Gabriele Cosmo
703 lines
27 KiB
Plaintext
703 lines
27 KiB
Plaintext
// Copyright (C) 2010, Guy Barrand. All rights reserved.
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// See the file tools.license for terms.
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#ifndef tools_spline
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#define tools_spline
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// From Federico Carminati code found in root-6.08.06/TSpline.h, TSpline.cxx.
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#include "mnmx"
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#include <cstddef>
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#include <vector>
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#include <ostream>
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#include <cmath>
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namespace tools {
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namespace spline {
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class base_poly {
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public:
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base_poly():fX(0),fY(0) {}
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base_poly(double x,double y):fX(x),fY(y) {}
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virtual ~base_poly(){}
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public:
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base_poly(base_poly const &a_from):fX(a_from.fX),fY(a_from.fY) {}
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base_poly& operator=(base_poly const &a_from) {
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if(this==&a_from) return *this;
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fX = a_from.fX;
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fY = a_from.fY;
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return *this;
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}
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public:
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const double& X() const {return fX;}
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const double& Y() const {return fY;}
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double &X() {return fX;}
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double &Y() {return fY;}
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protected:
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double fX; // abscissa
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double fY; // constant term
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};
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class cubic_poly : public base_poly {
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public:
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cubic_poly():fB(0), fC(0), fD(0) {}
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cubic_poly(double x, double y, double b, double c, double d):base_poly(x,y), fB(b), fC(c), fD(d) {}
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public:
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cubic_poly(cubic_poly const &a_from)
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:base_poly(a_from), fB(a_from.fB), fC(a_from.fC), fD(a_from.fD) {}
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cubic_poly& operator=(cubic_poly const &a_from) {
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if(this==&a_from) return *this;
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base_poly::operator=(a_from);
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fB = a_from.fB;
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fC = a_from.fC;
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fD = a_from.fD;
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return *this;
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}
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public:
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double &B() {return fB;}
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double &C() {return fC;}
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double &D() {return fD;}
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double eval(double x) const {double dx=x-fX;return (fY+dx*(fB+dx*(fC+dx*fD)));}
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protected:
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double fB; // first order expansion coefficient : fB*1! is the first derivative at x
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double fC; // second order expansion coefficient : fC*2! is the second derivative at x
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double fD; // third order expansion coefficient : fD*3! is the third derivative at x
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};
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class quintic_poly : public base_poly {
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public:
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quintic_poly():fB(0), fC(0), fD(0), fE(0), fF(0) {}
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quintic_poly(double x, double y, double b, double c, double d, double e, double f)
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:base_poly(x,y), fB(b), fC(c), fD(d), fE(e), fF(f) {}
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public:
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quintic_poly(quintic_poly const &a_from)
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:base_poly(a_from)
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,fB(a_from.fB),fC(a_from.fC),fD(a_from.fD),fE(a_from.fE),fF(a_from.fF) {}
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quintic_poly& operator=(quintic_poly const &a_from) {
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if(this==&a_from) return *this;
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base_poly::operator=(a_from);
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fB = a_from.fB;
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fC = a_from.fC;
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fD = a_from.fD;
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fE = a_from.fE;
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fF = a_from.fF;
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return *this;
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}
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public:
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double &B() {return fB;}
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double &C() {return fC;}
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double &D() {return fD;}
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double &E() {return fE;}
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double &F() {return fF;}
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double eval(double x) const {double dx=x-fX;return (fY+dx*(fB+dx*(fC+dx*(fD+dx*(fE+dx*fF)))));}
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protected:
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double fB; // first order expansion coefficient : fB*1! is the first derivative at x
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double fC; // second order expansion coefficient : fC*2! is the second derivative at x
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double fD; // third order expansion coefficient : fD*3! is the third derivative at x
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double fE; // fourth order expansion coefficient : fE*4! is the fourth derivative at x
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double fF; // fifth order expansion coefficient : fF*5! is the fifth derivative at x
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};
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////////////////////////////////////////////////////////////////////////////////////
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////////////////////////////////////////////////////////////////////////////////////
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////////////////////////////////////////////////////////////////////////////////////
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class base_spline {
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protected:
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base_spline(std::ostream& a_out):m_out(a_out), fDelta(-1), fXmin(0), fXmax(0), fNp(0), fKstep(false) {}
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public:
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base_spline(std::ostream& a_out,double delta, double xmin, double xmax, size_t np, bool step)
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:m_out(a_out),fDelta(delta), fXmin(xmin),fXmax(xmax), fNp(np), fKstep(step)
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{}
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virtual ~base_spline() {}
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protected:
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base_spline(const base_spline& a_from)
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:m_out(a_from.m_out)
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,fDelta(a_from.fDelta),fXmin(a_from.fXmin),fXmax(a_from.fXmax),fNp(a_from.fNp),fKstep(a_from.fKstep) {}
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base_spline& operator=(const base_spline& a_from) {
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if(this==&a_from) return *this;
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fDelta=a_from.fDelta;
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fXmin=a_from.fXmin;
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fXmax=a_from.fXmax;
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fNp=a_from.fNp;
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fKstep=a_from.fKstep;
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return *this;
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}
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protected:
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std::ostream& m_out;
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double fDelta; // Distance between equidistant knots
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double fXmin; // Minimum value of abscissa
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double fXmax; // Maximum value of abscissa
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size_t fNp; // Number of knots
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bool fKstep; // True of equidistant knots
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};
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//////////////////////////////////////////////////////////////////////////
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// //
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// cubic //
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// //
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// Class to create third splines to interpolate knots //
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// Arbitrary conditions can be introduced for first and second //
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// derivatives at beginning and ending points //
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// //
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//////////////////////////////////////////////////////////////////////////
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class cubic : public base_spline {
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protected:
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cubic(std::ostream& a_out) : base_spline(a_out) , fPoly(0), fValBeg(0), fValEnd(0), fBegCond(-1), fEndCond(-1) {}
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public:
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cubic(std::ostream& a_out,size_t a_n,double a_x[], double a_y[], double a_valbeg = 0, double a_valend = 0)
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:base_spline(a_out,-1,0,0,a_n,false)
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,fValBeg(a_valbeg), fValEnd(a_valend), fBegCond(0), fEndCond(0)
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{
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if(!a_n) {
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m_out << "tools::spline::cubic : a_np is null." << std::endl;
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return;
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}
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fXmin = a_x[0];
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fXmax = a_x[a_n-1];
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fPoly.resize(a_n);
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for (size_t i=0; i<a_n; ++i) {
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fPoly[i].X() = a_x[i];
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fPoly[i].Y() = a_y[i];
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}
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build_coeff(); // Build the spline coefficients
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}
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public:
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cubic(const cubic& a_from)
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:base_spline(a_from)
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,fPoly(a_from.fPoly),fValBeg(a_from.fValBeg),fValEnd(a_from.fValEnd),fBegCond(a_from.fBegCond),fEndCond(a_from.fEndCond)
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{}
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cubic& operator=(const cubic& a_from) {
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if(this==&a_from) return *this;
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base_spline::operator=(a_from);
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fPoly = a_from.fPoly;
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fValBeg=a_from.fValBeg;
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fValEnd=a_from.fValEnd;
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fBegCond=a_from.fBegCond;
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fEndCond=a_from.fEndCond;
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return *this;
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}
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public:
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double eval(double x) const {
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if(!fNp) return 0;
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// Eval this spline at x
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size_t klow = find_x(x);
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if ( (fNp > 1) && (klow >= (fNp-1))) klow = fNp-2; //see: https://savannah.cern.ch/bugs/?71651
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return fPoly[klow].eval(x);
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}
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protected:
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template<typename T>
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static int TMath_Nint(T x) {
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// Round to nearest integer. Rounds half integers to the nearest even integer.
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int i;
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if (x >= 0) {
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i = int(x + 0.5);
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if ( i & 1 && x + 0.5 == T(i) ) i--;
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} else {
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i = int(x - 0.5);
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if ( i & 1 && x - 0.5 == T(i) ) i++;
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}
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return i;
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}
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static int TMath_FloorNint(double x) { return TMath_Nint(::floor(x)); }
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size_t find_x(double x) const {
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int klow=0, khig=fNp-1;
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//
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// If out of boundaries, extrapolate
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// It may be badly wrong
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if(x<=fXmin) klow=0;
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else if(x>=fXmax) klow=khig;
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else {
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if(fKstep) { // Equidistant knots, use histogramming :
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klow = TMath_FloorNint((x-fXmin)/fDelta);
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// Correction for rounding errors
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if (x < fPoly[klow].X())
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klow = mx<int>(klow-1,0);
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else if (klow < khig) {
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if (x > fPoly[klow+1].X()) ++klow;
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}
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} else {
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int khalf;
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//
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// Non equidistant knots, binary search
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while((khig-klow)>1) {
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khalf = (klow+khig)/2;
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if(x>fPoly[khalf].X()) klow=khalf;
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else khig=khalf;
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}
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//
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// This could be removed, sanity check
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if( (x<fPoly[klow].X()) || (fPoly[klow+1].X()<x) ) {
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m_out << "tools::spline::cubic::find_x : Binary search failed"
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<< " x(" << klow << ") = " << fPoly[klow].X() << " < x= " << x
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<< " < x(" << klow+1 << ") = " << fPoly[klow+1].X() << "."
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<< "." << std::endl;
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}
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}
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}
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return klow;
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}
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void build_coeff() {
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/// subroutine cubspl ( tau, c, n, ibcbeg, ibcend )
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/// from * a practical guide to splines * by c. de boor
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/// ************************ input ***************************
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/// n = number of data points. assumed to be .ge. 2.
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/// (tau(i), c(1,i), i=1,...,n) = abscissae and ordinates of the
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/// data points. tau is assumed to be strictly increasing.
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/// ibcbeg, ibcend = boundary condition indicators, and
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/// c(2,1), c(2,n) = boundary condition information. specifically,
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/// ibcbeg = 0 means no boundary condition at tau(1) is given.
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/// in this case, the not-a-knot condition is used, i.e. the
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/// jump in the third derivative across tau(2) is forced to
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/// zero, thus the first and the second cubic polynomial pieces
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/// are made to coincide.)
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/// ibcbeg = 1 means that the slope at tau(1) is made to equal
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/// c(2,1), supplied by input.
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/// ibcbeg = 2 means that the second derivative at tau(1) is
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/// made to equal c(2,1), supplied by input.
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/// ibcend = 0, 1, or 2 has analogous meaning concerning the
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/// boundary condition at tau(n), with the additional infor-
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/// mation taken from c(2,n).
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/// *********************** output **************************
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/// c(j,i), j=1,...,4; i=1,...,l (= n-1) = the polynomial coefficients
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/// of the cubic interpolating spline with interior knots (or
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/// joints) tau(2), ..., tau(n-1). precisely, in the interval
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/// (tau(i), tau(i+1)), the spline f is given by
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/// f(x) = c(1,i)+h*(c(2,i)+h*(c(3,i)+h*c(4,i)/3.)/2.)
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/// where h = x - tau(i). the function program *ppvalu* may be
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/// used to evaluate f or its derivatives from tau,c, l = n-1,
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/// and k=4.
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int j, l;
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double divdf1,divdf3,dtau,g=0;
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// ***** a tridiagonal linear system for the unknown slopes s(i) of
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// f at tau(i), i=1,...,n, is generated and then solved by gauss elim-
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// ination, with s(i) ending up in c(2,i), all i.
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// c(3,.) and c(4,.) are used initially for temporary storage.
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l = fNp-1;
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// compute first differences of x sequence and store in C also,
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// compute first divided difference of data and store in D.
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{for (size_t m=1; m<fNp ; ++m) {
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fPoly[m].C() = fPoly[m].X() - fPoly[m-1].X();
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fPoly[m].D() = (fPoly[m].Y() - fPoly[m-1].Y())/fPoly[m].C();
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}}
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// construct first equation from the boundary condition, of the form
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// D[0]*s[0] + C[0]*s[1] = B[0]
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if(fBegCond==0) {
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if(fNp == 2) {
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// no condition at left end and n = 2.
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fPoly[0].D() = 1.;
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fPoly[0].C() = 1.;
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fPoly[0].B() = 2.*fPoly[1].D();
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} else {
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// not-a-knot condition at left end and n .gt. 2.
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fPoly[0].D() = fPoly[2].C();
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fPoly[0].C() = fPoly[1].C() + fPoly[2].C();
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fPoly[0].B() = ((fPoly[1].C()+2.*fPoly[0].C())*fPoly[1].D()*fPoly[2].C()+
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fPoly[1].C()*fPoly[1].C()*fPoly[2].D())/fPoly[0].C();
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}
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} else if (fBegCond==1) {
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// slope prescribed at left end.
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fPoly[0].B() = fValBeg;
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fPoly[0].D() = 1.;
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fPoly[0].C() = 0.;
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} else if (fBegCond==2) {
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// second derivative prescribed at left end.
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fPoly[0].D() = 2.;
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fPoly[0].C() = 1.;
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fPoly[0].B() = 3.*fPoly[1].D() - fPoly[1].C()/2.*fValBeg;
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}
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bool forward_gauss_elimination = true;
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if(fNp > 2) {
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// if there are interior knots, generate the corresp. equations and car-
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// ry out the forward pass of gauss elimination, after which the m-th
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// equation reads D[m]*s[m] + C[m]*s[m+1] = B[m].
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{for (int m=1; m<l; ++m) {
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g = -fPoly[m+1].C()/fPoly[m-1].D();
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fPoly[m].B() = g*fPoly[m-1].B() + 3.*(fPoly[m].C()*fPoly[m+1].D()+fPoly[m+1].C()*fPoly[m].D());
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fPoly[m].D() = g*fPoly[m-1].C() + 2.*(fPoly[m].C() + fPoly[m+1].C());
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}}
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// construct last equation from the second boundary condition, of the form
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// (-g*D[n-2])*s[n-2] + D[n-1]*s[n-1] = B[n-1]
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// if slope is prescribed at right end, one can go directly to back-
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// substitution, since c array happens to be set up just right for it
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// at this point.
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if(fEndCond == 0) {
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if (fNp > 3 || fBegCond != 0) {
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// not-a-knot and n .ge. 3, and either n.gt.3 or also not-a-knot at
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// left end point.
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g = fPoly[fNp-2].C() + fPoly[fNp-1].C();
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fPoly[fNp-1].B() = ((fPoly[fNp-1].C()+2.*g)*fPoly[fNp-1].D()*fPoly[fNp-2].C()
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+ fPoly[fNp-1].C()*fPoly[fNp-1].C()*(fPoly[fNp-2].Y()-fPoly[fNp-3].Y())/fPoly[fNp-2].C())/g;
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g = -g/fPoly[fNp-2].D();
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fPoly[fNp-1].D() = fPoly[fNp-2].C();
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} else {
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// either (n=3 and not-a-knot also at left) or (n=2 and not not-a-
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// knot at left end point).
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fPoly[fNp-1].B() = 2.*fPoly[fNp-1].D();
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fPoly[fNp-1].D() = 1.;
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g = -1./fPoly[fNp-2].D();
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}
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} else if (fEndCond == 1) {
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fPoly[fNp-1].B() = fValEnd;
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forward_gauss_elimination = false;
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} else if (fEndCond == 2) {
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// second derivative prescribed at right endpoint.
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fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D() + fPoly[fNp-1].C()/2.*fValEnd;
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fPoly[fNp-1].D() = 2.;
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g = -1./fPoly[fNp-2].D();
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}
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} else {
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if(fEndCond == 0) {
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if (fBegCond > 0) {
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// either (n=3 and not-a-knot also at left) or (n=2 and not not-a-
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// knot at left end point).
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fPoly[fNp-1].B() = 2.*fPoly[fNp-1].D();
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fPoly[fNp-1].D() = 1.;
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g = -1./fPoly[fNp-2].D();
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} else {
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// not-a-knot at right endpoint and at left endpoint and n = 2.
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fPoly[fNp-1].B() = fPoly[fNp-1].D();
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forward_gauss_elimination = false;
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}
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} else if(fEndCond == 1) {
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fPoly[fNp-1].B() = fValEnd;
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forward_gauss_elimination = false;
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} else if(fEndCond == 2) {
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// second derivative prescribed at right endpoint.
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fPoly[fNp-1].B() = 3.*fPoly[fNp-1].D() + fPoly[fNp-1].C()/2.*fValEnd;
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fPoly[fNp-1].D() = 2.;
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g = -1./fPoly[fNp-2].D();
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}
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}
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// complete forward pass of gauss elimination.
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if(forward_gauss_elimination) {
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fPoly[fNp-1].D() = g*fPoly[fNp-2].C() + fPoly[fNp-1].D();
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fPoly[fNp-1].B() = (g*fPoly[fNp-2].B() + fPoly[fNp-1].B())/fPoly[fNp-1].D();
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}
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// carry out back substitution
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j = l-1;
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do {
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fPoly[j].B() = (fPoly[j].B() - fPoly[j].C()*fPoly[j+1].B())/fPoly[j].D();
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--j;
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} while (j>=0);
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// ****** generate cubic coefficients in each interval, i.e., the deriv.s
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// at its left endpoint, from value and slope at its endpoints.
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for (size_t i=1; i<fNp; ++i) {
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dtau = fPoly[i].C();
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divdf1 = (fPoly[i].Y() - fPoly[i-1].Y())/dtau;
|
|
divdf3 = fPoly[i-1].B() + fPoly[i].B() - 2.*divdf1;
|
|
fPoly[i-1].C() = (divdf1 - fPoly[i-1].B() - divdf3)/dtau;
|
|
fPoly[i-1].D() = (divdf3/dtau)/dtau;
|
|
}
|
|
}
|
|
|
|
protected:
|
|
std::vector<cubic_poly> fPoly; //[fNp] Array of polynomial terms
|
|
double fValBeg; // Initial value of first or second derivative
|
|
double fValEnd; // End value of first or second derivative
|
|
int fBegCond; // 0=no beg cond, 1=first derivative, 2=second derivative
|
|
int fEndCond; // 0=no end cond, 1=first derivative, 2=second derivative
|
|
};
|
|
|
|
//////////////////////////////////////////////////////////////////////////
|
|
// //
|
|
// quintic //
|
|
// //
|
|
// Class to create quintic natural splines to interpolate knots //
|
|
// Arbitrary conditions can be introduced for first and second //
|
|
// derivatives using double knots (see build_coeff) for more on this. //
|
|
// Double knots are automatically introduced at ending points //
|
|
// //
|
|
//////////////////////////////////////////////////////////////////////////
|
|
class quintic : public base_spline {
|
|
protected:
|
|
quintic(std::ostream& a_out):base_spline(a_out),fPoly() {}
|
|
public:
|
|
quintic(std::ostream& a_out,size_t a_n ,double a_x[], double a_y[])
|
|
:base_spline(a_out,-1,0,0,a_n,false) {
|
|
if(!a_n) {
|
|
m_out << "tools::spline::quintic : a_np is null." << std::endl;
|
|
return;
|
|
}
|
|
fXmin = a_x[0];
|
|
fXmax = a_x[a_n-1];
|
|
fPoly.resize(fNp);
|
|
for (size_t i=0; i<a_n; ++i) {
|
|
fPoly[i].X() = a_x[i];
|
|
fPoly[i].Y() = a_y[i];
|
|
}
|
|
build_coeff(); // Build the spline coefficients.
|
|
}
|
|
public:
|
|
quintic(const quintic& a_from):base_spline(a_from),fPoly(a_from.fPoly) {}
|
|
quintic& operator=(const quintic& a_from) {
|
|
if(this==&a_from) return *this;
|
|
base_spline::operator=(a_from);
|
|
fPoly = a_from.fPoly;
|
|
return *this;
|
|
}
|
|
public:
|
|
double eval(double x) const {if(!fNp) return 0;size_t klow=find_x(x);return fPoly[klow].eval(x);}
|
|
|
|
protected:
|
|
size_t find_x(double x) const {
|
|
int klow=0;
|
|
|
|
// If out of boundaries, extrapolate
|
|
// It may be badly wrong
|
|
if(x<=fXmin) klow=0;
|
|
else if(x>=fXmax) klow=fNp-1;
|
|
else {
|
|
if(fKstep) { // Equidistant knots, use histogramming :
|
|
klow = mn<int>(int((x-fXmin)/fDelta),fNp-1);
|
|
} else {
|
|
int khig=fNp-1;
|
|
int khalf;
|
|
// Non equidistant knots, binary search
|
|
while((khig-klow)>1) {
|
|
khalf = (klow+khig)/2;
|
|
if(x>fPoly[khalf].X()) klow=khalf;
|
|
else khig=khalf;
|
|
}
|
|
}
|
|
|
|
// This could be removed, sanity check
|
|
if( (x<fPoly[klow].X()) || (fPoly[klow+1].X()<x) ) {
|
|
m_out << "tools::spline::quintic::find_x : Binary search failed"
|
|
<< " x(" << klow << ") = " << fPoly[klow].X() << " < x= " << x
|
|
<< " < x(" << klow+1<< ") = " << fPoly[klow+1].X() << "."
|
|
<< std::endl;
|
|
}
|
|
}
|
|
return klow;
|
|
}
|
|
|
|
void build_coeff() {
|
|
////////////////////////////////////////////////////////////////////////////////
|
|
/// algorithm 600, collected algorithms from acm.
|
|
/// algorithm appeared in acm-trans. math. software, vol.9, no. 2,
|
|
/// jun., 1983, p. 258-259.
|
|
///
|
|
/// quintic computes the coefficients of a quintic natural quintic spli
|
|
/// s(x) with knots x(i) interpolating there to given function values:
|
|
/// s(x(i)) = y(i) for i = 1,2, ..., n.
|
|
/// in each interval (x(i),x(i+1)) the spline function s(xx) is a
|
|
/// polynomial of fifth degree:
|
|
/// s(xx) = ((((f(i)*p+e(i))*p+d(i))*p+c(i))*p+b(i))*p+y(i) (*)
|
|
/// = ((((-f(i)*q+e(i+1))*q-d(i+1))*q+c(i+1))*q-b(i+1))*q+y(i+1)
|
|
/// where p = xx - x(i) and q = x(i+1) - xx.
|
|
/// (note the first subscript in the second expression.)
|
|
/// the different polynomials are pieced together so that s(x) and
|
|
/// its derivatives up to s"" are continuous.
|
|
///
|
|
/// input:
|
|
///
|
|
/// n number of data points, (at least three, i.e. n > 2)
|
|
/// x(1:n) the strictly increasing or decreasing sequence of
|
|
/// knots. the spacing must be such that the fifth power
|
|
/// of x(i+1) - x(i) can be formed without overflow or
|
|
/// underflow of exponents.
|
|
/// y(1:n) the prescribed function values at the knots.
|
|
///
|
|
/// output:
|
|
///
|
|
/// b,c,d,e,f the computed spline coefficients as in (*).
|
|
/// (1:n) specifically
|
|
/// b(i) = s'(x(i)), c(i) = s"(x(i))/2, d(i) = s"'(x(i))/6,
|
|
/// e(i) = s""(x(i))/24, f(i) = s""'(x(i))/120.
|
|
/// f(n) is neither used nor altered. the five arrays
|
|
/// b,c,d,e,f must always be distinct.
|
|
///
|
|
/// option:
|
|
///
|
|
/// it is possible to specify values for the first and second
|
|
/// derivatives of the spline function at arbitrarily many knots.
|
|
/// this is done by relaxing the requirement that the sequence of
|
|
/// knots be strictly increasing or decreasing. specifically:
|
|
///
|
|
/// if x(j) = x(j+1) then s(x(j)) = y(j) and s'(x(j)) = y(j+1),
|
|
/// if x(j) = x(j+1) = x(j+2) then in addition s"(x(j)) = y(j+2).
|
|
///
|
|
/// note that s""(x) is discontinuous at a double knot and, in
|
|
/// addition, s"'(x) is discontinuous at a triple knot. the
|
|
/// subroutine assigns y(i) to y(i+1) in these cases and also to
|
|
/// y(i+2) at a triple knot. the representation (*) remains
|
|
/// valid in each open interval (x(i),x(i+1)). at a double knot,
|
|
/// x(j) = x(j+1), the output coefficients have the following values:
|
|
/// y(j) = s(x(j)) = y(j+1)
|
|
/// b(j) = s'(x(j)) = b(j+1)
|
|
/// c(j) = s"(x(j))/2 = c(j+1)
|
|
/// d(j) = s"'(x(j))/6 = d(j+1)
|
|
/// e(j) = s""(x(j)-0)/24 e(j+1) = s""(x(j)+0)/24
|
|
/// f(j) = s""'(x(j)-0)/120 f(j+1) = s""'(x(j)+0)/120
|
|
/// at a triple knot, x(j) = x(j+1) = x(j+2), the output
|
|
/// coefficients have the following values:
|
|
/// y(j) = s(x(j)) = y(j+1) = y(j+2)
|
|
/// b(j) = s'(x(j)) = b(j+1) = b(j+2)
|
|
/// c(j) = s"(x(j))/2 = c(j+1) = c(j+2)
|
|
/// d(j) = s"'((x(j)-0)/6 d(j+1) = 0 d(j+2) = s"'(x(j)+0)/6
|
|
/// e(j) = s""(x(j)-0)/24 e(j+1) = 0 e(j+2) = s""(x(j)+0)/24
|
|
/// f(j) = s""'(x(j)-0)/120 f(j+1) = 0 f(j+2) = s""'(x(j)+0)/120
|
|
|
|
size_t i, m;
|
|
double pqqr, p, q, r, s, t, u, v,
|
|
b1, p2, p3, q2, q3, r2, pq, pr, qr;
|
|
|
|
if (fNp <= 2) return;
|
|
|
|
// coefficients of a positive definite, pentadiagonal matrix,
|
|
// stored in D, E, F from 1 to n-3.
|
|
m = fNp-2;
|
|
q = fPoly[1].X()-fPoly[0].X();
|
|
r = fPoly[2].X()-fPoly[1].X();
|
|
q2 = q*q;
|
|
r2 = r*r;
|
|
qr = q+r;
|
|
fPoly[0].D() = fPoly[0].E() = 0;
|
|
if (q) fPoly[1].D() = q*6.*q2/(qr*qr);
|
|
else fPoly[1].D() = 0;
|
|
|
|
if (m > 1) {
|
|
for (i = 1; i < m; ++i) {
|
|
p = q;
|
|
q = r;
|
|
r = fPoly[i+2].X()-fPoly[i+1].X();
|
|
p2 = q2;
|
|
q2 = r2;
|
|
r2 = r*r;
|
|
pq = qr;
|
|
qr = q+r;
|
|
if (q) {
|
|
q3 = q2*q;
|
|
pr = p*r;
|
|
pqqr = pq*qr;
|
|
fPoly[i+1].D() = q3*6./(qr*qr);
|
|
fPoly[i].D() += (q+q)*(pr*15.*pr+(p+r)*q
|
|
*(pr* 20.+q2*7.)+q2*
|
|
((p2+r2)*8.+pr*21.+q2+q2))/(pqqr*pqqr);
|
|
fPoly[i-1].D() += q3*6./(pq*pq);
|
|
fPoly[i].E() = q2*(p*qr+pq*3.*(qr+r+r))/(pqqr*qr);
|
|
fPoly[i-1].E() += q2*(r*pq+qr*3.*(pq+p+p))/(pqqr*pq);
|
|
fPoly[i-1].F() = q3/pqqr;
|
|
} else
|
|
fPoly[i+1].D() = fPoly[i].E() = fPoly[i-1].F() = 0;
|
|
}
|
|
}
|
|
if (r) fPoly[m-1].D() += r*6.*r2/(qr*qr);
|
|
|
|
// First and second order divided differences of the given function
|
|
// values, stored in b from 2 to n and in c from 3 to n
|
|
// respectively. care is taken of double and triple knots.
|
|
for (i = 1; i < fNp; ++i) {
|
|
if (fPoly[i].X() != fPoly[i-1].X()) {
|
|
fPoly[i].B() =
|
|
(fPoly[i].Y()-fPoly[i-1].Y())/(fPoly[i].X()-fPoly[i-1].X());
|
|
} else {
|
|
fPoly[i].B() = fPoly[i].Y();
|
|
fPoly[i].Y() = fPoly[i-1].Y();
|
|
}
|
|
}
|
|
for (i = 2; i < fNp; ++i) {
|
|
if (fPoly[i].X() != fPoly[i-2].X()) {
|
|
fPoly[i].C() =
|
|
(fPoly[i].B()-fPoly[i-1].B())/(fPoly[i].X()-fPoly[i-2].X());
|
|
} else {
|
|
fPoly[i].C() = fPoly[i].B()*.5;
|
|
fPoly[i].B() = fPoly[i-1].B();
|
|
}
|
|
}
|
|
|
|
// Solve the linear system with c(i+2) - c(i+1) as right-hand side.
|
|
if (m > 1) {
|
|
p=fPoly[0].C()=fPoly[m-1].E()=fPoly[0].F()
|
|
=fPoly[m-2].F()=fPoly[m-1].F()=0;
|
|
fPoly[1].C() = fPoly[3].C()-fPoly[2].C();
|
|
fPoly[1].D() = 1./fPoly[1].D();
|
|
|
|
if (m > 2) {
|
|
for (i = 2; i < m; ++i) {
|
|
q = fPoly[i-1].D()*fPoly[i-1].E();
|
|
fPoly[i].D() = 1./(fPoly[i].D()-p*fPoly[i-2].F()-q*fPoly[i-1].E());
|
|
fPoly[i].E() -= q*fPoly[i-1].F();
|
|
fPoly[i].C() = fPoly[i+2].C()-fPoly[i+1].C()-p*fPoly[i-2].C()
|
|
-q*fPoly[i-1].C();
|
|
p = fPoly[i-1].D()*fPoly[i-1].F();
|
|
}
|
|
}
|
|
}
|
|
|
|
fPoly[fNp-2].C() = fPoly[fNp-1].C() = 0;
|
|
if (fNp > 3)
|
|
for (i=fNp-3; i > 0; --i)
|
|
fPoly[i].C() = (fPoly[i].C()-fPoly[i].E()*fPoly[i+1].C()
|
|
-fPoly[i].F()*fPoly[i+2].C())*fPoly[i].D();
|
|
|
|
// Integrate the third derivative of s(x)
|
|
m = fNp-1;
|
|
q = fPoly[1].X()-fPoly[0].X();
|
|
r = fPoly[2].X()-fPoly[1].X();
|
|
b1 = fPoly[1].B();
|
|
q3 = q*q*q;
|
|
qr = q+r;
|
|
if (qr) {
|
|
v = fPoly[1].C()/qr;
|
|
t = v;
|
|
} else
|
|
v = t = 0;
|
|
if (q) fPoly[0].F() = v/q;
|
|
else fPoly[0].F() = 0;
|
|
for (i = 1; i < m; ++i) {
|
|
p = q;
|
|
q = r;
|
|
if (i != m-1) r = fPoly[i+2].X()-fPoly[i+1].X();
|
|
else r = 0;
|
|
p3 = q3;
|
|
q3 = q*q*q;
|
|
pq = qr;
|
|
qr = q+r;
|
|
s = t;
|
|
if (qr) t = (fPoly[i+1].C()-fPoly[i].C())/qr;
|
|
else t = 0;
|
|
u = v;
|
|
v = t-s;
|
|
if (pq) {
|
|
fPoly[i].F() = fPoly[i-1].F();
|
|
if (q) fPoly[i].F() = v/q;
|
|
fPoly[i].E() = s*5.;
|
|
fPoly[i].D() = (fPoly[i].C()-q*s)*10;
|
|
fPoly[i].C() =
|
|
fPoly[i].D()*(p-q)+(fPoly[i+1].B()-fPoly[i].B()+(u-fPoly[i].E())*
|
|
p3-(v+fPoly[i].E())*q3)/pq;
|
|
fPoly[i].B() = (p*(fPoly[i+1].B()-v*q3)+q*(fPoly[i].B()-u*p3))/pq-p
|
|
*q*(fPoly[i].D()+fPoly[i].E()*(q-p));
|
|
} else {
|
|
fPoly[i].C() = fPoly[i-1].C();
|
|
fPoly[i].D() = fPoly[i].E() = fPoly[i].F() = 0;
|
|
}
|
|
}
|
|
|
|
// End points x(1) and x(n)
|
|
p = fPoly[1].X()-fPoly[0].X();
|
|
s = fPoly[0].F()*p*p*p;
|
|
fPoly[0].E() = fPoly[0].D() = 0;
|
|
fPoly[0].C() = fPoly[1].C()-s*10;
|
|
fPoly[0].B() = b1-(fPoly[0].C()+s)*p;
|
|
|
|
q = fPoly[fNp-1].X()-fPoly[fNp-2].X();
|
|
t = fPoly[fNp-2].F()*q*q*q;
|
|
fPoly[fNp-1].E() = fPoly[fNp-1].D() = 0;
|
|
fPoly[fNp-1].C() = fPoly[fNp-2].C()+t*10;
|
|
fPoly[fNp-1].B() += (fPoly[fNp-1].C()-t)*q;
|
|
}
|
|
protected:
|
|
std::vector<quintic_poly> fPoly; //[fNp] Array of polynomial terms
|
|
};
|
|
|
|
}}
|
|
|
|
#endif
|