From 7b09710392d35fb55b52031d447a542d99fc6b4b Mon Sep 17 00:00:00 2001 From: Tavian Barnes Date: Tue, 19 Aug 2014 17:10:03 -0400 Subject: Modularize the libdimension codebase. --- libdimension/math/polynomial.c | 443 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 443 insertions(+) create mode 100644 libdimension/math/polynomial.c (limited to 'libdimension/math/polynomial.c') diff --git a/libdimension/math/polynomial.c b/libdimension/math/polynomial.c new file mode 100644 index 0000000..09e9603 --- /dev/null +++ b/libdimension/math/polynomial.c @@ -0,0 +1,443 @@ +/************************************************************************* + * Copyright (C) 2010-2011 Tavian Barnes * + * * + * This file is part of The Dimension Library. * + * * + * The Dimension Library is free software; you can redistribute it and/ * + * or modify it under the terms of the GNU Lesser General Public License * + * as published by the Free Software Foundation; either version 3 of the * + * License, or (at your option) any later version. * + * * + * The Dimension Library 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 * + * Lesser General Public License for more details. * + * * + * You should have received a copy of the GNU Lesser General Public * + * License along with this program. If not, see * + * . * + *************************************************************************/ + +/** + * @file + * Real root isolation algorithm based on work by Vincent, Uspensky, Collins and + * Akritas, Johnson, Krandick, and Rouillier and Zimmerman. + */ + +#include "internal.h" +#include "internal/polynomial.h" +#include "dimension/math.h" +#include + +/// Get the real degree of a polynomial, ignoring leading zeros. +static inline size_t +dmnsn_real_degree(const double poly[], size_t degree) +{ + for (size_t i = degree + 1; i-- > 0;) { + if (dmnsn_likely(fabs(poly[i]) >= dmnsn_epsilon)) { + return i; + } + } + + return 0; +} + +/// Divide each coefficient by the leading coefficient. +static inline void +dmnsn_polynomial_normalize(double poly[], size_t degree) +{ + for (size_t i = 0; i < degree; ++i) { + poly[i] /= poly[degree]; + } + poly[degree] = 1.0; +} + +/// Eliminate trivial zero roots from \p poly[]. +static inline void +dmnsn_eliminate_zero_roots(double **poly, size_t *degree) +{ + size_t i; + for (i = 0; i <= *degree; ++i) { + if (dmnsn_likely(fabs((*poly)[i]) >= dmnsn_epsilon)) { + break; + } + } + + *poly += i; + *degree -= i; +} + +/// Calculate a finite upper bound on the roots of a normalized polynomial. +static inline double +dmnsn_root_bound(const double poly[], size_t degree) +{ + double bound = fabs(poly[0]); + for (size_t i = 1; i < degree; ++i) { + bound = dmnsn_max(bound, fabs(poly[i])); + } + bound += 1.0; + return bound; +} + +/// Copy a polynomial. +static inline void +dmnsn_polynomial_copy(double dest[], const double src[], size_t degree) +{ + for (size_t i = 0; i <= degree; ++i) { + dest[i] = src[i]; + } +} + +/// Transform a polynomial by P'(x) = P(x + 1). +static inline void +dmnsn_polynomial_translate(double poly[], size_t degree) +{ + for (size_t i = 0; i <= degree; ++i) { + for (size_t j = degree - i; j <= degree - 1; ++j) { + poly[j] += poly[j + 1]; + } + } +} + +/// Transform a polynomial by P'(x) = P(c*x). +static inline void +dmnsn_polynomial_scale(double poly[], size_t degree, double c) +{ + double factor = c; + for (size_t i = 1; i <= degree; ++i) { + poly[i] *= factor; + factor *= c; + } +} + +/// Returns the result of Descartes' rule on x^degree * poly(1/(x + 1)). +static size_t +dmnsn_descartes_bound(const double poly[], size_t degree) +{ + // Copy the polynomial so we can be destructive + double p[degree + 1]; + dmnsn_polynomial_copy(p, poly, degree); + + // Calculate poly(1/(1/x + 1)) which avoids reversal + for (size_t i = 1; i <= degree; ++i) { + for (size_t j = i; j >= 1; --j) { + p[j] += p[j - 1]; + } + } + + // Find the number of sign changes in p[] + size_t changes = 0; + int lastsign = dmnsn_sgn(p[0]); + for (size_t i = 1; changes <= 1 && i <= degree; ++i) { + int sign = dmnsn_sgn(p[i]); + if (sign != 0 && sign != lastsign) { + ++changes; + lastsign = sign; + } + } + + return changes; +} + +/// Depth-first search of possible isolating intervals. +static size_t +dmnsn_root_bounds_recursive(double poly[], size_t degree, double *c, double *k, + double bounds[][2], size_t nbounds) +{ + size_t s = dmnsn_descartes_bound(poly, degree); + if (s >= 2) { + // Get the left child + dmnsn_polynomial_scale(poly, degree, 1.0/2.0); + *c *= 2.0; + *k /= 2.0; + double currc = *c, currk = *k; + + // Test the left child + size_t n = dmnsn_root_bounds_recursive(poly, degree, c, k, bounds, nbounds); + if (nbounds == n) { + return n; + } + bounds += n; + nbounds -= n; + + // Get the right child from the last tested polynomial + dmnsn_polynomial_translate(poly, degree); + dmnsn_polynomial_scale(poly, degree, currk/(*k)); + *c = currc + 1.0; + *k = currk; + + // Test the right child + n += dmnsn_root_bounds_recursive(poly, degree, c, k, bounds, nbounds); + return n; + } else if (s == 1) { + bounds[0][0] = (*c)*(*k); + bounds[0][1] = (*c + 1.0)*(*k); + return 1; + } else { + return 0; + } +} + +/// Find ranges that contain a single root. +static size_t +dmnsn_root_bounds(const double poly[], size_t degree, double bounds[][2], + size_t nbounds) +{ + // Copy the polynomial so we can be destructive + double p[degree + 1]; + dmnsn_polynomial_copy(p, poly, degree); + + // Scale the roots to within (0, 1] + double bound = dmnsn_root_bound(p, degree); + dmnsn_polynomial_scale(p, degree, bound); + + // Bounding intervals are of the form (c*k, (c + 1)*k) + double c = 0.0, k = 1.0; + + // Isolate the roots + size_t n = dmnsn_root_bounds_recursive(p, degree, &c, &k, bounds, nbounds); + + // Scale the roots back to within (0, bound] + for (size_t i = 0; i < n; ++i) { + bounds[i][0] *= bound; + bounds[i][1] *= bound; + } + + return n; +} + +/// Maximum number of iterations in dmnsn_bisect_root() before bailout. +#define DMNSN_BISECT_ITERATIONS 64 + +/// Use the false position method to find a root in a range that contains +/// exactly one root. +static inline double +dmnsn_bisect_root(const double poly[], size_t degree, double min, double max) +{ + double evmin = dmnsn_polynomial_evaluate(poly, degree, min); + double evmax = dmnsn_polynomial_evaluate(poly, degree, max); + + // Handle equal bounds, and equal values at the bounds. + if (dmnsn_unlikely(fabs(evmax - evmin) < dmnsn_epsilon)) { + return (min + max)/2.0; + } + + double evinitial = dmnsn_min(fabs(evmin), fabs(evmax)); + double mid, evmid; + int lastsign = 0; + + for (size_t i = 0; i < DMNSN_BISECT_ITERATIONS; ++i) { + mid = (min*evmax - max*evmin)/(evmax - evmin); + evmid = dmnsn_polynomial_evaluate(poly, degree, mid); + int sign = dmnsn_sgn(evmid); + + if ((fabs(evmid) < fabs(mid)*dmnsn_epsilon + // This condition improves stability when one of the bounds is close to + // a different root than we are trying to find + && fabs(evmid) <= evinitial) + || max - min < fabs(mid)*dmnsn_epsilon) + { + break; + } + + if (mid < min) { + // This can happen due to numerical instability in the root bounding + // algorithm, so behave like the normal secant method + max = min; + evmax = evmin; + min = mid; + evmin = evmid; + } else if (mid > max) { + min = max; + evmin = evmax; + max = mid; + evmax = evmid; + } else if (sign == dmnsn_sgn(evmax)) { + max = mid; + evmax = evmid; + if (sign == lastsign) { + // Don't allow the algorithm to keep the same endpoint for three + // iterations in a row; this ensures superlinear convergence + evmin /= 2.0; + } + } else { + min = mid; + evmin = evmid; + if (sign == lastsign) { + evmax /= 2.0; + } + } + + lastsign = sign; + } + + return mid; +} + +/// Use synthetic division to eliminate the root \p r from \p poly[]. +static inline size_t +dmnsn_eliminate_root(double poly[], size_t degree, double r) +{ + double rem = poly[degree]; + for (size_t i = degree; i-- > 0;) { + double temp = poly[i]; + poly[i] = rem; + rem = temp + r*rem; + } + return degree - 1; +} + +/// Solve a normalized linear polynomial algebraically. +static inline size_t +dmnsn_solve_linear(const double poly[2], double x[1]) +{ + x[0] = -poly[0]; + if (x[0] >= dmnsn_epsilon) + return 1; + else + return 0; +} + +/// Solve a normalized quadratic polynomial algebraically. +static inline size_t +dmnsn_solve_quadratic(const double poly[3], double x[2]) +{ + double disc = poly[1]*poly[1] - 4.0*poly[0]; + if (disc >= 0.0) { + double s = sqrt(disc); + x[0] = (-poly[1] + s)/2.0; + x[1] = (-poly[1] - s)/2.0; + + if (x[1] >= dmnsn_epsilon) + return 2; + else if (x[0] >= dmnsn_epsilon) + return 1; + else + return 0; + } else { + return 0; + } +} + +/// Solve a normalized cubic polynomial algebraically. +static inline size_t +dmnsn_solve_cubic(double poly[4], double x[3]) +{ + // Reduce to a monic trinomial (t^3 + p*t + q, t = x + b/3) + double b2 = poly[2]*poly[2]; + double p = poly[1] - b2/3.0; + double q = poly[0] - poly[2]*(9.0*poly[1] - 2.0*b2)/27.0; + + double disc = 4.0*p*p*p + 27.0*q*q; + double bdiv3 = poly[2]/3.0; + + if (disc < 0.0) { + // Three real roots -- this implies p < 0 + double msqrtp3 = -sqrt(-p/3.0); + double theta = acos(3*q/(2*p*msqrtp3))/3.0; + + // Store the roots in order from largest to smallest + x[2] = 2.0*msqrtp3*cos(theta) - bdiv3; + x[0] = -2.0*msqrtp3*cos(4.0*atan(1.0)/3.0 - theta) - bdiv3; + x[1] = -(x[0] + x[2] + poly[2]); + + if (x[2] >= dmnsn_epsilon) + return 3; + else if (x[1] >= dmnsn_epsilon) + return 2; + } else if (disc > 0.0) { + // One real root + double cbrtdiscq = cbrt(sqrt(disc/108.0) + fabs(q)/2.0); + double abst = cbrtdiscq - p/(3.0*cbrtdiscq); + + if (q >= 0) { + x[0] = -abst - bdiv3; + } else { + x[0] = abst - bdiv3; + } + } else if (fabs(p) < dmnsn_epsilon) { + // Equation is a perfect cube + x[0] = -bdiv3; + } else { + // Two real roots; one duplicate + double t1 = -(3.0*q)/(2.0*p), t2 = -2.0*t1; + x[0] = dmnsn_max(t1, t2) - bdiv3; + x[1] = dmnsn_min(t1, t2) - bdiv3; + if (x[1] >= dmnsn_epsilon) + return 2; + } + + if (x[0] >= dmnsn_epsilon) + return 1; + else + return 0; +} + +// Solve a polynomial +DMNSN_HOT size_t +dmnsn_polynomial_solve(const double poly[], size_t degree, double x[]) +{ + // Copy the polynomial so we can be destructive + double copy[degree + 1], *p = copy; + dmnsn_polynomial_copy(p, poly, degree); + + // Index into x[] + size_t i = 0; + + // Account for leading zero coefficients + degree = dmnsn_real_degree(p, degree); + // Normalize the leading coefficient to 1.0 + dmnsn_polynomial_normalize(p, degree); + // Eliminate simple zero roots + dmnsn_eliminate_zero_roots(&p, °ree); + + static const size_t max_algebraic = 3; + if (degree > max_algebraic) { + // Find isolating intervals for (degree - max_algebraic) roots of p[] + double ranges[degree - max_algebraic][2]; + size_t n = dmnsn_root_bounds(p, degree, ranges, degree - max_algebraic); + + for (size_t j = 0; j < n; ++j) { + // Bisect within the found range + double r = dmnsn_bisect_root(p, degree, ranges[j][0], ranges[j][1]); + + // Use synthetic division to eliminate the root `r' + degree = dmnsn_eliminate_root(p, degree, r); + + // Store the found root + x[i] = r; + ++i; + } + } + + switch (degree) { + case 1: + i += dmnsn_solve_linear(p, x + i); + break; + case 2: + i += dmnsn_solve_quadratic(p, x + i); + break; + case 3: + i += dmnsn_solve_cubic(p, x + i); + break; + } + + return i; +} + +// Print a polynomial +void +dmnsn_polynomial_print(FILE *file, const double poly[], size_t degree) +{ + for (size_t i = degree + 1; i-- > 0;) { + if (i < degree) { + fprintf(file, (poly[i] >= 0.0) ? " + " : " - "); + } + fprintf(file, "%.17g", fabs(poly[i])); + if (i >= 2) { + fprintf(file, "*x^%zu", i); + } else if (i == 1) { + fprintf(file, "*x"); + } + } +} -- cgit v1.2.3