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https://github.com/paboyle/Grid.git
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96671bbb24
Added a general implementation of the Remez algorithm for producing arbitrary rational polynomial approximation with optional restriction to even/odd polynomials Added implementation of computation of ZMobius parameters Added Test_zMADWF_prec to test ZMobius in MADWF
474 lines
11 KiB
C++
474 lines
11 KiB
C++
#include<math.h>
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#include<stdio.h>
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#include<stdlib.h>
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#include<string>
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#include<iostream>
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#include<iomanip>
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#include<cassert>
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#include<Grid/algorithms/approx/RemezGeneral.h>
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// Constructor
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AlgRemezGeneral::AlgRemezGeneral(double lower, double upper, long precision,
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bigfloat (*f)(bigfloat x, void *data), void *data): f(f),
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data(data),
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prec(precision),
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apstrt(lower), apend(upper), apwidt(upper - lower),
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n(0), d(0), pow_n(0), pow_d(0)
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{
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bigfloat::setDefaultPrecision(prec);
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std::cout<<"Approximation bounds are ["<<apstrt<<","<<apend<<"]\n";
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std::cout<<"Precision of arithmetic is "<<precision<<std::endl;
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}
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//Determine the properties of the numerator and denominator polynomials
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void AlgRemezGeneral::setupPolyProperties(int num_degree, int den_degree, PolyType num_type_in, PolyType den_type_in){
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pow_n = num_degree;
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pow_d = den_degree;
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if(pow_n % 2 == 0 && num_type_in == PolyType::Odd) assert(0);
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if(pow_n % 2 == 1 && num_type_in == PolyType::Even) assert(0);
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if(pow_d % 2 == 0 && den_type_in == PolyType::Odd) assert(0);
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if(pow_d % 2 == 1 && den_type_in == PolyType::Even) assert(0);
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num_type = num_type_in;
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den_type = den_type_in;
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num_pows.resize(pow_n+1);
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den_pows.resize(pow_d+1);
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int n_in = 0;
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bool odd = num_type == PolyType::Full || num_type == PolyType::Odd;
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bool even = num_type == PolyType::Full || num_type == PolyType::Even;
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for(int i=0;i<=pow_n;i++){
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num_pows[i] = -1;
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if(i % 2 == 0 && even) num_pows[i] = n_in++;
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if(i % 2 == 1 && odd) num_pows[i] = n_in++;
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}
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std::cout << n_in << " terms in numerator" << std::endl;
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--n_in; //power is 1 less than the number of terms, eg pow=1 a x^1 + b x^0
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int d_in = 0;
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odd = den_type == PolyType::Full || den_type == PolyType::Odd;
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even = den_type == PolyType::Full || den_type == PolyType::Even;
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for(int i=0;i<=pow_d;i++){
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den_pows[i] = -1;
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if(i % 2 == 0 && even) den_pows[i] = d_in++;
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if(i % 2 == 1 && odd) den_pows[i] = d_in++;
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}
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std::cout << d_in << " terms in denominator" << std::endl;
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--d_in;
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n = n_in;
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d = d_in;
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}
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//Setup algorithm
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void AlgRemezGeneral::reinitializeAlgorithm(){
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spread = 1.0e37;
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iter = 0;
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neq = n + d + 1; //not +2 because highest-power term in denominator is fixed to 1
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param.resize(neq);
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yy.resize(neq+1);
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//Initialize linear equation temporaries
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A.resize(neq*neq);
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B.resize(neq);
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IPS.resize(neq);
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//Initialize maximum and minimum errors
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xx.resize(neq+2);
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mm.resize(neq+1);
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initialGuess();
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//Initialize search steps
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step.resize(neq+1);
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stpini();
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}
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double AlgRemezGeneral::generateApprox(const int num_degree, const int den_degree,
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const PolyType num_type_in, const PolyType den_type_in,
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const double _tolerance, const int report_freq){
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//Setup the properties of the polynomial
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setupPolyProperties(num_degree, den_degree, num_type_in, den_type_in);
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//Setup the algorithm
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reinitializeAlgorithm();
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bigfloat tolerance = _tolerance;
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//Iterate until convergance
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while (spread > tolerance) {
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if (iter++ % report_freq==0)
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std::cout<<"Iteration " <<iter-1<<" spread "<<(double)spread<<" delta "<<(double)delta << std::endl;
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equations();
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if (delta < tolerance) {
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std::cout<<"Iteration " << iter-1 << " delta too small (" << delta << "<" << tolerance << "), try increasing precision\n";
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assert(0);
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};
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assert( delta>= tolerance );
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search();
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}
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int sign;
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double error = (double)getErr(mm[0],&sign);
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std::cout<<"Converged at "<<iter<<" iterations; error = "<<error<<std::endl;
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// Return the maximum error in the approximation
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return error;
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}
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// Initial values of maximal and minimal errors
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void AlgRemezGeneral::initialGuess(){
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// Supply initial guesses for solution points
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long ncheb = neq; // Degree of Chebyshev error estimate
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// Find ncheb+1 extrema of Chebyshev polynomial
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bigfloat a = ncheb;
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bigfloat r;
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mm[0] = apstrt;
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for (long i = 1; i < ncheb; i++) {
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r = 0.5 * (1 - cos((M_PI * i)/(double) a));
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//r *= sqrt_bf(r);
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r = (exp((double)r)-1.0)/(exp(1.0)-1.0);
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mm[i] = apstrt + r * apwidt;
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}
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mm[ncheb] = apend;
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a = 2.0 * ncheb;
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for (long i = 0; i <= ncheb; i++) {
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r = 0.5 * (1 - cos(M_PI * (2*i+1)/(double) a));
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//r *= sqrt_bf(r); // Squeeze to low end of interval
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r = (exp((double)r)-1.0)/(exp(1.0)-1.0);
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xx[i] = apstrt + r * apwidt;
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}
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}
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// Initialise step sizes
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void AlgRemezGeneral::stpini(){
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xx[neq+1] = apend;
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delta = 0.25;
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step[0] = xx[0] - apstrt;
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for (int i = 1; i < neq; i++) step[i] = xx[i] - xx[i-1];
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step[neq] = step[neq-1];
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}
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// Search for error maxima and minima
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void AlgRemezGeneral::search(){
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bigfloat a, q, xm, ym, xn, yn, xx1;
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int emsign, ensign, steps;
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int meq = neq + 1;
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bigfloat eclose = 1.0e30;
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bigfloat farther = 0l;
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bigfloat xx0 = apstrt;
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for (int i = 0; i < meq; i++) {
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steps = 0;
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xx1 = xx[i]; // Next zero
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if (i == meq-1) xx1 = apend;
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xm = mm[i];
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ym = getErr(xm,&emsign);
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q = step[i];
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xn = xm + q;
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if (xn < xx0 || xn >= xx1) { // Cannot skip over adjacent boundaries
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q = -q;
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xn = xm;
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yn = ym;
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ensign = emsign;
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} else {
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yn = getErr(xn,&ensign);
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if (yn < ym) {
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q = -q;
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xn = xm;
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yn = ym;
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ensign = emsign;
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}
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}
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while(yn >= ym) { // March until error becomes smaller.
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if (++steps > 10)
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break;
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ym = yn;
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xm = xn;
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emsign = ensign;
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a = xm + q;
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if (a == xm || a <= xx0 || a >= xx1)
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break;// Must not skip over the zeros either side.
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xn = a;
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yn = getErr(xn,&ensign);
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}
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mm[i] = xm; // Position of maximum
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yy[i] = ym; // Value of maximum
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if (eclose > ym) eclose = ym;
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if (farther < ym) farther = ym;
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xx0 = xx1; // Walk to next zero.
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} // end of search loop
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q = (farther - eclose); // Decrease step size if error spread increased
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if (eclose != 0.0) q /= eclose; // Relative error spread
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if (q >= spread)
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delta *= 0.5; // Spread is increasing; decrease step size
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spread = q;
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for (int i = 0; i < neq; i++) {
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q = yy[i+1];
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if (q != 0.0) q = yy[i] / q - (bigfloat)1l;
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else q = 0.0625;
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if (q > (bigfloat)0.25) q = 0.25;
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q *= mm[i+1] - mm[i];
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step[i] = q * delta;
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}
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step[neq] = step[neq-1];
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for (int i = 0; i < neq; i++) { // Insert new locations for the zeros.
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xm = xx[i] - step[i];
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if (xm <= apstrt)
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continue;
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if (xm >= apend)
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continue;
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if (xm <= mm[i])
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xm = (bigfloat)0.5 * (mm[i] + xx[i]);
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if (xm >= mm[i+1])
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xm = (bigfloat)0.5 * (mm[i+1] + xx[i]);
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xx[i] = xm;
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}
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}
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// Solve the equations
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void AlgRemezGeneral::equations(){
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bigfloat x, y, z;
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bigfloat *aa;
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for (int i = 0; i < neq; i++) { // set up the equations for solution by simq()
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int ip = neq * i; // offset to 1st element of this row of matrix
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x = xx[i]; // the guess for this row
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y = func(x); // right-hand-side vector
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z = (bigfloat)1l;
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aa = A.data()+ip;
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int t = 0;
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for (int j = 0; j <= pow_n; j++) {
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if(num_pows[j] != -1){ *aa++ = z; t++; }
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z *= x;
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}
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assert(t == n+1);
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z = (bigfloat)1l;
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t = 0;
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for (int j = 0; j < pow_d; j++) {
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if(den_pows[j] != -1){ *aa++ = -y * z; t++; }
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z *= x;
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}
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assert(t == d);
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B[i] = y * z; // Right hand side vector
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}
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// Solve the simultaneous linear equations.
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if (simq()){
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std::cout<<"simq failed\n";
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exit(0);
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}
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}
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// Evaluate the rational form P(x)/Q(x) using coefficients
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// from the solution vector param
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bigfloat AlgRemezGeneral::approx(const bigfloat x) const{
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// Work backwards toward the constant term.
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int c = n;
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bigfloat yn = param[c--]; // Highest order numerator coefficient
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for (int i = pow_n-1; i >= 0; i--) yn = x * yn + (num_pows[i] != -1 ? param[c--] : bigfloat(0l));
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c = n+d;
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bigfloat yd = 1l; //Highest degree coefficient is 1.0
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for (int i = pow_d-1; i >= 0; i--) yd = x * yd + (den_pows[i] != -1 ? param[c--] : bigfloat(0l));
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return(yn/yd);
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}
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// Compute size and sign of the approximation error at x
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bigfloat AlgRemezGeneral::getErr(bigfloat x, int *sign) const{
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bigfloat f = func(x);
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bigfloat e = approx(x) - f;
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if (f != 0) e /= f;
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if (e < (bigfloat)0.0) {
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*sign = -1;
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e = -e;
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}
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else *sign = 1;
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return(e);
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}
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// Solve the system AX=B
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int AlgRemezGeneral::simq(){
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int ip, ipj, ipk, ipn;
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int idxpiv;
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int kp, kp1, kpk, kpn;
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int nip, nkp;
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bigfloat em, q, rownrm, big, size, pivot, sum;
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bigfloat *aa;
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bigfloat *X = param.data();
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int n = neq;
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int nm1 = n - 1;
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// Initialize IPS and X
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int ij = 0;
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for (int i = 0; i < n; i++) {
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IPS[i] = i;
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rownrm = 0.0;
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for(int j = 0; j < n; j++) {
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q = abs_bf(A[ij]);
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if(rownrm < q) rownrm = q;
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++ij;
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}
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if (rownrm == (bigfloat)0l) {
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std::cout<<"simq rownrm=0\n";
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return(1);
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}
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X[i] = (bigfloat)1.0 / rownrm;
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}
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for (int k = 0; k < nm1; k++) {
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big = 0.0;
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idxpiv = 0;
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for (int i = k; i < n; i++) {
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ip = IPS[i];
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ipk = n*ip + k;
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size = abs_bf(A[ipk]) * X[ip];
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if (size > big) {
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big = size;
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idxpiv = i;
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}
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}
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if (big == (bigfloat)0l) {
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std::cout<<"simq big=0\n";
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return(2);
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}
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if (idxpiv != k) {
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int j = IPS[k];
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IPS[k] = IPS[idxpiv];
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IPS[idxpiv] = j;
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}
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kp = IPS[k];
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kpk = n*kp + k;
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pivot = A[kpk];
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kp1 = k+1;
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for (int i = kp1; i < n; i++) {
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ip = IPS[i];
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ipk = n*ip + k;
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em = -A[ipk] / pivot;
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A[ipk] = -em;
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nip = n*ip;
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nkp = n*kp;
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aa = A.data()+nkp+kp1;
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for (int j = kp1; j < n; j++) {
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ipj = nip + j;
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A[ipj] = A[ipj] + em * *aa++;
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}
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}
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}
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kpn = n * IPS[n-1] + n - 1; // last element of IPS[n] th row
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if (A[kpn] == (bigfloat)0l) {
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std::cout<<"simq A[kpn]=0\n";
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return(3);
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}
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ip = IPS[0];
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X[0] = B[ip];
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for (int i = 1; i < n; i++) {
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ip = IPS[i];
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ipj = n * ip;
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sum = 0.0;
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for (int j = 0; j < i; j++) {
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sum += A[ipj] * X[j];
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++ipj;
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}
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X[i] = B[ip] - sum;
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}
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ipn = n * IPS[n-1] + n - 1;
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X[n-1] = X[n-1] / A[ipn];
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for (int iback = 1; iback < n; iback++) {
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//i goes (n-1),...,1
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int i = nm1 - iback;
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ip = IPS[i];
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nip = n*ip;
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sum = 0.0;
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aa = A.data()+nip+i+1;
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for (int j= i + 1; j < n; j++)
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sum += *aa++ * X[j];
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X[i] = (X[i] - sum) / A[nip+i];
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}
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return(0);
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}
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void AlgRemezGeneral::csv(std::ostream & os) const{
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os << "Numerator" << std::endl;
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for(int i=0;i<=pow_n;i++){
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os << getCoeffNum(i) << "*x^" << i;
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if(i!=pow_n) os << " + ";
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}
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os << std::endl;
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os << "Denominator" << std::endl;
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for(int i=0;i<=pow_d;i++){
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os << getCoeffDen(i) << "*x^" << i;
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if(i!=pow_d) os << " + ";
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}
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os << std::endl;
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//For a true minimax solution the errors should all be equal and the signs should oscillate +-+-+- etc
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int sign;
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os << "Errors at maxima: coordinate, error, (sign)" << std::endl;
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for(int i=0;i<neq+1;i++){
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os << mm[i] << " " << getErr(mm[i],&sign) << " (" << sign << ")" << std::endl;
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}
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os << "Scan over range:" << std::endl;
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int npt = 60;
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bigfloat dlt = (apend - apstrt)/bigfloat(npt-1);
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for (bigfloat x=apstrt; x<=apend; x = x + dlt) {
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double f = evaluateFunc(x);
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double r = evaluateApprox(x);
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os<< x<<","<<r<<","<<f<<","<<r-f<<std::endl;
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}
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return;
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}
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