Gabriele Cosmo
375 lines
14 KiB
Plaintext
375 lines
14 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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////////////////////////////////////////////////////////////////////////////////////
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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=int(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 = max_of<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 = int(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;
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divdf3 = fPoly[i-1].B() + fPoly[i].B() - 2.*divdf1;
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fPoly[i-1].C() = (divdf1 - fPoly[i-1].B() - divdf3)/dtau;
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fPoly[i-1].D() = (divdf3/dtau)/dtau;
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}
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}
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protected:
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std::vector<cubic_poly> fPoly; //[fNp] Array of polynomial terms
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double fValBeg; // Initial value of first or second derivative
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double fValEnd; // End value of first or second derivative
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int fBegCond; // 0=no beg cond, 1=first derivative, 2=second derivative
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int fEndCond; // 0=no end cond, 1=first derivative, 2=second derivative
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};
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}}
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#endif
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