Merge pull request #566 from stan-dev/feature/issue-210-numerical-int…

…egration

Feature/issue 210 numerical integration

stan/math/rev/mat.hpp

@@ -54,6 +54,7 @@
 #include <stan/math/rev/mat/fun/variance.hpp>
 #include <stan/math/rev/mat/functor/algebra_solver.hpp>
 #include <stan/math/rev/mat/functor/gradient.hpp>
+#include <stan/math/rev/mat/functor/integrate_1d_tsc.hpp>
 #include <stan/math/rev/mat/functor/jacobian.hpp>
 #include <stan/math/rev/mat/functor/ode_system.hpp>
 #include <stan/math/rev/mat/functor/cvodes_utils.hpp>

stan/math/rev/mat/functor/de_integrator.hpp

@@ -0,0 +1,147 @@
+/*
+ * The original code on which this file is based is from John Cook,
+ * which is licensed under the 2-clause BSD license, reproduced below:
+ *
+ * Copyright (c) 2015, John D. Cook
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ *
+ * Redistributions of source code must retain the above copyright
+ * notice, this list of conditions and the following disclaimer.
+ * Redistributions in binary form must reproduce the above copyright
+ * notice, this list of conditions and the following disclaimer in the
+ * documentation and/or other materials provided with the distribution.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+ * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+ * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
+ * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
+ * COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
+ * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
+ * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+ * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
+ * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
+ * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+ * POSSIBILITY OF SUCH DAMAGE.
+ *
+ */
+#ifndef STAN_MATH_REV_MAT_FUNCTOR_DEINTEGRATOR_HPP
+#define STAN_MATH_REV_MAT_FUNCTOR_DEINTEGRATOR_HPP
+
+#include <stan/math/rev/mat/functor/de_integrator_constants.hpp>
+#include <cmath>
+#include <cfloat>
+
+namespace stan {
+
+namespace math {
+
+/**
+ * Double Exponential Integrator.
+ *
+ * @tparam T Type of f.
+ * @param f a functor with signature double (double).
+ * @param a lower limit of integration, must be double type.
+ * @param b upper limit of integration, must be double type.
+ * @param tre target relative error.
+ * @param tae target absolute error.
+ * @return numeric integral of function f.
+ */
+template <typename F>
+inline double de_integrator(const F& f, double a, double b, double tre,
+                            double tae) {
+  using std::fabs;
+  using std::log;
+
+  // Apply the linear change of variables x = ct + d
+  // $$\int_a^b f(x) dx = c \int_{-1}^1 f( ct + d ) dt$$
+  // c = (b-a)/2, d = (a+b)/2
+  double c = 0.5 * (b - a);
+  double d = 0.5 * (a + b);
+  int num_function_evaluations;
+
+  tae /= c;
+
+  // Offsets to where each level's integration constants start.
+  // The last element is not a beginning but an end.
+  static int offsets[] = {1, 4, 7, 13, 25, 49, 97, 193};
+  int num_levels = sizeof(offsets) / sizeof(int) - 1;
+
+  double new_contribution = 0.0;
+  double integral = 0.0;
+  double previous_integral = 0.0;
+  double error_estimate = DBL_MAX;
+  double h = 1.0;
+  double previous_delta, current_delta = DBL_MAX;
+
+  integral = f(c * de_abcissas[0] + d) * de_weights[0];
+
+  int i;
+  for (i = offsets[0]; i != offsets[1]; ++i)
+    integral += de_weights[i]
+                * (f(c * de_abcissas[i] + d) + f(-c * de_abcissas[i] + d));
+
+  for (int level = 1; level != num_levels; ++level) {
+    h *= 0.5;
+    new_contribution = 0.0;
+    for (i = offsets[level]; i != offsets[level + 1]; ++i)
+      new_contribution
+          += de_weights[i]
+             * (f(c * de_abcissas[i] + d) + f(-c * de_abcissas[i] + d));
+    new_contribution *= h;
+
+    // difference in consecutive integral estimates
+    previous_delta = current_delta;
+    current_delta = fabs(0.5 * integral - new_contribution);
+    previous_integral = integral;
+    integral = 0.5 * integral + new_contribution;
+
+    // Once convergence kicks in, error is approximately squared
+    // at each step.
+    // Determine whether we're in the convergent region by looking
+    // at the trend in the error.
+    if (level == 1)
+      // previous_delta meaningless, so cannot check
+      // convergence.
+      continue;
+
+    // Exact comparison with zero is harmless here. Could possibly
+    // be replaced with a small positive upper limit on the size
+    // of current_delta, but determining that upper limit would be
+    // difficult. At worse, the loop is executed more times than
+    // necessary. But no infinite loop can result since there is
+    // an upper bound on the loop variable.
+    if (current_delta == 0.0)
+      break;
+    // previous_delta != 0 or would have been kicked out
+    // previously
+    double r = log(current_delta) / log(previous_delta);
+
+    if (r > 1.9 && r < 2.1) {
+      // If convergence theory applied perfectly, r would be 2 in
+      // the convergence region. r close to 2 is good enough. We
+      // expect the difference between this integral estimate and
+      // the next one to be roughly delta^2.
+      error_estimate = current_delta * current_delta;
+    } else {
+      // Not in the convergence region. Assume only that error is
+      // decreasing.
+      error_estimate = current_delta;
+    }
+
+    if (error_estimate < 0.1 * tae
+        && fabs(1 - integral / previous_integral) < tre)
+      break;
+  }
+
+  num_function_evaluations = 2 * i - 1;
+  error_estimate *= c;
+  return c * integral;
+}
+}  // namespace math
+}  // namespace stan
+#endif

stan/math/rev/mat/functor/de_integrator_constants.hpp

@@ -0,0 +1,202 @@
+/*
+ * The original code on which this file is based is from John Cook,
+ * which is licensed under the 2-clause BSD license, reproduced below:
+ *
+ * Copyright (c) 2015, John D. Cook
+ * All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions are met:
+ *
+ * Redistributions of source code must retain the above copyright notice, this
+ * list of conditions and the following disclaimer. Redistributions in binary
+ * form must reproduce the above copyright notice, this list of conditions and
+ * the following disclaimer in the documentation and/or other materials provided
+ * with the distribution.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+ * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+ * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+ * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+ * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+ * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+ * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+ * POSSIBILITY OF SUCH DAMAGE.
+ *
+ */
+#ifndef STAN_MATH_REV_MAT_FUNCTOR_DEINTEGRATOR_CONSTANTS_HPP
+#define STAN_MATH_REV_MAT_FUNCTOR_DEINTEGRATOR_CONSTANTS_HPP
+
+namespace stan {
+
+namespace math {
+
+static const double de_abcissas[] = {
+    // 1st layer abcissas: transformed 0, 1, 2, 3
+    0.00000000000000000000, 0.95136796407274694573, 0.99997747719246159286,
+    0.99999999999995705839,
+    // 2nd layer abcissas: transformed 1/2, 3/2, 5/2
+    0.67427149224843582608, 0.99751485645722438683, 0.99999998887566488198,
+    // 3rd layer abcissas: transformed 1/4, 3/4, ...
+    0.37720973816403417379, 0.85956905868989663517, 0.98704056050737689169,
+    0.99968826402835320905, 0.99999920473711471266, 0.99999999995285644818,
+    // 4th layer abcissas: transformed 1/8, 3/8, ...
+    0.19435700332493543161, 0.53914670538796776905, 0.78060743898320029925,
+    0.91487926326457461091, 0.97396686819567744856, 0.99405550663140214329,
+    0.99906519645578584642, 0.99990938469514399984, 0.99999531604122052843,
+    0.99999989278161241838, 0.99999999914270509218, 0.99999999999823216531,
+    // 5th layer abcissa: transformed 1/16, 3/16, ...
+    0.097923885287832333262, 0.28787993274271591456, 0.46125354393958570440,
+    0.61027365750063894488, 0.73101803479256151149, 0.82331700550640237006,
+    0.88989140278426019808, 0.93516085752198468323, 0.96411216422354729193,
+    0.98145482667733517003, 0.99112699244169880223, 0.99610866543750854254,
+    0.99845420876769773751, 0.99945143443527460584, 0.99982882207287494166,
+    0.99995387100562796075, 0.99998948201481850361, 0.99999801714059543208,
+    0.99999969889415261122, 0.99999996423908091534, 0.99999999678719909830,
+    0.99999999978973286224, 0.99999999999039393352, 0.99999999999970809734,
+    // 6th layer abcissas: transformed 1/32, 3/32, ...
+    0.049055967305077886315, 0.14641798429058794053, 0.24156631953888365838,
+    0.33314226457763809244, 0.41995211127844715849, 0.50101338937930910152,
+    0.57558449063515165995, 0.64317675898520470128, 0.70355000514714201566,
+    0.75669390863372994941, 0.80279874134324126576, 0.84221924635075686382,
+    0.87543539763040867837, 0.90301328151357387064, 0.92556863406861266645,
+    0.94373478605275715685, 0.95813602271021369012, 0.96936673289691733517,
+    0.97797623518666497298, 0.98445883116743083087, 0.98924843109013389601,
+    0.99271699719682728538, 0.99517602615532735426, 0.99688031812819187372,
+    0.99803333631543375402, 0.99879353429880589929, 0.99928111192179195541,
+    0.99958475035151758732, 0.99976797159956083506, 0.99987486504878034648,
+    0.99993501992508242369, 0.99996759306794345976, 0.99998451990227082442,
+    0.99999293787666288565, 0.99999693244919035751, 0.99999873547186590954,
+    0.99999950700571943689, 0.99999981889371276701, 0.99999993755407837378,
+    0.99999997987450320175, 0.99999999396413420165, 0.99999999832336194826,
+    0.99999999957078777261, 0.99999999989927772326, 0.99999999997845533741,
+    0.99999999999582460688, 0.99999999999927152627, 0.99999999999988636130,
+    // 7th layer abcissas: transformed 1/64, 3/64, ...
+    0.024539763574649160379, 0.073525122985671294475, 0.12222912220155764235,
+    0.17046797238201051811, 0.21806347346971200463, 0.26484507658344795046,
+    0.31065178055284596083, 0.35533382516507453330, 0.39875415046723775644,
+    0.44078959903390086627, 0.48133184611690504422, 0.52028805069123015958,
+    0.55758122826077823080, 0.59315035359195315880, 0.62695020805104287950,
+    0.65895099174335012438, 0.68913772506166767176, 0.71750946748732412721,
+    0.74407838354734739913, 0.76886868676824658459, 0.79191549237614211447,
+    0.81326360850297385168, 0.83296629391941087564, 0.85108400798784873261,
+    0.86768317577564598669, 0.88283498824466895513, 0.89661425428007602579,
+    0.90909831816302043511, 0.92036605303195280235, 0.93049693799715340631,
+    0.93957022393327475539, 0.94766419061515309734, 0.95485549580502268541,
+    0.96121861515111640753, 0.96682537031235585284, 0.97174454156548730892,
+    0.97604156025657673933, 0.97977827580061576265, 0.98301279148110110558,
+    0.98579936302528343597, 0.98818835380074264243, 0.99022624046752774694,
+    0.99195566300267761562, 0.99341551316926403900, 0.99464105571251119672,
+    0.99566407681695316965, 0.99651305464025377317, 0.99721334704346870224,
+    0.99778739195890653083, 0.99825491617199629344, 0.99863314864067747762,
+    0.99893703483351217373, 0.99917944893488591716, 0.99937140114093768690,
+    0.99952223765121720422, 0.99963983134560036519, 0.99973076151980848263,
+    0.99980048143113838630, 0.99985347277311141171, 0.99989338654759256426,
+    0.99992317012928932869, 0.99994518061445869309, 0.99996128480785666613,
+    0.99997294642523223656, 0.99998130127012072679, 0.99998722128200062811,
+    0.99999136844834487344, 0.99999423962761663478, 0.99999620334716617675,
+    0.99999752962380516793, 0.99999841381096473542, 0.99999899541068996962,
+    0.99999937270733536947, 0.99999961398855024275, 0.99999976602333243312,
+    0.99999986037121459941, 0.99999991800479471056, 0.99999995264266446185,
+    0.99999997311323594362, 0.99999998500307631173, 0.99999999178645609907,
+    0.99999999558563361584, 0.99999999767323673790, 0.99999999879798350040,
+    0.99999999939177687583, 0.99999999969875436925, 0.99999999985405611550,
+    0.99999999993088839501, 0.99999999996803321674, 0.99999999998556879008,
+    0.99999999999364632387, 0.99999999999727404948, 0.99999999999886126543,
+    0.99999999999953722654, 0.99999999999981720098, 0.99999999999992987953};
+
+static const double de_weights[] = {
+    // First layer weights
+    1.5707963267948966192, 0.230022394514788685, 0.00026620051375271690866,
+    1.3581784274539090834e-12,
+    // 2nd layer weights
+    0.96597657941230114801, 0.018343166989927842087, 2.1431204556943039358e-7,
+    // 3rd layer weights
+    1.3896147592472563229, 0.53107827542805397476, 0.076385743570832304188,
+    0.0029025177479013135936, 0.000011983701363170720047,
+    1.1631165814255782766e-9,
+    // 4th layer weights
+    1.5232837186347052132, 1.1934630258491569639, 0.73743784836154784136,
+    0.36046141846934367417, 0.13742210773316772341, 0.039175005493600779072,
+    0.0077426010260642407123, 0.00094994680428346871691,
+    0.000062482559240744082891, 1.8263320593710659699e-6,
+    1.8687282268736410132e-8, 4.9378538776631926964e-11,
+    //  5th layer weights
+    1.5587733555333301451, 1.466014426716965781, 1.297475750424977998,
+    1.0816349854900704074, 0.85017285645662006895, 0.63040513516474369106,
+    0.44083323627385823707, 0.290240679312454185, 0.17932441211072829296,
+    0.10343215422333290062, 0.055289683742240583845, 0.027133510013712003219,
+    0.012083543599157953493, 0.0048162981439284630173, 0.0016908739981426396472,
+    0.00051339382406790336017, 0.00013205234125609974879,
+    0.000028110164327940134749, 4.8237182032615502124e-6,
+    6.4777566035929719908e-7, 6.5835185127183396672e-8,
+    4.8760060974240625869e-9, 2.5216347918530148572e-10,
+    8.6759314149796046502e-12,
+    // 6th layer weights
+    1.5677814313072218572, 1.5438811161769592204, 1.4972262225410362896,
+    1.4300083548722996676, 1.3452788847662516615, 1.2467012074518577048,
+    1.1382722433763053734, 1.0240449331118114483, 0.90787937915489531693,
+    0.79324270082051671787, 0.68306851634426375464, 0.57967810308778764708,
+    0.48475809121475539287, 0.39938474152571713515, 0.32408253961152890402,
+    0.258904639514053516, 0.20352399885860174519, 0.15732620348436615027,
+    0.11949741128869592428, 0.089103139240941462841, 0.065155533432536205042,
+    0.046668208054846613644, 0.032698732726609031113, 0.022379471063648476483,
+    0.014937835096050129696, 0.0097072237393916892692, 0.0061300376320830301252,
+    0.0037542509774318343023, 0.0022250827064786427022,
+    0.0012733279447082382027, 0.0007018595156842422708,
+    0.00037166693621677760301, 0.00018856442976700318572,
+    0.000091390817490710122732, 0.000042183183841757600604,
+    0.000018481813599879217116, 7.6595758525203162562e-6,
+    2.9916615878138787094e-6, 1.0968835125901264732e-6,
+    3.7595411862360630091e-7, 1.1992442782902770219e-7,
+    3.5434777171421953043e-8, 9.6498888961089633609e-9,
+    2.4091773256475940779e-9, 5.482835779709497755e-10,
+    1.1306055347494680536e-10, 2.0989335404511469109e-11,
+    3.4841937670261059685e-12,
+    // 7th layer weights
+    1.5700420292795931467, 1.5640214037732320999, 1.5520531698454121192,
+    1.5342817381543034316, 1.5109197230741697127, 1.48224329788553807,
+    1.4485862549613225916, 1.4103329714462590129, 1.3679105116808964881,
+    1.3217801174437728579, 1.2724283455378627082, 1.2203581095793582207,
+    1.1660798699324345766, 1.1101031939653403796, 1.0529288799552666556,
+    0.99504180404613271514, 0.93690461274566793366, 0.87895234555278212039,
+    0.82158803526696470334, 0.7651792989089561367, 0.71005590120546898385,
+    0.65650824613162753076, 0.60478673057840362158, 0.55510187800363350959,
+    0.5076251588319080997, 0.4624903980553677613, 0.41979566844501548066,
+    0.37960556938665160999, 0.3419537959230168323, 0.30684590941791694932,
+    0.27426222968906810637, 0.24416077786983990868, 0.21648020911729617038,
+    0.19114268413342749532, 0.16805663794826916233, 0.14711941325785693248,
+    0.12821973363120098675, 0.11123999898874453035, 0.096058391865189467849,
+    0.082550788110701737654, 0.070592469906866999352, 0.060059642358636300319,
+    0.05083075757257047107, 0.042787652157725676034, 0.035816505604196436523,
+    0.029808628117310126969, 0.024661087314753282511, 0.020277183817500123926,
+    0.016566786254247575375, 0.013446536605285730674, 0.010839937168255907211,
+    0.0086773307495391815854, 0.0068957859690660035329,
+    0.0054388997976239984331, 0.0042565295990178580165,
+    0.0033044669940348302363, 0.0025440657675291729678,
+    0.0019418357759843675814, 0.0014690143599429791058,
+    0.0011011261134519383862, 0.00081754101332469493115,
+    0.00060103987991147422573, 0.00043739495615911687786,
+    0.00031497209186021200274, 0.00022435965205008549104,
+    0.00015802788400701191949, 0.00011002112846666697224,
+    0.000075683996586201477788, 0.000051421497447658802092,
+    0.0000344921247593431977, 0.000022832118109036146591,
+    0.000014908514031870608449, 9.5981941283784710776e-6,
+    6.0899100320949039256e-6, 3.8061983264644899045e-6,
+    2.3421667208528096843e-6, 1.4183067155493917523e-6,
+    8.4473756384859863469e-7, 4.9458288702754198508e-7,
+    2.8449923659159806339e-7, 1.6069394579076224911e-7,
+    8.9071395140242387124e-8, 4.8420950198072369669e-8, 2.579956822953589238e-8,
+    1.3464645522302038796e-8, 6.8784610955899001111e-9,
+    3.4371856744650090511e-9, 1.6788897682161906807e-9,
+    8.0099784479729665356e-10, 3.7299501843052790038e-10,
+    1.6939457789411646876e-10, 7.4967397573818224522e-11,
+    3.230446433325236576e-11, 1.3542512912336274432e-11,
+    5.5182369468174885821e-12, 2.1835922099233609052e-12};
+}  // namespace math
+}  // namespace stan
+
+#endif