Finite difference checks.ipynb

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  {
   "cell_type": "markdown",
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   "source": [
    "# Hand-calculated derivative rules: Error prone\n",
    "\n",
    "Hand-computation of derivatives of complicated functions involving vectors and matrices is **error-prone**, and is a common source of bugs when performing numerical optimization, root-finding, sensitivity-analysis, etcetera.\n",
    "\n",
    "The ideal alternative is to use **automatic differentiation (AD)**:\n",
    "* AD lets software/compilers perform the derivatives for you.  This is extremely reliable and, with modern AD software, can be very efficient.\n",
    "* Unfortunately, there is still lots of code, e.g. code calling external libraries in other languages, that AD tools can't comprehend.\n",
    "* And there are other cases where AD \"needs help\" — for example, if you are computing an answer *approximately* (e.g. solving a nonlinear equation by Newton's method), AD can waste a lot of effort trying to *exactly differentiate the error* in your approximation.   Often, you can compute the approximate answer (to the same accuracy) much more efficiently.\n",
    "* Even in cases where AD falls down, often you only need to give it a **little help**: define a differentiation rule for a *small piece* of your program and let AD handle the rest.  In Julia, this is done with by defining a [\"ChainRule\"](https://github.com/JuliaDiff/ChainRulesCore.jl), and in Python autograd/JAX it is done by defining a custom \"vJp\" (row-vector—Jacobian product) and/or \"Jvp\" (Jacobian–vector product).\n",
    "\n",
    "If you do have to work out your own derivative rule, you *really* need to **check** it against something \"brute force\" and hard to get wrong.  The usual candidate is…"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2d90dbc",
   "metadata": {},
   "source": [
    "# Finite-difference approximations: Easy version\n",
    "\n",
    "The simplest way to check a derivative is to recall that the definition of a differential:\n",
    "$$\n",
    "df = f(x+dx) - f(x) = f'(x) dx\n",
    "$$\n",
    "came from dropping higher-order terms from a small but finite difference:\n",
    "$$\n",
    "\\delta f = f(x+\\delta x) - f(x) = f'(x) \\delta x + o(\\Vert \\delta x \\Vert)\n",
    "$$\n",
    "So, we can just compare the **finite difference** $\\boxed{f(x+\\delta x) - f(x)}$ to our **(directional) derivative operator** $f'(x) \\delta x$ (i.e. the derivative in the direction $\\delta x$).\n",
    "\n",
    "$f(x+\\delta x) - f(x)$ is also called a **forward difference** approximation.  We'll look at other possibilities below.\n",
    "\n",
    "* The antonym of a forward diffence is a **backward difference** approximation $f(x) - f(x - \\delta x) \\approx f'(x) \\delta x$ … if you just want to compute a derivative, there is not much practical distinction between forward and backward differences.  The distinction becomes more important when discretizing (approximating) differential equations.\n",
    "\n",
    "* Finite-difference approximations come in many forms, but they are generally a **last resort** in cases where it's too much effort to work out an analytical derivative and AD fails.  But they are also useful to **check** your analytical derivatives and to quickly **explore**."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a4818b6",
   "metadata": {},
   "source": [
    "## example: f(A)=A²\n",
    "\n",
    "Let's try it for $f(A) = A^2$ (a matrix-valued function of a square matrix $A \\in \\mathbb{R}^{m\\times m}$).\n",
    "\n",
    "First, we'll do it by hand via the product rule:\n",
    "\n",
    "$$\n",
    "df = A dA + dA A\n",
    "$$\n",
    "i.e. $f'(A)$ is the **linear operator** $\\boxed{f'(A)[\\delta A] = A \\delta A + \\delta A A}$.  This is **≠ 2AδA** because because **A and δA do not commute** in general.\n",
    "\n",
    "Let's check it against a finite difference.   We'll try it for a *random* input A and a *random small* perturbation $\\delta A$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "9d0e4943",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -2.1088     0.734105   0.239804     0.769526\n",
       " -0.50642    1.3228     0.820155    -0.155089\n",
       " -0.57114   -0.600107  -0.00180516  -1.41079\n",
       "  0.160461   0.874805  -0.642604    -0.5685"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(A) = A^2\n",
    "\n",
    "A = randn(4,4) # 4×4 matrix with Gaussian random (bell curve) entries of magnitude ~ 1 (std. dev = 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3daece4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " 1.07507e-8   7.14782e-9  -1.93596e-9   1.54683e-9\n",
       " 1.38533e-8   1.54251e-8   1.97596e-8  -1.5794e-8\n",
       " 9.02758e-9  -2.33086e-8   1.87726e-9   2.47228e-9\n",
       " 2.16736e-9   7.34033e-9   6.54572e-9   8.20985e-9"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dA = randn(4,4) * 1e-8 # a small random perturbation ~ 10⁻⁸"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "542cc00d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -3.36055e-8   1.61716e-8   3.15253e-8   1.07036e-9\n",
       " -3.08965e-8   1.42894e-9   5.37296e-8  -3.15509e-8\n",
       " -2.54365e-8  -4.68219e-8  -3.85341e-8   3.51576e-9\n",
       " -3.89827e-9   4.00008e-8   1.33e-8     -3.3197e-8"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "approx = f(A + dA) - f(A) # finite difference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e955820e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -3.36055e-8   1.61716e-8   3.15253e-8   1.07036e-9\n",
       " -3.08965e-8   1.42894e-9   5.37296e-8  -3.15509e-8\n",
       " -2.54365e-8  -4.68219e-8  -3.85341e-8   3.51576e-9\n",
       " -3.89827e-9   4.00008e-8   1.33e-8     -3.3197e-8"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exact = A*dA + dA*A # exact directional derivative (linearization)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "43cb4db7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -1.73372e-8  -7.38125e-9   4.81509e-8  -1.58918e-8\n",
       "  3.98974e-8  -6.9411e-9    5.52859e-8  -4.18427e-8\n",
       " -3.50552e-8  -4.73053e-8  -3.99804e-8  -5.98437e-9\n",
       "  1.36214e-8   5.08921e-8   2.40952e-8  -3.9649e-8"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2A*dA # wrong! matrices don't commute"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "20818794",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       "  1.62684e-8  -2.35529e-8    1.66256e-8  -1.69621e-8\n",
       "  7.07939e-8  -8.37003e-9    1.5563e-9   -1.02918e-8\n",
       " -9.61869e-9  -4.83447e-10  -1.44632e-9  -9.50013e-9\n",
       "  1.75197e-8   1.08913e-8    1.07951e-8  -6.452e-9"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2A*dA - approx"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "630c56ec",
   "metadata": {},
   "source": [
    "To be more quantitative, we might compute the \"norm\" $\\Vert \\text{approx} - \\text{exact} \\Vert$, which we want to be small.   But small **compared to what?**   The natural answer is **small compared to the correct answer**.  This is called the [relative error](https://en.wikipedia.org/wiki/Approximation_error) (or \"fractional\" error):\n",
    "$$\n",
    "\\text{relative error} = \\frac{\\Vert \\text{approx} - \\text{exact} \\Vert}{\\Vert \\text{exact}\\Vert}\n",
    "$$\n",
    "Here, $\\Vert \\cdot \\Vert$ is a [norm](https://en.wikipedia.org/wiki/Norm_(mathematics)), like the length of a vector. In Julia, the `LinearAlgebra` package provides a `norm` function, and we can use that to define a relative error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "45931cc4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "relative_error (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "relative_error(approx, exact) = norm(approx - exact) / norm(exact)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7dc766e",
   "metadata": {},
   "source": [
    "Let's try it on our case above:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "1fb85dc0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.201810486737118e-8"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "relative_error(approx, exact)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa60621b",
   "metadata": {},
   "source": [
    "Pretty good — we are matching to **about eight significant digits**!\n",
    "\n",
    "Getting a good match for a random input and a random small displacement isn't a proof of correctness, but it is always a good thing to check.   This kind of randomized comparison will almost always **catch major bugs** where you have calculated the symbolic derivative incorrectly, like our $2AdA$ example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c4ed374c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.7126976615760792"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "relative_error(2A*dA, exact)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a234dc8e",
   "metadata": {},
   "source": [
    "Here, for our buggy derivative, the relative error was **close to 50%**."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "117f334b",
   "metadata": {},
   "source": [
    "### aside: norm of a matrix?\n",
    "\n",
    "Wait, what do we mean by the norm $\\Vert A \\Vert$ of a matrix $A$?\n",
    "\n",
    "The ordinary Euclidean (\"L2\") norm of a column-vector $x \\in \\mathbb{R}^n$ is $\\Vert x \\Vert = \\sqrt{x^T x} = \\sqrt{\\sum_i x_i^2}$.\n",
    "\n",
    "For a matrix, the natural analogue is the **sqrt of the sum of the squares of the entries**.  This is called the [Frobenius norm](https://mathworld.wolfram.com/FrobeniusNorm.html), and is what `norm(A)` does in Julia by default."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a34d0cb4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{ccc}\n",
       "m11 & m12 & m13 \\\\\n",
       "m21 & m22 & m23 \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "2×3 Matrix{Num}:\n",
       " m11  m12  m13\n",
       " m21  m22  m23"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Symbolics\n",
    "\n",
    "@variables m11 m12 m13 m21 m22 m23\n",
    "M = [m11 m12 m13\n",
    "     m21 m22 m23]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f8846473",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "\\sqrt{\\mathrm{abs2}\\left( m11 \\right) + \\mathrm{abs2}\\left( m12 \\right) + \\mathrm{abs2}\\left( m13 \\right) + \\mathrm{abs2}\\left( m21 \\right) + \\mathrm{abs2}\\left( m22 \\right) + \\mathrm{abs2}\\left( m23 \\right)}\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "sqrt(abs2(m11) + abs2(m12) + abs2(m13) + abs2(m21) + abs2(m22) + abs2(m23))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab0e0512",
   "metadata": {},
   "source": [
    "(`abs2(x)` in Julia is the squared absolute value $|x|^2$.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a8981b83",
   "metadata": {},
   "source": [
    "But we can write this in linear algebra notation!\n",
    "\n",
    "Recall that the (Euclidean) \"inner product\" of two matrices is $\\text{tr}(B^T A)$.   Then we can define the Frobenius norm of a matrix as the sqrt of the dot product with itself!\n",
    "\n",
    "$$\n",
    "\\Vert A \\Vert_F = \\sqrt{\\text{trace}(A^T A)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "625bb0ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.541526593233751"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "b1dba485",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.5415265932337516"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(tr(A'A))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "e61e275b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.541526593233751"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(sum(abs2, A))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "e2440768",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.541526593233751"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(sum(A[i,j]^2 for i = 1:4, j = 1:4))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ada58d4",
   "metadata": {},
   "source": [
    "(There are also other ways to define a norm of matrix, most famously by \"induced norms\", but we will stick with the Frobenius norm here.)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f6347b0",
   "metadata": {},
   "source": [
    "# Accuracy of finite differences\n",
    "\n",
    "How accurate is our finite-difference approximation above?  How should we choose the size of δx?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd265557",
   "metadata": {},
   "source": [
    "Let's try our example $f(A) = A^2$ above, and plot the relative error as a function of $\\Vert \\delta A \\Vert$ as we scale $\\delta A$ by a factor $s$, making it larger or smaller."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "675c58b9",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots # load a plotting package; there are several to choose from in Julia; this is a popular one\n",
    "using LaTeXStrings # makes it easy to put LaTeX equations in L\"...\" strings"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0dc9de16",
   "metadata": {},
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      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = 10 .^ range(-8,8,50)\n",
    "errors = [relative_error(f(A+s*dA)-f(A), exact*s) for s in s]\n",
    "plot(s * norm(dA), errors, xaxis=:log, yaxis=:log, markershape=:circle,\n",
    "     title = L\"forward-difference accuracy for $f(A)=A^2$\", xlabel=L\"\\Vert \\delta A\\Vert\", ylabel=\"relative error\",\n",
    "     label = \"error\")\n",
    "plot!(s, s, label=L\"linear $\\Vert \\delta A\\Vert$ reference\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77479cce",
   "metadata": {},
   "source": [
    "We observe two main features:\n",
    "\n",
    "* The relative error decreases *linearly* with $\\Vert \\delta A \\Vert$.  This is called **first-order accuracy**.  Why?\n",
    "* When δA gets too small, the error *increases*.  Why?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "860ad8bb",
   "metadata": {},
   "source": [
    "## Order of accuracy\n",
    "\n",
    "If the relative error scales with $\\Vert \\delta A \\Vert^n$, we call the approximation **n-th order accurate**.  For forward differences, here, the order is **n=1**.  Why?\n",
    "\n",
    "For any $f(x)$ with a nonzero second derivative (think of the Taylor series), we have\n",
    "$$\n",
    "    f(x + \\delta x) = f(x) + f'(x) \\delta x + (\\text{terms proportional to }\\Vert \\delta x \\Vert^2) + \\underbrace{o(\\Vert \\delta x \\Vert^2)}_\\text{i.e. higher-order terms}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1675f2b3",
   "metadata": {},
   "source": [
    "That is, the terms we *dropped* in our forward-difference approximations are proportional to $\\Vert \\delta x\\Vert^2$.  But that means that the **relative error is linear**:\n",
    "$$\n",
    "\\begin{align}\n",
    "  \\text{relative error} &= \\frac{\\Vert f(x+\\delta x) - f(x) - f'(x) \\delta x \\Vert}{\\Vert f'(x) \\delta x \\Vert} \\\\\n",
    "   &= \\frac{(\\text{terms proportional to }\\Vert \\delta x \\Vert^2) + o(\\Vert \\delta x \\Vert^2)}{\\text{proportional to }\\Vert \\delta x \\Vert} = (\\text{terms proportional to }\\Vert \\delta x \\Vert) + o(\\Vert \\delta x \\Vert)\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc1ef94d",
   "metadata": {},
   "source": [
    "## Roundoff error\n",
    "\n",
    "The reason why the error *increased* for very small $\\delta A$ was due to **roundoff errors**.  The computer only stores a **finite number of significant digits** (about 15 decimal digits) for each real number and rounds off the rest on each operation — this is called [floating-point arithmetic](https://en.wikipedia.org/wiki/Floating-point_arithmetic).\n",
    "\n",
    "If δx is too small, then the difference $f(x+\\delta x) - f(x)$ gets rounded off to zero (some or all of the *significant digits cancel*).  This is called [catastrophic cancellation](https://en.wikipedia.org/wiki/Catastrophic_cancellation).\n",
    "\n",
    "Let's compute the derivative of $\\sin(x)$ at $x=1$ with finite differences.   For a scalar $f(x)$, we have $f(x+\\delta x) - f(x) \\approx f'(x) \\delta x$, or equivalently:\n",
    "\n",
    "$$\n",
    "f'(x) = \\frac{f(x+\\delta x) - f(x)}{\\delta x} + \\underbrace{O(\\delta x)}_{\\text{terms proportional to }|\\delta x|\\text{ or smaller}}\n",
    "$$\n",
    "\n",
    "(This is the usual way you see a forward difference approximation written, but it only works when $\\delta x$ is a scalar.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "60dd9d6d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5402980985058647"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx = 1e-5\n",
    "(sin(1 + dx) - sin(1))/dx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "2d2587ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5403023058681398"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos(1) # correct answer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "4fc600b4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.787052228587771e-6"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "relative_error((sin(1 + dx) - sin(1))/dx, cos(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f9e44dc",
   "metadata": {},
   "source": [
    "So far, so good.  For $\\delta x = 10^{-5}$, we got about 5–6 significant digits correct.  Now let's make $\\delta x$ much smaller, $\\delta x = 10^{-100}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "504d0d5f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx = 1e-100\n",
    "(sin(1 + dx) - sin(1))/dx"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35440ed2",
   "metadata": {},
   "source": [
    "What happened?   The computer can represent small numbers just fine, because floating-point corresponds to \"scientific notation\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "173b2d27",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0e-100"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1e-100"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a7d8a53",
   "metadata": {},
   "source": [
    "But look at this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "c8093692",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 + 1e-100"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1a5fdedb",
   "metadata": {},
   "source": [
    "The correct answer would be\n",
    "$$\n",
    "1 + 10^{-100} = 1.\\underbrace{000000\\cdots000000}_{\\text{99 zeros}}1\n",
    "$$\n",
    "but the computer only stores about 15 decimal digits (by default … you can ask for more, at the expense of speed), so it rounds this to $1.0$.\n",
    "\n",
    "Hence, our finite difference gave zero.\n",
    "\n",
    "Even if we do something less extreme, we have problems.   This looks okay:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "bdb27010",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000001"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx = 1e-13\n",
    "1 + dx"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d2510c60",
   "metadata": {},
   "source": [
    "because the computer has enough digits, right?  But compare:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "ee8e581d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8414709848079505"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(1 + dx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "173807c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8414709848078965"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9852d20f",
   "metadata": {},
   "source": [
    "They only differ in the **last 4 significant digits stored**.  So when you subtract them, you only have **4 digits** left:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "6bbe34e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001358343083756518"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "relative_error((sin(1 + dx) - sin(1))/dx, cos(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a4220af",
   "metadata": {},
   "source": [
    "and the error for $\\delta x = 10^{-13}$ is now $10^{-3}$!!"
   ]
  }
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