#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 SLASD4 computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one modification to a positive diagonal matrix. Used by sbdsdc. */ /* =========== DOCUMENTATION =========== */ /* Online html documentation available at */ /* http://www.netlib.org/lapack/explore-html/ */ /* > \htmlonly */ /* > Download SLASD4 + dependencies */ /* > */ /* > [TGZ] */ /* > */ /* > [ZIP] */ /* > */ /* > [TXT] */ /* > \endhtmlonly */ /* Definition: */ /* =========== */ /* SUBROUTINE SLASD4( N, I, D, Z, DELTA, RHO, SIGMA, WORK, INFO ) */ /* INTEGER I, INFO, N */ /* REAL RHO, SIGMA */ /* REAL D( * ), DELTA( * ), WORK( * ), Z( * ) */ /* > \par Purpose: */ /* ============= */ /* > */ /* > \verbatim */ /* > */ /* > This subroutine computes the square root of the I-th updated */ /* > eigenvalue of a positive symmetric rank-one modification to */ /* > a positive diagonal matrix whose entries are given as the squares */ /* > of the corresponding entries in the array d, and that */ /* > */ /* > 0 <= D(i) < D(j) for i < j */ /* > */ /* > and that RHO > 0. This is arranged by the calling routine, and is */ /* > no loss in generality. The rank-one modified system is thus */ /* > */ /* > diag( D ) * diag( D ) + RHO * Z * Z_transpose. */ /* > */ /* > where we assume the Euclidean norm of Z is 1. */ /* > */ /* > The method consists of approximating the rational functions in the */ /* > secular equation by simpler interpolating rational functions. */ /* > \endverbatim */ /* Arguments: */ /* ========== */ /* > \param[in] N */ /* > \verbatim */ /* > N is INTEGER */ /* > The length of all arrays. */ /* > \endverbatim */ /* > */ /* > \param[in] I */ /* > \verbatim */ /* > I is INTEGER */ /* > The index of the eigenvalue to be computed. 1 <= I <= N. */ /* > \endverbatim */ /* > */ /* > \param[in] D */ /* > \verbatim */ /* > D is REAL array, dimension ( N ) */ /* > The original eigenvalues. It is assumed that they are in */ /* > order, 0 <= D(I) < D(J) for I < J. */ /* > \endverbatim */ /* > */ /* > \param[in] Z */ /* > \verbatim */ /* > Z is REAL array, dimension ( N ) */ /* > The components of the updating vector. */ /* > \endverbatim */ /* > */ /* > \param[out] DELTA */ /* > \verbatim */ /* > DELTA is REAL array, dimension ( N ) */ /* > If N .ne. 1, DELTA contains (D(j) - sigma_I) in its j-th */ /* > component. If N = 1, then DELTA(1) = 1. The vector DELTA */ /* > contains the information necessary to construct the */ /* > (singular) eigenvectors. */ /* > \endverbatim */ /* > */ /* > \param[in] RHO */ /* > \verbatim */ /* > RHO is REAL */ /* > The scalar in the symmetric updating formula. */ /* > \endverbatim */ /* > */ /* > \param[out] SIGMA */ /* > \verbatim */ /* > SIGMA is REAL */ /* > The computed sigma_I, the I-th updated eigenvalue. */ /* > \endverbatim */ /* > */ /* > \param[out] WORK */ /* > \verbatim */ /* > WORK is REAL array, dimension ( N ) */ /* > If N .ne. 1, WORK contains (D(j) + sigma_I) in its j-th */ /* > component. If N = 1, then WORK( 1 ) = 1. */ /* > \endverbatim */ /* > */ /* > \param[out] INFO */ /* > \verbatim */ /* > INFO is INTEGER */ /* > = 0: successful exit */ /* > > 0: if INFO = 1, the updating process failed. */ /* > \endverbatim */ /* > \par Internal Parameters: */ /* ========================= */ /* > */ /* > \verbatim */ /* > Logical variable ORGATI (origin-at-i?) is used for distinguishing */ /* > whether D(i) or D(i+1) is treated as the origin. */ /* > */ /* > ORGATI = .true. origin at i */ /* > ORGATI = .false. origin at i+1 */ /* > */ /* > Logical variable SWTCH3 (switch-for-3-poles?) is for noting */ /* > if we are working with THREE poles! */ /* > */ /* > MAXIT is the maximum number of iterations allowed for each */ /* > eigenvalue. */ /* > \endverbatim */ /* Authors: */ /* ======== */ /* > \author Univ. of Tennessee */ /* > \author Univ. of California Berkeley */ /* > \author Univ. of Colorado Denver */ /* > \author NAG Ltd. */ /* > \date December 2016 */ /* > \ingroup OTHERauxiliary */ /* > \par Contributors: */ /* ================== */ /* > */ /* > Ren-Cang Li, Computer Science Division, University of California */ /* > at Berkeley, USA */ /* > */ /* ===================================================================== */ /* Subroutine */ int slasd4_(integer *n, integer *i__, real *d__, real *z__, real *delta, real *rho, real *sigma, real *work, integer *info) { /* System generated locals */ integer i__1; real r__1; /* Local variables */ real dphi, sglb, dpsi, sgub; integer iter; real temp, prew, temp1, temp2, a, b, c__; integer j; real w, dtiim, delsq, dtiip; integer niter; real dtisq; logical swtch; real dtnsq; extern /* Subroutine */ int slaed6_(integer *, logical *, real *, real *, real *, real *, real *, integer *); real delsq2; extern /* Subroutine */ int slasd5_(integer *, real *, real *, real *, real *, real *, real *); real dd[3], dtnsq1; logical swtch3; integer ii; real dw; extern real slamch_(char *); real zz[3]; logical orgati; real erretm, dtipsq, rhoinv; integer ip1; real sq2, eta, phi, eps, tau, psi; logical geomavg; integer iim1, iip1; real tau2; /* -- 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 */ /* ===================================================================== */ /* Since this routine is called in an inner loop, we do no argument */ /* checking. */ /* Quick return for N=1 and 2. */ /* Parameter adjustments */ --work; --delta; --z__; --d__; /* Function Body */ *info = 0; if (*n == 1) { /* Presumably, I=1 upon entry */ *sigma = sqrt(d__[1] * d__[1] + *rho * z__[1] * z__[1]); delta[1] = 1.f; work[1] = 1.f; return 0; } if (*n == 2) { slasd5_(i__, &d__[1], &z__[1], &delta[1], rho, sigma, &work[1]); return 0; } /* Compute machine epsilon */ eps = slamch_("Epsilon"); rhoinv = 1.f / *rho; tau2 = 0.f; /* The case I = N */ if (*i__ == *n) { /* Initialize some basic variables */ ii = *n - 1; niter = 1; /* Calculate initial guess */ temp = *rho / 2.f; /* If ||Z||_2 is not one, then TEMP should be set to */ /* RHO * ||Z||_2^2 / TWO */ temp1 = temp / (d__[*n] + sqrt(d__[*n] * d__[*n] + temp)); i__1 = *n; for (j = 1; j <= i__1; ++j) { work[j] = d__[j] + d__[*n] + temp1; delta[j] = d__[j] - d__[*n] - temp1; /* L10: */ } psi = 0.f; i__1 = *n - 2; for (j = 1; j <= i__1; ++j) { psi += z__[j] * z__[j] / (delta[j] * work[j]); /* L20: */ } c__ = rhoinv + psi; w = c__ + z__[ii] * z__[ii] / (delta[ii] * work[ii]) + z__[*n] * z__[* n] / (delta[*n] * work[*n]); if (w <= 0.f) { temp1 = sqrt(d__[*n] * d__[*n] + *rho); temp = z__[*n - 1] * z__[*n - 1] / ((d__[*n - 1] + temp1) * (d__[* n] - d__[*n - 1] + *rho / (d__[*n] + temp1))) + z__[*n] * z__[*n] / *rho; /* The following TAU2 is to approximate */ /* SIGMA_n^2 - D( N )*D( N ) */ if (c__ <= temp) { tau = *rho; } else { delsq = (d__[*n] - d__[*n - 1]) * (d__[*n] + d__[*n - 1]); a = -c__ * delsq + z__[*n - 1] * z__[*n - 1] + z__[*n] * z__[* n]; b = z__[*n] * z__[*n] * delsq; if (a < 0.f) { tau2 = b * 2.f / (sqrt(a * a + b * 4.f * c__) - a); } else { tau2 = (a + sqrt(a * a + b * 4.f * c__)) / (c__ * 2.f); } tau = tau2 / (d__[*n] + sqrt(d__[*n] * d__[*n] + tau2)); } /* It can be proved that */ /* D(N)^2+RHO/2 <= SIGMA_n^2 < D(N)^2+TAU2 <= D(N)^2+RHO */ } else { delsq = (d__[*n] - d__[*n - 1]) * (d__[*n] + d__[*n - 1]); a = -c__ * delsq + z__[*n - 1] * z__[*n - 1] + z__[*n] * z__[*n]; b = z__[*n] * z__[*n] * delsq; /* The following TAU2 is to approximate */ /* SIGMA_n^2 - D( N )*D( N ) */ if (a < 0.f) { tau2 = b * 2.f / (sqrt(a * a + b * 4.f * c__) - a); } else { tau2 = (a + sqrt(a * a + b * 4.f * c__)) / (c__ * 2.f); } tau = tau2 / (d__[*n] + sqrt(d__[*n] * d__[*n] + tau2)); /* It can be proved that */ /* D(N)^2 < D(N)^2+TAU2 < SIGMA(N)^2 < D(N)^2+RHO/2 */ } /* The following TAU is to approximate SIGMA_n - D( N ) */ /* TAU = TAU2 / ( D( N )+SQRT( D( N )*D( N )+TAU2 ) ) */ *sigma = d__[*n] + tau; i__1 = *n; for (j = 1; j <= i__1; ++j) { delta[j] = d__[j] - d__[*n] - tau; work[j] = d__[j] + d__[*n] + tau; /* L30: */ } /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = ii; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (delta[j] * work[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L40: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ temp = z__[*n] / (delta[*n] * work[*n]); phi = z__[*n] * temp; dphi = temp * temp; erretm = (-phi - psi) * 8.f + erretm - phi + rhoinv; /* $ + ABS( TAU2 )*( DPSI+DPHI ) */ w = rhoinv + phi + psi; /* Test for convergence */ if (abs(w) <= eps * erretm) { goto L240; } /* Calculate the new step */ ++niter; dtnsq1 = work[*n - 1] * delta[*n - 1]; dtnsq = work[*n] * delta[*n]; c__ = w - dtnsq1 * dpsi - dtnsq * dphi; a = (dtnsq + dtnsq1) * w - dtnsq * dtnsq1 * (dpsi + dphi); b = dtnsq * dtnsq1 * w; if (c__ < 0.f) { c__ = abs(c__); } if (c__ == 0.f) { eta = *rho - *sigma * *sigma; } else if (a >= 0.f) { eta = (a + sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))) / ( c__ * 2.f); } else { eta = b * 2.f / (a - sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1) ))); } /* Note, eta should be positive if w is negative, and */ /* eta should be negative otherwise. However, */ /* if for some reason caused by roundoff, eta*w > 0, */ /* we simply use one Newton step instead. This way */ /* will guarantee eta*w < 0. */ if (w * eta > 0.f) { eta = -w / (dpsi + dphi); } temp = eta - dtnsq; if (temp > *rho) { eta = *rho + dtnsq; } eta /= *sigma + sqrt(eta + *sigma * *sigma); tau += eta; *sigma += eta; i__1 = *n; for (j = 1; j <= i__1; ++j) { delta[j] -= eta; work[j] += eta; /* L50: */ } /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = ii; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (work[j] * delta[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L60: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ tau2 = work[*n] * delta[*n]; temp = z__[*n] / tau2; phi = z__[*n] * temp; dphi = temp * temp; erretm = (-phi - psi) * 8.f + erretm - phi + rhoinv; /* $ + ABS( TAU2 )*( DPSI+DPHI ) */ w = rhoinv + phi + psi; /* Main loop to update the values of the array DELTA */ iter = niter + 1; for (niter = iter; niter <= 400; ++niter) { /* Test for convergence */ if (abs(w) <= eps * erretm) { goto L240; } /* Calculate the new step */ dtnsq1 = work[*n - 1] * delta[*n - 1]; dtnsq = work[*n] * delta[*n]; c__ = w - dtnsq1 * dpsi - dtnsq * dphi; a = (dtnsq + dtnsq1) * w - dtnsq1 * dtnsq * (dpsi + dphi); b = dtnsq1 * dtnsq * w; if (a >= 0.f) { eta = (a + sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))) / (c__ * 2.f); } else { eta = b * 2.f / (a - sqrt((r__1 = a * a - b * 4.f * c__, abs( r__1)))); } /* Note, eta should be positive if w is negative, and */ /* eta should be negative otherwise. However, */ /* if for some reason caused by roundoff, eta*w > 0, */ /* we simply use one Newton step instead. This way */ /* will guarantee eta*w < 0. */ if (w * eta > 0.f) { eta = -w / (dpsi + dphi); } temp = eta - dtnsq; if (temp <= 0.f) { eta /= 2.f; } eta /= *sigma + sqrt(eta + *sigma * *sigma); tau += eta; *sigma += eta; i__1 = *n; for (j = 1; j <= i__1; ++j) { delta[j] -= eta; work[j] += eta; /* L70: */ } /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = ii; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (work[j] * delta[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L80: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ tau2 = work[*n] * delta[*n]; temp = z__[*n] / tau2; phi = z__[*n] * temp; dphi = temp * temp; erretm = (-phi - psi) * 8.f + erretm - phi + rhoinv; /* $ + ABS( TAU2 )*( DPSI+DPHI ) */ w = rhoinv + phi + psi; /* L90: */ } /* Return with INFO = 1, NITER = MAXIT and not converged */ *info = 1; goto L240; /* End for the case I = N */ } else { /* The case for I < N */ niter = 1; ip1 = *i__ + 1; /* Calculate initial guess */ delsq = (d__[ip1] - d__[*i__]) * (d__[ip1] + d__[*i__]); delsq2 = delsq / 2.f; sq2 = sqrt((d__[*i__] * d__[*i__] + d__[ip1] * d__[ip1]) / 2.f); temp = delsq2 / (d__[*i__] + sq2); i__1 = *n; for (j = 1; j <= i__1; ++j) { work[j] = d__[j] + d__[*i__] + temp; delta[j] = d__[j] - d__[*i__] - temp; /* L100: */ } psi = 0.f; i__1 = *i__ - 1; for (j = 1; j <= i__1; ++j) { psi += z__[j] * z__[j] / (work[j] * delta[j]); /* L110: */ } phi = 0.f; i__1 = *i__ + 2; for (j = *n; j >= i__1; --j) { phi += z__[j] * z__[j] / (work[j] * delta[j]); /* L120: */ } c__ = rhoinv + psi + phi; w = c__ + z__[*i__] * z__[*i__] / (work[*i__] * delta[*i__]) + z__[ ip1] * z__[ip1] / (work[ip1] * delta[ip1]); geomavg = FALSE_; if (w > 0.f) { /* d(i)^2 < the ith sigma^2 < (d(i)^2+d(i+1)^2)/2 */ /* We choose d(i) as origin. */ orgati = TRUE_; ii = *i__; sglb = 0.f; sgub = delsq2 / (d__[*i__] + sq2); a = c__ * delsq + z__[*i__] * z__[*i__] + z__[ip1] * z__[ip1]; b = z__[*i__] * z__[*i__] * delsq; if (a > 0.f) { tau2 = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, abs( r__1)))); } else { tau2 = (a - sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))) / (c__ * 2.f); } /* TAU2 now is an estimation of SIGMA^2 - D( I )^2. The */ /* following, however, is the corresponding estimation of */ /* SIGMA - D( I ). */ tau = tau2 / (d__[*i__] + sqrt(d__[*i__] * d__[*i__] + tau2)); temp = sqrt(eps); if (d__[*i__] <= temp * d__[ip1] && (r__1 = z__[*i__], abs(r__1)) <= temp && d__[*i__] > 0.f) { /* Computing MIN */ r__1 = d__[*i__] * 10.f; tau = f2cmin(r__1,sgub); geomavg = TRUE_; } } else { /* (d(i)^2+d(i+1)^2)/2 <= the ith sigma^2 < d(i+1)^2/2 */ /* We choose d(i+1) as origin. */ orgati = FALSE_; ii = ip1; sglb = -delsq2 / (d__[ii] + sq2); sgub = 0.f; a = c__ * delsq - z__[*i__] * z__[*i__] - z__[ip1] * z__[ip1]; b = z__[ip1] * z__[ip1] * delsq; if (a < 0.f) { tau2 = b * 2.f / (a - sqrt((r__1 = a * a + b * 4.f * c__, abs( r__1)))); } else { tau2 = -(a + sqrt((r__1 = a * a + b * 4.f * c__, abs(r__1)))) / (c__ * 2.f); } /* TAU2 now is an estimation of SIGMA^2 - D( IP1 )^2. The */ /* following, however, is the corresponding estimation of */ /* SIGMA - D( IP1 ). */ tau = tau2 / (d__[ip1] + sqrt((r__1 = d__[ip1] * d__[ip1] + tau2, abs(r__1)))); } *sigma = d__[ii] + tau; i__1 = *n; for (j = 1; j <= i__1; ++j) { work[j] = d__[j] + d__[ii] + tau; delta[j] = d__[j] - d__[ii] - tau; /* L130: */ } iim1 = ii - 1; iip1 = ii + 1; /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = iim1; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (work[j] * delta[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L150: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ dphi = 0.f; phi = 0.f; i__1 = iip1; for (j = *n; j >= i__1; --j) { temp = z__[j] / (work[j] * delta[j]); phi += z__[j] * temp; dphi += temp * temp; erretm += phi; /* L160: */ } w = rhoinv + phi + psi; /* W is the value of the secular function with */ /* its ii-th element removed. */ swtch3 = FALSE_; if (orgati) { if (w < 0.f) { swtch3 = TRUE_; } } else { if (w > 0.f) { swtch3 = TRUE_; } } if (ii == 1 || ii == *n) { swtch3 = FALSE_; } temp = z__[ii] / (work[ii] * delta[ii]); dw = dpsi + dphi + temp * temp; temp = z__[ii] * temp; w += temp; erretm = (phi - psi) * 8.f + erretm + rhoinv * 2.f + abs(temp) * 3.f; /* $ + ABS( TAU2 )*DW */ /* Test for convergence */ if (abs(w) <= eps * erretm) { goto L240; } if (w <= 0.f) { sglb = f2cmax(sglb,tau); } else { sgub = f2cmin(sgub,tau); } /* Calculate the new step */ ++niter; if (! swtch3) { dtipsq = work[ip1] * delta[ip1]; dtisq = work[*i__] * delta[*i__]; if (orgati) { /* Computing 2nd power */ r__1 = z__[*i__] / dtisq; c__ = w - dtipsq * dw + delsq * (r__1 * r__1); } else { /* Computing 2nd power */ r__1 = z__[ip1] / dtipsq; c__ = w - dtisq * dw - delsq * (r__1 * r__1); } a = (dtipsq + dtisq) * w - dtipsq * dtisq * dw; b = dtipsq * dtisq * w; if (c__ == 0.f) { if (a == 0.f) { if (orgati) { a = z__[*i__] * z__[*i__] + dtipsq * dtipsq * (dpsi + dphi); } else { a = z__[ip1] * z__[ip1] + dtisq * dtisq * (dpsi + dphi); } } eta = b / a; } else if (a <= 0.f) { eta = (a - sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))) / (c__ * 2.f); } else { eta = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, abs( r__1)))); } } else { /* Interpolation using THREE most relevant poles */ dtiim = work[iim1] * delta[iim1]; dtiip = work[iip1] * delta[iip1]; temp = rhoinv + psi + phi; if (orgati) { temp1 = z__[iim1] / dtiim; temp1 *= temp1; c__ = temp - dtiip * (dpsi + dphi) - (d__[iim1] - d__[iip1]) * (d__[iim1] + d__[iip1]) * temp1; zz[0] = z__[iim1] * z__[iim1]; if (dpsi < temp1) { zz[2] = dtiip * dtiip * dphi; } else { zz[2] = dtiip * dtiip * (dpsi - temp1 + dphi); } } else { temp1 = z__[iip1] / dtiip; temp1 *= temp1; c__ = temp - dtiim * (dpsi + dphi) - (d__[iip1] - d__[iim1]) * (d__[iim1] + d__[iip1]) * temp1; if (dphi < temp1) { zz[0] = dtiim * dtiim * dpsi; } else { zz[0] = dtiim * dtiim * (dpsi + (dphi - temp1)); } zz[2] = z__[iip1] * z__[iip1]; } zz[1] = z__[ii] * z__[ii]; dd[0] = dtiim; dd[1] = delta[ii] * work[ii]; dd[2] = dtiip; slaed6_(&niter, &orgati, &c__, dd, zz, &w, &eta, info); if (*info != 0) { /* If INFO is not 0, i.e., SLAED6 failed, switch back */ /* to 2 pole interpolation. */ swtch3 = FALSE_; *info = 0; dtipsq = work[ip1] * delta[ip1]; dtisq = work[*i__] * delta[*i__]; if (orgati) { /* Computing 2nd power */ r__1 = z__[*i__] / dtisq; c__ = w - dtipsq * dw + delsq * (r__1 * r__1); } else { /* Computing 2nd power */ r__1 = z__[ip1] / dtipsq; c__ = w - dtisq * dw - delsq * (r__1 * r__1); } a = (dtipsq + dtisq) * w - dtipsq * dtisq * dw; b = dtipsq * dtisq * w; if (c__ == 0.f) { if (a == 0.f) { if (orgati) { a = z__[*i__] * z__[*i__] + dtipsq * dtipsq * ( dpsi + dphi); } else { a = z__[ip1] * z__[ip1] + dtisq * dtisq * (dpsi + dphi); } } eta = b / a; } else if (a <= 0.f) { eta = (a - sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1))) ) / (c__ * 2.f); } else { eta = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))); } } } /* Note, eta should be positive if w is negative, and */ /* eta should be negative otherwise. However, */ /* if for some reason caused by roundoff, eta*w > 0, */ /* we simply use one Newton step instead. This way */ /* will guarantee eta*w < 0. */ if (w * eta >= 0.f) { eta = -w / dw; } eta /= *sigma + sqrt(*sigma * *sigma + eta); temp = tau + eta; if (temp > sgub || temp < sglb) { if (w < 0.f) { eta = (sgub - tau) / 2.f; } else { eta = (sglb - tau) / 2.f; } if (geomavg) { if (w < 0.f) { if (tau > 0.f) { eta = sqrt(sgub * tau) - tau; } } else { if (sglb > 0.f) { eta = sqrt(sglb * tau) - tau; } } } } prew = w; tau += eta; *sigma += eta; i__1 = *n; for (j = 1; j <= i__1; ++j) { work[j] += eta; delta[j] -= eta; /* L170: */ } /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = iim1; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (work[j] * delta[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L180: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ dphi = 0.f; phi = 0.f; i__1 = iip1; for (j = *n; j >= i__1; --j) { temp = z__[j] / (work[j] * delta[j]); phi += z__[j] * temp; dphi += temp * temp; erretm += phi; /* L190: */ } tau2 = work[ii] * delta[ii]; temp = z__[ii] / tau2; dw = dpsi + dphi + temp * temp; temp = z__[ii] * temp; w = rhoinv + phi + psi + temp; erretm = (phi - psi) * 8.f + erretm + rhoinv * 2.f + abs(temp) * 3.f; /* $ + ABS( TAU2 )*DW */ swtch = FALSE_; if (orgati) { if (-w > abs(prew) / 10.f) { swtch = TRUE_; } } else { if (w > abs(prew) / 10.f) { swtch = TRUE_; } } /* Main loop to update the values of the array DELTA and WORK */ iter = niter + 1; for (niter = iter; niter <= 400; ++niter) { /* Test for convergence */ if (abs(w) <= eps * erretm) { /* $ .OR. (SGUB-SGLB).LE.EIGHT*ABS(SGUB+SGLB) ) THEN */ goto L240; } if (w <= 0.f) { sglb = f2cmax(sglb,tau); } else { sgub = f2cmin(sgub,tau); } /* Calculate the new step */ if (! swtch3) { dtipsq = work[ip1] * delta[ip1]; dtisq = work[*i__] * delta[*i__]; if (! swtch) { if (orgati) { /* Computing 2nd power */ r__1 = z__[*i__] / dtisq; c__ = w - dtipsq * dw + delsq * (r__1 * r__1); } else { /* Computing 2nd power */ r__1 = z__[ip1] / dtipsq; c__ = w - dtisq * dw - delsq * (r__1 * r__1); } } else { temp = z__[ii] / (work[ii] * delta[ii]); if (orgati) { dpsi += temp * temp; } else { dphi += temp * temp; } c__ = w - dtisq * dpsi - dtipsq * dphi; } a = (dtipsq + dtisq) * w - dtipsq * dtisq * dw; b = dtipsq * dtisq * w; if (c__ == 0.f) { if (a == 0.f) { if (! swtch) { if (orgati) { a = z__[*i__] * z__[*i__] + dtipsq * dtipsq * (dpsi + dphi); } else { a = z__[ip1] * z__[ip1] + dtisq * dtisq * ( dpsi + dphi); } } else { a = dtisq * dtisq * dpsi + dtipsq * dtipsq * dphi; } } eta = b / a; } else if (a <= 0.f) { eta = (a - sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1))) ) / (c__ * 2.f); } else { eta = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))); } } else { /* Interpolation using THREE most relevant poles */ dtiim = work[iim1] * delta[iim1]; dtiip = work[iip1] * delta[iip1]; temp = rhoinv + psi + phi; if (swtch) { c__ = temp - dtiim * dpsi - dtiip * dphi; zz[0] = dtiim * dtiim * dpsi; zz[2] = dtiip * dtiip * dphi; } else { if (orgati) { temp1 = z__[iim1] / dtiim; temp1 *= temp1; temp2 = (d__[iim1] - d__[iip1]) * (d__[iim1] + d__[ iip1]) * temp1; c__ = temp - dtiip * (dpsi + dphi) - temp2; zz[0] = z__[iim1] * z__[iim1]; if (dpsi < temp1) { zz[2] = dtiip * dtiip * dphi; } else { zz[2] = dtiip * dtiip * (dpsi - temp1 + dphi); } } else { temp1 = z__[iip1] / dtiip; temp1 *= temp1; temp2 = (d__[iip1] - d__[iim1]) * (d__[iim1] + d__[ iip1]) * temp1; c__ = temp - dtiim * (dpsi + dphi) - temp2; if (dphi < temp1) { zz[0] = dtiim * dtiim * dpsi; } else { zz[0] = dtiim * dtiim * (dpsi + (dphi - temp1)); } zz[2] = z__[iip1] * z__[iip1]; } } dd[0] = dtiim; dd[1] = delta[ii] * work[ii]; dd[2] = dtiip; slaed6_(&niter, &orgati, &c__, dd, zz, &w, &eta, info); if (*info != 0) { /* If INFO is not 0, i.e., SLAED6 failed, switch */ /* back to two pole interpolation */ swtch3 = FALSE_; *info = 0; dtipsq = work[ip1] * delta[ip1]; dtisq = work[*i__] * delta[*i__]; if (! swtch) { if (orgati) { /* Computing 2nd power */ r__1 = z__[*i__] / dtisq; c__ = w - dtipsq * dw + delsq * (r__1 * r__1); } else { /* Computing 2nd power */ r__1 = z__[ip1] / dtipsq; c__ = w - dtisq * dw - delsq * (r__1 * r__1); } } else { temp = z__[ii] / (work[ii] * delta[ii]); if (orgati) { dpsi += temp * temp; } else { dphi += temp * temp; } c__ = w - dtisq * dpsi - dtipsq * dphi; } a = (dtipsq + dtisq) * w - dtipsq * dtisq * dw; b = dtipsq * dtisq * w; if (c__ == 0.f) { if (a == 0.f) { if (! swtch) { if (orgati) { a = z__[*i__] * z__[*i__] + dtipsq * dtipsq * (dpsi + dphi); } else { a = z__[ip1] * z__[ip1] + dtisq * dtisq * (dpsi + dphi); } } else { a = dtisq * dtisq * dpsi + dtipsq * dtipsq * dphi; } } eta = b / a; } else if (a <= 0.f) { eta = (a - sqrt((r__1 = a * a - b * 4.f * c__, abs( r__1)))) / (c__ * 2.f); } else { eta = b * 2.f / (a + sqrt((r__1 = a * a - b * 4.f * c__, abs(r__1)))); } } } /* Note, eta should be positive if w is negative, and */ /* eta should be negative otherwise. However, */ /* if for some reason caused by roundoff, eta*w > 0, */ /* we simply use one Newton step instead. This way */ /* will guarantee eta*w < 0. */ if (w * eta >= 0.f) { eta = -w / dw; } eta /= *sigma + sqrt(*sigma * *sigma + eta); temp = tau + eta; if (temp > sgub || temp < sglb) { if (w < 0.f) { eta = (sgub - tau) / 2.f; } else { eta = (sglb - tau) / 2.f; } if (geomavg) { if (w < 0.f) { if (tau > 0.f) { eta = sqrt(sgub * tau) - tau; } } else { if (sglb > 0.f) { eta = sqrt(sglb * tau) - tau; } } } } prew = w; tau += eta; *sigma += eta; i__1 = *n; for (j = 1; j <= i__1; ++j) { work[j] += eta; delta[j] -= eta; /* L200: */ } /* Evaluate PSI and the derivative DPSI */ dpsi = 0.f; psi = 0.f; erretm = 0.f; i__1 = iim1; for (j = 1; j <= i__1; ++j) { temp = z__[j] / (work[j] * delta[j]); psi += z__[j] * temp; dpsi += temp * temp; erretm += psi; /* L210: */ } erretm = abs(erretm); /* Evaluate PHI and the derivative DPHI */ dphi = 0.f; phi = 0.f; i__1 = iip1; for (j = *n; j >= i__1; --j) { temp = z__[j] / (work[j] * delta[j]); phi += z__[j] * temp; dphi += temp * temp; erretm += phi; /* L220: */ } tau2 = work[ii] * delta[ii]; temp = z__[ii] / tau2; dw = dpsi + dphi + temp * temp; temp = z__[ii] * temp; w = rhoinv + phi + psi + temp; erretm = (phi - psi) * 8.f + erretm + rhoinv * 2.f + abs(temp) * 3.f; /* $ + ABS( TAU2 )*DW */ if (w * prew > 0.f && abs(w) > abs(prew) / 10.f) { swtch = ! swtch; } /* L230: */ } /* Return with INFO = 1, NITER = MAXIT and not converged */ *info = 1; } L240: return 0; /* End of SLASD4 */ } /* slasd4_ */