geodesic.ipynb

Manifest.toml

[[Base64]]
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[[BinDeps]]
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[[ForwardDiff]]
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[[GR]]
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[[IJulia]]
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[[InteractiveUtils]]
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[[MbedTLS]]
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[[Measures]]
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[[Missings]]
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[[NaNMath]]
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[[PlotThemes]]
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[[Plots]]
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[[Reexport]]
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[[ZMQ]]
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Paths.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra, Printf, Statistics, ForwardDiff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "path2 (generic function with 5 methods)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function path2(logπ, M, T=100, ϵ=0.01, x=[sqrt(1/2), sqrt(1/2)], z=[1,0])\n",
    "    E = zeros(T)\n",
    "    \n",
    "    tol = sqrt(eps())\n",
    "        \n",
    "    Π = I - x*x'\n",
    "    A = Π * M * Π\n",
    "\n",
    "    Λ,U = eigen(A)\n",
    "    Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
    "    invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
    "\n",
    "    A⁺ = U*Diagonal(invΛ)*U'\n",
    "    v = U * (sqrt.(invΛ).*z)\n",
    "\n",
    "    e = -logπ(x) + v'*A*v/2\n",
    "\n",
    "    for t = 1:T\n",
    "        # 5\n",
    "        v += ϵ/2 * A⁺ * (ForwardDiff.gradient(logπ, x) + A⁺*(Π*M*x))\n",
    "        # 6\n",
    "        vv = U * Diagonal(sqrt.(Λ)) * U' * v\n",
    "        # 7\n",
    "        α = norm(vv)\n",
    "        nvv = vv / α            \n",
    "        x,nvv = cos(α*ϵ)*x+sin(α*ϵ)*nvv, -sin(α*ϵ)*x+cos(α*ϵ)*nvv\n",
    "        vv = nvv * α\n",
    "\n",
    "        # recompute quantities\n",
    "        Π = I - x*x'\n",
    "        A = Π * M * Π\n",
    "\n",
    "        Λ,U = eigen(A)\n",
    "        Λ = [abs(λ) < tol ? 0.0 : λ for λ in Λ]\n",
    "        invΛ = [abs(λ) < tol ? 0.0 : 1/λ for λ in Λ]\n",
    "\n",
    "        A⁺ = U*Diagonal(invΛ)*U'\n",
    "\n",
    "        # 8\n",
    "        v = U * Diagonal(sqrt.(invΛ)) * U' * vv\n",
    "        # 9\n",
    "        v += ϵ/2 * A⁺ * ( +A⁺*(Π*M*x))\n",
    "\n",
    "        ee = -logπ(x) + v'*A*v/2\n",
    "        E[t] = e-ee\n",
    "    end\n",
    "    E\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2150.43, 269.224)\" x=\"2150.43\" y=\"269.224\">y2</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = 400; ϵ = 0.01\n",
    "E = path2(x->0.0, [1.0 0.0; 0.0 4.0],T, ϵ)\n",
    "p = plot((1:T)*ϵ, E)\n",
    "\n",
    "T = 200; ϵ = 0.02\n",
    "E = path2(x->0.0, [1.0 0.0; 0.0 4.0],T, ϵ)\n",
    "plot!(p, (1:T)*ϵ, E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.0.1",
   "language": "julia",
   "name": "julia-1.0"
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   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
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Project.toml

[deps]
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"