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Added ability to pass callback to MADWF that is called every inner iteration and allows user to, for example, adjust the inner solver tolerance depending on residual
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
This commit is contained in:
Christopher Kelly
473
Grid/algorithms/approx/RemezGeneral.cc
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473
Grid/algorithms/approx/RemezGeneral.cc
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@ -0,0 +1,473 @@
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#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) {
|
||||
double f = evaluateFunc(x);
|
||||
double r = evaluateApprox(x);
|
||||
os<< x<<","<<r<<","<<f<<","<<r-f<<std::endl;
|
||||
}
|
||||
return;
|
||||
}
|
||||
170
Grid/algorithms/approx/RemezGeneral.h
Normal file
170
Grid/algorithms/approx/RemezGeneral.h
Normal file
@ -0,0 +1,170 @@
|
||||
/*
|
||||
C.Kelly Jan 2020 based on implementation by M. Clark May 2005
|
||||
|
||||
AlgRemezGeneral is an implementation of the Remez algorithm for approximating an arbitrary function by a rational polynomial
|
||||
It includes optional restriction to odd/even polynomials for the numerator and/or denominator
|
||||
*/
|
||||
|
||||
#ifndef INCLUDED_ALG_REMEZ_GENERAL_H
|
||||
#define INCLUDED_ALG_REMEZ_GENERAL_H
|
||||
|
||||
#include <stddef.h>
|
||||
#include <Grid/GridStd.h>
|
||||
|
||||
#ifdef HAVE_LIBGMP
|
||||
#include "bigfloat.h"
|
||||
#else
|
||||
#include "bigfloat_double.h"
|
||||
#endif
|
||||
|
||||
|
||||
class AlgRemezGeneral{
|
||||
public:
|
||||
enum PolyType { Even, Odd, Full };
|
||||
|
||||
private:
|
||||
|
||||
// In GSL-style, pass the function as a function pointer. Any data required to evaluate the function is passed in as a void pointer
|
||||
bigfloat (*f)(bigfloat x, void *data);
|
||||
void *data;
|
||||
|
||||
// The approximation parameters
|
||||
std::vector<bigfloat> param;
|
||||
bigfloat norm;
|
||||
|
||||
// The number of non-zero terms in the numerator and denominator
|
||||
int n, d;
|
||||
// The numerator and denominator degree (i.e. the largest power)
|
||||
int pow_n, pow_d;
|
||||
|
||||
// Specify if the numerator and/or denominator are odd/even polynomials
|
||||
PolyType num_type;
|
||||
PolyType den_type;
|
||||
std::vector<int> num_pows; //contains the mapping, with -1 if not present
|
||||
std::vector<int> den_pows;
|
||||
|
||||
// The bounds of the approximation
|
||||
bigfloat apstrt, apwidt, apend;
|
||||
|
||||
// Variables used to calculate the approximation
|
||||
int nd1, iter;
|
||||
std::vector<bigfloat> xx;
|
||||
std::vector<bigfloat> mm;
|
||||
std::vector<bigfloat> step;
|
||||
|
||||
bigfloat delta, spread;
|
||||
|
||||
// Variables used in search
|
||||
std::vector<bigfloat> yy;
|
||||
|
||||
// Variables used in solving linear equations
|
||||
std::vector<bigfloat> A;
|
||||
std::vector<bigfloat> B;
|
||||
std::vector<int> IPS;
|
||||
|
||||
// The number of equations we must solve at each iteration (n+d+1)
|
||||
int neq;
|
||||
|
||||
// The precision of the GNU MP library
|
||||
long prec;
|
||||
|
||||
// Initialize member variables associated with the polynomial's properties
|
||||
void setupPolyProperties(int num_degree, int den_degree, PolyType num_type_in, PolyType den_type_in);
|
||||
|
||||
// Initial values of maximal and minmal errors
|
||||
void initialGuess();
|
||||
|
||||
// Initialise step sizes
|
||||
void stpini();
|
||||
|
||||
// Initialize the algorithm
|
||||
void reinitializeAlgorithm();
|
||||
|
||||
// Solve the equations
|
||||
void equations();
|
||||
|
||||
// Search for error maxima and minima
|
||||
void search();
|
||||
|
||||
// Calculate function required for the approximation
|
||||
inline bigfloat func(bigfloat x) const{
|
||||
return f(x, data);
|
||||
}
|
||||
|
||||
// Compute size and sign of the approximation error at x
|
||||
bigfloat getErr(bigfloat x, int *sign) const;
|
||||
|
||||
// Solve the system AX=B where X = param
|
||||
int simq();
|
||||
|
||||
// Evaluate the rational form P(x)/Q(x) using coefficients from the solution vector param
|
||||
bigfloat approx(bigfloat x) const;
|
||||
|
||||
public:
|
||||
|
||||
AlgRemezGeneral(double lower, double upper, long prec,
|
||||
bigfloat (*f)(bigfloat x, void *data), void *data);
|
||||
|
||||
inline int getDegree(void) const{
|
||||
assert(n==d);
|
||||
return n;
|
||||
}
|
||||
// Reset the bounds of the approximation
|
||||
inline void setBounds(double lower, double upper) {
|
||||
apstrt = lower;
|
||||
apend = upper;
|
||||
apwidt = apend - apstrt;
|
||||
}
|
||||
|
||||
// Get the bounds of the approximation
|
||||
inline void getBounds(double &lower, double &upper) const{
|
||||
lower=(double)apstrt;
|
||||
upper=(double)apend;
|
||||
}
|
||||
|
||||
// Run the algorithm to generate the rational approximation
|
||||
double generateApprox(int num_degree, int den_degree,
|
||||
PolyType num_type, PolyType den_type,
|
||||
const double tolerance = 1e-15, const int report_freq = 1000);
|
||||
|
||||
inline double generateApprox(int num_degree, int den_degree,
|
||||
const double tolerance = 1e-15, const int report_freq = 1000){
|
||||
return generateApprox(num_degree, den_degree, Full, Full, tolerance, report_freq);
|
||||
}
|
||||
|
||||
// Evaluate the rational form P(x)/Q(x) using coefficients from the
|
||||
// solution vector param
|
||||
inline double evaluateApprox(double x) const{
|
||||
return (double)approx((bigfloat)x);
|
||||
}
|
||||
|
||||
// Evaluate the rational form Q(x)/P(x) using coefficients from the solution vector param
|
||||
inline double evaluateInverseApprox(double x) const{
|
||||
return 1.0/(double)approx((bigfloat)x);
|
||||
}
|
||||
|
||||
// Calculate function required for the approximation
|
||||
inline double evaluateFunc(double x) const{
|
||||
return (double)func((bigfloat)x);
|
||||
}
|
||||
|
||||
// Calculate inverse function required for the approximation
|
||||
inline double evaluateInverseFunc(double x) const{
|
||||
return 1.0/(double)func((bigfloat)x);
|
||||
}
|
||||
|
||||
// Dump csv of function, approx and error
|
||||
void csv(std::ostream &os = std::cout) const;
|
||||
|
||||
// Get the coefficient of the term x^i in the numerator
|
||||
inline double getCoeffNum(const int i) const{
|
||||
return num_pows[i] == -1 ? 0. : double(param[num_pows[i]]);
|
||||
}
|
||||
// Get the coefficient of the term x^i in the denominator
|
||||
inline double getCoeffDen(const int i) const{
|
||||
if(i == pow_d) return 1.0;
|
||||
else return den_pows[i] == -1 ? 0. : double(param[den_pows[i]+n+1]);
|
||||
}
|
||||
};
|
||||
|
||||
#endif
|
||||
183
Grid/algorithms/approx/ZMobius.cc
Normal file
183
Grid/algorithms/approx/ZMobius.cc
Normal file
@ -0,0 +1,183 @@
|
||||
/*************************************************************************************
|
||||
|
||||
Grid physics library, www.github.com/paboyle/Grid
|
||||
|
||||
Source file: ./lib/algorithms/approx/ZMobius.cc
|
||||
|
||||
Copyright (C) 2015
|
||||
|
||||
Author: Christopher Kelly <ckelly@phys.columbia.edu>
|
||||
|
||||
This program is free software; you can redistribute it and/or modify
|
||||
it under the terms of the GNU General Public License as published by
|
||||
the Free Software Foundation; either version 2 of the License, or
|
||||
(at your option) any later version.
|
||||
|
||||
This program is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
||||
GNU General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU General Public License along
|
||||
with this program; if not, write to the Free Software Foundation, Inc.,
|
||||
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
|
||||
|
||||
See the full license in the file "LICENSE" in the top level distribution directory
|
||||
*************************************************************************************/
|
||||
/* END LEGAL */
|
||||
|
||||
#include <Grid/algorithms/approx/ZMobius.h>
|
||||
#include <Grid/algorithms/approx/RemezGeneral.h>
|
||||
|
||||
NAMESPACE_BEGIN(Grid);
|
||||
NAMESPACE_BEGIN(Approx);
|
||||
|
||||
//Compute the tanh approximation
|
||||
inline double epsilonMobius(const double x, const std::vector<ComplexD> &w){
|
||||
int Ls = w.size();
|
||||
|
||||
ComplexD fxp = 1., fmp = 1.;
|
||||
for(int i=0;i<Ls;i++){
|
||||
fxp = fxp * ( w[i] + x );
|
||||
fmp = fmp * ( w[i] - x );
|
||||
}
|
||||
return ((fxp - fmp)/(fxp + fmp)).real();
|
||||
}
|
||||
inline double epsilonMobius(const double x, const std::vector<RealD> &w){
|
||||
int Ls = w.size();
|
||||
|
||||
double fxp = 1., fmp = 1.;
|
||||
for(int i=0;i<Ls;i++){
|
||||
fxp = fxp * ( w[i] + x );
|
||||
fmp = fmp * ( w[i] - x );
|
||||
}
|
||||
return (fxp - fmp)/(fxp + fmp);
|
||||
}
|
||||
|
||||
|
||||
|
||||
//Compute the tanh approximation in a form suitable for the Remez
|
||||
bigfloat epsilonMobius(bigfloat x, void* data){
|
||||
const std::vector<RealD> &omega = *( (std::vector<RealD> const*)data );
|
||||
bigfloat fxp(1.0);
|
||||
bigfloat fmp(1.0);
|
||||
|
||||
for(int i=0;i<omega.size();i++){
|
||||
fxp = fxp * ( bigfloat(omega[i]) + x);
|
||||
fmp = fmp * ( bigfloat(omega[i]) - x);
|
||||
}
|
||||
return (fxp - fmp)/(fxp + fmp);
|
||||
}
|
||||
|
||||
//Compute the Zmobius Omega parameters suitable for eigenvalue range -lambda_bound <= lambda <= lambda_bound
|
||||
//Note omega_i = 1/(b_i + c_i) where b_i and c_i are the Mobius parameters
|
||||
void computeZmobiusOmega(std::vector<ComplexD> &omega_out, const int Ls_out,
|
||||
const std::vector<RealD> &omega_in, const int Ls_in,
|
||||
const RealD lambda_bound){
|
||||
assert(omega_in.size() == Ls_in);
|
||||
omega_out.resize(Ls_out);
|
||||
|
||||
//Use the Remez algorithm to generate the appropriate rational polynomial
|
||||
//For odd polynomial, to satisfy Haar condition must take either positive or negative half of range (cf https://arxiv.org/pdf/0803.0439.pdf page 6)
|
||||
AlgRemezGeneral remez(0, lambda_bound, 64, &epsilonMobius, (void*)&omega_in);
|
||||
remez.generateApprox(Ls_out-1, Ls_out,AlgRemezGeneral::Odd, AlgRemezGeneral::Even, 1e-15, 100);
|
||||
remez.csv(std::cout);
|
||||
|
||||
//The rational approximation has the form [ f(x) - f(-x) ] / [ f(x) + f(-x) ] where f(x) = \Prod_{i=0}^{L_s-1} ( \omega_i + x )
|
||||
//cf https://academiccommons.columbia.edu/doi/10.7916/D8T72HD7 pg 102
|
||||
//omega_i are therefore the negative of the complex roots of f(x)
|
||||
|
||||
//We can find the roots by recognizing that the eigenvalues of a matrix A are the roots of the characteristic polynomial
|
||||
// \rho(\lambda) = det( A - \lambda I ) where I is the unit matrix
|
||||
//The matrix whose characteristic polynomial is an arbitrary monic polynomial a0 + a1 x + a2 x^2 + ... x^n is the companion matrix
|
||||
// A = | 0 1 0 0 0 .... 0 |
|
||||
// | 0 0 1 0 0 .... 0 |
|
||||
// | : : : : : : |
|
||||
// | 0 0 0 0 0 1
|
||||
// | -a0 -a1 -a2 ... ... -an|
|
||||
|
||||
|
||||
//Note the Remez defines the largest power to have unit coefficient
|
||||
std::vector<RealD> coeffs(Ls_out+1);
|
||||
for(int i=0;i<Ls_out+1;i+=2) coeffs[i] = coeffs[i] = remez.getCoeffDen(i); //even powers
|
||||
for(int i=1;i<Ls_out+1;i+=2) coeffs[i] = coeffs[i] = remez.getCoeffNum(i); //odd powers
|
||||
|
||||
std::vector<std::complex<RealD> > roots(Ls_out);
|
||||
|
||||
//Form the companion matrix
|
||||
Eigen::MatrixXd compn(Ls_out,Ls_out);
|
||||
for(int i=0;i<Ls_out-1;i++) compn(i,0) = 0.;
|
||||
compn(Ls_out - 1, 0) = -coeffs[0];
|
||||
|
||||
for(int j=1;j<Ls_out;j++){
|
||||
for(int i=0;i<Ls_out-1;i++) compn(i,j) = i == j-1 ? 1. : 0.;
|
||||
compn(Ls_out - 1, j) = -coeffs[j];
|
||||
}
|
||||
|
||||
//Eigensolve
|
||||
Eigen::EigenSolver<Eigen::MatrixXd> slv(compn, false);
|
||||
|
||||
const auto & ev = slv.eigenvalues();
|
||||
for(int i=0;i<Ls_out;i++)
|
||||
omega_out[i] = -ev(i);
|
||||
|
||||
//Sort ascending (smallest at start of vector!)
|
||||
std::sort(omega_out.begin(), omega_out.end(),
|
||||
[&](const ComplexD &a, const ComplexD &b){ return a.real() < b.real() || (a.real() == b.real() && a.imag() < b.imag()); });
|
||||
|
||||
//McGlynn thesis pg 122 suggest improved iteration counts if magnitude of omega diminishes towards the center of the 5th dimension
|
||||
std::vector<ComplexD> omega_tmp = omega_out;
|
||||
int s_low=0, s_high=Ls_out-1, ss=0;
|
||||
for(int s_from = Ls_out-1; s_from >= 0; s_from--){ //loop from largest omega
|
||||
int s_to;
|
||||
if(ss % 2 == 0){
|
||||
s_to = s_low++;
|
||||
}else{
|
||||
s_to = s_high--;
|
||||
}
|
||||
omega_out[s_to] = omega_tmp[s_from];
|
||||
++ss;
|
||||
}
|
||||
|
||||
std::cout << "Resulting omega_i:" << std::endl;
|
||||
for(int i=0;i<Ls_out;i++)
|
||||
std::cout << omega_out[i] << std::endl;
|
||||
|
||||
std::cout << "Test result matches the approximate polynomial found by the Remez" << std::endl;
|
||||
std::cout << "<x> <remez approx> <poly approx> <diff poly approx remez approx> <exact> <diff poly approx exact>\n";
|
||||
|
||||
int npt = 60;
|
||||
double dlt = lambda_bound/double(npt-1);
|
||||
|
||||
for (int i =0; i<npt; i++){
|
||||
double x = i*dlt;
|
||||
double r = remez.evaluateApprox(x);
|
||||
double p = epsilonMobius(x, omega_out);
|
||||
double e = epsilonMobius(x, omega_in);
|
||||
|
||||
std::cout << x<< " " << r << " " << p <<" " <<r-p << " " << e << " " << e-p << std::endl;
|
||||
}
|
||||
|
||||
}
|
||||
|
||||
//mobius_param = b+c with b-c=1
|
||||
void computeZmobiusOmega(std::vector<ComplexD> &omega_out, const int Ls_out, const RealD mobius_param, const int Ls_in, const RealD lambda_bound){
|
||||
std::vector<RealD> omega_in(Ls_in, 1./mobius_param);
|
||||
computeZmobiusOmega(omega_out, Ls_out, omega_in, Ls_in, lambda_bound);
|
||||
}
|
||||
|
||||
//ZMobius class takes gamma_i = (b+c) omega_i as its input, where b, c are factored out
|
||||
void computeZmobiusGamma(std::vector<ComplexD> &gamma_out,
|
||||
const RealD mobius_param_out, const int Ls_out,
|
||||
const RealD mobius_param_in, const int Ls_in,
|
||||
const RealD lambda_bound){
|
||||
computeZmobiusOmega(gamma_out, Ls_out, mobius_param_in, Ls_in, lambda_bound);
|
||||
for(int i=0;i<Ls_out;i++) gamma_out[i] = gamma_out[i] * mobius_param_out;
|
||||
}
|
||||
//Assumes mobius_param_out == mobius_param_in
|
||||
void computeZmobiusGamma(std::vector<ComplexD> &gamma_out, const int Ls_out, const RealD mobius_param, const int Ls_in, const RealD lambda_bound){
|
||||
computeZmobiusGamma(gamma_out, mobius_param, Ls_out, mobius_param, Ls_in, lambda_bound);
|
||||
}
|
||||
|
||||
NAMESPACE_END(Approx);
|
||||
NAMESPACE_END(Grid);
|
||||
57
Grid/algorithms/approx/ZMobius.h
Normal file
57
Grid/algorithms/approx/ZMobius.h
Normal file
@ -0,0 +1,57 @@
|
||||
/*************************************************************************************
|
||||
|
||||
Grid physics library, www.github.com/paboyle/Grid
|
||||
|
||||
Source file: ./lib/algorithms/approx/ZMobius.h
|
||||
|
||||
Copyright (C) 2015
|
||||
|
||||
Author: Christopher Kelly <ckelly@phys.columbia.edu>
|
||||
|
||||
This program is free software; you can redistribute it and/or modify
|
||||
it under the terms of the GNU General Public License as published by
|
||||
the Free Software Foundation; either version 2 of the License, or
|
||||
(at your option) any later version.
|
||||
|
||||
This program is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
||||
GNU General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU General Public License along
|
||||
with this program; if not, write to the Free Software Foundation, Inc.,
|
||||
51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
|
||||
|
||||
See the full license in the file "LICENSE" in the top level distribution directory
|
||||
*************************************************************************************/
|
||||
/* END LEGAL */
|
||||
#ifndef GRID_ZMOBIUS_APPROX_H
|
||||
#define GRID_ZMOBIUS_APPROX_H
|
||||
|
||||
#include <Grid/GridCore.h>
|
||||
|
||||
NAMESPACE_BEGIN(Grid);
|
||||
NAMESPACE_BEGIN(Approx);
|
||||
|
||||
//Compute the Zmobius Omega parameters suitable for eigenvalue range -lambda_bound <= lambda <= lambda_bound
|
||||
//Note omega_i = 1/(b_i + c_i) where b_i and c_i are the Mobius parameters
|
||||
void computeZmobiusOmega(std::vector<ComplexD> &omega_out, const int Ls_out,
|
||||
const std::vector<RealD> &omega_in, const int Ls_in,
|
||||
const RealD lambda_bound);
|
||||
|
||||
//mobius_param = b+c with b-c=1
|
||||
void computeZmobiusOmega(std::vector<ComplexD> &omega_out, const int Ls_out, const RealD mobius_param, const int Ls_in, const RealD lambda_bound);
|
||||
|
||||
//ZMobius class takes gamma_i = (b+c) omega_i as its input, where b, c are factored out
|
||||
void computeZmobiusGamma(std::vector<ComplexD> &gamma_out,
|
||||
const RealD mobius_param_out, const int Ls_out,
|
||||
const RealD mobius_param_in, const int Ls_in,
|
||||
const RealD lambda_bound);
|
||||
|
||||
//Assumes mobius_param_out == mobius_param_in
|
||||
void computeZmobiusGamma(std::vector<ComplexD> &gamma_out, const int Ls_out, const RealD mobius_param, const int Ls_in, const RealD lambda_bound);
|
||||
|
||||
NAMESPACE_END(Approx);
|
||||
NAMESPACE_END(Grid);
|
||||
|
||||
#endif
|
||||
Reference in New Issue
Block a user