piecewise.h revision 0c670557d9bedbc7e9990341751b4a25f6c248a5
/*
* piecewise.h - Piecewise function class
*
* Copyright 2007 Michael Sloan <mgsloan@gmail.com>
*
* This library is free software; you can redistribute it and/or
* modify it either under the terms of the GNU Lesser General Public
* License version 2.1 as published by the Free Software Foundation
* (the "LGPL") or, at your option, under the terms of the Mozilla
* Public License Version 1.1 (the "MPL"). If you do not alter this
* notice, a recipient may use your version of this file under either
* the MPL or the LGPL.
*
* You should have received a copy of the LGPL along with this library
* in the file COPYING-LGPL-2.1; if not, output to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* You should have received a copy of the MPL along with this library
* in the file COPYING-MPL-1.1
*
* The contents of this file are subject to the Mozilla Public License
* Version 1.1 (the "License"); you may not use this file except in
* compliance with the License. You may obtain a copy of the License at
* http://www.mozilla.org/MPL/
*
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
* OF ANY KIND, either express or implied. See the LGPL or the MPL for
* the specific language governing rights and limitations.
*
*/
#ifndef SEEN_GEOM_PW_SB_H
#define SEEN_GEOM_PW_SB_H
#include "sbasis.h"
#include <vector>
#include <map>
#include "concepts.h"
#include "isnan.h"
#include <boost/concept_check.hpp>
namespace Geom {
template <typename T>
class Piecewise {
BOOST_CLASS_REQUIRE(T, Geom, FragmentConcept);
public:
std::vector<double> cuts;
std::vector<T> segs;
//segs[i] stretches from cuts[i] to cuts[i+1].
Piecewise() {}
explicit Piecewise(const T &s) {
push_cut(0.);
push_seg(s);
push_cut(1.);
}
typedef typename T::output_type output_type;
explicit Piecewise(const output_type & v) {
push_cut(0.);
push_seg(T(v));
push_cut(1.);
}
inline T operator[](unsigned i) const { return segs[i]; }
inline T &operator[](unsigned i) { return segs[i]; }
inline output_type operator()(double t) const { return valueAt(t); }
inline output_type valueAt(double t) const {
unsigned n = segN(t);
return segs[n](segT(t, n));
}
inline output_type firstValue() const {
return valueAt(cuts.front());
}
inline output_type lastValue() const {
return valueAt(cuts.back());
}
std::vector<output_type> valueAndDerivatives(double t, unsigned cnt) const {
unsigned n = segN(t);
std::vector<output_type> ret, val = segs[n].valueAndDerivatives(segT(t, n), cnt);
double mult = 1;
for(unsigned i = 0; i < val.size(); i++) {
ret.push_back(val[i] * mult);
mult /= cuts[n+1] - cuts[n];
}
return ret;
}
//TODO: maybe it is not a good idea to have this?
Piecewise<T> operator()(SBasis f);
Piecewise<T> operator()(Piecewise<SBasis>f);
inline unsigned size() const { return segs.size(); }
inline bool empty() const { return segs.empty(); }
inline void clear() {
segs.clear();
cuts.clear();
}
/**Convenience/implementation hiding function to add segment/cut pairs.
* Asserts that basic size and order invariants are correct
*/
inline void push(const T &s, double to) {
assert(cuts.size() - segs.size() == 1);
push_seg(s);
push_cut(to);
}
//Convenience/implementation hiding function to add cuts.
inline void push_cut(double c) {
ASSERT_INVARIANTS(cuts.empty() || c > cuts.back());
cuts.push_back(c);
}
//Convenience/implementation hiding function to add segments.
inline void push_seg(const T &s) { segs.push_back(s); }
/**Returns the segment index which corresponds to a 'global' piecewise time.
* Also takes optional low/high parameters to expedite the search for the segment.
*/
inline unsigned segN(double t, int low = 0, int high = -1) const {
high = (high == -1) ? size() : high;
if(t < cuts[0]) return 0;
if(t >= cuts[size()]) return size() - 1;
while(low < high) {
int mid = (high + low) / 2; //Lets not plan on having huge (> INT_MAX / 2) cut sequences
double mv = cuts[mid];
if(mv < t) {
if(t < cuts[mid + 1]) return mid; else low = mid + 1;
} else if(t < mv) {
if(cuts[mid - 1] < t) return mid - 1; else high = mid - 1;
} else {
return mid;
}
}
return low;
}
/**Returns the time within a segment, given the 'global' piecewise time.
* Also takes an optional index parameter which may be used for efficiency or to find the time on a
* segment outside its range. If it is left to its default, -1, it will call segN to find the index.
*/
inline double segT(double t, int i = -1) const {
if(i == -1) i = segN(t);
assert(i >= 0);
return (t - cuts[i]) / (cuts[i+1] - cuts[i]);
}
inline double mapToDomain(double t, unsigned i) const {
return (1-t)*cuts[i] + t*cuts[i+1]; //same as: t * (cuts[i+1] - cuts[i]) + cuts[i]
}
//Offsets the piecewise domain
inline void offsetDomain(double o) {
assert(IS_FINITE(o));
if(o != 0)
for(unsigned i = 0; i <= size(); i++)
cuts[i] += o;
}
//Scales the domain of the function by a value. 0 will result in an empty Piecewise.
inline void scaleDomain(double s) {
assert(s > 0);
if(s == 0) {
cuts.clear(); segs.clear();
return;
}
for(unsigned i = 0; i <= size(); i++)
cuts[i] *= s;
}
//Retrieves the domain in interval form
inline Interval domain() const { return Interval(cuts.front(), cuts.back()); }
//Transforms the domain into another interval
inline void setDomain(Interval dom) {
if(empty()) return;
if(dom.isEmpty()) {
cuts.clear(); segs.clear();
return;
}
double cf = cuts.front();
double o = dom.min() - cf, s = dom.extent() / (cuts.back() - cf);
for(unsigned i = 0; i <= size(); i++)
cuts[i] = (cuts[i] - cf) * s + o;
}
//Concatenates this Piecewise function with another, offseting time of the other to match the end.
inline void concat(const Piecewise<T> &other) {
if(other.empty()) return;
if(empty()) {
cuts = other.cuts; segs = other.segs;
return;
}
segs.insert(segs.end(), other.segs.begin(), other.segs.end());
double t = cuts.back() - other.cuts.front();
for(unsigned i = 0; i < other.size(); i++)
push_cut(other.cuts[i + 1] + t);
}
//Like concat, but ensures continuity.
inline void continuousConcat(const Piecewise<T> &other) {
boost::function_requires<AddableConcept<typename T::output_type> >();
if(other.empty()) return;
typename T::output_type y = segs.back().at1() - other.segs.front().at0();
if(empty()) {
for(unsigned i = 0; i < other.size(); i++)
push_seg(other[i] + y);
cuts = other.cuts;
return;
}
double t = cuts.back() - other.cuts.front();
for(unsigned i = 0; i < other.size(); i++)
push(other[i] + y, other.cuts[i + 1] + t);
}
//returns true if the Piecewise<T> meets some basic invariants.
inline bool invariants() const {
// segs between cuts
if(!(segs.size() + 1 == cuts.size() || (segs.empty() && cuts.empty())))
return false;
// cuts in order
for(unsigned i = 0; i < segs.size(); i++)
if(cuts[i] >= cuts[i+1])
return false;
return true;
}
};
template<typename T>
inline typename FragmentConcept<T>::BoundsType bounds_fast(const Piecewise<T> &f) {
boost::function_requires<FragmentConcept<T> >();
if(f.empty()) return typename FragmentConcept<T>::BoundsType();
typename FragmentConcept<T>::BoundsType ret(bounds_fast(f[0]));
for(unsigned i = 1; i < f.size(); i++)
ret.unionWith(bounds_fast(f[i]));
return ret;
}
template<typename T>
inline typename FragmentConcept<T>::BoundsType bounds_exact(const Piecewise<T> &f) {
boost::function_requires<FragmentConcept<T> >();
if(f.empty()) return typename FragmentConcept<T>::BoundsType();
typename FragmentConcept<T>::BoundsType ret(bounds_exact(f[0]));
for(unsigned i = 1; i < f.size(); i++)
ret.unionWith(bounds_exact(f[i]));
return ret;
}
template<typename T>
inline typename FragmentConcept<T>::BoundsType bounds_local(const Piecewise<T> &f, const Interval &m) {
boost::function_requires<FragmentConcept<T> >();
if(f.empty()) return typename FragmentConcept<T>::BoundsType();
if(m.isEmpty()) return typename FragmentConcept<T>::BoundsType(f(m.min()));
unsigned fi = f.segN(m.min()), ti = f.segN(m.max());
double ft = f.segT(m.min(), fi), tt = f.segT(m.max(), ti);
if(fi == ti) return bounds_local(f[fi], Interval(ft, tt));
typename FragmentConcept<T>::BoundsType ret(bounds_local(f[fi], Interval(ft, 1.)));
for(unsigned i = fi + 1; i < ti; i++)
ret.unionWith(bounds_exact(f[i]));
if(tt != 0.) ret.unionWith(bounds_local(f[ti], Interval(0., tt)));
return ret;
}
//returns a portion of a piece of a Piecewise<T>, given the piece's index and a to/from time.
template<typename T>
T elem_portion(const Piecewise<T> &a, unsigned i, double from, double to) {
assert(i < a.size());
double rwidth = 1 / (a.cuts[i+1] - a.cuts[i]);
return portion( a[i], (from - a.cuts[i]) * rwidth, (to - a.cuts[i]) * rwidth );
}
/**Piecewise<T> partition(const Piecewise<T> &pw, std::vector<double> const &c);
* Further subdivides the Piecewise<T> such that there is a cut at every value in c.
* Precondition: c sorted lower to higher.
*
* //Given Piecewise<T> a and b:
* Piecewise<T> ac = a.partition(b.cuts);
* Piecewise<T> bc = b.partition(a.cuts);
* //ac.cuts should be equivalent to bc.cuts
*/
template<typename T>
Piecewise<T> partition(const Piecewise<T> &pw, std::vector<double> const &c) {
assert(pw.invariants());
if(c.empty()) return Piecewise<T>(pw);
Piecewise<T> ret = Piecewise<T>();
ret.cuts.reserve(c.size() + pw.cuts.size());
ret.segs.reserve(c.size() + pw.cuts.size() - 1);
if(pw.empty()) {
ret.cuts = c;
for(unsigned i = 0; i < c.size() - 1; i++)
ret.push_seg(T());
return ret;
}
unsigned si = 0, ci = 0; //Segment index, Cut index
//if the cuts have something earlier than the Piecewise<T>, add portions of the first segment
while(c[ci] < pw.cuts.front() && ci < c.size()) {
bool isLast = (ci == c.size()-1 || c[ci + 1] >= pw.cuts.front());
ret.push_cut(c[ci]);
ret.push_seg( elem_portion(pw, 0, c[ci], isLast ? pw.cuts.front() : c[ci + 1]) );
ci++;
}
ret.push_cut(pw.cuts[0]);
double prev = pw.cuts[0]; //previous cut
//Loop which handles cuts within the Piecewise<T> domain
//Should have the cuts = segs + 1 invariant
while(si < pw.size() && ci <= c.size()) {
if(ci == c.size() && prev <= pw.cuts[si]) { //cuts exhausted, straight copy the rest
ret.segs.insert(ret.segs.end(), pw.segs.begin() + si, pw.segs.end());
ret.cuts.insert(ret.cuts.end(), pw.cuts.begin() + si + 1, pw.cuts.end());
return ret;
}else if(ci == c.size() || c[ci] >= pw.cuts[si + 1]) { //no more cuts within this segment, finalize
if(prev > pw.cuts[si]) { //segment already has cuts, so portion is required
ret.push_seg(portion(pw[si], pw.segT(prev, si), 1.0));
} else { //plain copy is fine
ret.push_seg(pw[si]);
}
ret.push_cut(pw.cuts[si + 1]);
prev = pw.cuts[si + 1];
si++;
} else if(c[ci] == pw.cuts[si]){ //coincident
//Already finalized the seg with the code immediately above
ci++;
} else { //plain old subdivision
ret.push(elem_portion(pw, si, prev, c[ci]), c[ci]);
prev = c[ci];
ci++;
}
}
//input cuts extend further than this Piecewise<T>, extend the last segment.
while(ci < c.size()) {
if(c[ci] > prev) {
ret.push(elem_portion(pw, pw.size() - 1, prev, c[ci]), c[ci]);
prev = c[ci];
}
ci++;
}
return ret;
}
/**Piecewise<T> portion(const Piecewise<T> &pw, double from, double to);
* Returns a Piecewise<T> with a defined domain of [min(from, to), max(from, to)].
*/
template<typename T>
Piecewise<T> portion(const Piecewise<T> &pw, double from, double to) {
if(pw.empty() || from == to) return Piecewise<T>();
Piecewise<T> ret;
double temp = from;
from = std::min(from, to);
to = std::max(temp, to);
unsigned i = pw.segN(from);
ret.push_cut(from);
if(i == pw.size() - 1 || to < pw.cuts[i + 1]) { //to/from inhabit the same segment
ret.push(elem_portion(pw, i, from, to), to);
return ret;
}
ret.push_seg(portion( pw[i], pw.segT(from, i), 1.0 ));
i++;
unsigned fi = pw.segN(to, i);
ret.segs.insert(ret.segs.end(), pw.segs.begin() + i, pw.segs.begin() + fi); //copy segs
ret.cuts.insert(ret.cuts.end(), pw.cuts.begin() + i, pw.cuts.begin() + fi + 1); //and their cuts
ret.push_seg( portion(pw[fi], 0.0, pw.segT(to, fi)));
if(to != ret.cuts.back()) ret.push_cut(to);
ret.invariants();
return ret;
}
template<typename T>
Piecewise<T> remove_short_cuts(Piecewise<T> const &f, double tol) {
if(f.empty()) return f;
Piecewise<T> ret;
ret.push_cut(f.cuts[0]);
for(unsigned i=0; i<f.size(); i++){
if (f.cuts[i+1]-f.cuts[i] >= tol || i==f.size()-1) {
ret.push(f[i], f.cuts[i+1]);
}
}
return ret;
}
template<typename T>
Piecewise<T> remove_short_cuts_extending(Piecewise<T> const &f, double tol) {
if(f.empty()) return f;
Piecewise<T> ret;
ret.push_cut(f.cuts[0]);
double last = f.cuts[0]; // last cut included
for(unsigned i=0; i<f.size(); i++){
if (f.cuts[i+1]-f.cuts[i] >= tol) {
ret.push(elem_portion(f, i, last, f.cuts[i+1]), f.cuts[i+1]);
last = f.cuts[i+1];
}
}
return ret;
}
template<typename T>
std::vector<double> roots(const Piecewise<T> &pw) {
std::vector<double> ret;
for(unsigned i = 0; i < pw.size(); i++) {
std::vector<double> sr = roots(pw[i]);
for (unsigned j = 0; j < sr.size(); j++) ret.push_back(sr[j] * (pw.cuts[i + 1] - pw.cuts[i]) + pw.cuts[i]);
}
return ret;
}
//IMPL: OffsetableConcept
template<typename T>
Piecewise<T> operator+(Piecewise<T> const &a, typename T::output_type b) {
boost::function_requires<OffsetableConcept<T> >();
//TODO:empty
Piecewise<T> ret = Piecewise<T>();
ret.cuts = a.cuts;
for(unsigned i = 0; i < a.size();i++)
ret.push_seg(a[i] + b);
return ret;
}
template<typename T>
Piecewise<T> operator-(Piecewise<T> const &a, typename T::output_type b) {
boost::function_requires<OffsetableConcept<T> >();
//TODO: empty
Piecewise<T> ret = Piecewise<T>();
ret.cuts = a.cuts;
for(unsigned i = 0; i < a.size();i++)
ret.push_seg(a[i] - b);
return ret;
}
template<typename T>
Piecewise<T> operator+=(Piecewise<T>& a, typename T::output_type b) {
boost::function_requires<OffsetableConcept<T> >();
if(a.empty()) { a.push_cut(0.); a.push(T(b), 1.); return a; }
for(unsigned i = 0; i < a.size();i++)
a[i] += b;
return a;
}
template<typename T>
Piecewise<T> operator-=(Piecewise<T>& a, typename T::output_type b) {
boost::function_requires<OffsetableConcept<T> >();
if(a.empty()) { a.push_cut(0.); a.push(T(b), 1.); return a; }
for(unsigned i = 0;i < a.size();i++)
a[i] -= b;
return a;
}
//IMPL: ScalableConcept
template<typename T>
Piecewise<T> operator-(Piecewise<T> const &a) {
boost::function_requires<ScalableConcept<T> >();
Piecewise<T> ret;
ret.cuts = a.cuts;
for(unsigned i = 0; i < a.size();i++)
ret.push_seg(- a[i]);
return ret;
}
template<typename T>
Piecewise<T> operator*(Piecewise<T> const &a, double b) {
boost::function_requires<ScalableConcept<T> >();
if(a.empty()) return Piecewise<T>();
Piecewise<T> ret;
ret.cuts = a.cuts;
for(unsigned i = 0; i < a.size();i++)
ret.push_seg(a[i] * b);
return ret;
}
template<typename T>
Piecewise<T> operator/(Piecewise<T> const &a, double b) {
boost::function_requires<ScalableConcept<T> >();
//FIXME: b == 0?
if(a.empty()) return Piecewise<T>();
Piecewise<T> ret;
ret.cuts = a.cuts;
for(unsigned i = 0; i < a.size();i++)
ret.push_seg(a[i] / b);
return ret;
}
template<typename T>
Piecewise<T> operator*=(Piecewise<T>& a, double b) {
boost::function_requires<ScalableConcept<T> >();
if(a.empty()) return Piecewise<T>();
for(unsigned i = 0; i < a.size();i++)
a[i] *= b;
return a;
}
template<typename T>
Piecewise<T> operator/=(Piecewise<T>& a, double b) {
boost::function_requires<ScalableConcept<T> >();
//FIXME: b == 0?
if(a.empty()) return Piecewise<T>();
for(unsigned i = 0; i < a.size();i++)
a[i] /= b;
return a;
}
//IMPL: AddableConcept
template<typename T>
Piecewise<T> operator+(Piecewise<T> const &a, Piecewise<T> const &b) {
boost::function_requires<AddableConcept<T> >();
Piecewise<T> pa = partition(a, b.cuts), pb = partition(b, a.cuts);
Piecewise<T> ret = Piecewise<T>();
assert(pa.size() == pb.size());
ret.cuts = pa.cuts;
for (unsigned i = 0; i < pa.size(); i++)
ret.push_seg(pa[i] + pb[i]);
return ret;
}
template<typename T>
Piecewise<T> operator-(Piecewise<T> const &a, Piecewise<T> const &b) {
boost::function_requires<AddableConcept<T> >();
Piecewise<T> pa = partition(a, b.cuts), pb = partition(b, a.cuts);
Piecewise<T> ret = Piecewise<T>();
assert(pa.size() == pb.size());
ret.cuts = pa.cuts;
for (unsigned i = 0; i < pa.size(); i++)
ret.push_seg(pa[i] - pb[i]);
return ret;
}
template<typename T>
inline Piecewise<T> operator+=(Piecewise<T> &a, Piecewise<T> const &b) {
a = a+b;
return a;
}
template<typename T>
inline Piecewise<T> operator-=(Piecewise<T> &a, Piecewise<T> const &b) {
a = a-b;
return a;
}
template<typename T1,typename T2>
Piecewise<T2> operator*(Piecewise<T1> const &a, Piecewise<T2> const &b) {
//function_requires<MultiplicableConcept<T1> >();
//function_requires<MultiplicableConcept<T2> >();
Piecewise<T1> pa = partition(a, b.cuts);
Piecewise<T2> pb = partition(b, a.cuts);
Piecewise<T2> ret = Piecewise<T2>();
assert(pa.size() == pb.size());
ret.cuts = pa.cuts;
for (unsigned i = 0; i < pa.size(); i++)
ret.push_seg(pa[i] * pb[i]);
return ret;
}
template<typename T>
inline Piecewise<T> operator*=(Piecewise<T> &a, Piecewise<T> const &b) {
a = a * b;
return a;
}
Piecewise<SBasis> divide(Piecewise<SBasis> const &a, Piecewise<SBasis> const &b, unsigned k);
//TODO: replace divide(a,b,k) by divide(a,b,tol,k)?
//TODO: atm, relative error is ~(tol/a)%. Find a way to make it independant of a.
//Nota: the result is 'truncated' where b is smaller than 'zero': ~ a/max(b,zero).
Piecewise<SBasis>
divide(Piecewise<SBasis> const &a, Piecewise<SBasis> const &b, double tol, unsigned k, double zero=1.e-3);
Piecewise<SBasis>
divide(SBasis const &a, Piecewise<SBasis> const &b, double tol, unsigned k, double zero=1.e-3);
Piecewise<SBasis>
divide(Piecewise<SBasis> const &a, SBasis const &b, double tol, unsigned k, double zero=1.e-3);
Piecewise<SBasis>
divide(SBasis const &a, SBasis const &b, double tol, unsigned k, double zero=1.e-3);
//Composition: functions called compose_* are pieces of compose that are factored out in pw.cpp.
std::map<double,unsigned> compose_pullback(std::vector<double> const &cuts, SBasis const &g);
int compose_findSegIdx(std::map<double,unsigned>::iterator const &cut,
std::map<double,unsigned>::iterator const &next,
std::vector<double> const &levels,
SBasis const &g);
//TODO: add concept check
template<typename T>
Piecewise<T> compose(Piecewise<T> const &f, SBasis const &g){
Piecewise<T> result;
if (f.empty()) return result;
if (g.isZero()) return Piecewise<T>(f(0));
if (f.size()==1){
double t0 = f.cuts[0], width = f.cuts[1] - t0;
return (Piecewise<T>) compose(f.segs[0],compose(Linear(-t0 / width, (1-t0) / width), g));
}
//first check bounds...
Interval bs = bounds_fast(g);
if (f.cuts.front() > bs.max() || bs.min() > f.cuts.back()){
int idx = (bs.max() < f.cuts[1]) ? 0 : f.cuts.size()-2;
double t0 = f.cuts[idx], width = f.cuts[idx+1] - t0;
return (Piecewise<T>) compose(f.segs[idx],compose(Linear(-t0 / width, (1-t0) / width), g));
}
std::vector<double> levels;//we can forget first and last cuts...
levels.insert(levels.begin(),f.cuts.begin()+1,f.cuts.end()-1);
//TODO: use a std::vector<pairs<double,unsigned> > instead of a map<double,unsigned>.
std::map<double,unsigned> cuts_pb = compose_pullback(levels,g);
//-- Compose each piece of g with the relevant seg of f.
result.cuts.push_back(0.);
std::map<double,unsigned>::iterator cut=cuts_pb.begin();
std::map<double,unsigned>::iterator next=cut; next++;
while(next!=cuts_pb.end()){
//assert(std::abs(int((*cut).second-(*next).second))<1);
//TODO: find a way to recover from this error? the root finder missed some root;
// the levels/variations of f might be too close/fast...
int idx = compose_findSegIdx(cut,next,levels,g);
double t0=(*cut).first;
double t1=(*next).first;
SBasis sub_g=compose(g, Linear(t0,t1));
sub_g=compose(Linear(-f.cuts[idx]/(f.cuts[idx+1]-f.cuts[idx]),
(1-f.cuts[idx])/(f.cuts[idx+1]-f.cuts[idx])),sub_g);
result.push(compose(f[idx],sub_g),t1);
cut++;
next++;
}
return(result);
}
//TODO: add concept check for following composition functions
template<typename T>
Piecewise<T> compose(Piecewise<T> const &f, Piecewise<SBasis> const &g){
Piecewise<T> result;
for(unsigned i = 0; i < g.segs.size(); i++){
Piecewise<T> fgi=compose(f, g.segs[i]);
fgi.setDomain(Interval(g.cuts[i], g.cuts[i+1]));
result.concat(fgi);
}
return result;
}
/*
Piecewise<D2<SBasis> > compose(D2<SBasis2d> const &sb2d, Piecewise<D2<SBasis> > const &pwd2sb){
Piecewise<D2<SBasis> > result;
result.push_cut(0.);
for(unsigned i = 0; i < pwd2sb.size(); i++){
result.push(compose_each(sb2d,pwd2sb[i]),i+1);
}
return result;
}*/
template <typename T>
Piecewise<T> Piecewise<T>::operator()(SBasis f){return compose((*this),f);}
template <typename T>
Piecewise<T> Piecewise<T>::operator()(Piecewise<SBasis>f){return compose((*this),f);}
template<typename T>
Piecewise<T> integral(Piecewise<T> const &a) {
Piecewise<T> result;
result.segs.resize(a.segs.size());
result.cuts = a.cuts;
typename T::output_type c = a.segs[0].at0();
for(unsigned i = 0; i < a.segs.size(); i++){
result.segs[i] = integral(a.segs[i])*(a.cuts[i+1]-a.cuts[i]);
result.segs[i]+= c-result.segs[i].at0();
c = result.segs[i].at1();
}
return result;
}
template<typename T>
Piecewise<T> derivative(Piecewise<T> const &a) {
Piecewise<T> result;
result.segs.resize(a.segs.size());
result.cuts = a.cuts;
for(unsigned i = 0; i < a.segs.size(); i++){
result.segs[i] = derivative(a.segs[i])/(a.cuts[i+1]-a.cuts[i]);
}
return result;
}
std::vector<double> roots(Piecewise<SBasis> const &f);
}
#endif //SEEN_GEOM_PW_SB_H
/*
Local Variables:
mode:c++
c-file-style:"stroustrup"
c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
indent-tabs-mode:nil
fill-column:99
End:
*/
// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :