Evaluate_polynomials.ipynb

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   "source": [
    "# What shall we learn in this course?\n",
    "\n",
    "\n",
    "We will learn some algorithms and analysis that address the following questions: \n",
    "\n",
    "\n",
    "How to solve equations using computers?\n",
    "\n",
    "solve --> approximately solve\n",
    "\n",
    "* How accurate?\n",
    "\n",
    "* How expensive?\n",
    "\n",
    "    How fast?\n",
    "\n",
    "    How much resource?\n",
    "    \n",
    "* How robust is the algorithm used to compute approximate solutions?\n",
    "    \n",
    "    Stability of the comptued solutions against different sorts of perturbations.\n",
    "\n",
    "Make decision on which algorithm to use.\n",
    "\n",
    "Decide if we trust the computed results.\n",
    "\n",
    "At the beginning of the semester, we will start with some examples to illustrate some essential concepts that we shall learn."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 1: Operation counts\n",
    "\n",
    "In numerical analysis and scientific computing, we need to know how \"expensive\" it is to run certain algorithm.\n",
    "\n",
    "* \"Expensiveness\" (cost) may bedefined in many different ways.\n",
    "* Here we show an example of the number of \"floating point operations\" relative to the size of data, using in evaluating a polynomial.\n",
    "\n",
    "\n",
    "## The big O notation\n",
    "\n",
    "This is a notation for asymptotics. It tells how fast a quantity goes to $\\infty$ or $0$ compared/relative to a parameter's going to $infty$ or $0$.\n",
    "\n",
    "Typically, you will see two sitations:\n",
    "\n",
    "* for the computation cost of an algorithm relative to the given (large) number of variables ($N$)\n",
    "\n",
    "$$O(N^p),~~~N\\rightarrow\\infty$$\n",
    "\n",
    "or\n",
    "\n",
    "* for the accuracy of an algorithm relative to a small parameter $\\epsilon$, which could be $1/N$, where $N$ is the number of variables involved in the algorithm:\n",
    "\n",
    "$$O(\\epsilon^q),~~~\\epsilon\\rightarrow 0$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evaluating polynomials\n",
    "\n",
    "$$p(x)= a_3 x^3 + a_2 x^2 + a_1 x +a_0$$\n",
    "\n",
    "What is the most efficient way to evalute polynomials?\n",
    "Obviosuly, we shall not evaluate $x^3$, $x^2$, $x$ separately.\n",
    "\n",
    "We introduce Horner's method that uses *nested multiplications*.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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     "outputs_hidden": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "p (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [1.0 2.0 3.0 1.0134]\n",
    "\n",
    "\n",
    "#the most straightforward way\n",
    "\n",
    "p(x)= a[4]*x.^3 .+ a[3]*x.^2 .+ a[2]*x .+ a[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×3 Array{Float64,2}:\n",
       " 9.47116  2.54713  -60.675"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "(9.4711552, 2.5471254655999997, -60.67500000000001)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x=1\n",
    "\n",
    "p(x)\n",
    "\n",
    "X = [ 1.2 0.44 -5.0]\n",
    "display(p(X))\n",
    "\n",
    "p(X[1]), p(X[2]), p(X[3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "p_smart (generic function with 1 method)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_smart(x) = ((a[4].*x.+a[3]).*x.+a[2]).*x.+a[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000370 seconds (22 allocations: 781.828 KiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "100000-element Array{Float64,1}:\n",
       " 2.900676407894041 \n",
       " 2.6376113085835566\n",
       " 1.623173751439899 \n",
       " 2.002224831971807 \n",
       " 2.8392150084222525\n",
       " 1.193834103018485 \n",
       " 1.2182205568993223\n",
       " 6.31760781807919  \n",
       " 1.8350299142065243\n",
       " 6.655691238112637 \n",
       " 2.3729633593039368\n",
       " 3.923919962957055 \n",
       " 2.936000673525715 \n",
       " ⋮                 \n",
       " 1.2659881546982608\n",
       " 5.391745086770009 \n",
       " 1.5784565302652234\n",
       " 6.084485705511977 \n",
       " 1.825791593313085 \n",
       " 2.1378120111586987\n",
       " 1.145936531817891 \n",
       " 2.677726580522279 \n",
       " 2.3593877455516488\n",
       " 2.8955555014146315\n",
       " 5.64643546734121  \n",
       " 2.3841316416999483"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time p_smart(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Homeowork: write a general function that evaluate polynomials efficiently for each of the elements in the input vector x\n",
    "Polynomials of the follow form:\n",
    "$$ P(x)= a_{n+1}x^n+a_{n}x^{n-1}+\\cdots+a_1x+a_1$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "jupyter": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "P (generic function with 1 method)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function P( a::Array{Float64}, x::Array{Float64})\n",
    "    \n",
    "    val=copy(x)\n",
    "    val.=a[end]\n",
    "    \n",
    "    for k in length(a)-1:-1:1\n",
    "        val .= (val.*x.+a[k])\n",
    "    end\n",
    "    \n",
    "    return val\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "jupyter": {
     "outputs_hidden": false
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.000644 seconds (6 allocations: 781.484 KiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "100000-element Array{Float64,1}:\n",
       " 2.900676407894041 \n",
       " 2.6376113085835566\n",
       " 1.623173751439899 \n",
       " 2.002224831971807 \n",
       " 2.8392150084222525\n",
       " 1.193834103018485 \n",
       " 1.2182205568993223\n",
       " 6.31760781807919  \n",
       " 1.8350299142065243\n",
       " 6.655691238112637 \n",
       " 2.3729633593039368\n",
       " 3.923919962957055 \n",
       " 2.936000673525715 \n",
       " ⋮                 \n",
       " 1.2659881546982608\n",
       " 5.391745086770009 \n",
       " 1.5784565302652234\n",
       " 6.084485705511977 \n",
       " 1.825791593313085 \n",
       " 2.1378120111586987\n",
       " 1.145936531817891 \n",
       " 2.677726580522279 \n",
       " 2.3593877455516488\n",
       " 2.8955555014146315\n",
       " 5.64643546734121  \n",
       " 2.3841316416999483"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@time P(a,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 2: Floating point numbers and loss of significant digits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Another example: solving a quadratic equation\n",
    "\n",
    "$$ ax^2+bx+c=0 $$\n",
    "\n",
    "Let's implement the formula:\n",
    "\n",
    "$$x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$\n",
    "\n",
    "The problem with this formula is that when $b^2$ is very large compared to $4ac$, we may lose a lot of information in computing $b^2-4ac$ due to *round off errors*.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "roots (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots(a,b,c) =  ((-b+sqrt(b^2-4a*c))/(2a), (-b-sqrt(b^2-4a*c))/(2a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "2.82429536481e11"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/plain": [
       "2.82429536481e11"
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     "metadata": {},
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   ],
   "source": [
    "a, b, c = 1.0, 9.0^12, -3.0\n",
    "\n",
    "Q(x)=a*x.^2 .+ b*x .+ c\n",
    "\n",
    "r1, r2 =roots(a,b,c)\n",
    "\n",
    "#println(\"The residual: \",Q([r1,r2]))\n",
    "\n",
    "display(r1)\n",
    "\n",
    "display(sqrt(b^2-4a*c))\n",
    "display(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.976644307687251e22"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "-12.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "7.976644307687251e22"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(b^2)\n",
    "display(4a*c)\n",
    "\n",
    "display(b^2-4a*c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0, 2.82429536481e+11, -3.0)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A,B,C = big.( (a,b,c) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.062211848441644930860055321254373665493827609316243536458177956524440072302496e-11"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R1, R2=roots(A,B,C)\n",
    "\n",
    "Q([R1, R2])\n",
    "\n",
    "R1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.062211848441645e-11"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "improved_root1(a,b,c) =  (-2c)/(b+sqrt(b^2-4a*c))\n",
    "\n",
    "improved_root1(a,b,c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An example about stability of an algorithm\n",
    "\n",
    "We solve $$ x^\\prime = \\lambda x,~~~x(0)=x_0.$$\n",
    "\n",
    "The solution is $ x(t)=x_0 e^{-2t}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "ename": "ErrorException",
     "evalue": "cannot define function x_exact; it already has a value",
     "output_type": "error",
     "traceback": [
      "cannot define function x_exact; it already has a value",
      "",
      "Stacktrace:",
      " [1] top-level scope at none:0",
      " [2] top-level scope at In[105]:1"
     ]
    }
   ],
   "source": [
    "x_exact(x0,T; λ=-1.0) = x0*exp.(λ*T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fe_steps (generic function with 1 method)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fe_steps(x0, Δt, nsteps; λ=-1.0)\n",
    "   \n",
    "    x=zeros(nsteps+1)\n",
    "    t=zeros(nsteps+1)\n",
    "    \n",
    "    x[1]=x0\n",
    "    t[1]=0.0\n",
    "    \n",
    "    for k=1:nsteps \n",
    "        x[k+1]=x[k]+Δt*(λ*x[k])\n",
    "        t[k+1]=t[k]+Δt\n",
    "    end\n",
    "    \n",
    "    return (x, t)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "be_steps (generic function with 1 method)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function be_steps(x0, Δt, nsteps; λ=-1.0)\n",
    "   \n",
    "    x=zeros(nsteps+1)\n",
    "    t=zeros(nsteps+1)\n",
    "    \n",
    "    x[1]=x0\n",
    "    t[1]=0.0\n",
    "    \n",
    "    for k=1:nsteps \n",
    "        x[k+1]=x[k]/(1-Δt*λ)\n",
    "        t[k+1]=t[k]+Δt\n",
    "    end\n",
    "    \n",
    "    return (x, t)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "([10.1, 9.898, 9.70004, 9.5060392, 9.315918416, 9.12960004768, 8.9470080467264, 8.768067885791872, 8.592706528076034, 8.420852397514514, 8.252435349564223], [0.0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.12000000000000001, 0.14, 0.16, 0.18, 0.19999999999999998])"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X,T=fe_steps(10.1, 0.02, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "-0.0020066003982286418"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using PyPlot\n",
    "\n",
    "plot(T,X)\n",
    "\n",
    "X[2]-x_exact[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.0019541839154975094"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X,T=be_steps(10.1, 0.02, 100)\n",
    "\n",
    "plot(T,X)\n",
    "\n",
    "X[2]-x_exact[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.1899999999999995"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "8.08-8.27"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.14000000000000057"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "8.41-8.27"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Julia 1.2.0",
   "language": "julia",
   "name": "julia-1.2"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}