Topic4a Scipy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Topic 4: Scipy and Numerical Techniques"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are now ready to apply some of our basic know-how to basic numerical problems like finding solutions of equations, ODEs, and integrating.   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we get started with examples, we have to learn a useful trick involving lambda functions.  There is some deep origin of this idea in computer science, but our superficial level, they give us a way of defining a function while we are doing something else"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here for example, is a function that adds x and y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def add(x,y):\n",
    "    return x+y\n",
    "add(1,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do the same wit a lambda function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function __main__.<lambda>(x, y)>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lambda x,y:x+y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So for I have just created the fucntion without giving it a name.  I can act on it too by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(lambda x,y:x+y)(3,5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we can think of lambda x,y: as the same thing as add(x,y) but we never stopped to define it.  At this moment, it might seem stange that we would need such a thing.  However, when it comes to solving equations, the arguements of our numerical technique is a function! We integrate functions and minimize functions, for example, so if we have a code that does that, the input is a function, not a piece of data.  lambda lets us do this easily."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Newton's method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The best way to understand both how these libraries work and what they do is to try to solve a simple equation numerically. For example, let's that the quadratic equation $x^2 +2x +1 =0$.  We know the solution, $x=-1$ already, but those will be good checks our method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**2+2*x+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999999805834525"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.newton(f,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, so it finds the solution pretty well.  We can see we could have done the same directly with our lambda function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999999805834525"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.newton(lambda x:x**2+2*x+1,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is why it is nice to have these lambda functions, because it saves us space so that we aren't defining all kinds of boring functions.  We save our functions for things that are actually useful"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Combining back to our problem: what do we think of this answer?  This is a pretty simple equation and while we are close, this is not just some machine precision error, since there are many incorret digits.  What is going on and how can we do better?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's make our own Newton's method code and figure out what's going on.  First, this piece code took a function as an input.  So we must be able to define a function where the name of the function is the input:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_func(func): #func is the name of some function that take 2 inputs\n",
    "    return func(1,2) #return the function evaluated at (1,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_func(add)  # Yay! It works"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When a one function uses another function as its arguement, that is when we can use the lambda function instead.  It"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "201"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_func(lambda x,y:x**2+100*y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we know how to feed a function to another function.  But what exactly are we trying to do? Newton's method is supposed to take our function and approximate it as a line based on the point we start $x_0$, \n",
    "$$y = f'(x_0)(x-x_0) + f(x_0)$$  \n",
    "It then finds the point where $y=0$, we'll call that $x_1$, whose solution is\n",
    "$$x_1 = x_0 - f(x_0)/f'(x_0)$$\n",
    "We can guess this is probably close to the solution than where we started, so let's use repeat with $x_1 \\to x_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Problem: if I want to make a code that does this, I need to tell it when to stop.  Otherwise it will just repeat forever.  I that two options: stop when my answer is close enough to zero or stop after some fixed number of steps (or what ever comes first)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_newton(f,fprime,x0,err):\n",
    "    x1=x0-f(x0)/float(fprime(x0)) # new point according to newton\n",
    "    if abs(f(x1))>err:  # need the absolute value here\n",
    "        print(f(x1))\n",
    "        return my_newton(f,fprime,x1,err) # repeat if the error is too large\n",
    "    else:\n",
    "        return x1 # return x1 when we are happy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30.25\n",
      "7.5625\n",
      "1.890625\n",
      "0.47265625\n",
      "0.1181640625\n",
      "0.029541015625\n",
      "0.00738525390625\n",
      "0.0018463134765625\n",
      "0.000461578369140625\n",
      "0.00011539459228515625\n",
      "2.8848648071289062e-05\n",
      "7.212162017822266e-06\n",
      "1.8030405044555664e-06\n",
      "4.507601261138916e-07\n",
      "1.126900315284729e-07\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.9998321533203125"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_newton(f,lambda x:2*x+2,10,0.0000001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wow!  That's pretty cool.  So what did we learn? This method needs two things that we didn't feed the scipy function: (a) the derivative of f and (b) a measure of when to stop.  Presumably those are options inside the function that we didn't use?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scipy isn't going to calculate derivatives for us, so it must be approximating the derivative some other way.  Indeed that is what's happening: if you don't provide it with the derivative, it uses something called the secant method.  To use the true Newton's method, you need to give it the derivative as well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.9999999893281472\n"
     ]
    }
   ],
   "source": [
    "newton1=optimize.newton(lambda x:x**2+2*x+1,10,fprime=lambda x:2*x+2)\n",
    "print(newton1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.999999984901414\n"
     ]
    }
   ],
   "source": [
    "sectan1=optimize.newton(lambda x:x**2+2*x+1,10)\n",
    "print(sectan1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.426733202222977e-09"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(newton1-sectan1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First,let's make a guess that the error is set at $10^{-16}$ because we are using 16 bit numbers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30.25\n",
      "7.5625\n",
      "1.890625\n",
      "0.47265625\n",
      "0.1181640625\n",
      "0.029541015625\n",
      "0.00738525390625\n",
      "0.0018463134765625\n",
      "0.000461578369140625\n",
      "0.00011539459228515625\n",
      "2.8848648071289062e-05\n",
      "7.212162017822266e-06\n",
      "1.8030405044555664e-06\n",
      "4.507601261138916e-07\n",
      "1.126900315284729e-07\n",
      "2.8172507882118225e-08\n",
      "7.043126970529556e-09\n",
      "1.760781742632389e-09\n",
      "4.4019543565809727e-10\n",
      "1.1004885891452432e-10\n",
      "2.751221472863108e-11\n",
      "6.87805368215777e-12\n",
      "1.7195134205394424e-12\n",
      "4.298783551348606e-13\n",
      "1.0746958878371515e-13\n",
      "2.6867397195928788e-14\n",
      "6.661338147750939e-15\n",
      "1.6653345369377348e-15\n",
      "4.440892098500626e-16\n",
      "1.1102230246251565e-16\n",
      "-0.9999999945297885\n"
     ]
    }
   ],
   "source": [
    "myresult=my_newton(f,lambda x:2*x+2,10,10**(-16))\n",
    "print(myresult)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.201641339880325e-09"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton1-myresult"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This seems like a good guess, but when we look at the documention, it seems more like what they did was stop when the difference between x1 and x0 gets small.  It takes an optional input called \"tol\" whose default is 1.48e-08.  Let's see what happens if we do that instead:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_newton2(f,fprime,x0,err):\n",
    "    x1=x0-f(x0)/float(fprime(x0)) # new point according to newton\n",
    "    if abs(x1-x0)>err:  # need the absolute value here\n",
    "        print(f(x1))\n",
    "        return my_newton2(f,fprime,x1,err) # repeat if the error is too large\n",
    "    else:\n",
    "        return x1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30.25\n",
      "7.5625\n",
      "1.890625\n",
      "0.47265625\n",
      "0.1181640625\n",
      "0.029541015625\n",
      "0.00738525390625\n",
      "0.0018463134765625\n",
      "0.000461578369140625\n",
      "0.00011539459228515625\n",
      "2.8848648071289062e-05\n",
      "7.212162017822266e-06\n",
      "1.8030405044555664e-06\n",
      "4.507601261138916e-07\n",
      "1.126900315284729e-07\n",
      "2.8172507882118225e-08\n",
      "7.043126970529556e-09\n",
      "1.760781742632389e-09\n",
      "4.4019543565809727e-10\n",
      "1.1004885891452432e-10\n",
      "2.751221472863108e-11\n",
      "6.87805368215777e-12\n",
      "1.7195134205394424e-12\n",
      "4.298783551348606e-13\n",
      "1.0746958878371515e-13\n",
      "2.6867397195928788e-14\n",
      "6.661338147750939e-15\n",
      "1.6653345369377348e-15\n",
      "4.440892098500626e-16\n",
      "-0.9999999893281472\n"
     ]
    }
   ],
   "source": [
    "myresult2=my_newton2(f,lambda x:2*x+2,10,1.48e-08)\n",
    "print(myresult2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.9999999893281472"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton1-myresult2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We hit exactly the same number as Scipy! We can even get back to our old answer in scipy by replacing "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "newton2=optimize.newton(lambda x:x**2+2*x+1,10,fprime=lambda x:2*x+2,tol=10**(-16))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton2-myresult"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the one thing that is nice about scipy is that it has more methods built in. We have a pretty good idea of what scipy is doing, although it is definitely coded up in a better way that is probably faster and can handle situations where we don't actually converge (e.g. scipy has a max number of iterations so it doesn't get stuck in an infinite loop).  Yet, based on what we have seen, we can imagine that if we gave it more information about the function, it could probably do better.  For example, we could improve Newton's method to use the second derivative too!  Scipy has this built in for us"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.9999999957639969\n"
     ]
    }
   ],
   "source": [
    "fprime2=optimize.newton(lambda x:x**2+2*x+1,10,fprime=lambda x:2*x+2,fprime2=lambda x:2.,tol=10**(-16))\n",
    "print(fprime2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4.236003103841313e-09\n",
      "1.0671852845867136e-08\n"
     ]
    }
   ],
   "source": [
    "print(fprime2+1)\n",
    "print(newton1+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For all our effort, it seems like we have hit a wall.  We can't do much better than what we have.  What's the problem?  Right now we have hit the point where the difference between x1 and x0 is 10^(-16) which is around the machine error for generic floating point numbers.  Presumably if we want to do better, we are going to have to make the working precision higher."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30.25\n",
      "7.5625\n",
      "1.890625\n",
      "0.47265625\n",
      "0.1181640625\n",
      "0.029541015625\n",
      "0.00738525390625\n",
      "0.0018463134765625\n",
      "0.000461578369140625\n",
      "0.00011539459228515625\n",
      "2.8848648071289062e-05\n",
      "7.212162017822266e-06\n",
      "1.8030405044555664e-06\n",
      "4.507601261138916e-07\n",
      "1.126900315284729e-07\n",
      "2.8172507882118225e-08\n",
      "7.043126970529556e-09\n",
      "1.760781742632389e-09\n",
      "4.4019543565809727e-10\n",
      "1.1004885891452432e-10\n",
      "2.751221472863108e-11\n",
      "6.87805368215777e-12\n",
      "1.7195134205394424e-12\n",
      "4.298783551348606e-13\n",
      "1.0746958878371515e-13\n",
      "2.6867397195928788e-14\n",
      "6.661338147750939e-15\n",
      "1.6653345369377348e-15\n",
      "4.440892098500626e-16\n",
      "1.1102230246251565e-16\n",
      "0.0\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.9999999945297885"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_newton2(f,lambda x:2*x+2,10,1.48e-64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try increasing the working precision of our code using high precision numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "from decimal import Decimal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Decimal('0.1000000000000000055511151231257827021181583404541015625')"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Decimal(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_newton3(f,fprime,x0,err):\n",
    "    xs=Decimal(x0)\n",
    "    x1=xs-Decimal(f(xs))/Decimal(fprime(xs)) # new point according to newton\n",
    "    if abs(Decimal(f(x1)))>Decimal(err):  # need the absolute value here\n",
    "        print(Decimal(f(x1)))\n",
    "        return my_newton3(f,fprime,x1,err) # repeat if the error is too large\n",
    "    else:\n",
    "        return x1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "30.25\n",
      "7.5625\n",
      "1.890625\n",
      "0.47265625\n",
      "0.1181640625\n",
      "0.029541015625\n",
      "0.00738525390625\n",
      "0.0018463134765625\n",
      "0.000461578369140625\n",
      "0.00011539459228515625\n",
      "0.0000288486480712890625\n",
      "0.000007212162017822265625\n",
      "0.00000180304050445556640625\n",
      "4.507601261138916015625E-7\n",
      "1.126900315284729003906E-7\n",
      "2.81725078821182250973E-8\n",
      "7.0431269705295562748E-9\n",
      "1.7607817426323890684E-9\n",
      "4.401954356580972670E-10\n",
      "1.100488589145243165E-10\n",
      "2.75122147286310790E-11\n",
      "6.8780536821577699E-12\n",
      "1.7195134205394430E-12\n",
      "4.298783551348607E-13\n",
      "1.074695887837150E-13\n",
      "2.68673971959283E-14\n",
      "6.7168492989822E-15\n",
      "1.6792123247455E-15\n",
      "4.198030811861E-16\n",
      "1.049507702966E-16\n",
      "2.62376925747E-17\n",
      "6.5594231434E-18\n",
      "1.6398557855E-18\n",
      "4.099639466E-19\n",
      "1.024909865E-19\n",
      "2.56227470E-20\n",
      "6.4056866E-21\n",
      "1.6014220E-21\n",
      "4.003550E-22\n",
      "1.000887E-22\n",
      "2.50223E-23\n",
      "6.2560E-24\n",
      "1.5641E-24\n",
      "3.905E-25\n",
      "9.78E-26\n",
      "2.46E-26\n",
      "5.6E-27\n",
      "1.5E-27\n",
      "6E-28\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Decimal('-0.9999999999999890669998056123')"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_newton3(f,lambda x:2*x+2,10,1.48e-64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Decimal('1.067185284586713578391936607658863067626953125E-8')"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Decimal(newton1+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Decimal('1.09330001943877E-14')"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Decimal('-0.9999999999999890669998056123')-Decimal(-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So what are the morals of this exercise: \n",
    "\n",
    "(1) you may not know exactly how things are coded inside numpy or scipy, you should be able to get a pretty good idea how they work \n",
    "\n",
    "(2) numerical techniques always have some error associated with them.  You can learn a lot about a code by understanding what controls the size of the error and trying to make it smaller \n",
    "\n",
    "(3) there are lots more options built into a scipy solver than you would want to code yourself.  But, the simplest implementations are often pretty close to what you do know how to code up yourself.  The complicated options are usually just the simple idea extended to higher order (more derivatives, more grid points, higher order polynomial extrolations, etc)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpolation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import interpolate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sometimes you don't actually have the analytic form of a function you want to use. Instead you might have some raw data.  Alternatively, you have the function but calculating it is so slow you need to pre-compute it at a bunch of points before you can use it in some other applications (like integrating, equation solving, etc). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is when interpolation is super helpful.  Interpolation is just a rule for filling in the gaps in your function based on the data you have.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "xI=np.linspace(0,np.pi,40)\n",
    "yI=np.sin(2*xI**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xI,yI)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So that is our data.  Now we want to make a function that is defined on the entire range between the endpoints of our data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's just try the 1d interpolate function out of the box and see what it does"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_try = interpolate.interp1d(xI, yI)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_fine=np.linspace(0,np.pi,400)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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RZdGlMW8qQqKkMPujOrJ4hXLixFFuGv4OJ2p/Df/64gnPaHpVenmo62m9PLRY+k6mi8r4Ny5/fyCLveU6ymWSi2eKd5+m52h6fyCw+dVL6te+89WklDBuwcCyibEopTKNKtOGQLNIlFIMP/i/UGJjzduLq0h8y/rtdNpaqejUK4SLJXHpIFPKSeuma3Ieq35rOktn8GTxBpaNXzpCQtlo37y0J6SKqmou2ZtxB63nQhoeTPvf2CtqCzK+NgQrkAPPPMKNU7/kzNq7KatpM1vOkulrfD2bpo8yPFj8boxG4B0+QqdrPU6XO+exmtduJUI5dO/PgzJzcITPMehsxO5c+t8jXLmJ+qmzpFLzecCbw2ioBwC3N/8xBKANwYojkUji/sXHGZYqNr/VmsFjC1Fz7W9iF0XHU98xW4rlScRjtMU6iPq25WU8sdnoKtlCbdR634oXS0O8i2jZ2mX1tTVsp4EQ5y9dWrixgWTTS5RVNxRkfG0IVhjP/uQb7Egep3fXH+IsrTJbzrJYt/1V9FGD6+xDZkuxPBdPv0SJxLA3528faKJ2N63JS5b1nrkSE5OTtKp+JquWZwgq29PLSeHzL+ZTVs7EogNAYdJLgDYEK4qJyUlaD/413fYWtv76vWbLWTZis9FVewubJw4wNhoxW46lGTr9LAB1m3PfKM5Svm5fxnum+PYJhrrP4pQkqer1y+pf1px2wU0FrVXDODU2CIA3oJ8INAuw/z8/Txu9TLzm44jdabacnKjY9RbcEuf0U98zW4q16TnICKU0rc3P0hBA246bSSlh7NwzeRvTKEYHLwLgCSxvb8xb00RKCanRgXzKyhnb+CBRyq5YYCen8QsyqsZwQqEg28/+A6c8u9h441vNlpMzm657A2EqUSd+ZLYUS1MdPc5F91XY7PmLGq/0+rlob6Z00FrLI4thMpRe26+oaV1Wf6fLTUQqsI9byxA4p4aIFCi9BGhDsGI4/sAnqZZRyn7900UTPHYl7A4HZ703snHkGWLTU2bLsSRTU5O0JjoZr87f00CWofKrqJu8kPdxC00ikvY08zcs31suYqvGNTWUL0l5wTM9zJhDGwLNFbh44TTX9d/PS7430rJtcdkWiwHX9juolAlOPvtjs6VYkgsnD+CSJJ623COKZxOvaCGgQiTisbyPXVBG+xhTJZRXLv+mOebyUxqzliEoT4aZchUmvQRoQ7Ai6P3uxxCg5a3FFTy2EJte9WbGlYepIz9YuPEqZPhs2te/cdO+vI9tr27DISmCvcX1VOCaHCBkz+2GOeWuoTIxvHBDA/GmIiQ82hBo5uHEwSfYN/owR1p+i0DT8jwlrIqnpIzTFXtZN/w4SV3C8jJSfUcYo4Sa1k15H7ukph2A4d7zeR+7kJROBRl1LlyR7EokS2uoVmHL1MWYnpqgknFSBUovAdoQFDUqlSL5048RppItb/u42XIKgmy4lQARzp4ovo3LQlMdPUGPZwPY8v/f2NuwDoCJgeIyBN7EEJOeHNMwlNfjkiSjYWtUQswWCrJX1BVsDm0IipiDj9zP9vgRzm25l7KqarPlFITmna8DIHhc1zKeyejEFGuSnUxU55Z6ej5qmtOGIDF8sSDjF4JUMolfDZMoyy0Ng6Mq3T8StEZ08ehQ2hC4qgqTXgK0ISha4vEYgWf/iku2Jnbd+QdmyykYNW2bGaYKV0/xF0vJJ73njlIq09ibdxVkfE9JGUN4sY8UjyEID/XilCRS0ZjTOCWZNA6jQz35kJUzE+G0ISjxaUOgmcWB736BdtVN+FV/iiMPycYsiwgXK3bSPHoYpayVCMxMRrsOAVDZdnXB5gg56iidLJ7Ef+H+tNFyV+dmCCoCzQBMha3x2aej6ajiCn9un+tK5MUQiMhtInJaRDpE5KNznH+NiERF5FDm588X21dzOSPRYTae/BInXdvY/tp3mS2n4MQb99LEIH2Ximu9upDEgx0A1K8pzNIQwFhJI96YtQKrrsTYUHopp9TfskDLK+OtTfdPRPtz1pQPkiPpa+CrLUyeIciDIRARO/Bl4HZgC/BOEdkyR9MnlVK7Mj9/scS+mhkce+Av8BPFedunkAJsFFoN/5Z0gZHuI4+ZrMQ6OCLn6SeAp7S8YHPEy5upTQVJFYnH1vRwNwBVdbmlXq+s9DKh3DBmDSMo40HGlSenetQLkY+7yHVAR6bsZAy4H7jDgL6rkv7u81zd/U0OVr6O9buXVoGpWGnbuo8J5SbR+azZUixDxfglQq7CfUMEEF8rLkkw1G+NTdOFSI30kVKCvz63JwKx2QjbvDgnBvOkLDcck0OEbd6CzpEPQ9AEzPyX0p05NpvrReSwiPxERLLPs4vti4jcIyIHRORAMGgNty4zuPidj2EnRcN/+bTZUgzD7nBy3rOF2rB2Ic1Sl+hhrLywRYc8L8cSdBR0nnxhH+tjWKpwOl05jxW1+/FMWyO62BMLMeYorFdgPgzBXIltZu/qvQi0KaV2Al8Cvr+EvumDSt2nlNqjlNpTU1O4wAorc+7oc+wJ/4QXG95BY/tVZssxlLHaPaxJXCBahDny8000PISPEVT18nLuLxZvfXr8sSKJJfBMDhK25xZMlmXS7ac8bo1/a2XxMJPOwuUZgvwYgm5g5rNYM/CK7Xal1IhSaizz+iHAKSKBxfTVZFCK6R/9ESNSxua3f9JsNYZTtuEm7KLoPPRLs6WYzmDnCQDctRsKOk9NSzpSPR7qKug8+aIiFmTMnZ8viTFPLd5UOC9j5UpVKkysJD8Gbj7yYQj2AxtEZI2IuIC7gAdnNhCRepF0SkwRuS4zb2gxfTVpTj7yNbZMH+bYVf8fVdWr74loza6bSSphoqP4cuTnm2jPKQB8LflPLTGT0nIvYSqwRYtjj8CbChEryU/0baqslkrGiU+N52W85ZKIx/CqUVKlhf0/78h1AKVUQkTuBX4G2IGvKqWOi8gHM+e/ArwV+G8ikgAmgbtU2il8zr65alppxCdHCDzzvzkja7n2N//QbDmmUF7p45xjDeXBg2ZLMZ344DkA6tsKawgAQvZaPBPWf0ifmhzHxyip8vwEXdkqs9HFPdS0bMzLmMshEuonIApbeY5pMxYgZ0MALy/3PDTr2FdmvP574O8X21fzSo7/x8fZpYY4f8uX2OjKfSOsWBny7WJb8CES8RiOPGwIFiuOaNp1tL6ArqNZRj2NVBdBXYJQ30WaALs3P55Ubm86ungk2G2qIYgGewkAzqrC5RkCHVlseUJdJ9hy4es8VXYr1918u9lyTMXRfj1lMsWFEwfMlmIqleMXGXI3GzJXrLyJ2uSgZTJxzsdIMFOisjo/f5fSTBTv+LC5aSYmwn0AeLyFSy8B2hBYnoEH/pAYTlrf/jlkBVQey4Xm7em4iaGTT5isxFxqEz2Mly2vFOOS8bZSIjGGh/qMmW+ZTAxlS1TmFkOQpbImbVBiEXOji6cz81f4C1O0Pos2BBbm9BMPsGX8OQ6suYfWtjVmyzGdutaNBPFh79lvthTTiA4P4WO04K6jWdyBdgCGus8ZMt9yiUfS39yr69vzMl51TWO6iP2IuYYgkUkvUVVT2CdAbQgsSnx6gopf/hkXpJm97/hTs+VYAxG6y3fQPHpk1SagG+hM+1IU2nU0S+XLsQTWNgSM9DGpXFR681PFy+N2MyyV2MZNji4eG2RaOanIofTmYtCGwKIcfuAvaUz1M3TTX1JS4jFbjmVINF1LI4P0dVt/A7MQjPSeBgrvOpol0JI2OLEha8cSOCcGCNn8ec29FbFV454y1xDYJ4cIi7fgOcW0IbAgQ90dbO34J14ouYk9t9xpthxLUb3pJgC6j6zOQjXxwQ5SSqhv32zIfJXeAKOUIFFr1yUomRpkJMcSlbMZc/opmTY3utg9Pcyow1vwebQhsCDd//E/AKh/+9+u+g3i2bRt3ceUchLvXJ2FapyR8wyKH09JmWFzBu11uMetHUtQlQgy4c6vr/2UO0BV0twi9qXxYSacha8+qA2BxTj59IPsGv0lB1vfT+ua1ZVPaDE4XB463VfhH37JbCmmUDFxyTDX0Swj7gaqpq3rNaRSKfypYeJl+fW1T5TW4k1FwETX2cpkmGl3fvY9roQ2BBYiHpum9NE/pUfquOadK7MYfT4YqdnN2kQHY2OjZksxnNpED+MFzjo6m+myJmqSA5bdoB8JB/FIHHIsUXkZFXU4Jcl41Jx9glQyiU9FSRY4vQRoQ2ApDn77r2lLXWLgVZ+gpNS4R/9io3Tdq3BJknOHnzJbiqFEh4Np11Gfwa7EVS1UyCQjYWukZZ7NcH8nAC5ffg2BozLtux8Z6M7ruItlJBxM12AucHoJ0IbAMgxdOs32M1/msOdarn7dXWbLsTTtu24BYOTM0yYrMZYBg7KOzsYVSD+BBLvPGjrvYhkdTAeTleVYonI2Hl/aEIyFzDEE0aH0voyjUhuCVYFKJgh/83dIIfje8eVVUX4yF8qr6+m2NVE6sLpSTYxkso5WtxjjMZSlsn5dev5+a9YlmA6ng8kqcyxROZvyQDpv0VTYnP2RsZAx6SVAGwJLcPQ7n2LD1FFe2PwneoN4kQx6d7Jm8hiJRHHU080H8WDadbSu3ZgYgiz+pnRdgmmLxhIkoulvzrmWqJyNr7Y5M745hmAqkp633J/nvY850IbAZPrPHGDTiS/wvOcGXvO23zdbTtEgLXupllHOnzlqthTDcEYuMCgBQ11HAar89UwqF0SsaQhsY32EqcTtKc3ruD5vNWPKgzKpiH08m14iUNja1KANgakkY1NMP/A7jFBGy3v/EbtdX47FUr8tnYBu8PjqSUBXMXGRIXfhbwqzEZuNQXsd7jFrxhK4JwYZtuffxdJmE8LixTFhTo10NTZIQtmoqi6SPQIRuU1ETotIh4h8dI7zvyUiRzI/z4jIzhnnOkXkqIgcEpFVteh7+Bt/RFuikzP7Pk1jY34fa1c69et2MEIZ0v282VIMozbRa7jraJaoq54Ki8YSlMcGGXMVxsUy6vDjmTLHENgnhghLFTa7veBz5WwIRMQOfBm4HdgCvFNEtsxqdgF4tVJqB/C/gftmnb9FKbVLKbUnVz3FwrkDP2fXxa/zVNWvc/1t7zJbTtEhNjuXSrfRMHLYbCmGEB0ezLiOGpN1dDZTZY0EkuYskSyENxli2lOYb80TLj9lcXOii13TIUbsXkPmyscTwXVAh1LqvFIqBtwP3DGzgVLqGaVUthL0c6SL1K9aJkcjlPz4Xnqkjm3v/5JOI7FMphr2sFZdon/A3FTBRvAr19H1psyfqmrFxyijI9Yo6J4lHpumWkVJlhcmX3/MU4MvZY4hKI2FGDcgvQTkxxA0ATOrW3dnjs3H7wA/mfFeAT8XkYMics98nUTkHhE5ICIHgkFzHtXyxYl//RB1qUHCb/wiXq8xF3ol4r3qRgA6D6/8BHQjPemso0a7jmZx+bOxBB2mzD8foYFL2ERhqyqMZ02qvJZyJkhNG1/EviJhTHoJyI8hmOvr7Jyx6CJyC2lD8MczDt+glNpNemnpQyJy81x9lVL3KaX2KKX21NQUPuS6UBx59FtcE/oRzza8hx3Xv9FsOUVN6/YbSSgb0+efMVtKwTHLdTRLeW06mjnaZ61YgshA2pPJU12YTXR7RdqHPxo0NqhMpVL4VIRkiTH3unwYgm5g5k5nM3CZe4GI7AD+GbhDKfVyblelVG/m9yDwPdJLTSuS8GAPzU/+ER22tex532fNllP0OEsquehai3foRbOlFJx01lHjXUez+JvT0cxWiyXIlqgsDxTG2cLpTT9pjAwZW7t4fCySzp9UXjyGYD+wQUTWiIgLuAt4cGYDEWkFvgu8Ryl1ZsbxMhGpyL4G3gAcy4Mmy6FSKbr+9QOUqwnkv/wjHk+J2ZJWBGH/btbHTzM5NW22lIKSzjpqvOtoluq6ZmLKgbJYLEEsnP7O6asvjDdVaXWmiL3BhiAymP5cdgPyDEEeDIFSKgHcC/wMOAk8oJQ6LiIfFJEPZpr9OeAH/u8sN9E64CkROQy8APxYKfXTXDVZkRe+9ZfsmniGgxt+n3XbVuxDj+HY2/ZRJtN0nnzBbCkFpS7Rw4RJrqOQ9tIatNXgHDX2hrgQaqSPmLLjLVBx92wR++mIsa6zY6G0IXAbkF4CwJGPQZRSDwEPzTr2lRmvPwB8YI5+54Gds4+vNF766b9w7ZnPc7D8Jva+88/MlrOiqN96MzyfSUB39U1myykI0dAAXsZImeQ6miXiqqd8ylqxBM7xPkJSTUOBgjH9NQ0klZAaNdYzbTJjeMqqC2PgZqNDWQvMqRd+xpZnP8Jp1ya2/N79hgSHrCbqmtcxTCW2vpUbTzDQdRIAT91GU3VMljYSSFjLVbdkapCos3Dr6GUlbkJ4DS9iH4sal14CtCEoKJfOHKL+od9mwFZD3T3fo6Ss3GxJKw6x2bjk2Yh/9ITZUgrGSHc666ivxdyEhKmqFgJEmJww3pVyPiriQ0y4C7uhGrH7cE0a67KeGk0bHm+NfiIoaob6L2H/1ttJYsf27v+k2qALuhoZ92+nNXGRqUnr3KDySTx4Lu062maO62gWhy+9RzFooVgCfypErLSw6+hjTj8lMWOL8tgmgoSpwOF0GTOfIbOsMibGooT/+S14UxGG3vx1mtfNzrihySfult04JMXFk/vNllIQnFFzXUezlNWlYwkivedM1ZFlNDpMmUxBRWENwZQ7QGXC2Ohi51SIqM1r2HzaEOSZRDzG2S+/jbXxDk7f9EWuuuY1Zkta8TRs3gdApGNleg5VTlwkaHDB+rmobkoXqJkMdporJMNwf9qV1eEt7Dp6vKQWr4pAyrjaFyWxEOMO47IOaEOQR1QqxcGvfICdk8+zf+vHuPr1uuSkETS0rCdMBdJ3yGwp+Ucp6hM9jJe3m60Ef8MaEspGKnLRbCkAjAymdZTkuUTlZZTX4iDF1Ihx+wQViTBTBqWXAG0I8srzX/8Ye0M/4NmG97Lv7R8xW86qQWw2Lrk3Uj1y0mwpeSca6qOScVS1OcnmZmJ3OAna/DhHzanhO5upTC3hytrWgs7jqMqkmRi8tEDL/OFNRYiXBAybTxuCPJBKxHnxK/+VfZ3/lwOVr2fvB/7ObEmrjjH/NloTXcSmJsyWklcGLhwHwF1vrutoloiznrJJaxSoSUTTwW2BhsIG2nl86ejiUYOii6cmxiiXSSg1LqeaNgQ5MhYZ4tTfvoHd/Q/wuP8d7LhXxwqYgat5N05JcunUyqptNJYtWN9qTtbR2UyUNFIdt0ZdAhntZ4QySsoqCjpPuT9bxN4YAxgOpg2OrcKY9BKgDUFO9HYcIfLFm1g/cZgnt3yCm+/9R1wup9myViX1m9IbxsNnV9aGcSJ4lriy09BmjSeCeGUzNSrE9PSU2VJwTQwQshV+Hd1bl96oj0eNCaYbDaWjil0GpZcAbQiWzfEnv0/5v91GaWqMU2/4d256+x/qAjMm0tS+kQjlsMI2jN3RC/Ta6nG73GZLAcBR3YZdFMGeC2ZLSZeodBZ+Hb3a62NUlYBBaSYmhtOGoNRnXOyRNgRLRKVSvHD/p7jqkfczZAsw8d6H2XHDbWbLWvWIzcZF90aqR1ZWhHHV5EVCbuvUsy6pSccShC0QS1CVCDFVUlfweZx2GyHxYTeoiH0s8+RRGShMsZ250IZgCcQmx9n/9+/lulN/zdHSfdT8weM0rzU32lPzK0Z9W2mJdxKfnjRbSn5IpahP9jBe0W62kpepbkzHEkwEzX0iSCYS+FWYRJkxyycjdh+eaWMMQWo0PY+3RhsCy3Hy2Z/Q87c3cN3wD3m28X3s+J8/oqLSZ7YszQycLbtxSZLu0ytjwzgy0ImHOPjNdx3NEmhKZ0BNDptblyA82INDUtgqjblZTrgClMVDCzfMB+ODjKoSQyPJtSG4AiqV4tjTP+TEp25k88/uoiIR5rnr/5Hr7/kCdu0ZZDnqMhvGoRWyYTzUmXYdLa03N9ncTJzuEoJUYx8xN5YgnClR6fIZYwhinhp8SWPSTDinhojYjP2SmZd6BCsNlUpx7Kkf4Hjyc2yLHyeIj+c2foRdd36YfaWFdVXTLJ+W9k1EVRn0HjJbSl4Y68u6jlpr+XHYWWd6LMFYtkRlTWGDybIky2opDU+hpkcRd2HvAZ7pEGMOYw1BXp4IROQ2ETktIh0i8tE5zouIfDFz/oiI7F5sXyNRqSTHHvk3zn5qH9t/8T6q4/08v/lPqPjj4+x71//Co42ApbHZbXS5N+KLrowN41SwgwnlprHF3II0sxkvacAXN7cuQWw47WvvqzOmaputMr0pPRYqfGGe8kSYSZdx6SUgD08EImIHvgzcSrqQ/X4ReVApNfN/4+3AhszPXuAfgL2L7FsQxkbC9HYcJtJ1lNTASUqiHTRMdrCNED1Sx/Nb/4xdb/499npKCy1Fk0dGfdvY1P/vJKYncbiLuy60e6STblsjG53WenCPV7RQE32cRDyOw2lO3ExqpJeEsuGrNaZwS7aIfTTYTUVjYWM6KlMR+j3GJZyD/CwNXQd0ZMpOIiL3A3cAM2/mdwBfV0op4DkR8YpIA9C+iL55Y/93v4D7zIPUTnVSzxDZyxlTDrrtzVyq2MG59bdzze3vp8llTB5wTX5xtFyNa+AbdJ19kbZtN5gtJyd8kxe56NlgtozLsPtacfUk6e3torHNnI1s+/gAw+Kl1mGMkXy5iH2osGkm4rFpfIySKjMuqhjyYwiagJnZmLpJf+tfqE3TIvsCICL3APcAtLYub10wEenBFw/TXXk1F/wb8TRsIbB2Jw1tV7HWoAIQmsJSe9VeOAChM88XtSFQiWlqU/2crrzVbCmX4cnEEgz3nTPNEHimBog4Ahh1u8yWjJwucJqJyFAfNYCtvPgMwVzhtGqRbRbTN31QqfuA+wD27NkzZ5uFuP63P7ucbpoionXtFiKqDFXkG8aR3g58pLAFrPdE4G1IxxKMD5gXS1ARCxL2GLNRDFBd05hOwT1a2DxL0WAPNYCzyrj0EpCfzeJuYGboYzMw22zO12YxfTWaRWO327jo3oA3UtwbxkNdadfR8gKvRy+HmmbzYwn8qRDxUuO+NVeWughRhYwX1hD8Kr1E8RmC/cAGEVkjIi7gLuDBWW0eBN6b8R7aB0SVUn2L7KvRLIkR31Za4hdIxqfNlrJsxntPA1DbvtVkJZfjLq1kmEpsI8bl55/J5NgIlYyTqjAuF4+IELZVF7yI/XQ0bWjKq42tcZ6zIVBKJYB7gZ8BJ4EHlFLHReSDIvLBTLOHgPNAB/BPwO9dqW+umjSrG3vT1bgkQe+ZF82WsmxU6BzDqpyGeuPSDCyFYUcdpRPmPLyH+s4DYPcatzQEMOqspmS6sEXsk5mlJ69B3lBZ8rLlrpR6iPTNfuaxr8x4rYAPLbavRpMLtVfthRdh6MzztGy93mw5y6Jk5AJ99iaq7dYM/h/1NFI93mHO3AOdAHgCxsQQZJl011A5VuDPPDbIpHJRVl5V2HlmYc1/ZRpNDrSt20pUlZHqfclsKcvGP32JSKmxN7qlECtvoi41SCqZMnzuqVC6VnGlQcFkWRIlNVSpaEGL2DsmhwjbvIjN2FuzNgSaFYfDYafLtR5vpDhXGZNTo9SoEHGvtSKKZ2LzteKROEMDxuccSoYvkVKCv6Hd0HlVeR12UsRHBws2h3s6xKjd+GSW2hBoViRR31ZaYhdIFeGGcbDrJADOGutkHZ2Nu6YdgFCv8ctD9tFegnipKDM26t9RmfbkGQkWzviVJYaZdBkbVQzaEGhWKNkN476O4lseGr6Ydn2tbLJGneK5qKpPP62MDZw3fG7PRB8hu3GF3bO4qwtfxL4qGWbaXfiqa7PRhkCzIqnZmA5QD5553mQlS2dq4CwAdWutawhqmtOBbvHMer2RVMQGGHUb62cPvypiPzlcGG+pVDyGn6hhxXZmog2BZkXSvmEbI6qUVE/xPRHYhjvoV35qfMYvESyW0spqRihDogbHEihFIDnIdKnxN8usS2ciWpgMpOHB9JKTrdLYGALQhkCzQnE67HS61lMVLr4N47KxLgZczYjMlYHFOgzZaykZL2wSttnERkN4iJGqbDZ0XoCA18uIKkWNFSa6ODKYfrpy+YyNIQBtCDQrmIh3K83xC6hEzGwpS6I2donRMuu6jmYZcTdSGTO2LsFw7zkAnL6WBVrmH4/TTki82McL4zU0Hko/XZUFtCHQaPKGvfFq3MTp7zhktpRFMxUNUsUYSZ91XUezxMqbqE0OolLGxRJEM8FkZbXths35ivnt1binChNdHMvsPXhrjf8SoA2BZsUSKMIN48FMnWJ3nfWSzV2Gt4VymWQ4VDi/+tlMDnUC4GtYY9icMxl3+SmPF8YQpEb6SCgb/jrj04poQ6BZsbRv3MqoKiFRRBvGke50nWJfyxaTlSyMy98OQKjbuFiCZLibaeWktt74PQKAaU8tVclwQca2jQ8wJD6cBhXbecXchs+o0RiE2+nkgnM9VeFjZktZNLGBMySUjaY11ipYPxeVmViCUQNjCRxjPQyKnxK3SSUyS2spYQqmR/M+tmdygKjd2FrFWbQh0KxoIt6tNMfOF82GsTNynl6po7zU+vWWA83pyOfpUKdhc5ZM9jPsMLZ610wkU8R+ogCxBOXxIcbdxgfKgTYEmhWOrWkXbuIMnDtstpRFUTnRRdBtvEfMcqisrmNCuZGIcbEEVbFBxj11hs03G7c37eMfHsz/Z/YmQ0yXmGPktCHQrGiyG8aDxbBhnEpRn+hhssL6rqMAiBC01+I2KpYglcSfCpEoN69GQ0UgvTcxGszvZ45PT+BljJQJUcWgDYFmhbNm43bGVAnxS9YvUjMSvEQJ06hq6yabm03U3UDldGEibWczOtSNQ1JQZc5GMYC/Lv20NjmcX0MQHkgHk9mrjI8qhhwNgYhUi8jDInI28/uy/Kki0iIij4nISRE5LiIfnnHuEyLSIyKHMj9vykWPRjOb9IbxOqqKICV11nW0tOEqk5UsnqmyJgIJY9xHs8Fkbr+xlclmUlNXT0zZSY7kN7o4mllqcpsQVQy5PxF8FHhUKbUBeDTzfjYJ4H8opTYD+4APichM37j/o5TalfnRlco0eSe9YXwOlYybLeWKjPakXUf9bdZ3HX2Zqha8MkY0MlzwqUYHOwEorzUnhgDSXyyGxQt5LmI/PpTOM1ReY87+UK6G4A7ga5nXXwPunN1AKdWnlHox83qUdG1ic8yeZlVia7oaD3H6O6y9YZwMnmVSuWhsWWe2lEXjzMQSDBkQSxDLZDoNNJobdT1ir8ad5yL28UgmqriuOA1BnVKqD9I3fOCKW94i0g5cDczcubtXRI6IyFfnWlqa0fceETkgIgeCwfxeBM3KJpuSeuD0cyYruTLukQv02BtxOY0PKFou5XXpb+fRvnMFn0tFexhRpQQCxufrn8m4u4byWH7vQWqkj2nloNpv0c1iEXlERI7N8XPHUiYSkXLgP4E/UEqNZA7/A7AO2AX0AX87X3+l1H1KqT1KqT01Neb42mqKk/aNOxhTHpIWjzD2TV4k7CkO19Es/ibjYglcYz0EbQHsNnOzssZK6/Gn8ptmwj4+QEiqsdvN8d9Z8KuHUur1850TkQERaVBK9YlIAzDnrpGIOEkbgW8qpb47Y+yBGW3+CfjRUsRrNIvB5XRw1rmOymHrbhirZJy6ZD+dla8zW8qS8NU2Ma2cqHDhYwnKpgaIuMyLIciSqmymcmic8dEIZRXevIzpmRok6vBjlmNsrubnQeDuzOu7gR/MbiDppOr/DziplPr8rHMzfaXeAhRPLgBNUZFOSW3dDeOh7g6cksQW2GC2lCUhNjtBWw3uscIXsfclBpksMWfpZCZOb3qLc6i3M29jVsSHmDApqhhyNwSfAW4VkbPArZn3iEijiGQ9gG4A3gO8dg430c+KyFEROQLcAvxhjno0mjmxNe2ihBi9Z625YRzsTH8HqmgqHtfRLGF3PeUFjiVITk/gY4RkhXkxBFlKa9IBfyMDF/I2pi8VIlZi3tNOTrtSSqkQcNmzrFKqF3hT5vVTwJyLekqp9+Qyv0azWAIb98JhGDzzAk2b9pgt5zIm+k4DULNmq8lKls5UaRMNw08WdI5wfycBwOY13xB469sBmAzlZzlsajxKBZOoCvMMgY4s1qwK2jfuZFy5SfRYNMI4dI6IKqO2xrz0CcslWdlMgAgT4/nPyJkl3JfOcFoaMC+YLIu/If1EkAznZzksPJA2KLYq87zqtSHQrAqcTiedzvWW3TAuHb1Av6MJm0leI7ng9KdvjIOXCudCOh7sAqDKpII0M3F7SglRhW00PxlIo5laxSXV5j3tFN+/Oo1mmUS9W2iJnSOVSJgt5TL805cYKS2SZHOzyEb6RvsLZwgSw+mbZU2j+YYAYNheQ8lkfuo1T4bSeYsqarQh0GgKjr3pakplmt5zR8yW8gpik+PUqSHiXuvXKZ4LX1Pa02ky2FmwOWSkh6DyUlleXrA5lsKYu47KWH7STGSjin115n0R0IZAs2rwZ1NSWyzCeKDrBACO2uJyHc0SqG8jruykwhcLNodnoo+QwzqBpLHSeqrzFFSmRvsYV2683nkTKxQcbQg0q4a27IZxt7UijMMXTwJQ1Wz98pRzYXM4GLL5cY4WLpagIjbAqNv8GIIsqrKJSiYYG8m9frFzfIBhWzViM+92rA2BZtXgdDrpcq6j0mI1jKf6066j9e3F5zqaZdhZT/lU/ss3AqAUgWSQ6VJzcvXPhcOXXs8P9eYeS1AyHWTEYW7+JG0INKuKqG8rrRbbMLaFzxPEh9dXbbaUZTNZ2kh1PD+bp7OZGh2mlClUpXWSFv8qqKwz57Eq40FTo4pBGwLNKsPWtItSmabbQhvGFWOdDDrND5TKhWRFCzUqzPT0ZN7HHupJp7h2VlsnIZ+3rh3IQ1CZUlSnhomXmZtDSRsCzaoim5I6eNo6NYxr4t2MlrebLSMnbNVt2EQx2J2/tAtZov3pGIKyGmu4jgIEGttJKSEZyW1fZGxkmBKJQbm5y17aEGhWFa0bdjGh3CS6rRFhPBYZopoRUr7idB3NUlrTDkCkN/8FaqZCaUPgs0AwWRaX28OwVGEfza12cThj5BxebQg0GsNwOJ1cdK2lMmyNCOOBC2kd7vriSzY3E1+mLsFkMP9PBKnIJWLKTk2DtZbPwo4aPJO5xRKMBtNLS6UmRhWDNgSaVUjEu5W2WAdJC2wYR7vTrqPVLZtNVpIbNU1r0kslw/mPJbCP9hIUP26nM+9j50I+gsqmhjNRxbXm7n9oQ6BZdWQjjLs7zN8wjg+eJamEhvbijCHI4nR5GBIf9gLEEpRO9RF2XrEKrinESuvxJ3MLKstGFfvrzU2mpw2BZtURyG4YnzE/wtgZOUe/rRZPSanZUnJm2FlP6WT+YwmqYoOMe6wTTJZFVTZRIZOMRoeXP8hoPyOqlPKKqvwJWwY5GQIRqRaRh0XkbOb3nDHSItKZKUBzSEQOLLW/RpNPWjbszGwYmx9hXDlxkSG3ddwic2G8JP+xBCqZIJAKkSizXnpupy993YZzCCpzTaajis0m1yeCjwKPKqU2AI9m3s/HLUqpXUqpmVVBltJfo8kLDqeLLtc6vCZHGKtUioZED5MV1vGGyYV4RTM1qRCJeP7KgUaGenFKErzWM5alNenlnGgOQWWl00FGneZGFUPuhuAO4GuZ118D7jS4v0azLKLebbTGzpm6YTw80E2ZTIF/nWka8onN14pTkgT7OvM25nBPOrW1x289Q5CtVDYVWv4GeWViiEmP+fsfuRqCOqVUH0Dm93yfSAE/F5GDInLPMvprNHnF1rw7vWF89pBpGgYzdYrLGorbdTRLSSaWINyTv1iCscFOACrqrPfUFGjIBpUtL5ZApVL4U8MkTI4qhkXULBaRR4C5dmo+toR5blBK9YpILfCwiJxSSj2xhP5kDMg9AK2t5per0xQ3tVfthZfSKanbNptTw3is9xQA/rbiTTY3E29D+skme/POB7HMt21/o/UC7pwuN0HxLjuobHR4kEpJIhXmb4QvaAiUUq+f75yIDIhIg1KqT0QagMF5xujN/B4Uke8B1wFPAIvqn+l7H3AfwJ49e9RCujWaK9GyfgfjykOqx7wN41TwLNPKSV3zylgaqmlOB5UlhrvyN+hID2OqhOpq89fR5yLsqMEztbxYguHBLioBp9f8ZHq5Lg09CNydeX038IPZDUSkTEQqsq+BNwDHFttfoykEdoeDLtd6vBHzNoy9kWNccLRjdyz4fawo8JSWE6IKx0iOidhm4BzvJWgLmJqr/0qMueuois37/fXKfTNRxWUB8yOmc/3rfga4VUTOArdm3iMijSLyUKZNHfCUiBwGXgB+rJT66ZX6azRGEPVtoy12jkQ8ZvjcKhmndeo0waodhs9dSEKOOkom8hdLUD7VT9Rl/hr6fMTLlh9Ulq1VXFlj/lJ3Tl9FlFIh4HVzHO8F3pR5fR7YuZT+Go0ROFp24xm8n84zL9G+da+hc/effYkGprG1mLM/USjGPA3UjJ/N23i+RJBguXWjrlVlE+WDk4xEQlR6/UvqmxzpA8Bfb75HlDWftzQaA6i9ah8AQ2eMT0k9cOLJtIbNNxs+dyGJVTRTmwqSSiZzHis+PYmfCKkK6wWTZflVUNn5Jfe1jfUTpsISUeXaEGhWLc3rtjGqSkh2HzR+8u4XGFJVrN2wxfi5C4h4W3FLnNBgbumZAYYyEbt2n/nfmOejLJBe1hkZWPoGuWtigLAFoopBGwLNKsZut3OhZBv1YeMNQSBylM6SzdjtK+u/oCfQDkCoO/floUhf2hCUBtpyHqtQeDM1EpYTVFYaG2LMZW6Jyiwr61+hRrNEJppvoC11iWBvp2FzTkaCNKd6mKi9xrA5jaLq5ViC3OsSTA51ZcZsz3msQuGvb00HlUWX/gTkTQwxZYGoYtCGQLPKCWx/AwBdB35i2JxdR9OxlGXr9hk2p1FkYwnioc6cx4pnahsELBhMlsXpcmfSby/NUyqVSFCtIpaIKgZtCDSrnLXb9hGmAnX+ccPmHOt4lqQS1uy40bA5jaKs0keUMmzR3GMJbKM9hKiirKw8D8oKR8QRoGRyaVlXw6FeHJLCVmFuicos2hBoVjU2u50L5VfTEtmPSqUMmbN08EUu2Nup9lljozDfBO31ePIQS+CZ6CNkt8Ya+pUYc9dTGV9aUFm4P/204/KZH1UM2hBoNMRab6aeIbrPF76OsUolaZ08yUDl9oLPZRajnnqqpvtyHqdieoAxt/l5eBYiVtZATTK4pC8S40PpSm5lAW0INBpL0Hj1GwHoffGnC7TMnf5zRyhnAlquLfhcZjFd1kxNcjDnJ6xAKsh0mTWWTq5IZSOlMs3IEiqVTYfTm8tVteZHFYM2BBoNLeu2MYAf58UlJcRdFv3Hs4FkK29/4GW8rZTJNJHQ8quVjUWHKWcSVWl+Hp6FcFZng8rOLbpPKtpHSgn+Omt8Pm0INKsesdm4WHUta8ZeyktE7JVIXXqBqCpjzcY5s66sCBx16foK4c6jyx4jlKlp4Ky2xo3ySrwcVDa4+FgC23g/w1KF0+UulKwloQ2BRgOw9tX4GOX8scIWtPdHjnDeswWHw17QecykvDm9/zHZs/zMrtnyj+WZYjdW5uWgsqHFGwL35CAR+9JyExUSbQg0GqD92jcBMHT05wWbY2o0TGviIuM1uwo2hxVoaFnLiColNXhi2WNkb6o+C8cQZAnUt5JUglpCUFlZbIgxl3VqLGhDoNEANY3tdNmaKe1+umBzdB19EpsoStZeX7A5rEBVqYtLjlZcodPLHiMVuURc2QnUWTfPUBaH00VIfNjHFu8y60uGmLZIVDFoQ6DRvEx/9XWsnzxCbHqqIONHzz5LSgntO1dWxtG5GPVtoWXqDPHY9LL628d6GZJqHE5nnpUVhrCjFs8ig8oSsWl8Kkqq3DqusdoQaDQZnBteS6lM0/HSLwsyvmfgIF22Zvx+6wdJ5Ypz3c2UyjTnjjy1rP5lk31EnNb5xrwQ455aqhYZVBYa7MYmCluldVxjczIEIlItIg+LyNnMb98cba4SkUMzfkZE5A8y5z4hIj0zzr0pFz0aTS6su/Y2kkqIHn8k72OrVIrWiRMrOpBsJu27bwUgfPwXy+pfFR9k3GOdG+VCxMoaCSSHFhU7ER1Mp99wV1sjmAxyfyL4KPCoUmoD8Gjm/StQSp1WSu1SSu0CrgEmgO/NaPJ/sueVUg/N7q/RGEVVdQ3nnevx9j+T97EHOk/gZZRU88oNJJuJv66ZTlsrZX1L98JKJZPUpIaIlxePIXg5qCwSWrBpNqq43AK1irPkagjuAL6Wef014M4F2r8OOKeUWnoVB43GAIZqrmd97BTjo5G8jtubCSSr2bSCA8lmMVC9h/WTR5e8TzAc7MElSWxe628UZ3H60rEEoUVUKotlooq9ddaIKobcDUGdUqoPIPN7oUW9u4BvzTp2r4gcEZGvzrW0lEVE7hGRAyJyIBgM5qZao5mH8s2vxSlJOg7kd3ko2fU8Y6qENZuuzuu4Vma5+wTZso9uv3VulAtRVpM2WqODC3/HTY30kVA2fAHrlOBc0BCIyCMicmyOnzuWMpGIuIDfAL494/A/AOuAXUAf8Lfz9VdK3aeU2qOU2lNTs/I32zTmsP6a1xNTDiZPL29tez6qI0c4776qaLxg8kHbMvcJxgY7ASiva8+zosLhW0KlMvv4AMPixe5wFFrWollQiVLq9fOdE5EBEWlQSvWJSANwpW3z24EXlVIDM8Z++bWI/BPwo8XJ1mgKQ0lZBcfdWwgE8xdhPDU+Qlv8As833Z23MYuBZe8T9LyYjiFovqowwgqAv66VhLKRWkRQmWdqkIgjsODyiZHkujT0IJD913038IMrtH0ns5aFMsYjy1uA5cekazR5YqThVaxNnCcytPykaTPpPPo0DklRsnblVSRbiP7qa5e8TxDo/QXHXNvxFlG9BofTSUh8OBZRqawiNsS4haKKIXdD8BngVhE5C9yaeY+INIrIyx5AIlKaOf/dWf0/KyJHReQIcAvwhznq0WhyxrftVmyiOL8/P+UrR86mvZBat6/8QLLZuNan9wnOH3lyUe0vnTlEa/ISk2veUGBl+SfsqKVkauEvD95UiFiJlZ4HcjQESqmQUup1SqkNmd/DmeO9Sqk3zWg3oZTyK6Wis/q/Rym1XSm1Qyn1G9mNZ43GTNbtupkxVUK845d5Gc/Vf5BL0kCgzjqbg0ax1H2C7ie/QUoJG17zW4WUVRAmPLVUxa4cVDY1OYGPUZSFoopBRxZrNJfhdLo4V7qDxuEXch5LpVI0jx+nr2J1BJLNxl/bxAVbG6WL2CdQqRQt3T/ihGcnNY3thReXZ2JljQRSVw4qGx5IB5PZq6z1pUAbAo1mDiabb6RF9dJ/qSOncQa6OwgQIdl4TZ6UFR+D1XtYP3lswX2C0y/+kmbVz8Sm3zRIWZ6pbKREYoyE53dvj2ZqFlgpqhi0IdBo5qR2Z7p85cUDue0T9B59HIDAphty1lSsODP7BOcOX3mfIPL8N5lWTjbdUnzLQgCuTKWyod4L87aZCKWjiitqrBUspw2BRjMH7Zv3MEwltguP5zROrOsFJpWL9q1786Ss+Hg579CJ+fcJ4vEYG4IPc7zieiq91inYshRKa9IBcGPBznnbxDNRxT6L1CrOog2BRjMHNrudCxXX0DZyIKci7L7hw1xwXYXT6cqjuuKiOrNPcKV4ghNPPYifKOx4h4HK8kt1Q7qIztTQpXnbeMKnCKtyqvx1RslaFNoQaDTzkGy7mRrCXDrz4rL6T02OsybeQTSwcusTL5aF9gniL32LKGVsfXWR7g8A/roWEsqGinbP26Z25Bin7BsRm7VuvdZSo9FYiKbr7mRaOQg9+oVl9b946DFcksTdvrIrki0G5/pXz7tPMDEWZUv0SU5Vvw63u8QEdfnB7nAQFD+lI5cnnlOpFINdJ6mf7qSrZIsJ6q6MdZJdaDQWo6l1LU8F7mDf4Pc4d/Q51m1fWmRw8sV/Y0SV0rzn1wqksHho330rPAeR47+Aa1+ZtebEY/ezR6apuPZdJqnLH4dK9vG60Z9y5JFvMN7XgT10hqqxczTGL1IrkwBEAtbzINOGQKO5Atvf9WlGvvQwyR/8PsnNzy46UVjnyQOsHXyEZypv4xb/vEl1Vw3VtY2ct7VT1vfsZeecJ75DPwE2XVd80cSzOVr7Zm7v+iE7nroXgCG8DLjaOFFzO9Rsoqx1F++75nUmq7wcbQg0mitQ5a/jwO6PsefFP+a57/wN++66rPbSZUxPTZD89gcYk1J2vPszBqgsDoL+PWwf/CHx2BROlweA4cEetk4cYH/ju6m3201WmDtvf/Ov89j+f6ahuoqG9TsJ+OuwVlahudF7BBrNAlzz6/dwxHMN207+HYM98/uIZ3nxX/8n61IXuHTTZ/HXWacKldlk8w6dO/SrfYKzj30Dh6Sov/HdJirLH+2BMm65/W1s2vsGy3kGXQltCDSaBRCbDf/bv4SDBN3//vtXbHvsqR+yt+/fed5/J7ted5dBCouD9t3ppZ/IicdePlZ19vtcsLWzZhXHWVgBbQg0mkXQtHYrL629h93jT3Lo4X+fs010OEjNIx+m297Ijt/+e4MVWh9fTQMXZuwT9Jw/wabESfrbf8NkZRptCDSaRbLnnR/ngq2Nhqf/F2Mj4VecU6kUZ//lv1KtIky/+SuUlFWYpNLaDPr3sG7qOLHpKS4+ni53vuY17zVZlUYbAo1mkThdbqZv/zw1aphj//bHrzh38Ef/yJ7Rxzi45oNsuHr11R1YLC/XJzj0BI2XfsQJ13bqWzeYLWvVow2BRrMENl37evYH7uDagQc4m9n07O08zVUHP8lJ51auffdfmKzQ2rRfk07mN/HEl2hLdTO64S0mK9JAjoZARN4mIsdFJCUie67Q7jYROS0iHSLy0RnHq0XkYRE5m/mtHa41lmfTez5PWKqQH36Y6akJot98PwJU/da/WKoguRXxBeo5Z1vD7vEniCs7m167MryFip1cnwiOAf8FeGK+BiJiB75Munj9FuCdIpKNsf4o8KhSagPwaOa9RmNpqrx+uq77BOuT51CfaWdz/Dgnd/85je3FU2zdTI60pNNMD7paisrFciWTa6nKk0qp0ws0uw7oUEqdV0rFgPuBOzLn7gC+lnn9NeDOXPRoNEax+7a7eankejxM81zlG9nz5g+aLaloKLv2XfxL4o103vg5s6VoMhjxHNsEzMzL2g1knYbrsnWKlVJ9IjJvRWcRuQe4B6C11Vq5vDWrD7HZaPyv9/PlJ17i3bfeYLlsklbm9VsaefruL/Oq9cUQc7s6WNAQiMgjwFyVlj+mlPrBIuaQOY6pRfR7ZQel7gPuA9izZ8+S+2s0+aau2suH7rzFbBlFh80m3LShxmwZmhksaAiUUq9fqM0CdAMz67I1A72Z1wMi0pB5GmgABnOcS6PRaDRLxIjn2f3ABhFZIyIu4C7gwcy5B4G7M6/vBhbzhKHRaDSaPJKr++hbRKQbuB74sYj8LHO8UUQeAlBKJYB7gZ8BJ4EHlFLHM0N8BrhVRM4Ct2beazQajcZARKniW27fs2ePOnDggNkyNBqNpqgQkYNKqctivrSrg0aj0axytCHQaDSaVY42BBqNRrPK0YZAo9FoVjlFuVksIkGga5ndA8BQHuWYQbF/Bq3ffIr9MxS7fjDnM7QppS6L5itKQ5ALInJgrl3zYqLYP4PWbz7F/hmKXT9Y6zPopSGNRqNZ5WhDoNFoNKuc1WgI7jNbQB4o9s+g9ZtPsX+GYtcPFvoMq26PQKPRaDSvZDU+EWg0Go1mBtoQaDQazSpnxRoCEblNRE6LSIeIXFYLWdJ8MXP+iIjsNkPnfCxC/2tEJCoihzI/f26GzvkQka+KyKCIHJvnvKX//rCoz2D1a9AiIo+JyEkROS4iH56jjWWvwyL1W/0aeETkBRE5nPkMn5yjjfnXQCm14n4AO3AOWAu4gMPAlllt3gT8hHQFtX3A82brXqL+1wA/MlvrFT7DzcBu4Ng85y3791/CZ7D6NWgAdmdeVwBniuz/wWL0W/0aCFCeee0Engf2We0arNQnguuADqXUeaVUDLgfuGNWmzuAr6s0zwHeTJU0K7AY/ZZGKfUEMHyFJlb++wOL+gyWRinVp5R6MfN6lHQ9kKZZzSx7HRap39Jk/q5jmbfOzM9sDx3Tr8FKNQRNwKUZ77u5/B/QYtqYxWK1XZ955PyJiGw1RlresPLffykUxTUQkXbgatLfSGdSFNfhCvrB4tdAROwicoh0Kd6HlVKWuwYL1iwuUmSOY7Ot8GLamMVitL1IOm/ImIi8Cfg+sKHQwvKIlf/+i6UoroGIlAP/CfyBUmpk9uk5uljqOiyg3/LXQCmVBHaJiBf4nohsU0rN3Hcy/Rqs1CeCbqBlxvtmoHcZbcxiQW1KqZHsI6dS6iHAKSIB4yTmjJX//ouiGK6BiDhJ30S/qZT67hxNLH0dFtJfDNcgi1IqAvwSuG3WKdOvwUo1BPuBDSKyRkRcwF3Ag7PaPAi8N7Njvw+IKqX6jBY6DwvqF5F6EZHM6+tIX8uQ4UqXj5X//ovC6tcgo+3/ASeVUp+fp5llr8Ni9BfBNajJPAkgIiXA64FTs5qZfg1W5NKQUiohIvcCPyPtgfNVpdRxEflg5vxXgIdI79Z3ABPA+83SO5tF6n8r8N9EJAFMAnepjAuCFRCRb5H26AiISDfwcdIbZZb/+2dZxGew9DUAbgDeAxzNrFED/CnQCkVxHRaj3+rXoAH4mojYSRupB5RSP7LavUinmNBoNJpVzkpdGtJoNBrNItGGQKPRaFY52hBoNBrNKkcbAo1Go1nlaEOg0Wg0qxxtCDQajWaVow2BRqPRrHL+fwNnW1DpgQgmAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xI,yI)\n",
    "plt.plot(x_fine,f_try(x_fine))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay - so it looks like our default function is just drawing straight lines between the points.  This is exaclty what our plot is doing, and so we get the same sharp corners.  But if we fit with a polynomial between several points, we will get a smoother looking result.  E.g. using a cubic polynomial:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_cube= interpolate.interp1d(xI, yI,kind='cubic')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xI,yI)\n",
    "plt.plot(x_fine,f_cube(x_fine))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_fine,f_try(x_fine)-np.sin(2*x_fine**2))\n",
    "plt.plot(x_fine,f_cube(x_fine)-np.sin(2*x_fine**2),color='darkorange')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_fine,f_cube(x_fine)-np.sin(2*x_fine**2),color='darkorange')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we see that our initial function was making about a 10% error.  Using a cubic instead, we get less than 1%.  In fact, for the early points we do really well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_fine,np.abs(f_cube(x_fine)-np.sin(2*x_fine**2)))\n",
    "plt.yscale('log')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we are getting 1 part in a 100 000 up to about $x=1$ using only 40 data points for the full range of $[0,\\pi]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the interpolation is a function.  So we can also use this function inside of our solving code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2532996656450621"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.newton(f_cube,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.254977768854082e-05"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sin(2*1.2532996656450606**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Operations on functions / interpolations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the central reasons to use a spline is to a approximate a quantity we don't know exactly.  For example, Scipy has a built in function that computes a numerical derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.misc import derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    return x**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.000000000097369"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivative(g,1,dx=1e-5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could make our own very simple version:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_derivative(g,x0,dx):\n",
    "    return (g(x0+dx)-g(x0-dx))/(2*dx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.000000000097369"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_derivative(g,1,dx=1e-5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We matched the result exactly.  Of course, that is just the default, derivative has many ways to improve. E.g. we can include more than just the difference of two poitns but include more points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.0000000000075104"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivative(g,1,dx=1e-5,order=7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Of course, we notice that this just returns a number for a single point.  However, if we wanted the derivative as a function we could just take a bunch of derivatives and make an interpolating function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "def h(x):\n",
    "    return np.sin(2*x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "dh=np.zeros(len(xI))\n",
    "for i in range(len(xI)):\n",
    "    dh[i]=derivative(h,xI[i],dx=1e-5,order=7)\n",
    "dh_fnc=interpolate.interp1d(xI,dh,kind='cubic')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x118b9edf0>]"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xI,h(xI))\n",
    "plt.plot(xI,dh_fnc(xI),color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can look for maxima"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.886226905815053\n"
     ]
    }
   ],
   "source": [
    "max_h=optimize.newton(dh_fnc,1.2)\n",
    "print(max_h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(xI,h(xI))\n",
    "plt.plot([max_h,max_h],[-1,1.5])\n",
    "plt.xlim(0.5,1.1)\n",
    "plt.ylim(0.5,1.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Special Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A common thing one encounters in physics is the need to use special functions.  Many of these are implemented in scipy for us:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import special as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11877cb20>]"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(-10,1,102)\n",
    "plt.plot(x,sp.gamma(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x118f36c40>]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(0,100,102)\n",
    "plt.plot(x,sp.y0(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We don't have a particular use for them right now, but this is something you will need in most scientific applications."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is also worth noting that these functions are defined in terms of complex arguments as well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.4177906356943205+9.86028723231441j)"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.gamma(-1+0.1j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The behavior of functions for complex arguments is a very important subject. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.collections.PolyCollection at 0x118903130>"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=y=np.linspace(-1,1,102)\n",
    "X,Y=np.meshgrid(x,y)\n",
    "plt.pcolor(X,Y,np.real(sp.hankel1(0.2,X+(0+1j)*Y)),cmap='RdBu',shading='auto')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scipy does not define all special functions for complex values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "ufunc '_riemann_zeta' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m/var/folders/vh/rcz751b57355h3qzj37sc3080000gp/T/ipykernel_35349/817459801.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mzeta\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1j\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m/opt/anaconda3/lib/python3.9/site-packages/scipy/special/_basic.py\u001b[0m in \u001b[0;36mzeta\u001b[0;34m(x, q, out)\u001b[0m\n\u001b[1;32m   2570\u001b[0m     \"\"\"\n\u001b[1;32m   2571\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mq\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2572\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mufuncs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_riemann_zeta\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mout\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2573\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2574\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mufuncs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_zeta\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mq\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mout\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: ufunc '_riemann_zeta' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''"
     ]
    }
   ],
   "source": [
    "sp.zeta(1j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are going to see that scipy has a lot of built in code that is extremely powerful.  It is essentially built on top of numpy so they tend to work very well together.  What we hopefully learned in these examples is that there are really two things going on inside scipy: (1) an algorithm and (2) an implementation of the algorithm in terms of code.  The algorithm works on some premise that we can and should understand.  We can try our own version of the algorithm to get a deeper understanding of what scipy is doing.  Fundamentally, scipy is not doing anything deeper or more sophisticed than what we can code ourselves.  What scipy has really done is code it in a very fast way (probably using some neat python/C tricks we don't know) and with lots of built in options that would be time consuming for us to do ourselves."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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