tsgCacheLagrange.hpp

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/*
 * Copyright (c) 2017, Miroslav Stoyanov
 *
 * This file is part of
 * Toolkit for Adaptive Stochastic Modeling And Non-Intrusive ApproximatioN: TASMANIAN
 *
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#ifndef __TSG_CACHE_LAGRANGE_HPP
#define __TSG_CACHE_LAGRANGE_HPP
 
#include "tsgGridCore.hpp"
 
namespace TasGrid{
 
template <typename T>
class CacheLagrange{
public:
    CacheLagrange(int num_dimensions, const std::vector<int> &max_levels, const OneDimensionalWrapper &rule, const double x[]) :
            cache(std::vector<std::vector<T>>(num_dimensions, std::vector<T>())), offsets(rule.getPointsCount()){
        for(int dim=0; dim<num_dimensions; dim++){
            cache[dim].resize(offsets[max_levels[dim] + 1]);
            for(int level=0; level <= max_levels[dim]; level++)
                cacheLevel(level, x[dim], rule, &(cache[dim][offsets[level]]));
        }
    }
    ~CacheLagrange() = default;
 
    static void cacheLevel(int level, double x, const OneDimensionalWrapper &rule, T *cc){
        const double *nodes = rule.getNodes(level);
        const double *coeff = rule.getCoefficients(level);
        int num_points = rule.getNumPoints(level);
 
        cc[0] = 1.0;
        T c = 1.0;
        for(int j=0; j<num_points-1; j++){
            c *= (x - nodes[j]);
            cc[j+1] = c;
        }
        c = (rule.getRule() == rule_clenshawcurtis0) ? (x * x - 1.0) : 1.0;
        cc[num_points-1] *= c * coeff[num_points-1];
        for(int j=num_points-2; j>=0; j--){
            c *= (x - nodes[j+1]);
            cc[j] *= c * coeff[j];
        }
    }
 
    T getLagrange(int dimension, int level, int local) const{
        return cache[dimension][offsets[level] + local];
    }
 
protected:
    std::vector<std::vector<T>> cache;
    std::vector<int> offsets;
};
 
 
template <typename T>
class CacheLagrangeDerivative {
public:
    CacheLagrangeDerivative(int num_dimensions, const std::vector<int> &max_levels, const OneDimensionalWrapper &rule, const double x[]) :
            cache(std::vector<std::vector<T>>(num_dimensions, std::vector<T>())), offsets(rule.getPointsCount()) {
        for(int dim=0; dim<num_dimensions; dim++){
            cache[dim].resize(offsets[max_levels[dim] + 1]);
            for(int level=0; level <= max_levels[dim]; level++)
                cacheDerivativeLevel(level, x[dim], rule, &(cache[dim][offsets[level]]));
        }
    }
    ~CacheLagrangeDerivative() = default;
 
    static void cacheDerivativeLevel(int level, double x, const OneDimensionalWrapper &rule, T *cc){
        /*
         * The usual j-th Lagrange derivative is of the form Lj(x) = cj * fj(x) * gj(x) where
         *
         *   fj(x) = (x - nodes[0])...(x - nodes[j-1]),
         *   gj(x) = (x - nodes[j+1])...(x - nodes[num_points-1]),
         *
         * and cj is the j-th Lagrange coefficient. The product rule gives Lj'(x) = cj * [fj'(x) * gj(x) + fj(x) * gj'(x)].
         * Hence, we do a forward pass for [fj(x), fj'(x)] and a backward pass for [gj(x), gj'(x)].
         */
        const double *nodes = rule.getNodes(level);
        const double *coeff = rule.getCoefficients(level);
        int num_points = rule.getNumPoints(level);
        // cc first stores fj'(x), aux_f stores fj(x), and aux_g stores gj(x).
        std::vector<T> aux_f(num_points), aux_g(num_points);
        cc[0] = (rule.getRule() == rule_clenshawcurtis0) ? 2.0 * x : 0.0;
        aux_f[0] = (rule.getRule() == rule_clenshawcurtis0) ? x * x - 1.0 : 1.0;
        aux_g[num_points-1] = 1.0;
        for(int i=1; i<num_points; i++) {
            aux_f[i] = aux_f[i-1] * (x - nodes[i-1]);
            aux_g[num_points-1-i] = aux_g[num_points-i] * (x - nodes[num_points-i]);
            cc[i] = aux_f[i-1] + (x - nodes[i-1]) * cc[i-1];
        }
        cc[num_points-1] *= coeff[num_points-1];
        T diff_gj = 0.0;
        for(int i=num_points-2; i>=0; i--) {
            diff_gj = aux_g[i+1] + diff_gj * (x - nodes[i+1]);
            cc[i] = coeff[i] * (cc[i] * aux_g[i] + aux_f[i] * diff_gj);
        }
    }
 
    T getLagrangeDerivative(int dimension, int level, int local) const {
        return cache[dimension][offsets[level] + local];
    }
 
protected:
    std::vector<std::vector<T>> cache;
    std::vector<int> offsets;
};
 
}
 
#endif