From e2cd2e72dc02c0d8ea995af60a7c19c046a3713c Mon Sep 17 00:00:00 2001 From: jzmaddock Date: Thu, 20 Nov 2014 09:56:21 +0000 Subject: [PATCH] [polygamma] Document new method for negative x in code comments, simply some code, change table to coefficients to store only non-zero values. --- .../special_functions/detail/polygamma.hpp | 74 ++++++++++--------- 1 file changed, 38 insertions(+), 36 deletions(-) diff --git a/include/boost/math/special_functions/detail/polygamma.hpp b/include/boost/math/special_functions/detail/polygamma.hpp index be80a60dd..27003fc3a 100644 --- a/include/boost/math/special_functions/detail/polygamma.hpp +++ b/include/boost/math/special_functions/detail/polygamma.hpp @@ -116,7 +116,6 @@ if(k > policies::get_max_series_iterations()) { return policies::raise_evaluation_error(function, "Series did not converge, closest value was %1%", sum, pol); - break; } } @@ -230,7 +229,7 @@ // Series acceleration: c = b - c; sum = sum + c * term; - b = (k + nd) * (k - nd) * b / ((k + 0.5) * (k + 1)); + b = (k + nd) * (k - nd) * b / ((k + 0.5f) * (k + 1)); // Termination condition: if(fabs(c * term) < (sum + prefix * d) * boost::math::policies::get_epsilon()) break; @@ -276,133 +275,125 @@ // // The general form of each derivative is: // - // pi^n * SUM{k=0, n} C[k,n] * cos(k * pi * x) * csc^(n+1)(pi * x) + // pi^n * SUM{k=0, n} C[k,n] * cos^k(pi * x) * csc^(n+1)(pi * x) // // With constant C[0,1] = -1 and all other C[k,n] = 0; // Then for each k < n+1: - // C[|1 - k|, n+1] += -(k + n + 2) * C[k, n] / 2; - // C[k + 1, n+1] += -(n + 2 - k) * C[k, n] / 2; + // C[k-1, n+1] -= k * C[k, n]; + // C[k+1, n+1] += (k-n-1) * C[k, n]; // - // It's worth noting however, that as well as requiring quite a bit - // of storage space, this method has no better accuracy than recursion - // on x to x > 0 when computing polygamma :-( + // Note that there are many different ways of representing this derivative thanks to + // the many trigomonetric identies available. In particular, the sum of powers of + // cosines could be replaced by a sum of cosine multiple angles, and indeed if you + // plug the derivative into Mathematica this is the form it will give. The two + // forms are related via the Chebeshev polynomials of the first kind and + // T_n(cos(x)) = cos(n x). The polynomial form has the great advantage that + // all the cosine terms are zero at half integer arguments - right where this + // function has it's minumum - thus avoiding cancellation error in this region. + // + // And finally, since every other term in the polynomials is zero, we can save + // space by only storing the non-zero terms. This greatly complexifies + // subscripting the tables in the calculation, but halves the storage space + // (and complexity for that matter). // T s = fabs(x) < fabs(xc) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(xc, pol); + T c = boost::math::cos_pi(x, pol); switch(n) { case 1: return -constants::pi() / (s * s); case 2: { - T c = boost::math::cos_pi(x, pol); return 2 * constants::pi() * constants::pi() * c / boost::math::pow<3>(s, pol); } case 3: { - T c = boost::math::cos_pi(x, pol); int P[] = { -2, -4 }; return boost::math::pow<3>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<4>(s, pol); } case 4: { - T c = boost::math::cos_pi(x, pol); int P[] = { 16, 8 }; return boost::math::pow<4>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<5>(s, pol); } case 5: { - T c = boost::math::cos_pi(x, pol); int P[] = { -16, -88, -16 }; return boost::math::pow<5>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<6>(s, pol); } case 6: { - T c = boost::math::cos_pi(x, pol); int P[] = { 272, 416, 32 }; return boost::math::pow<6>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<7>(s, pol); } case 7: { - T c = boost::math::cos_pi(x, pol); int P[] = { -272, -2880, -1824, -64 }; return boost::math::pow<7>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<8>(s, pol); } case 8: { - T c = boost::math::cos_pi(x, pol); int P[] = { 7936, 24576, 7680, 128 }; return boost::math::pow<8>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<9>(s, pol); } case 9: { - T c = boost::math::cos_pi(x, pol); int P[] = { -7936, -137216, -185856, -31616, -256 }; return boost::math::pow<9>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<10>(s, pol); } case 10: { - T c = boost::math::cos_pi(x, pol); int P[] = { 353792, 1841152, 1304832, 128512, 512 }; return boost::math::pow<10>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<11>(s, pol); } case 11: { - T c = boost::math::cos_pi(x, pol); int P[] = { -353792, -9061376, -21253376, -8728576, -518656, -1024}; return boost::math::pow<11>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<12>(s, pol); } case 12: { - T c = boost::math::cos_pi(x, pol); int P[] = { 22368256, 175627264, 222398464, 56520704, 2084864, 2048 }; return boost::math::pow<12>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<13>(s, pol); } #ifndef BOOST_NO_LONG_LONG case 13: { - T c = boost::math::cos_pi(x, pol); long long P[] = { -22368256LL, -795300864LL, -2868264960LL, -2174832640LL, -357888000LL, -8361984LL, -4096 }; return boost::math::pow<13>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<14>(s, pol); } case 14: { - T c = boost::math::cos_pi(x, pol); long long P[] = { 1903757312LL, 21016670208LL, 41731645440LL, 20261765120LL, 2230947840LL, 33497088LL, 8192 }; return boost::math::pow<14>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<15>(s, pol); } case 15: { - T c = boost::math::cos_pi(x, pol); long long P[] = { -1903757312LL, -89702612992LL, -460858269696LL, -559148810240LL, -182172651520LL, -13754155008LL, -134094848LL, -16384 }; return boost::math::pow<15>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<16>(s, pol); } case 16: { - T c = boost::math::cos_pi(x, pol); long long P[] = { 209865342976LL, 3099269660672LL, 8885192097792LL, 7048869314560LL, 1594922762240LL, 84134068224LL, 536608768LL, 32768 }; return boost::math::pow<16>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<17>(s, pol); } case 17: { - T c = boost::math::cos_pi(x, pol); long long P[] = { -209865342976LL, -12655654469632LL, -87815735738368LL, -155964390375424LL, -84842998005760LL, -13684856848384LL, -511780323328LL, -2146926592LL, -65536 }; return boost::math::pow<17>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<18>(s, pol); } case 18: { - T c = boost::math::cos_pi(x, pol); long long P[] = { 29088885112832LL, 553753414467584LL, 2165206642589696LL, 2550316668551168LL, 985278548541440LL, 115620218667008LL, 3100738912256LL, 8588754944LL, 131072 }; return boost::math::pow<18>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<19>(s, pol); } case 19: { - T c = boost::math::cos_pi(x, pol); long long P[] = { -29088885112832LL, -2184860175433728LL, -19686087844429824LL, -48165109676113920LL, -39471306959486976LL, -11124607890751488LL, -965271355195392LL, -18733264797696LL, -34357248000LL, -262144 }; return boost::math::pow<19>(constants::pi(), pol) * tools::evaluate_even_polynomial(P, c) / boost::math::pow<20>(s, pol); } case 20: { - T c = boost::math::cos_pi(x, pol); long long P[] = { 4951498053124096LL, 118071834535526400LL, 603968063567560704LL, 990081991141490688LL, 584901762421358592LL, 122829335169859584LL, 7984436548730880LL, 112949304754176LL, 137433710592LL, 524288 }; return boost::math::pow<20>(constants::pi(), pol) * c * tools::evaluate_even_polynomial(P, c) / boost::math::pow<21>(s, pol); } @@ -410,7 +401,7 @@ } // - // We'll have to compute the corefficients up to n: + // We'll have to compute the coefficients up to n: // #ifdef BOOST_HAS_THREADS static boost::detail::lightweight_mutex m; @@ -424,18 +415,29 @@ { for(int i = (int)table.size() - 1; i < index; ++i) { - int sin_order = i + 2; - table.push_back(std::vector(i + 2, T(0))); - table[i + 1][1] -= sin_order * table[i][0] / (sin_order - 1); - for(int cos_order = 1; cos_order < i + 1; ++cos_order) + int offset = i & 1; // 1 if the first cos power is 0, otherwise 0. + int sin_order = i + 2; // order of the sin term + int max_cos_order = sin_order - 1; // largest order of the polynomial of cos terms + int max_columns = (max_cos_order - offset) / 2; // How many entries there are in the current row. + int next_offset = offset ? 0 : 1; + int next_max_columns = (max_cos_order + 1 - next_offset) / 2; // How many entries there will be in the next row + table.push_back(std::vector(next_max_columns + 1, T(0))); + + for(int column = 0; column <= max_columns; ++column) { - table[i + 1][cos_order + 1] += ((cos_order - sin_order) * table[i][cos_order]) / (sin_order - 1); - table[i + 1][cos_order - 1] += (-cos_order * table[i][cos_order]) / (sin_order - 1); + int cos_order = 2 * column + offset; // order of the cosine term in entry "column" + BOOST_ASSERT(column < table[i].size()); + BOOST_ASSERT((cos_order + 1) / 2 < table[i + 1].size()); + table[i + 1][(cos_order + 1) / 2] += ((cos_order - sin_order) * table[i][column]) / (sin_order - 1); + if(cos_order) + table[i + 1][(cos_order - 1) / 2] += (-cos_order * table[i][column]) / (sin_order - 1); } } } - T sum = boost::math::tools::evaluate_polynomial(&table[index][0], boost::math::cos_pi(x, pol), n); + T sum = boost::math::tools::evaluate_even_polynomial(&table[index][0], c, table[index].size()); + if(index & 1) + sum *= c; // First coeffient is order 1, and really an odd polynomial. if(sum == 0) return sum; // @@ -463,7 +465,7 @@ init() { // Forces initialization of our table of coefficients and mutex: - boost::math::polygamma(30, T(-2.5), Policy()); + boost::math::polygamma(30, T(-2.5f), Policy()); } void force_instantiate()const{} };