507 lines
13 KiB
C++
507 lines
13 KiB
C++
#ifndef _POLYNOMIAL_HPP
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#define _POLYNOMIAL_HPP
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#include <iostream>
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#include <complex> // std::complex
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#include <mp_complex.hpp>
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#include <icomplex.hpp>
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#include <mp_icomplex.hpp>
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#define MIN(a,b) (((a) < (b))? (a) : (b))
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#define MAX(a,b) (((a) > (b))? (a) : (b))
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class _Polynomial {
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public:
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_Polynomial() {}
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~_Polynomial() {}
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static unsigned int *combinatorial;
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static void startCombinatorialNumbers (int degree);
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static void releaseCombinatorialNumbers () {
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delete [] _Polynomial::combinatorial;
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}
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};
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template <class T>
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class Polynomial : _Polynomial {
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public:
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// CONSTRUCTOR
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Polynomial (int = 0);
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Polynomial (int, const T *);
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Polynomial (const Polynomial<T> &);
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~Polynomial () {delete [] coef;}
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// ASSIGNMENT
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Polynomial & operator= (const Polynomial<T> &);
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Polynomial & operator= (const T &);
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Polynomial & operator= (double);
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T operator() (const T &, const T &) const;
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T diffX (const T &, const T &) const;
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T diffY (const T &, const T &) const;
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T diffXX (const T &, const T &) const;
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T diffXY (const T &, const T &) const;
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T diffYY (const T &, const T &) const;
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// MISC FUNCTIONS FOR HORNER
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T evalPolClassic (const T &, const T &) const;
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T evalIPolHornerXY (const T &, const T &) const {return T();}
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T evalIPolHornerYX (const T &, const T &) const {return T();}
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T evalPolYClassic (const T &, const T &) const;
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T evalIPolYHornerXY (const T &, const T &) const {return T();}
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T evalIPolYHornerYX (const T &, const T &) const {return T();}
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// CHANGE OF VARIABLE
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Polynomial<T> IlinearVarChange (const T &a) const;
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Polynomial<T> IlinearVarChange2 (const T &x0) const;
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// MEMBERS
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T *coef;
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int degree;
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};
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template <class T>
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std::ostream & operator<< (std::ostream &output, const Polynomial<T> &op) {
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int allZero = 1;
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for (int i=0; i<=op.degree; ++i)
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for (int j=0; j<=i; ++j){
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if (op.coef[i*(i+1)/2 + j] == 0.0) continue;
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allZero = 0;
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if (op.coef[i*(i+1)/2 + j] == -1.0)
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output << "-";
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else if (op.coef[i*(i+1)/2 + j] != 1.0)
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output << "+(" << op.coef[i*(i+1)/2 + j] << ")*";
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else
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output << "+";
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if (i == 0)
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if (op.coef[0] == T(1))
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output << "1";
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if (i != 0)
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if (i-j == 0)
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if (j == 1)
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output << "y";
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else
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output << "y^" << j;
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else if (i-j == 1)
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if (j == 0)
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output << "x";
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else if (j == 1)
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output << "x*y";
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else
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output << "x*y^" << j;
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else
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if (j == 0)
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output << "x^" << i;
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else if (j == 1)
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output << "x^" << (i-j) << "*y";
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else
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output << "x^" << (i-j) << "*y^" << j;
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}
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if (allZero) output << "0";
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return output;
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}
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// CONSTRUCTOR
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// DEFAULT CONSTRUCTOR FOR ZERO POLYNOMIAL
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template <class T>
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Polynomial<T>::Polynomial (int op) {
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degree = op;
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coef = new T [((op+1)*(op+2))/2];
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for (int i=0; i<((op+1)*(op+2))/2; ++i)
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coef[i] = 0.0;
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}
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// CONSTRUCTOR FROM ARRAY OF COMPLEX or ICOMPLEX
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template <class T>
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Polynomial<T>::Polynomial (int op1, const T *op2) {
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degree = op1;
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int nCoef = ((degree+1)*(degree+2)) / 2;
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coef = new T [nCoef];
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for (int i=0; i<nCoef; ++i)
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coef[i] = op2[i];
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}
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// COPY CONSTRUCTOR
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template <class T>
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Polynomial<T>::Polynomial (const Polynomial<T> &op) {
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degree = op.degree;
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int nCoef = ((degree+1)*(degree+2)) / 2;
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coef = new T [nCoef];
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for (int i=0; i<nCoef; ++i)
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coef[i] = op.coef[i];
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}
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// ASSIGNMENT
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template <class T>
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Polynomial<T> & Polynomial<T>::operator= (const Polynomial<T> &op) {
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// CHECKS WHETHER ASSIGNED POL IS TARGET POL
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if (this == &op) return (*this);
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// GETS FINAL NUMBER OF COEFICIENTS
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int nCoef = ((op.degree+1)*(op.degree+2)) / 2;
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// IF NUMBER OF COEFFICIENTS ARE THE SAME, NO NEED TO REALLOCATE
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if (this->degree != op.degree) {
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delete [] this->coef;
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this->coef = new T[nCoef];
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this->degree = op.degree;
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}
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// UPDATE VALUES OF COEFICIENTS
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for (int i=0; i<nCoef; ++i)
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this->coef[i] = op.coef[i];
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return (*this);
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}
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template <class T>
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Polynomial<T> & Polynomial<T>::operator= (const T &op) {
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// CHECKS IF TARGET POL IS ALSO A CONSTANT POL
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// IN THAT CASE THERE IS NO NEED TO REALLOCATE MEMORY
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if (this->degree != 0) {
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delete [] this->coef;
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this->coef = new T[1];
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this->degree = 0;
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}
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// UPDATE INDEPENDENT TERM
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this->coef[0] = op;
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return (*this);
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}
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template <class T>
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Polynomial<T> & Polynomial<T>::operator= (double op) {
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// USE PREVIOUS DEFINITION
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return ((*this) = T(op));
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}
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// EVALUATION
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template <class T>
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T Polynomial<T>::evalPolClassic (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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// compute all powers of x and y (from 0 to degree)
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T xPower[this->degree+1], yPower[this->degree+1];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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// loop for each monomial of each degree
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for (int i=0; i<=this->degree; ++i)
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for (int j=0; j<=i; ++j)
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rop = rop +
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((this->coef)[(i*(i+1))/2 + j]) * xPower[i-j] * yPower[j];
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return rop;
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}
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template <class T>
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T Polynomial<T>::operator() (const T &x, const T &y) const {
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return this->evalPolClassic (x,y);
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}
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template <class T>
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T Polynomial<T>::diffX (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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// compute all powers of x and y
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// (from 0 to degree-1) (from 0 to degree) resp
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T xPower[this->degree], yPower[this->degree+1];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree-1; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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yPower[this->degree] = yPower[this->degree-1] * y;
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// loop for each monomial of each degree
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for (int i=1; i<=this->degree; ++i)
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for (int j=0; j<i; ++j)
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rop = rop +
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T(i-j)*((this->coef)[(i*(i+1))/2 + j]) *
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xPower[i-j-1] * yPower[j];
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return rop;
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}
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template <class T>
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T Polynomial<T>::evalPolYClassic (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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// compute all powers of x and y
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// (from 0 to degree) (from 0 to degree-1) resp
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T xPower[this->degree], yPower[this->degree];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree-1; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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// loop for each monomial of each degree
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for (int i=1; i<=this->degree; ++i)
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for (int j=1; j<=i; ++j)
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rop = rop +
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T(j)*((this->coef)[(i*(i+1))/2 + j]) *
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xPower[i-j] * yPower[j-1];
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return rop;
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}
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template <class T>
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T Polynomial<T>::diffY (const T &x, const T &y) const {
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return this->evalPolYClassic (x,y);
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}
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template <class T>
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T Polynomial<T>::diffXX (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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if (this->degree == 1){
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return rop;
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}
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if (this->degree == 2)
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{
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rop = T(2)*(this->coef)[3];
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return rop;
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}
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// compute all powers of x and y
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// (from 0 to degree-2) (from 0 to degree-2) resp
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T xPower[this->degree-1], yPower[this->degree-1];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree-2; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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// loop for each monomial of each degree
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for (int i=2; i<=this->degree; ++i)
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for (int j=0; j<i-1; ++j)
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rop = rop +
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T(i-j)*(i-j-1.0)*((this->coef)[(i*(i+1))/2 + j]) *
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xPower[i-j-2] * yPower[j];
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return rop;
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}
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template <class T>
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T Polynomial<T>::diffXY (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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if (this->degree == 1){
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return rop;
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}
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if (this->degree == 2)
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{
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rop = T(2)*(this->coef)[4];
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return rop;
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}
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// compute all powers of x and y
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// (from 0 to degree-2) (from 0 to degree-2) resp
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T xPower[this->degree-1], yPower[this->degree-1];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree-2; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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// loop for each monomial of each degree
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for (int i=2; i<=this->degree; ++i)
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for (int j=1; j<i; ++j)
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rop = rop +
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T(j)*T(i-j)*((this->coef)[(i*(i+1))/2 + j]) *
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xPower[i-j-1] * yPower[j-1];
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return rop;
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}
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template <class T>
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T Polynomial<T>::diffYY (const T &x, const T &y) const {
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// coef[(i*(i+1))/2 + j] is coeficient of monomial of degree 'i',
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// with degree of second variable 'j' -> coef of x^(i-j) y^j
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// i; // counter along degree of polynomial
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// j; // counter for monomials of equal degree
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// sets output to 0+0i
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T rop;
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if (this->degree == 1){
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return rop;
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}
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if (this->degree == 2)
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{
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rop = T(2)*(this->coef)[5];
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return rop;
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}
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// compute all powers of x and y
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// (from 0 to degree-2) (from 0 to degree-2) resp
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T xPower[this->degree-1];
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T yPower[this->degree-1];
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xPower[0] = yPower[0] = 1.0;
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xPower[1] = x; yPower[1] = y;
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for (int j=1; j<this->degree-2; ++j) {
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xPower[j+1] = xPower[j] * x;
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yPower[j+1] = yPower[j] * y;
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}
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// loop for each monomial of each degree
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for (int i=2; i<=this->degree; ++i)
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for (int j=2; j<=i; ++j)
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rop = rop +
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T(j)*T(j-1)*((this->coef)[(i*(i+1))/2 + j]) *
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xPower[i-j] * yPower[j-2];
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return rop;
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}
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// CHANGE OF VARIABLE
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// RETURNS q = p(x, y + ax);
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// q = sum D(i,j)x^(i-j)y^j
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template <class T>
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Polynomial<T> Polynomial<T>::IlinearVarChange (const T &a) const {
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int i; // counter for degree
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int j; // counter for exponent of y
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int k; // counter for elements modifying x^(i-j) y^j
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// for each degree i, only coefficients of monomials with j in {0, ..., i-1} are changed
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// for each j, only monomials with degree i and k in {j+1, ..., i} modify it
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// for each k, coef[(i*(i+1))/2+k] x^(i-k) (y+ax)^k
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// from binomial expansion, the term with x^(i-j) y^j has multiplier:
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// coef[(i*(i+1))/2+k] a^(k-j) (k over j)
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T aPower[this->degree+1];
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aPower[0] = 1;
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aPower[1] = a;
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for (i=1; i<this->degree; ++i)
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aPower[i+1] = aPower[i] * a;
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unsigned int *combinatorial = _Polynomial::combinatorial;
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Polynomial<T> rop(this->degree);
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// rop = sum_{i=0}^{degree} sum_{j=0}^{i} D(i,j)x^{i-j}y^j
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// D(i,j) = sum_{k=j}^{i} C(i,k)(k over j) a^{k-j}
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for (i=0; i<=this->degree; ++i) for (j=0; j<=i; ++j) {
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rop.coef[(i*(i+1))/2 + j] = this->coef[(i*(i+1))/2 + j];
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for (k=j+1; k<=i; ++k)
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rop.coef[(i*(i+1))/2 + j] = rop.coef[(i*(i+1))/2 + j] +
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combinatorial[(k*(k+1))/2+j] * aPower[k-j] * this->coef[(i*(i+1))/2+k];
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}
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return rop;
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}
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// RETURNS q = p(x+x0, y);
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template <class T>
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Polynomial<T> Polynomial<T>::IlinearVarChange2 (const T &x0) const {
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int i; // counter for degree
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int j; // counter for exponent of y
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int k; // counter for elements modifying x^(i-j) y^j
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// for each degree i, only coefficients of monomials with j in {0, ..., i-1} are changed
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// for each j, only monomials with degree i and k in {j+1, ..., i} modify it
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// for each k, coef[(i*(i+1))/2+k] x^(i-k) (y+ax)^k
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// from binomial expansion, the term with x^(i-j) y^j has multiplier:
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// coef[(i*(i+1))/2+k] a^(k-j) (k over j)
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T x0Power[this->degree+1];
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x0Power[0] = 1;
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x0Power[1] = x0;
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for (i=1; i<this->degree; ++i)
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x0Power[i+1] = x0Power[i] * x0;
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unsigned int *combinatorial = _Polynomial::combinatorial;
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Polynomial<T> rop(this->degree);
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// rop = sum_{i=0}^{degree} sum_{j=0}^{i} D(i,j)x^{i-j}y^j
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// D(i,j) = sum_{k=i}^{degree} (k-j over i-j) C(k,j) x0^(k-i)
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for (i=0; i<=this->degree; ++i) for (j=0; j<=i; ++j) {
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rop.coef[(i*(i+1))/2+j] = this->coef[(i*(i+1))/2+j];
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for (k=i+1; k<=this->degree; ++k)
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rop.coef[(i*(i+1))/2+j] = rop.coef[(i*(i+1))/2+j] +
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combinatorial[((k-j)*(k-j+1))/2 + i-j] * this->coef[(k*(k+1))/2+j] * x0Power[k-i];
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}
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return rop;
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}
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#endif
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