ /* . ==== */ kv = *n - nw + 1; kt = nw + 1; nho = *n - nw - 1 - kt + 1; kwv = nw + 2; nve = *n - nw - kwv + 1; /* ==== Aggressive early deflation ==== */ dlaqr2_(wantt, wantz, n, &ktop, &kbot, &nw, &h__[h_offset], ldh, iloz, ihiz, &z__[z_offset], ldz, &ls, &ld, &wr[1], &wi[1], &h__[kv + h_dim1], ldh, &nho, &h__[kv + kt * h_dim1], ldh, &nve, &h__[kwv + h_dim1], ldh, &work[1], lwork); /* ==== Adjust KBOT accounting for new deflations. ==== */ kbot -= ld; /* ==== KS points to the shifts. ==== */ ks = kbot - ls + 1; /* ==== Skip an expensive QR sweep if there is a (partly */ /* . heuristic) reason to expect that many eigenvalues */ /* . will deflate without it. Here, the QR sweep is */ /* . skipped if many eigenvalues have just been deflated */ /* . or if the remaining active block is small. */ if (ld == 0 || ld * 100 <= nw * nibble && kbot - ktop + 1 > f2cmin( nmin,nwmax)) { /* ==== NS = nominal number of simultaneous shifts. */ /* . This may be lowered (slightly) if DLAQR2 */ /* . did not provide that many shifts. ==== */ /* Computing MIN */ /* Computing MAX */ i__4 = 2, i__5 = kbot - ktop; i__2 = f2cmin(nsmax,nsr), i__3 = f2cmax(i__4,i__5); ns = f2cmin(i__2,i__3); ns -= ns % 2; /* ==== If there have been no deflations */ /* . in a multiple of KEXSH iterations, */ /* . then try exceptional shifts. */ /* . Otherwise use shifts provided by */ /* . DLAQR2 above or from the eigenvalues */ /* . of a trailing principal submatrix. ==== */ if (ndfl % 6 == 0) { ks = kbot - ns + 1; /* Computing MAX */ i__3 = ks + 1, i__4 = ktop + 2; i__2 = f2cmax(i__3,i__4); for (i__ = kbot; i__ >= i__2; i__ += -2) { ss = (d__1 = h__[i__ + (i__ - 1) * h_dim1], abs(d__1)) + (d__2 = h__[i__ - 1 + (i__ - 2) * h_dim1], abs(d__2)); aa = ss * .75 + h__[i__ + i__ * h_dim1]; bb = ss; cc = ss * -.4375; dd = aa; dlanv2_(&aa, &bb, &cc, &dd, &wr[i__ - 1], &wi[i__ - 1] , &wr[i__], &wi[i__], &cs, &sn); /* L30: */ } if (ks == ktop) { wr[ks + 1] = h__[ks + 1 + (ks + 1) * h_dim1]; wi[ks + 1] = 0.; wr[ks] = wr[ks + 1]; wi[ks] = wi[ks + 1]; } } else { /* ==== Got NS/2 or fewer shifts? Use DLAHQR */ /* . on a trailing principal submatrix to */ /* . get more. (Since NS.LE.NSMAX.LE.(N-3)/6, */ /* . there is enough space below the subdiagonal */ /* . to fit an NS-by-NS scratch array.) ==== */ if (kbot - ks + 1 <= ns / 2) { ks = kbot - ns + 1; kt = *n - ns + 1; dlacpy_("A", &ns, &ns, &h__[ks + ks * h_dim1], ldh, & h__[kt + h_dim1], ldh); dlahqr_(&c_false, &c_false, &ns, &c__1, &ns, &h__[kt + h_dim1], ldh, &wr[ks], &wi[ks], &c__1, & c__1, zdum, &c__1, &inf); ks += inf; /* ==== In case of a rare QR failure use */ /* . eigenvalues of the trailing 2-by-2 */ /* . principal submatrix. ==== */ if (ks >= kbot) { aa = h__[kbot - 1 + (kbot - 1) * h_dim1]; cc = h__[kbot + (kbot - 1) * h_dim1]; bb = h__[kbot - 1 + kbot * h_dim1]; dd = h__[kbot + kbot * h_dim1]; dlanv2_(&aa, &bb, &cc, &dd, &wr[kbot - 1], &wi[ kbot - 1], &wr[kbot], &wi[kbot], &cs, &sn) ; ks = kbot - 1; } } if (kbot - ks + 1 > ns) { /* ==== Sort the shifts (Helps a little) */ /* . Bubble sort keeps complex conjugate */ /* . pairs together. ==== */ sorted = FALSE_; i__2 = ks + 1; for (k = kbot; k >= i__2; --k) { if (sorted) { goto L60; } sorted = TRUE_; i__3 = k - 1; for (i__ = ks; i__ <= i__3; ++i__) { if ((d__1 = wr[i__], abs(d__1)) + (d__2 = wi[ i__], abs(d__2)) < (d__3 = wr[i__ + 1] , abs(d__3)) + (d__4 = wi[i__ + 1], abs(d__4))) { sorted = FALSE_; swap = wr[i__]; wr[i__] = wr[i__ + 1]; wr[i__ + 1] = swap; swap = wi[i__]; wi[i__] = wi[i__ + 1]; wi[i__ + 1] = swap; } /* L40: */ } /* L50: */ } L60: ; } /* ==== Shuffle shifts into pairs of real shifts */ /* . and pairs of complex conjugate shifts */ /* . assuming complex conjugate shifts are */ /* . already adjacent to one another. (Yes, */ /* . they are.) ==== */ i__2 = ks + 2; for (i__ = kbot; i__ >= i__2; i__ += -2) { if (wi[i__] != -wi[i__ - 1]) { swap = wr[i__]; wr[i__] = wr[i__ - 1]; wr[i__ - 1] = wr[i__ - 2]; wr[i__ - 2] = swap; swap = wi[i__]; wi[i__] = wi[i__ - 1]; wi[i__ - 1] = wi[i__ - 2]; wi[i__ - 2] = swap; } /* L70: */ } } /* ==== If there are only two shifts and both are */ /* . real, then use only one. ==== */ if (kbot - ks + 1 == 2) { if (wi[kbot] == 0.) { if ((d__1 = wr[kbot] - h__[kbot + kbot * h_dim1], abs( d__1)) < (d__2 = wr[kbot - 1] - h__[kbot + kbot * h_dim1], abs(d__2))) { wr[kbot - 1] = wr[kbot]; } else { wr[kbot] = wr[kbot - 1]; } } } /* ==== Use up to NS of the the smallest magnitude */ /* . shifts. If there aren't NS shifts available, */ /* . then use them all, possibly dropping one to */ /* . make the number of shifts even. ==== */ /* Computing MIN */ i__2 = ns, i__3 = kbot - ks + 1; ns = f2cmin(i__2,i__3); ns -= ns % 2; ks = kbot - ns + 1; /* ==== Small-bulge multi-shift QR sweep: */ /* . split workspace under the subdiagonal into */ /* . - a KDU-by-KDU work array U in the lower */ /* . left-hand-corner, */ /* . - a KDU-by-at-least-KDU-but-more-is-better */ /* . (KDU-by-NHo) horizontal work array WH along */ /* . the bottom edge, */ /* . - and an at-least-KDU-but-more-is-better-by-KDU */ /* . (NVE-by-KDU) vertical work WV arrow along */ /* . the left-hand-edge. ==== */ kdu = ns << 1; ku = *n - kdu + 1; kwh = kdu + 1; nho = *n - kdu - 3 - (kdu + 1) + 1; kwv = kdu + 4; nve = *n - kdu - kwv + 1; /* ==== Small-bulge multi-shift QR sweep ==== */ dlaqr5_(wantt, wantz, &kacc22, n, &ktop, &kbot, &ns, &wr[ks], &wi[ks], &h__[h_offset], ldh, iloz, ihiz, &z__[ z_offset], ldz, &work[1], &c__3, &h__[ku + h_dim1], ldh, &nve, &h__[kwv + h_dim1], ldh, &nho, &h__[ku + kwh * h_dim1], ldh); } /* ==== Note progress (or the lack of it). ==== */ if (ld > 0) { ndfl = 1; } else { ++ndfl; } /* ==== End of main loop ==== */ /* L80: */ } /* ==== Iteration limit exceeded. Set INFO to show where */ /* . the problem occurred and exit. ==== */ *info = kbot; L90: ; } /* ==== Return the optimal value of LWORK. ==== */ work[1] = (doublereal) lwkopt; /* ==== End of DLAQR4 ==== */ return 0; } /* dlaqr4_ */