| Geant4 Cross Reference |
1 // Copyright (C) 2010, Guy Barrand. All rights
2 // See the file tools.license for terms.
3
4 #ifndef tools_spline
5 #define tools_spline
6
7 // From Federico Carminati code found in root-
8
9 #include "mnmx"
10 #include <cstddef>
11 #include <vector>
12 #include <ostream>
13 #include <cmath>
14
15 namespace tools {
16 namespace spline {
17
18 class base_poly {
19 public:
20 base_poly():fX(0),fY(0) {}
21 base_poly(double x,double y):fX(x),fY(y) {}
22 virtual ~base_poly(){}
23 public:
24 base_poly(base_poly const &a_from):fX(a_from
25 base_poly& operator=(base_poly const &a_from
26 if(this==&a_from) return *this;
27 fX = a_from.fX;
28 fY = a_from.fY;
29 return *this;
30 }
31 public:
32 const double& X() const {return fX;}
33 const double& Y() const {return fY;}
34 double &X() {return fX;}
35 double &Y() {return fY;}
36 protected:
37 double fX; // abscissa
38 double fY; // constant term
39 };
40
41 class cubic_poly : public base_poly {
42 public:
43 cubic_poly():fB(0), fC(0), fD(0) {}
44 cubic_poly(double x, double y, double b, dou
45 public:
46 cubic_poly(cubic_poly const &a_from)
47 :base_poly(a_from), fB(a_from.fB), fC(a_from
48 cubic_poly& operator=(cubic_poly const &a_fr
49 if(this==&a_from) return *this;
50 base_poly::operator=(a_from);
51 fB = a_from.fB;
52 fC = a_from.fC;
53 fD = a_from.fD;
54 return *this;
55 }
56 public:
57 double &B() {return fB;}
58 double &C() {return fC;}
59 double &D() {return fD;}
60 double eval(double x) const {double dx=x-fX;
61 protected:
62 double fB; // first order expansion coeffici
63 double fC; // second order expansion coeffic
64 double fD; // third order expansion coeffici
65 };
66
67 class quintic_poly : public base_poly {
68 public:
69 quintic_poly():fB(0), fC(0), fD(0), fE(0), f
70 quintic_poly(double x, double y, double b, d
71 :base_poly(x,y), fB(b), fC(c), fD(d), fE(e),
72 public:
73 quintic_poly(quintic_poly const &a_from)
74 :base_poly(a_from)
75 ,fB(a_from.fB),fC(a_from.fC),fD(a_from.fD),f
76 quintic_poly& operator=(quintic_poly const &
77 if(this==&a_from) return *this;
78 base_poly::operator=(a_from);
79 fB = a_from.fB;
80 fC = a_from.fC;
81 fD = a_from.fD;
82 fE = a_from.fE;
83 fF = a_from.fF;
84 return *this;
85 }
86 public:
87 double &B() {return fB;}
88 double &C() {return fC;}
89 double &D() {return fD;}
90 double &E() {return fE;}
91 double &F() {return fF;}
92 double eval(double x) const {double dx=x-fX;
93 protected:
94 double fB; // first order expansion coeffici
95 double fC; // second order expansion coeffic
96 double fD; // third order expansion coeffici
97 double fE; // fourth order expansion coeffic
98 double fF; // fifth order expansion coeffici
99 };
100
101 //////////////////////////////////////////////
102 //////////////////////////////////////////////
103 //////////////////////////////////////////////
104 class base_spline {
105 protected:
106 base_spline(std::ostream& a_out):m_out(a_out
107 public:
108 base_spline(std::ostream& a_out,double delta
109 :m_out(a_out),fDelta(delta), fXmin(xmin),fXm
110 {}
111 virtual ~base_spline() {}
112 protected:
113 base_spline(const base_spline& a_from)
114 :m_out(a_from.m_out)
115 ,fDelta(a_from.fDelta),fXmin(a_from.fXmin),f
116 base_spline& operator=(const base_spline& a_
117 if(this==&a_from) return *this;
118 fDelta=a_from.fDelta;
119 fXmin=a_from.fXmin;
120 fXmax=a_from.fXmax;
121 fNp=a_from.fNp;
122 fKstep=a_from.fKstep;
123 return *this;
124 }
125 protected:
126 std::ostream& m_out;
127 double fDelta; // Distance between equi
128 double fXmin; // Minimum value of absc
129 double fXmax; // Maximum value of absc
130 size_t fNp; // Number of knots
131 bool fKstep; // True of equidistant k
132 };
133
134
135 //////////////////////////////////////////////
136 //
137 // cubic
138 //
139 // Class to create third splines to interpolat
140 // Arbitrary conditions can be introduced for
141 // derivatives at beginning and ending points
142 //
143 //////////////////////////////////////////////
144
145 class cubic : public base_spline {
146 protected:
147 cubic(std::ostream& a_out) : base_spline(a_o
148 public:
149 cubic(std::ostream& a_out,size_t a_n,double
150 :base_spline(a_out,-1,0,0,a_n,false)
151 ,fValBeg(a_valbeg), fValEnd(a_valend), fBegC
152 {
153 if(!a_n) {
154 m_out << "tools::spline::cubic : a_np is
155 return;
156 }
157 fXmin = a_x[0];
158 fXmax = a_x[a_n-1];
159 fPoly.resize(a_n);
160 for (size_t i=0; i<a_n; ++i) {
161 fPoly[i].X() = a_x[i];
162 fPoly[i].Y() = a_y[i];
163 }
164 build_coeff(); // Build the spline coeffic
165 }
166 public:
167 cubic(const cubic& a_from)
168 :base_spline(a_from)
169 ,fPoly(a_from.fPoly),fValBeg(a_from.fValBeg)
170 {}
171 cubic& operator=(const cubic& a_from) {
172 if(this==&a_from) return *this;
173 base_spline::operator=(a_from);
174 fPoly = a_from.fPoly;
175 fValBeg=a_from.fValBeg;
176 fValEnd=a_from.fValEnd;
177 fBegCond=a_from.fBegCond;
178 fEndCond=a_from.fEndCond;
179 return *this;
180 }
181 public:
182 double eval(double x) const {
183 if(!fNp) return 0;
184 // Eval this spline at x
185 size_t klow = find_x(x);
186 if ( (fNp > 1) && (klow >= (fNp-1))) klow
187 return fPoly[klow].eval(x);
188 }
189 protected:
190 template<typename T>
191 static int TMath_Nint(T x) {
192 // Round to nearest integer. Rounds half i
193 int i;
194 if (x >= 0) {
195 i = int(x + 0.5);
196 if ( i & 1 && x + 0.5 == T(i) ) i--;
197 } else {
198 i = int(x - 0.5);
199 if ( i & 1 && x - 0.5 == T(i) ) i++;
200 }
201 return i;
202 }
203 static int TMath_FloorNint(double x) { retur
204
205 size_t find_x(double x) const {
206 int klow=0, khig=int(fNp-1);
207 //
208 // If out of boundaries, extrapolate
209 // It may be badly wrong
210 if(x<=fXmin) klow=0;
211 else if(x>=fXmax) klow=khig;
212 else {
213 if(fKstep) { // Equidistant knots, use h
214 klow = TMath_FloorNint((x-fXmin)/fDel
215 // Correction for rounding errors
216 if (x < fPoly[klow].X())
217 klow = mx<int>(klow-1,0);
218 else if (klow < khig) {
219 if (x > fPoly[klow+1].X()) ++klow;
220 }
221 } else {
222 int khalf;
223 //
224 // Non equidistant knots, binary sear
225 while((khig-klow)>1) {
226 khalf = (klow+khig)/2;
227 if(x>fPoly[khalf].X()) klow=khalf;
228 else khig=khalf;
229 }
230 //
231 // This could be removed, sanity chec
232 if( (x<fPoly[klow].X()) || (fPoly[klo
233 m_out << "tools::spline::cubic::fi
234 << " x(" << klow << ") = " <
235 << " < x(" << klow+1 << ") =
236 << "." << std::endl;
237 }
238 }
239 }
240 return klow;
241 }
242
243 void build_coeff() {
244 /// subroutine cubspl ( tau, c, n, ib
245 /// from * a practical guide to splines
246 /// ************************ input *
247 /// n = number of data points. assumed
248 /// (tau(i), c(1,i), i=1,...,n) = absc
249 /// data points. tau is assumed to
250 /// ibcbeg, ibcend = boundary conditio
251 /// c(2,1), c(2,n) = boundary conditio
252 /// ibcbeg = 0 means no boundary c
253 /// in this case, the not-a-knot
254 /// jump in the third derivative
255 /// zero, thus the first and the
256 /// are made to coincide.)
257 /// ibcbeg = 1 means that the slop
258 /// c(2,1), supplied by input.
259 /// ibcbeg = 2 means that the seco
260 /// made to equal c(2,1), suppli
261 /// ibcend = 0, 1, or 2 has analogo
262 /// boundary condition at tau(n)
263 /// mation taken from c(2,n).
264 /// *********************** output *
265 /// c(j,i), j=1,...,4; i=1,...,l (= n-
266 /// of the cubic interpolating spli
267 /// joints) tau(2), ..., tau(n-1).
268 /// (tau(i), tau(i+1)), the spline
269 /// f(x) = c(1,i)+h*(c(2,i)+h*(c
270 /// where h = x - tau(i). the funct
271 /// used to evaluate f or its deriv
272 /// and k=4.
273
274 int j, l;
275 double divdf1,divdf3,dtau,g=0;
276 // ***** a tridiagonal linear system for t
277 // f at tau(i), i=1,...,n, is generated
278 // ination, with s(i) ending up in c(2,i)
279 // c(3,.) and c(4,.) are used initiall
280 l = int(fNp-1);
281 // compute first differences of x sequence
282 // compute first divided difference of dat
283 {for (size_t m=1; m<fNp ; ++m) {
284 fPoly[m].C() = fPoly[m].X() - fPoly[m-1
285 fPoly[m].D() = (fPoly[m].Y() - fPoly[m-
286 }}
287 // construct first equation from the bound
288 // D[0]*s[0] + C[0]*s[1] = B[0
289 if(fBegCond==0) {
290 if(fNp == 2) {
291 // no condition at left end and n
292 fPoly[0].D() = 1.;
293 fPoly[0].C() = 1.;
294 fPoly[0].B() = 2.*fPoly[1].D();
295 } else {
296 // not-a-knot condition at left e
297 fPoly[0].D() = fPoly[2].C();
298 fPoly[0].C() = fPoly[1].C() + fPoly[2
299 fPoly[0].B() = ((fPoly[1].C()+2.*fPol
300 fPoly[1].C()*fPoly[1].C()*fP
301 }
302 } else if (fBegCond==1) {
303 // slope prescribed at left end.
304 fPoly[0].B() = fValBeg;
305 fPoly[0].D() = 1.;
306 fPoly[0].C() = 0.;
307 } else if (fBegCond==2) {
308 // second derivative prescribed at l
309 fPoly[0].D() = 2.;
310 fPoly[0].C() = 1.;
311 fPoly[0].B() = 3.*fPoly[1].D() - fPoly[1
312 }
313 bool forward_gauss_elimination = true;
314 if(fNp > 2) {
315 // if there are interior knots, generat
316 // ry out the forward pass of gauss eli
317 // equation reads D[m]*s[m] + C[m]*s
318 {for (int m=1; m<l; ++m) {
319 g = -fPoly[m+1].C()/fPoly[m-1].D();
320 fPoly[m].B() = g*fPoly[m-1].B() + 3.*
321 fPoly[m].D() = g*fPoly[m-1].C() + 2.*
322 }}
323 // construct last equation from the seco
324 // (-g*D[n-2])*s[n-2] + D[n-1]
325 // if slope is prescribed at right e
326 // substitution, since c array happe
327 // at this point.
328 if(fEndCond == 0) {
329 if (fNp > 3 || fBegCond != 0) {
330 // not-a-knot and n .ge. 3, an
331 // left end point.
332 g = fPoly[fNp-2].C() + fPoly[fNp-1
333 fPoly[fNp-1].B() = ((fPoly[fNp-1].
334 + fPoly[fNp-1].C()*fP
335 g = -g/fPoly[fNp-2].D();
336 fPoly[fNp-1].D() = fPoly[fNp-2].C(
337 } else {
338 // either (n=3 and not-a-knot
339 // knot at left end point).
340 fPoly[fNp-1].B() = 2.*fPoly[fNp-1]
341 fPoly[fNp-1].D() = 1.;
342 g = -1./fPoly[fNp-2].D();
343 }
344 } else if (fEndCond == 1) {
345 fPoly[fNp-1].B() = fValEnd;
346 forward_gauss_elimination = false;
347 } else if (fEndCond == 2) {
348 // second derivative prescribed a
349 fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D(
350 fPoly[fNp-1].D() = 2.;
351 g = -1./fPoly[fNp-2].D();
352 }
353 } else {
354 if(fEndCond == 0) {
355 if (fBegCond > 0) {
356 // either (n=3 and not-a-knot
357 // knot at left end point).
358 fPoly[fNp-1].B() = 2.*fPoly[fNp-1]
359 fPoly[fNp-1].D() = 1.;
360 g = -1./fPoly[fNp-2].D();
361 } else {
362 // not-a-knot at right endpoin
363 fPoly[fNp-1].B() = fPoly[fNp-1].D(
364 forward_gauss_elimination = false;
365 }
366 } else if(fEndCond == 1) {
367 fPoly[fNp-1].B() = fValEnd;
368 forward_gauss_elimination = false;
369 } else if(fEndCond == 2) {
370 // second derivative prescribed a
371 fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D(
372 fPoly[fNp-1].D() = 2.;
373 g = -1./fPoly[fNp-2].D();
374 }
375 }
376 // complete forward pass of gauss eliminat
377 if(forward_gauss_elimination) {
378 fPoly[fNp-1].D() = g*fPoly[fNp-2].C() +
379 fPoly[fNp-1].B() = (g*fPoly[fNp-2].B() +
380 }
381 // carry out back substitution
382 j = l-1;
383 do {
384 fPoly[j].B() = (fPoly[j].B() - fPoly[j]
385 --j;
386 } while (j>=0);
387 // ****** generate cubic coefficients in e
388 // at its left endpoint, from value and s
389 for (size_t i=1; i<fNp; ++i) {
390 dtau = fPoly[i].C();
391 divdf1 = (fPoly[i].Y() - fPoly[i-1].Y()
392 divdf3 = fPoly[i-1].B() + fPoly[i].B()
393 fPoly[i-1].C() = (divdf1 - fPoly[i-1].B
394 fPoly[i-1].D() = (divdf3/dtau)/dtau;
395 }
396 }
397
398 protected:
399 std::vector<cubic_poly> fPoly; //[fNp] Array
400 double fValBeg; // Initial value
401 double fValEnd; // End value of f
402 int fBegCond; // 0=no beg cond,
403 int fEndCond; // 0=no end cond,
404 };
405
406 //////////////////////////////////////////////
407 //
408 // quintic
409 //
410 // Class to create quintic natural splines to
411 // Arbitrary conditions can be introduced for
412 // derivatives using double knots (see build_c
413 // Double knots are automatically introduced a
414 //
415 //////////////////////////////////////////////
416 class quintic : public base_spline {
417 protected:
418 quintic(std::ostream& a_out):base_spline(a_o
419 public:
420 quintic(std::ostream& a_out,size_t a_n ,doub
421 :base_spline(a_out,-1,0,0,a_n,false) {
422 if(!a_n) {
423 m_out << "tools::spline::quintic : a_np
424 return;
425 }
426 fXmin = a_x[0];
427 fXmax = a_x[a_n-1];
428 fPoly.resize(fNp);
429 for (size_t i=0; i<a_n; ++i) {
430 fPoly[i].X() = a_x[i];
431 fPoly[i].Y() = a_y[i];
432 }
433 build_coeff(); // Build the spline coeffic
434 }
435 public:
436 quintic(const quintic& a_from):base_spline(a
437 quintic& operator=(const quintic& a_from) {
438 if(this==&a_from) return *this;
439 base_spline::operator=(a_from);
440 fPoly = a_from.fPoly;
441 return *this;
442 }
443 public:
444 double eval(double x) const {if(!fNp) return
445
446 protected:
447 size_t find_x(double x) const {
448 int klow=0;
449
450 // If out of boundaries, extrapolate
451 // It may be badly wrong
452 if(x<=fXmin) klow=0;
453 else if(x>=fXmax) klow=int(fNp-1);
454 else {
455 if(fKstep) { // Equidistant knots, use h
456 klow = mn<int>(int((x-fXmin)/fDelta),i
457 } else {
458 int khig=int(fNp-1);
459 int khalf;
460 // Non equidistant knots, binary searc
461 while((khig-klow)>1) {
462 khalf = (klow+khig)/2;
463 if(x>fPoly[khalf].X()) klow=khalf;
464 else khig=khalf;
465 }
466 }
467
468 // This could be removed, sanity check
469 if( (x<fPoly[klow].X()) || (fPoly[klow+1
470 m_out << "tools::spline::quintic::find
471 << " x(" << klow << ") = " << fP
472 << " < x(" << klow+1<< ") = " <<
473 << std::endl;
474 }
475 }
476 return klow;
477 }
478
479 void build_coeff() {
480 //////////////////////////////////////////
481 /// algorithm 600, collected algorithm
482 /// algorithm appeared in acm-trans. m
483 /// jun., 1983, p. 258-259.
484 ///
485 /// quintic computes the coefficients
486 /// s(x) with knots x(i) interpolating
487 /// s(x(i)) = y(i) for i =
488 /// in each interval (x(i),x(i+1)) the
489 /// polynomial of fifth degree:
490 /// s(xx) = ((((f(i)*p+e(i))*p+d(i))*p
491 /// = ((((-f(i)*q+e(i+1))*q-d(i+
492 /// where p = xx - x(i) and q = x(i
493 /// (note the first subscript in the s
494 /// the different polynomials are piec
495 /// its derivatives up to s"" are cont
496 ///
497 /// input:
498 ///
499 /// n number of data points,
500 /// x(1:n) the strictly increasing
501 /// knots. the spacing mus
502 /// of x(i+1) - x(i) can be
503 /// underflow of exponents.
504 /// y(1:n) the prescribed function
505 ///
506 /// output:
507 ///
508 /// b,c,d,e,f the computed spline coe
509 /// (1:n) specifically
510 /// b(i) = s'(x(i)), c(i) =
511 /// e(i) = s""(x(i))/24, f
512 /// f(n) is neither used no
513 /// b,c,d,e,f must always b
514 ///
515 /// option:
516 ///
517 /// it is possible to specify values f
518 /// derivatives of the spline function
519 /// this is done by relaxing the requi
520 /// knots be strictly increasing or de
521 ///
522 /// if x(j) = x(j+1) then s(x(j)) = y(
523 /// if x(j) = x(j+1) = x(j+2) then in
524 ///
525 /// note that s""(x) is discontinuous
526 /// addition, s"'(x) is discontinuous
527 /// subroutine assigns y(i) to y(i+1)
528 /// y(i+2) at a triple knot. the repr
529 /// valid in each open interval (x(i),
530 /// x(j) = x(j+1), the output coeffici
531 /// y(j) = s(x(j)) = y(j+1)
532 /// b(j) = s'(x(j)) = b(j+1)
533 /// c(j) = s"(x(j))/2 = c(j+1)
534 /// d(j) = s"'(x(j))/6 = d(j+1)
535 /// e(j) = s""(x(j)-0)/24 e(j+1)
536 /// f(j) = s""'(x(j)-0)/120 f(j+1)
537 /// at a triple knot, x(j) = x(j+1) =
538 /// coefficients have the following va
539 /// y(j) = s(x(j)) = y(j+1)
540 /// b(j) = s'(x(j)) = b(j+1)
541 /// c(j) = s"(x(j))/2 = c(j+1)
542 /// d(j) = s"'((x(j)-0)/6 d(j+1)
543 /// e(j) = s""(x(j)-0)/24 e(j+1)
544 /// f(j) = s""'(x(j)-0)/120 f(j+1)
545
546 size_t i, _m;
547 double pqqr, p, q, r, _s, t, u, v,
548 b1, p2, p3, q2, q3, r2, pq, pr, qr;
549
550 if (fNp <= 2) return;
551
552 // coefficients of a positive definite
553 // stored in D, E, F from 1 to n-3.
554 _m = fNp-2;
555 q = fPoly[1].X()-fPoly[0].X();
556 r = fPoly[2].X()-fPoly[1].X();
557 q2 = q*q;
558 r2 = r*r;
559 qr = q+r;
560 fPoly[0].D() = fPoly[0].E() = 0;
561 if (q) fPoly[1].D() = q*6.*q2/(qr*qr);
562 else fPoly[1].D() = 0;
563
564 if (_m > 1) {
565 for (i = 1; i < _m; ++i) {
566 p = q;
567 q = r;
568 r = fPoly[i+2].X()-fPoly[i+1].X();
569 p2 = q2;
570 q2 = r2;
571 r2 = r*r;
572 pq = qr;
573 qr = q+r;
574 if (q) {
575 q3 = q2*q;
576 pr = p*r;
577 pqqr = pq*qr;
578 fPoly[i+1].D() = q3*6./(qr*qr);
579 fPoly[i].D() += (q+q)*(pr*15.*pr+
580 *(pr* 20.+q2
581 ((p2+r2)*8.+
582 fPoly[i-1].D() += q3*6./(pq*pq);
583 fPoly[i].E() = q2*(p*qr+pq*3.*(qr
584 fPoly[i-1].E() += q2*(r*pq+qr*3.*
585 fPoly[i-1].F() = q3/pqqr;
586 } else
587 fPoly[i+1].D() = fPoly[i].E() = f
588 }
589 }
590 if (r) fPoly[_m-1].D() += r*6.*r2/(qr*qr);
591
592 // First and second order divided diff
593 // values, stored in b from 2 to n and
594 // respectively. care is taken of doub
595 for (i = 1; i < fNp; ++i) {
596 if (fPoly[i].X() != fPoly[i-1].X()) {
597 fPoly[i].B() =
598 (fPoly[i].Y()-fPoly[i-1].Y())/(fP
599 } else {
600 fPoly[i].B() = fPoly[i].Y();
601 fPoly[i].Y() = fPoly[i-1].Y();
602 }
603 }
604 for (i = 2; i < fNp; ++i) {
605 if (fPoly[i].X() != fPoly[i-2].X()) {
606 fPoly[i].C() =
607 (fPoly[i].B()-fPoly[i-1].B())/(fP
608 } else {
609 fPoly[i].C() = fPoly[i].B()*.5;
610 fPoly[i].B() = fPoly[i-1].B();
611 }
612 }
613
614 // Solve the linear system with c(i+2)
615 if (_m > 1) {
616 p=fPoly[0].C()=fPoly[_m-1].E()=fPoly[0]
617 =fPoly[_m-2].F()=fPoly[_m-1].F()=0;
618 fPoly[1].C() = fPoly[3].C()-fPoly[2].C(
619 fPoly[1].D() = 1./fPoly[1].D();
620
621 if (_m > 2) {
622 for (i = 2; i < _m; ++i) {
623 q = fPoly[i-1].D()*fPoly[i-1].E()
624 fPoly[i].D() = 1./(fPoly[i].D()-p
625 fPoly[i].E() -= q*fPoly[i-1].F();
626 fPoly[i].C() = fPoly[i+2].C()-fPo
627 -q*fPoly[i-1].C();
628 p = fPoly[i-1].D()*fPoly[i-1].F()
629 }
630 }
631 }
632
633 fPoly[fNp-2].C() = fPoly[fNp-1].C() = 0;
634 if (fNp > 3)
635 for (i=fNp-3; i > 0; --i)
636 fPoly[i].C() = (fPoly[i].C()-fPoly[i
637 -fPoly[i].F()*fPoly[i
638
639 // Integrate the third derivative of s
640 _m = fNp-1;
641 q = fPoly[1].X()-fPoly[0].X();
642 r = fPoly[2].X()-fPoly[1].X();
643 b1 = fPoly[1].B();
644 q3 = q*q*q;
645 qr = q+r;
646 if (qr) {
647 v = fPoly[1].C()/qr;
648 t = v;
649 } else
650 v = t = 0;
651 if (q) fPoly[0].F() = v/q;
652 else fPoly[0].F() = 0;
653 for (i = 1; i < _m; ++i) {
654 p = q;
655 q = r;
656 if (i != _m-1) r = fPoly[i+2].X()-fPoly
657 else r = 0;
658 p3 = q3;
659 q3 = q*q*q;
660 pq = qr;
661 qr = q+r;
662 _s = t;
663 if (qr) t = (fPoly[i+1].C()-fPoly[i].C(
664 else t = 0;
665 u = v;
666 v = t-_s;
667 if (pq) {
668 fPoly[i].F() = fPoly[i-1].F();
669 if (q) fPoly[i].F() = v/q;
670 fPoly[i].E() = _s*5.;
671 fPoly[i].D() = (fPoly[i].C()-q*_s)*1
672 fPoly[i].C() =
673 fPoly[i].D()*(p-q)+(fPoly[i+1].B()-f
674 p3-(v+fPoly[i].E(
675 fPoly[i].B() = (p*(fPoly[i+1].B()-v*
676 *q*(fPoly[i].D()+fPoly[i].E()*(q-p))
677 } else {
678 fPoly[i].C() = fPoly[i-1].C();
679 fPoly[i].D() = fPoly[i].E() = fPoly[
680 }
681 }
682
683 // End points x(1) and x(n)
684 p = fPoly[1].X()-fPoly[0].X();
685 _s = fPoly[0].F()*p*p*p;
686 fPoly[0].E() = fPoly[0].D() = 0;
687 fPoly[0].C() = fPoly[1].C()-_s*10;
688 fPoly[0].B() = b1-(fPoly[0].C()+_s)*p;
689
690 q = fPoly[fNp-1].X()-fPoly[fNp-2].X();
691 t = fPoly[fNp-2].F()*q*q*q;
692 fPoly[fNp-1].E() = fPoly[fNp-1].D() = 0;
693 fPoly[fNp-1].C() = fPoly[fNp-2].C()+t*10;
694 fPoly[fNp-1].B() += (fPoly[fNp-1].C()-t)*q
695 }
696 protected:
697 std::vector<quintic_poly> fPoly; //[fNp]
698 };
699
700 }}
701
702 #endif