14 typedef long long BLASLONG;
15 typedef unsigned long long BLASULONG;
17 typedef long BLASLONG;
18 typedef unsigned long BLASULONG;
22 typedef BLASLONG blasint;
24 #define blasabs(x) llabs(x)
26 #define blasabs(x) labs(x)
30 #define blasabs(x) abs(x)
33 typedef blasint integer;
35 typedef unsigned int uinteger;
36 typedef char *address;
37 typedef short int shortint;
39 typedef double doublereal;
40 typedef struct { real r, i; } complex;
41 typedef struct { doublereal r, i; } doublecomplex;
43 static inline _Fcomplex Cf(complex *z) {_Fcomplex zz={z->r , z->i}; return zz;}
44 static inline _Dcomplex Cd(doublecomplex *z) {_Dcomplex zz={z->r , z->i};return zz;}
45 static inline _Fcomplex * _pCf(complex *z) {return (_Fcomplex*)z;}
46 static inline _Dcomplex * _pCd(doublecomplex *z) {return (_Dcomplex*)z;}
48 static inline _Complex float Cf(complex *z) {return z->r + z->i*_Complex_I;}
49 static inline _Complex double Cd(doublecomplex *z) {return z->r + z->i*_Complex_I;}
50 static inline _Complex float * _pCf(complex *z) {return (_Complex float*)z;}
51 static inline _Complex double * _pCd(doublecomplex *z) {return (_Complex double*)z;}
53 #define pCf(z) (*_pCf(z))
54 #define pCd(z) (*_pCd(z))
56 typedef short int shortlogical;
57 typedef char logical1;
58 typedef char integer1;
63 /* Extern is for use with -E */
74 /*external read, write*/
83 /*internal read, write*/
113 /*rewind, backspace, endfile*/
125 ftnint *inex; /*parameters in standard's order*/
151 union Multitype { /* for multiple entry points */
162 typedef union Multitype Multitype;
164 struct Vardesc { /* for Namelist */
170 typedef struct Vardesc Vardesc;
177 typedef struct Namelist Namelist;
179 #define abs(x) ((x) >= 0 ? (x) : -(x))
180 #define dabs(x) (fabs(x))
181 #define f2cmin(a,b) ((a) <= (b) ? (a) : (b))
182 #define f2cmax(a,b) ((a) >= (b) ? (a) : (b))
183 #define dmin(a,b) (f2cmin(a,b))
184 #define dmax(a,b) (f2cmax(a,b))
185 #define bit_test(a,b) ((a) >> (b) & 1)
186 #define bit_clear(a,b) ((a) & ~((uinteger)1 << (b)))
187 #define bit_set(a,b) ((a) | ((uinteger)1 << (b)))
189 #define abort_() { sig_die("Fortran abort routine called", 1); }
190 #define c_abs(z) (cabsf(Cf(z)))
191 #define c_cos(R,Z) { pCf(R)=ccos(Cf(Z)); }
193 #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]);}
194 #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]);}
196 #define c_div(c, a, b) {pCf(c) = Cf(a)/Cf(b);}
197 #define z_div(c, a, b) {pCd(c) = Cd(a)/Cd(b);}
199 #define c_exp(R, Z) {pCf(R) = cexpf(Cf(Z));}
200 #define c_log(R, Z) {pCf(R) = clogf(Cf(Z));}
201 #define c_sin(R, Z) {pCf(R) = csinf(Cf(Z));}
202 //#define c_sqrt(R, Z) {*(R) = csqrtf(Cf(Z));}
203 #define c_sqrt(R, Z) {pCf(R) = csqrtf(Cf(Z));}
204 #define d_abs(x) (fabs(*(x)))
205 #define d_acos(x) (acos(*(x)))
206 #define d_asin(x) (asin(*(x)))
207 #define d_atan(x) (atan(*(x)))
208 #define d_atn2(x, y) (atan2(*(x),*(y)))
209 #define d_cnjg(R, Z) { pCd(R) = conj(Cd(Z)); }
210 #define r_cnjg(R, Z) { pCf(R) = conjf(Cf(Z)); }
211 #define d_cos(x) (cos(*(x)))
212 #define d_cosh(x) (cosh(*(x)))
213 #define d_dim(__a, __b) ( *(__a) > *(__b) ? *(__a) - *(__b) : 0.0 )
214 #define d_exp(x) (exp(*(x)))
215 #define d_imag(z) (cimag(Cd(z)))
216 #define r_imag(z) (cimagf(Cf(z)))
217 #define d_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
218 #define r_int(__x) (*(__x)>0 ? floor(*(__x)) : -floor(- *(__x)))
219 #define d_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
220 #define r_lg10(x) ( 0.43429448190325182765 * log(*(x)) )
221 #define d_log(x) (log(*(x)))
222 #define d_mod(x, y) (fmod(*(x), *(y)))
223 #define u_nint(__x) ((__x)>=0 ? floor((__x) + .5) : -floor(.5 - (__x)))
224 #define d_nint(x) u_nint(*(x))
225 #define u_sign(__a,__b) ((__b) >= 0 ? ((__a) >= 0 ? (__a) : -(__a)) : -((__a) >= 0 ? (__a) : -(__a)))
226 #define d_sign(a,b) u_sign(*(a),*(b))
227 #define r_sign(a,b) u_sign(*(a),*(b))
228 #define d_sin(x) (sin(*(x)))
229 #define d_sinh(x) (sinh(*(x)))
230 #define d_sqrt(x) (sqrt(*(x)))
231 #define d_tan(x) (tan(*(x)))
232 #define d_tanh(x) (tanh(*(x)))
233 #define i_abs(x) abs(*(x))
234 #define i_dnnt(x) ((integer)u_nint(*(x)))
235 #define i_len(s, n) (n)
236 #define i_nint(x) ((integer)u_nint(*(x)))
237 #define i_sign(a,b) ((integer)u_sign((integer)*(a),(integer)*(b)))
238 #define pow_dd(ap, bp) ( pow(*(ap), *(bp)))
239 #define pow_si(B,E) spow_ui(*(B),*(E))
240 #define pow_ri(B,E) spow_ui(*(B),*(E))
241 #define pow_di(B,E) dpow_ui(*(B),*(E))
242 #define pow_zi(p, a, b) {pCd(p) = zpow_ui(Cd(a), *(b));}
243 #define pow_ci(p, a, b) {pCf(p) = cpow_ui(Cf(a), *(b));}
244 #define pow_zz(R,A,B) {pCd(R) = cpow(Cd(A),*(B));}
245 #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++ = ' '; }
246 #define s_cmp(a,b,c,d) ((integer)strncmp((a),(b),f2cmin((c),(d))))
247 #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]; }
248 #define sig_die(s, kill) { exit(1); }
249 #define s_stop(s, n) {exit(0);}
250 static char junk[] = "\n@(#)LIBF77 VERSION 19990503\n";
251 #define z_abs(z) (cabs(Cd(z)))
252 #define z_exp(R, Z) {pCd(R) = cexp(Cd(Z));}
253 #define z_sqrt(R, Z) {pCd(R) = csqrt(Cd(Z));}
254 #define myexit_() break;
255 #define mycycle() continue;
256 #define myceiling(w) {ceil(w)}
257 #define myhuge(w) {HUGE_VAL}
258 //#define mymaxloc_(w,s,e,n) {if (sizeof(*(w)) == sizeof(double)) dmaxloc_((w),*(s),*(e),n); else dmaxloc_((w),*(s),*(e),n);}
259 #define mymaxloc(w,s,e,n) {dmaxloc_(w,*(s),*(e),n)}
261 /* procedure parameter types for -A and -C++ */
263 #define F2C_proc_par_types 1
265 typedef logical (*L_fp)(...);
267 typedef logical (*L_fp)();
270 static float spow_ui(float x, integer n) {
271 float pow=1.0; unsigned long int u;
273 if(n < 0) n = -n, x = 1/x;
282 static double dpow_ui(double x, integer n) {
283 double pow=1.0; unsigned long int u;
285 if(n < 0) n = -n, x = 1/x;
295 static _Fcomplex cpow_ui(complex x, integer n) {
296 complex pow={1.0,0.0}; unsigned long int u;
298 if(n < 0) n = -n, x.r = 1/x.r, x.i=1/x.i;
300 if(u & 01) pow.r *= x.r, pow.i *= x.i;
301 if(u >>= 1) x.r *= x.r, x.i *= x.i;
305 _Fcomplex p={pow.r, pow.i};
309 static _Complex float cpow_ui(_Complex float x, integer n) {
310 _Complex float pow=1.0; unsigned long int u;
312 if(n < 0) n = -n, x = 1/x;
323 static _Dcomplex zpow_ui(_Dcomplex x, integer n) {
324 _Dcomplex pow={1.0,0.0}; unsigned long int u;
326 if(n < 0) n = -n, x._Val[0] = 1/x._Val[0], x._Val[1] =1/x._Val[1];
328 if(u & 01) pow._Val[0] *= x._Val[0], pow._Val[1] *= x._Val[1];
329 if(u >>= 1) x._Val[0] *= x._Val[0], x._Val[1] *= x._Val[1];
333 _Dcomplex p = {pow._Val[0], pow._Val[1]};
337 static _Complex double zpow_ui(_Complex double x, integer n) {
338 _Complex double pow=1.0; unsigned long int u;
340 if(n < 0) n = -n, x = 1/x;
350 static integer pow_ii(integer x, integer n) {
351 integer pow; unsigned long int u;
353 if (n == 0 || x == 1) pow = 1;
354 else if (x != -1) pow = x == 0 ? 1/x : 0;
357 if ((n > 0) || !(n == 0 || x == 1 || x != -1)) {
367 static integer dmaxloc_(double *w, integer s, integer e, integer *n)
369 double m; integer i, mi;
370 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
371 if (w[i-1]>m) mi=i ,m=w[i-1];
374 static integer smaxloc_(float *w, integer s, integer e, integer *n)
376 float m; integer i, mi;
377 for(m=w[s-1], mi=s, i=s+1; i<=e; i++)
378 if (w[i-1]>m) mi=i ,m=w[i-1];
381 static inline void cdotc_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
382 integer n = *n_, incx = *incx_, incy = *incy_, i;
384 _Fcomplex zdotc = {0.0, 0.0};
385 if (incx == 1 && incy == 1) {
386 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
387 zdotc._Val[0] += conjf(Cf(&x[i]))._Val[0] * Cf(&y[i])._Val[0];
388 zdotc._Val[1] += conjf(Cf(&x[i]))._Val[1] * Cf(&y[i])._Val[1];
391 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
392 zdotc._Val[0] += conjf(Cf(&x[i*incx]))._Val[0] * Cf(&y[i*incy])._Val[0];
393 zdotc._Val[1] += conjf(Cf(&x[i*incx]))._Val[1] * Cf(&y[i*incy])._Val[1];
399 _Complex float zdotc = 0.0;
400 if (incx == 1 && incy == 1) {
401 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
402 zdotc += conjf(Cf(&x[i])) * Cf(&y[i]);
405 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
406 zdotc += conjf(Cf(&x[i*incx])) * Cf(&y[i*incy]);
412 static inline void zdotc_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
413 integer n = *n_, incx = *incx_, incy = *incy_, i;
415 _Dcomplex zdotc = {0.0, 0.0};
416 if (incx == 1 && incy == 1) {
417 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
418 zdotc._Val[0] += conj(Cd(&x[i]))._Val[0] * Cd(&y[i])._Val[0];
419 zdotc._Val[1] += conj(Cd(&x[i]))._Val[1] * Cd(&y[i])._Val[1];
422 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
423 zdotc._Val[0] += conj(Cd(&x[i*incx]))._Val[0] * Cd(&y[i*incy])._Val[0];
424 zdotc._Val[1] += conj(Cd(&x[i*incx]))._Val[1] * Cd(&y[i*incy])._Val[1];
430 _Complex double zdotc = 0.0;
431 if (incx == 1 && incy == 1) {
432 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
433 zdotc += conj(Cd(&x[i])) * Cd(&y[i]);
436 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
437 zdotc += conj(Cd(&x[i*incx])) * Cd(&y[i*incy]);
443 static inline void cdotu_(complex *z, integer *n_, complex *x, integer *incx_, complex *y, integer *incy_) {
444 integer n = *n_, incx = *incx_, incy = *incy_, i;
446 _Fcomplex zdotc = {0.0, 0.0};
447 if (incx == 1 && incy == 1) {
448 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
449 zdotc._Val[0] += Cf(&x[i])._Val[0] * Cf(&y[i])._Val[0];
450 zdotc._Val[1] += Cf(&x[i])._Val[1] * Cf(&y[i])._Val[1];
453 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
454 zdotc._Val[0] += Cf(&x[i*incx])._Val[0] * Cf(&y[i*incy])._Val[0];
455 zdotc._Val[1] += Cf(&x[i*incx])._Val[1] * Cf(&y[i*incy])._Val[1];
461 _Complex float zdotc = 0.0;
462 if (incx == 1 && incy == 1) {
463 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
464 zdotc += Cf(&x[i]) * Cf(&y[i]);
467 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
468 zdotc += Cf(&x[i*incx]) * Cf(&y[i*incy]);
474 static inline void zdotu_(doublecomplex *z, integer *n_, doublecomplex *x, integer *incx_, doublecomplex *y, integer *incy_) {
475 integer n = *n_, incx = *incx_, incy = *incy_, i;
477 _Dcomplex zdotc = {0.0, 0.0};
478 if (incx == 1 && incy == 1) {
479 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
480 zdotc._Val[0] += Cd(&x[i])._Val[0] * Cd(&y[i])._Val[0];
481 zdotc._Val[1] += Cd(&x[i])._Val[1] * Cd(&y[i])._Val[1];
484 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
485 zdotc._Val[0] += Cd(&x[i*incx])._Val[0] * Cd(&y[i*incy])._Val[0];
486 zdotc._Val[1] += Cd(&x[i*incx])._Val[1] * Cd(&y[i*incy])._Val[1];
492 _Complex double zdotc = 0.0;
493 if (incx == 1 && incy == 1) {
494 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
495 zdotc += Cd(&x[i]) * Cd(&y[i]);
498 for (i=0;i<n;i++) { /* zdotc = zdotc + dconjg(x(i))* y(i) */
499 zdotc += Cd(&x[i*incx]) * Cd(&y[i*incy]);
505 /* -- translated by f2c (version 20000121).
506 You must link the resulting object file with the libraries:
507 -lf2c -lm (in that order)
513 /* > \brief \b DLAED4 used by sstedc. Finds a single root of the secular equation. */
515 /* =========== DOCUMENTATION =========== */
517 /* Online html documentation available at */
518 /* http://www.netlib.org/lapack/explore-html/ */
521 /* > Download DLAED4 + dependencies */
522 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaed4.
525 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaed4.
528 /* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaed4.
536 /* SUBROUTINE DLAED4( N, I, D, Z, DELTA, RHO, DLAM, INFO ) */
538 /* INTEGER I, INFO, N */
539 /* DOUBLE PRECISION DLAM, RHO */
540 /* DOUBLE PRECISION D( * ), DELTA( * ), Z( * ) */
543 /* > \par Purpose: */
548 /* > This subroutine computes the I-th updated eigenvalue of a symmetric */
549 /* > rank-one modification to a diagonal matrix whose elements are */
550 /* > given in the array d, and that */
552 /* > D(i) < D(j) for i < j */
554 /* > and that RHO > 0. This is arranged by the calling routine, and is */
555 /* > no loss in generality. The rank-one modified system is thus */
557 /* > diag( D ) + RHO * Z * Z_transpose. */
559 /* > where we assume the Euclidean norm of Z is 1. */
561 /* > The method consists of approximating the rational functions in the */
562 /* > secular equation by simpler interpolating rational functions. */
571 /* > The length of all arrays. */
577 /* > The index of the eigenvalue to be computed. 1 <= I <= N. */
582 /* > D is DOUBLE PRECISION array, dimension (N) */
583 /* > The original eigenvalues. It is assumed that they are in */
584 /* > order, D(I) < D(J) for I < J. */
589 /* > Z is DOUBLE PRECISION array, dimension (N) */
590 /* > The components of the updating vector. */
593 /* > \param[out] DELTA */
595 /* > DELTA is DOUBLE PRECISION array, dimension (N) */
596 /* > If N > 2, DELTA contains (D(j) - lambda_I) in its j-th */
597 /* > component. If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5 */
598 /* > for detail. The vector DELTA contains the information necessary */
599 /* > to construct the eigenvectors by DLAED3 and DLAED9. */
602 /* > \param[in] RHO */
604 /* > RHO is DOUBLE PRECISION */
605 /* > The scalar in the symmetric updating formula. */
608 /* > \param[out] DLAM */
610 /* > DLAM is DOUBLE PRECISION */
611 /* > The computed lambda_I, the I-th updated eigenvalue. */
614 /* > \param[out] INFO */
616 /* > INFO is INTEGER */
617 /* > = 0: successful exit */
618 /* > > 0: if INFO = 1, the updating process failed. */
621 /* > \par Internal Parameters: */
622 /* ========================= */
625 /* > Logical variable ORGATI (origin-at-i?) is used for distinguishing */
626 /* > whether D(i) or D(i+1) is treated as the origin. */
628 /* > ORGATI = .true. origin at i */
629 /* > ORGATI = .false. origin at i+1 */
631 /* > Logical variable SWTCH3 (switch-for-3-poles?) is for noting */
632 /* > if we are working with THREE poles! */
634 /* > MAXIT is the maximum number of iterations allowed for each */
641 /* > \author Univ. of Tennessee */
642 /* > \author Univ. of California Berkeley */
643 /* > \author Univ. of Colorado Denver */
644 /* > \author NAG Ltd. */
646 /* > \date December 2016 */
648 /* > \ingroup auxOTHERcomputational */
650 /* > \par Contributors: */
651 /* ================== */
653 /* > Ren-Cang Li, Computer Science Division, University of California */
654 /* > at Berkeley, USA */
656 /* ===================================================================== */
657 /* Subroutine */ int dlaed4_(integer *n, integer *i__, doublereal *d__,
658 doublereal *z__, doublereal *delta, doublereal *rho, doublereal *dlam,
661 /* System generated locals */
665 /* Local variables */
666 doublereal dphi, dpsi;
668 doublereal temp, prew, temp1, a, b, c__;
670 doublereal w, dltlb, dltub, midpt;
673 extern /* Subroutine */ int dlaed5_(integer *, doublereal *, doublereal *,
674 doublereal *, doublereal *, doublereal *), dlaed6_(integer *,
675 logical *, doublereal *, doublereal *, doublereal *, doublereal *,
676 doublereal *, integer *);
679 extern doublereal dlamch_(char *);
680 doublereal dw, zz[3];
682 doublereal erretm, rhoinv;
684 doublereal del, eta, phi, eps, tau, psi;
688 /* -- LAPACK computational routine (version 3.7.0) -- */
689 /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
690 /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
694 /* ===================================================================== */
697 /* Since this routine is called in an inner loop, we do no argument */
700 /* Quick return for N=1 and 2. */
702 /* Parameter adjustments */
711 /* Presumably, I=1 upon entry */
713 *dlam = d__[1] + *rho * z__[1] * z__[1];
718 dlaed5_(i__, &d__[1], &z__[1], &delta[1], rho, dlam);
722 /* Compute machine epsilon */
724 eps = dlamch_("Epsilon");
731 /* Initialize some basic variables */
736 /* Calculate initial guess */
740 /* If ||Z||_2 is not one, then TEMP should be set to */
741 /* RHO * ||Z||_2^2 / TWO */
744 for (j = 1; j <= i__1; ++j) {
745 delta[j] = d__[j] - d__[*i__] - midpt;
751 for (j = 1; j <= i__1; ++j) {
752 psi += z__[j] * z__[j] / delta[j];
757 w = c__ + z__[ii] * z__[ii] / delta[ii] + z__[*n] * z__[*n] / delta[*
761 temp = z__[*n - 1] * z__[*n - 1] / (d__[*n] - d__[*n - 1] + *rho)
762 + z__[*n] * z__[*n] / *rho;
766 del = d__[*n] - d__[*n - 1];
767 a = -c__ * del + z__[*n - 1] * z__[*n - 1] + z__[*n] * z__[*n]
769 b = z__[*n] * z__[*n] * del;
771 tau = b * 2. / (sqrt(a * a + b * 4. * c__) - a);
773 tau = (a + sqrt(a * a + b * 4. * c__)) / (c__ * 2.);
777 /* It can be proved that */
778 /* D(N)+RHO/2 <= LAMBDA(N) < D(N)+TAU <= D(N)+RHO */
783 del = d__[*n] - d__[*n - 1];
784 a = -c__ * del + z__[*n - 1] * z__[*n - 1] + z__[*n] * z__[*n];
785 b = z__[*n] * z__[*n] * del;
787 tau = b * 2. / (sqrt(a * a + b * 4. * c__) - a);
789 tau = (a + sqrt(a * a + b * 4. * c__)) / (c__ * 2.);
792 /* It can be proved that */
793 /* D(N) < D(N)+TAU < LAMBDA(N) < D(N)+RHO/2 */
800 for (j = 1; j <= i__1; ++j) {
801 delta[j] = d__[j] - d__[*i__] - tau;
805 /* Evaluate PSI and the derivative DPSI */
811 for (j = 1; j <= i__1; ++j) {
812 temp = z__[j] / delta[j];
813 psi += z__[j] * temp;
818 erretm = abs(erretm);
820 /* Evaluate PHI and the derivative DPHI */
822 temp = z__[*n] / delta[*n];
823 phi = z__[*n] * temp;
825 erretm = (-phi - psi) * 8. + erretm - phi + rhoinv + abs(tau) * (dpsi
828 w = rhoinv + phi + psi;
830 /* Test for convergence */
832 if (abs(w) <= eps * erretm) {
833 *dlam = d__[*i__] + tau;
838 dltlb = f2cmax(dltlb,tau);
840 dltub = f2cmin(dltub,tau);
843 /* Calculate the new step */
846 c__ = w - delta[*n - 1] * dpsi - delta[*n] * dphi;
847 a = (delta[*n - 1] + delta[*n]) * w - delta[*n - 1] * delta[*n] * (
849 b = delta[*n - 1] * delta[*n] * w;
855 /* ETA = RHO - TAU */
857 } else if (a >= 0.) {
858 eta = (a + sqrt((d__1 = a * a - b * 4. * c__, abs(d__1)))) / (c__
861 eta = b * 2. / (a - sqrt((d__1 = a * a - b * 4. * c__, abs(d__1)))
865 /* Note, eta should be positive if w is negative, and */
866 /* eta should be negative otherwise. However, */
867 /* if for some reason caused by roundoff, eta*w > 0, */
868 /* we simply use one Newton step instead. This way */
869 /* will guarantee eta*w < 0. */
872 eta = -w / (dpsi + dphi);
875 if (temp > dltub || temp < dltlb) {
877 eta = (dltub - tau) / 2.;
879 eta = (dltlb - tau) / 2.;
883 for (j = 1; j <= i__1; ++j) {
890 /* Evaluate PSI and the derivative DPSI */
896 for (j = 1; j <= i__1; ++j) {
897 temp = z__[j] / delta[j];
898 psi += z__[j] * temp;
903 erretm = abs(erretm);
905 /* Evaluate PHI and the derivative DPHI */
907 temp = z__[*n] / delta[*n];
908 phi = z__[*n] * temp;
910 erretm = (-phi - psi) * 8. + erretm - phi + rhoinv + abs(tau) * (dpsi
913 w = rhoinv + phi + psi;
915 /* Main loop to update the values of the array DELTA */
919 for (niter = iter; niter <= 30; ++niter) {
921 /* Test for convergence */
923 if (abs(w) <= eps * erretm) {
924 *dlam = d__[*i__] + tau;
929 dltlb = f2cmax(dltlb,tau);
931 dltub = f2cmin(dltub,tau);
934 /* Calculate the new step */
936 c__ = w - delta[*n - 1] * dpsi - delta[*n] * dphi;
937 a = (delta[*n - 1] + delta[*n]) * w - delta[*n - 1] * delta[*n] *
939 b = delta[*n - 1] * delta[*n] * w;
941 eta = (a + sqrt((d__1 = a * a - b * 4. * c__, abs(d__1)))) / (
944 eta = b * 2. / (a - sqrt((d__1 = a * a - b * 4. * c__, abs(
948 /* Note, eta should be positive if w is negative, and */
949 /* eta should be negative otherwise. However, */
950 /* if for some reason caused by roundoff, eta*w > 0, */
951 /* we simply use one Newton step instead. This way */
952 /* will guarantee eta*w < 0. */
955 eta = -w / (dpsi + dphi);
958 if (temp > dltub || temp < dltlb) {
960 eta = (dltub - tau) / 2.;
962 eta = (dltlb - tau) / 2.;
966 for (j = 1; j <= i__1; ++j) {
973 /* Evaluate PSI and the derivative DPSI */
979 for (j = 1; j <= i__1; ++j) {
980 temp = z__[j] / delta[j];
981 psi += z__[j] * temp;
986 erretm = abs(erretm);
988 /* Evaluate PHI and the derivative DPHI */
990 temp = z__[*n] / delta[*n];
991 phi = z__[*n] * temp;
993 erretm = (-phi - psi) * 8. + erretm - phi + rhoinv + abs(tau) * (
996 w = rhoinv + phi + psi;
1000 /* Return with INFO = 1, NITER = MAXIT and not converged */
1003 *dlam = d__[*i__] + tau;
1006 /* End for the case I = N */
1010 /* The case for I < N */
1015 /* Calculate initial guess */
1017 del = d__[ip1] - d__[*i__];
1020 for (j = 1; j <= i__1; ++j) {
1021 delta[j] = d__[j] - d__[*i__] - midpt;
1027 for (j = 1; j <= i__1; ++j) {
1028 psi += z__[j] * z__[j] / delta[j];
1034 for (j = *n; j >= i__1; --j) {
1035 phi += z__[j] * z__[j] / delta[j];
1038 c__ = rhoinv + psi + phi;
1039 w = c__ + z__[*i__] * z__[*i__] / delta[*i__] + z__[ip1] * z__[ip1] /
1044 /* d(i)< the ith eigenvalue < (d(i)+d(i+1))/2 */
1046 /* We choose d(i) as origin. */
1049 a = c__ * del + z__[*i__] * z__[*i__] + z__[ip1] * z__[ip1];
1050 b = z__[*i__] * z__[*i__] * del;
1052 tau = b * 2. / (a + sqrt((d__1 = a * a - b * 4. * c__, abs(
1055 tau = (a - sqrt((d__1 = a * a - b * 4. * c__, abs(d__1)))) / (
1062 /* (d(i)+d(i+1))/2 <= the ith eigenvalue < d(i+1) */
1064 /* We choose d(i+1) as origin. */
1067 a = c__ * del - z__[*i__] * z__[*i__] - z__[ip1] * z__[ip1];
1068 b = z__[ip1] * z__[ip1] * del;
1070 tau = b * 2. / (a - sqrt((d__1 = a * a + b * 4. * c__, abs(
1073 tau = -(a + sqrt((d__1 = a * a + b * 4. * c__, abs(d__1)))) /
1082 for (j = 1; j <= i__1; ++j) {
1083 delta[j] = d__[j] - d__[*i__] - tau;
1088 for (j = 1; j <= i__1; ++j) {
1089 delta[j] = d__[j] - d__[ip1] - tau;
1101 /* Evaluate PSI and the derivative DPSI */
1107 for (j = 1; j <= i__1; ++j) {
1108 temp = z__[j] / delta[j];
1109 psi += z__[j] * temp;
1110 dpsi += temp * temp;
1114 erretm = abs(erretm);
1116 /* Evaluate PHI and the derivative DPHI */
1121 for (j = *n; j >= i__1; --j) {
1122 temp = z__[j] / delta[j];
1123 phi += z__[j] * temp;
1124 dphi += temp * temp;
1129 w = rhoinv + phi + psi;
1131 /* W is the value of the secular function with */
1132 /* its ii-th element removed. */
1144 if (ii == 1 || ii == *n) {
1148 temp = z__[ii] / delta[ii];
1149 dw = dpsi + dphi + temp * temp;
1150 temp = z__[ii] * temp;
1152 erretm = (phi - psi) * 8. + erretm + rhoinv * 2. + abs(temp) * 3. +
1155 /* Test for convergence */
1157 if (abs(w) <= eps * erretm) {
1159 *dlam = d__[*i__] + tau;
1161 *dlam = d__[ip1] + tau;
1167 dltlb = f2cmax(dltlb,tau);
1169 dltub = f2cmin(dltub,tau);
1172 /* Calculate the new step */
1177 /* Computing 2nd power */
1178 d__1 = z__[*i__] / delta[*i__];
1179 c__ = w - delta[ip1] * dw - (d__[*i__] - d__[ip1]) * (d__1 *
1182 /* Computing 2nd power */
1183 d__1 = z__[ip1] / delta[ip1];
1184 c__ = w - delta[*i__] * dw - (d__[ip1] - d__[*i__]) * (d__1 *
1187 a = (delta[*i__] + delta[ip1]) * w - delta[*i__] * delta[ip1] *
1189 b = delta[*i__] * delta[ip1] * w;
1193 a = z__[*i__] * z__[*i__] + delta[ip1] * delta[ip1] *
1196 a = z__[ip1] * z__[ip1] + delta[*i__] * delta[*i__] *
1201 } else if (a <= 0.) {
1202 eta = (a - sqrt((d__1 = a * a - b * 4. * c__, abs(d__1)))) / (
1205 eta = b * 2. / (a + sqrt((d__1 = a * a - b * 4. * c__, abs(
1210 /* Interpolation using THREE most relevant poles */
1212 temp = rhoinv + psi + phi;
1214 temp1 = z__[iim1] / delta[iim1];
1216 c__ = temp - delta[iip1] * (dpsi + dphi) - (d__[iim1] - d__[
1218 zz[0] = z__[iim1] * z__[iim1];
1219 zz[2] = delta[iip1] * delta[iip1] * (dpsi - temp1 + dphi);
1221 temp1 = z__[iip1] / delta[iip1];
1223 c__ = temp - delta[iim1] * (dpsi + dphi) - (d__[iip1] - d__[
1225 zz[0] = delta[iim1] * delta[iim1] * (dpsi + (dphi - temp1));
1226 zz[2] = z__[iip1] * z__[iip1];
1228 zz[1] = z__[ii] * z__[ii];
1229 dlaed6_(&niter, &orgati, &c__, &delta[iim1], zz, &w, &eta, info);
1235 /* Note, eta should be positive if w is negative, and */
1236 /* eta should be negative otherwise. However, */
1237 /* if for some reason caused by roundoff, eta*w > 0, */
1238 /* we simply use one Newton step instead. This way */
1239 /* will guarantee eta*w < 0. */
1241 if (w * eta >= 0.) {
1245 if (temp > dltub || temp < dltlb) {
1247 eta = (dltub - tau) / 2.;
1249 eta = (dltlb - tau) / 2.;
1256 for (j = 1; j <= i__1; ++j) {
1261 /* Evaluate PSI and the derivative DPSI */
1267 for (j = 1; j <= i__1; ++j) {
1268 temp = z__[j] / delta[j];
1269 psi += z__[j] * temp;
1270 dpsi += temp * temp;
1274 erretm = abs(erretm);
1276 /* Evaluate PHI and the derivative DPHI */
1281 for (j = *n; j >= i__1; --j) {
1282 temp = z__[j] / delta[j];
1283 phi += z__[j] * temp;
1284 dphi += temp * temp;
1289 temp = z__[ii] / delta[ii];
1290 dw = dpsi + dphi + temp * temp;
1291 temp = z__[ii] * temp;
1292 w = rhoinv + phi + psi + temp;
1293 erretm = (phi - psi) * 8. + erretm + rhoinv * 2. + abs(temp) * 3. + (
1294 d__1 = tau + eta, abs(d__1)) * dw;
1298 if (-w > abs(prew) / 10.) {
1302 if (w > abs(prew) / 10.) {
1309 /* Main loop to update the values of the array DELTA */
1313 for (niter = iter; niter <= 30; ++niter) {
1315 /* Test for convergence */
1317 if (abs(w) <= eps * erretm) {
1319 *dlam = d__[*i__] + tau;
1321 *dlam = d__[ip1] + tau;
1327 dltlb = f2cmax(dltlb,tau);
1329 dltub = f2cmin(dltub,tau);
1332 /* Calculate the new step */
1337 /* Computing 2nd power */
1338 d__1 = z__[*i__] / delta[*i__];
1339 c__ = w - delta[ip1] * dw - (d__[*i__] - d__[ip1]) * (
1342 /* Computing 2nd power */
1343 d__1 = z__[ip1] / delta[ip1];
1344 c__ = w - delta[*i__] * dw - (d__[ip1] - d__[*i__]) *
1348 temp = z__[ii] / delta[ii];
1350 dpsi += temp * temp;
1352 dphi += temp * temp;
1354 c__ = w - delta[*i__] * dpsi - delta[ip1] * dphi;
1356 a = (delta[*i__] + delta[ip1]) * w - delta[*i__] * delta[ip1]
1358 b = delta[*i__] * delta[ip1] * w;
1363 a = z__[*i__] * z__[*i__] + delta[ip1] *
1364 delta[ip1] * (dpsi + dphi);
1366 a = z__[ip1] * z__[ip1] + delta[*i__] * delta[
1367 *i__] * (dpsi + dphi);
1370 a = delta[*i__] * delta[*i__] * dpsi + delta[ip1]
1371 * delta[ip1] * dphi;
1375 } else if (a <= 0.) {
1376 eta = (a - sqrt((d__1 = a * a - b * 4. * c__, abs(d__1))))
1379 eta = b * 2. / (a + sqrt((d__1 = a * a - b * 4. * c__,
1384 /* Interpolation using THREE most relevant poles */
1386 temp = rhoinv + psi + phi;
1388 c__ = temp - delta[iim1] * dpsi - delta[iip1] * dphi;
1389 zz[0] = delta[iim1] * delta[iim1] * dpsi;
1390 zz[2] = delta[iip1] * delta[iip1] * dphi;
1393 temp1 = z__[iim1] / delta[iim1];
1395 c__ = temp - delta[iip1] * (dpsi + dphi) - (d__[iim1]
1396 - d__[iip1]) * temp1;
1397 zz[0] = z__[iim1] * z__[iim1];
1398 zz[2] = delta[iip1] * delta[iip1] * (dpsi - temp1 +
1401 temp1 = z__[iip1] / delta[iip1];
1403 c__ = temp - delta[iim1] * (dpsi + dphi) - (d__[iip1]
1404 - d__[iim1]) * temp1;
1405 zz[0] = delta[iim1] * delta[iim1] * (dpsi + (dphi -
1407 zz[2] = z__[iip1] * z__[iip1];
1410 dlaed6_(&niter, &orgati, &c__, &delta[iim1], zz, &w, &eta,
1417 /* Note, eta should be positive if w is negative, and */
1418 /* eta should be negative otherwise. However, */
1419 /* if for some reason caused by roundoff, eta*w > 0, */
1420 /* we simply use one Newton step instead. This way */
1421 /* will guarantee eta*w < 0. */
1423 if (w * eta >= 0.) {
1427 if (temp > dltub || temp < dltlb) {
1429 eta = (dltub - tau) / 2.;
1431 eta = (dltlb - tau) / 2.;
1436 for (j = 1; j <= i__1; ++j) {
1444 /* Evaluate PSI and the derivative DPSI */
1450 for (j = 1; j <= i__1; ++j) {
1451 temp = z__[j] / delta[j];
1452 psi += z__[j] * temp;
1453 dpsi += temp * temp;
1457 erretm = abs(erretm);
1459 /* Evaluate PHI and the derivative DPHI */
1464 for (j = *n; j >= i__1; --j) {
1465 temp = z__[j] / delta[j];
1466 phi += z__[j] * temp;
1467 dphi += temp * temp;
1472 temp = z__[ii] / delta[ii];
1473 dw = dpsi + dphi + temp * temp;
1474 temp = z__[ii] * temp;
1475 w = rhoinv + phi + psi + temp;
1476 erretm = (phi - psi) * 8. + erretm + rhoinv * 2. + abs(temp) * 3.
1478 if (w * prew > 0. && abs(w) > abs(prew) / 10.) {
1485 /* Return with INFO = 1, NITER = MAXIT and not converged */
1489 *dlam = d__[*i__] + tau;
1491 *dlam = d__[ip1] + tau;