{
"cells": [
{
"cell_type": "markdown",
"id": "0c8a9458-2bc1-4fdc-a411-a5f836b9bb3c",
"metadata": {},
"source": [
"# 2025-08-27 Function Errors\n",
"\n",
"* Review last time\n",
"\n",
"* Conditioning\n",
"\n",
"* Well posedness\n",
"\n",
"* Absolute & relative errors\n",
"\n",
"* Condition numbers"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "3ee388cc-4119-4152-ba23-f8b326bf07a1",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Bad behavior for a function\n",
"using Plots\n",
"default(lw=4, ms=5, legendfontsize=12, xtickfontsize=12, ytickfontsize=12)\n",
"\n",
"a = 1e-15\n",
"plot(x -> (1 + x) - 1, xlim=(-a, a))\n",
"plot!(x -> x)"
]
},
{
"cell_type": "markdown",
"id": "104179d4-1e55-4da6-b543-b158a88236fd",
"metadata": {},
"source": [
"## Machine precision\n",
"\n",
"Floating point numbers do not exactly represent continuous values.\n",
"\n",
"There exists $\\epsilon_\\text{machine}$ such that\n",
"\n",
"$$ 1 \\oplus x = 1 $$ for all $$ \\lvert x \\rvert < \\epsilon_\\text{machine} $$\n",
"\n",
"Note: $\\oplus, \\ominus, \\otimes, \\oslash$ are the floating point arithmatic versions of $+, -, \\times, /$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "fa8c7d90-82e5-4f2b-af0b-c1143f11fdb2",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ϵ = 1.1102230246251565e-16\n",
"eps() = 2.220446049250313e-16\n"
]
}
],
"source": [
"# Computing machine precision\n",
"ϵ = 1\n",
"while 1 + ϵ != 1\n",
" ϵ = ϵ / 2\n",
"end\n",
"\n",
"# And lets ask Julia what ϵ actually is\n",
"@show ϵ\n",
"@show eps();"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "b4358aff-7735-4803-ac4f-9c41c45baeb1",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ϵ = 5.9604645f-8\n",
"eps(Float32) = 1.1920929f-7\n"
]
}
],
"source": [
"# Number type matters!\n",
"ϵ = 1.0f0\n",
"while 1 + ϵ != 1\n",
" ϵ = ϵ / 2\n",
"end\n",
"\n",
"# And lets ask Julia what ϵ actually is\n",
"@show ϵ\n",
"@show eps(Float32);"
]
},
{
"cell_type": "markdown",
"id": "2092a3e4-cd78-4f0e-b496-422b7c413c15",
"metadata": {},
"source": [
"### Remember Taylor Series?\n",
"\n",
"$$ log \\left( 1 + x \\right) = x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\cdots $$\n",
"\n",
"Which is more accurate for computing $log \\left( 1 + x \\right)$?\n",
"\n",
"Note: This issue shows up in Ratel!"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "2514bda4-18a0-4a8c-be45-fc1372b5cf8b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"absolute error: -6.07747099184471e-17\n",
"relative error: -6.077471022232065e-9\n"
]
}
],
"source": [
"# Lets compare\n",
"f1(x) = log(1 + x)\n",
"f2(x) = x - x^2 / 2 + x^3 / 3 # Taylor series\n",
"# Question - how many terms do I need here?\n",
"\n",
"y1 = f1(1e-8)\n",
"y2 = f2(1e-8)\n",
"\n",
"# Absolute\n",
"println(\"absolute error: $(y1 - y2)\")\n",
"\n",
"# Relative\n",
"println(\"relative error: $((y1 - y2) / y2)\")"
]
},
{
"cell_type": "markdown",
"id": "d7d9e33d-3cf8-415c-9169-a3eeb5947bcc",
"metadata": {},
"source": [
"## Conditioning\n",
"\n",
"A function $f \\left( x \\right)$ is **well conditioned** if a small change in the input $x$ results in a small change in the output $f \\left( x \\right)$. (ToDo: We should be precise about what 'small' means here!)\n",
"\n",
"The function $f \\left( x \\right)$ may be a simple expression:\n",
"\n",
"* $f \\left( x \\right) = 2 x$\n",
"\n",
"* $f \\left( x \\right) = \\sqrt{x}$\n",
"\n",
"* $f \\left( x \\right) = \\text{log} \\left( x \\right)$\n",
"\n",
"* $f \\left( x \\right) = x - 1$ "
]
},
{
"cell_type": "markdown",
"id": "34a22747-5a4c-4e7c-abf7-ed3f10dadcc6",
"metadata": {},
"source": [
"The function may also be much, much more complex:\n",
"\n",
"* Find the positive root of the function $t^2 + \\left( 1 - x \\right) t - x$\n",
"\n",
"* Find the eigenvectors of the matrix\n",
"$$ A \\left( x \\right) = \\left[ \\begin{array}{cc} 1 & 1 \\\\ 0 & x \\end{array} \\right] $$\n",
"\n",
"* Find the deflection of the bridge when a truck of mass $x$ drives over it\n",
"\n",
"* Determine the force required to fracture a composite material as a function of the manufacturing temperature $x$\n",
"\n",
"* Determine the probability of component failure as a function of the age $x$\n",
"\n",
"* Project the wind power generation given the current weather observations $x$"
]
},
{
"cell_type": "markdown",
"id": "2108674b-c1e9-4fd9-a140-7fe792e35fb8",
"metadata": {},
"source": [
"## Specification\n",
"\n",
"Do we have enough information to solve the problem?\n",
"\n",
"* Some of the previous problems are fully specified\n",
"\n",
"* Others involve active, ongoing research\n",
"\n",
"* The function modeling reality may be more or less smooth, ore or less behaved\n",
"\n",
"* Is the problem **well-posed**?"
]
},
{
"cell_type": "markdown",
"id": "971cfe2e-3069-4643-87cc-a7017b13912f",
"metadata": {},
"source": [
"## Well-posed Problems\n",
"\n",
"A problem is **well-posed** if\n",
"\n",
"1) The problem has a solution\n",
"\n",
"2) The solution is unique\n",
"\n",
"3) The solution's behavior changes continuously with initial conditions\n",
"\n",
"For 3), the variation may be continuous but rapid, and there may be real-world sources of error or variance. **Conditioning** focuses on this part of **well-possedness**."
]
},
{
"cell_type": "markdown",
"id": "6a0bb345-9fe8-4120-9fe2-1b1409e2903e",
"metadata": {},
"source": [
"## Test case: $e^x$\n",
"\n",
"$$ e^x = \\sum_{k = 0}^\\infty x^k / k! $$"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "daa824ce-a213-4f42-9ab1-dbc5dbab1fe9",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"myexp(1) - exp(1) = 1.445646891729250136554224997792468345749669676277240766303534163549513721620773e-16\n"
]
}
],
"source": [
"# Let's try to code this up\n",
"function myexp(x)\n",
" sum = 1\n",
" for k in 1:100\n",
" sum += x^k / factorial(big(k))\n",
" end\n",
" sum\n",
"end\n",
"@show myexp(1) - exp(1);"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "37cb8c7b-49cd-4503-8d24-7c53bff45a8b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"myexp(1) - exp(1) = 4.440892098500626e-16\n"
]
}
],
"source": [
"# Lets pick our with ϵ in mind\n",
"function myexp(x)\n",
" sum = 0\n",
" term = 1\n",
" n = 1\n",
" while sum + term != sum\n",
" sum += term\n",
" term *= x / n\n",
" n += 1\n",
" end\n",
" sum\n",
"end\n",
"@show myexp(1) - exp(1);"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "622a8f7b-ec54-4373-ada6-3ba716b7ed71",
"metadata": {},
"outputs": [
{
"data": {
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# looking good here\n",
"a = 2.5\n",
"plot([myexp, exp], xlims=[-a, a], label=[\"myexp\" \"exp\"])"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "8688f0ee-1f5f-4240-97fd-33055fdbe831",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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",
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"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"GKS: Possible loss of precision in routine SET_WINDOW\n"
]
}
],
"source": [
"# but something is bad here\n",
"a = 1e-15\n",
"plot([exp, myexp], xlims=[-a, a], label=[\"myexp\" \"exp\"])"
]
},
{
"cell_type": "markdown",
"id": "cbd5f0c0-b478-4a3b-b87b-d1601cdf7dfa",
"metadata": {},
"source": [
"## This looks fishy...\n",
"\n",
"* We're investigating the output of $f \\left( x \\right) = e^x$ near $x = 0$\n",
"\n",
"* Taylor series suggests the function is approximated by $1 + x$\n",
"\n",
"* Output has error on the order of $\\epsilon_\\text{machine}$"
]
},
{
"cell_type": "markdown",
"id": "47efbf11-c657-4370-a15d-4b28df96d3c7",
"metadata": {},
"source": [
"## Error\n",
"\n",
"### Absolute Error\n",
"\n",
"$$ \\lvert \\tilde{f} \\left( x \\right) - f \\left( x \\right) \\rvert $$\n",
"\n",
"### Relative Error\n",
"\n",
"$$ \\frac{\\lvert \\tilde{f} \\left( x \\right) - f \\left( x \\right) \\rvert}{\\lvert f \\left( x \\right) \\rvert} $$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "9d5521bd-a84e-4ecc-857c-b7e3e78405f9",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# What does the Taylor series show us?\n",
"a = 1e-15\n",
"plot([x -> myexp(x) - 1, x -> x], xlims=[-a, a], label=[\"myexp - 1\" \"y = x\"])"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "78d59d52-dcd4-47a6-bfb3-26f0d21f8c86",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"absolute_error = abs((myexp(a) - 1) - a) = 1.1022302462515646e-16\n",
"relative_error = abs((myexp(a) - 1) - a) / abs(a) = 0.11022302462515646\n"
]
}
],
"source": [
"# Checking error\n",
"a = 1e-15\n",
"@show absolute_error = abs((myexp(a) - 1) - a)\n",
"@show relative_error = abs((myexp(a) - 1) - a) / abs(a);"
]
},
{
"cell_type": "markdown",
"id": "48b790c4-4e3e-4e39-ba10-704b074e0901",
"metadata": {},
"source": [
"## Path forward?\n",
"\n",
"* We know we should be able to recover 16 digits of accuracy\n",
"\n",
"* $\\text{myexp} \\left( x \\right) - 1$ can't find the accuracy\n",
"\n",
"* Lets rebuild our algorithm from the start"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "fbceef78-d444-4a8e-8e8e-84eb66a048e4",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"myexpm1(1.0e-15) - expm1(1.0e-15) = 0.0\n"
]
}
],
"source": [
"function myexpm1(x)\n",
" sum = 0\n",
" term = x\n",
" n = 2\n",
" while sum + term != sum\n",
" sum += term\n",
" term *= x / n\n",
" n += 1\n",
" end\n",
" sum\n",
"end\n",
"@show myexpm1(1e-15) - expm1(1e-15);"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "4403e0a5-1c11-497a-a59b-fb8e573b91c7",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Checking the plot\n",
"a = 1e-15\n",
"plot([myexpm1], xlims=[-a, a], label=\"myexpm1\")"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "f6ee01f0-0960-4c10-b805-a5b9e3d496f2",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's compute relative error\n",
"function relativeerror(f, f_true, x)\n",
" fx = f(x)\n",
" fx_true = f_true(x)\n",
" max(abs(fx - fx_true) / abs(fx_true), eps())\n",
"end\n",
"\n",
"# and let's plot it\n",
"bad_expm1(x) = exp(x) - 1\n",
"\n",
"a = 1e-15\n",
"plot(\n",
" x -> relativeerror(bad_expm1, expm1, x),\n",
" yscale=:log10,\n",
" xlims=[-a, a],\n",
" label=\"relative_error(exp - 1)\",\n",
" legend=:bottomright\n",
")"
]
},
{
"attachments": {},
"cell_type": "markdown",
"id": "9619349c-c4e6-4943-98fc-f436267aebea",
"metadata": {},
"source": [
"## Floating point\n",
"\n",
"Floating point is *relative* (see [https://float.exposed](https://float.exposed))\n",
"\n",
"If we define $fl \\left ( x \\right)$ as the operatoion that rounds to the next floating point number, then\n",
"\n",
"$fl \\left( x \\right) = x \\left( 1 + \\epsilon \\right)$, where $\\lvert \\epsilon \\rvert \\leq \\epsilon_\\text{machine}$\n",
"\n",
"So relative error is small\n",
"\n",
"$$ \\frac{\\lvert fl \\left( x \\right) - x \\rvert}{\\lvert x \\rvert} \\leq \\epsilon_\\text{machine} $$"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "3b6b7a54-0ba4-427b-9130-13f067f3b967",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = 1e-15\n",
"plot([x -> x, x -> (1 + x) - 1], xlims=[-a, a], label=[\"x\" \"fl(x)\"])"
]
},
{
"cell_type": "markdown",
"id": "e1ada566-c321-433d-86a3-0f78ce51c7fc",
"metadata": {},
"source": [
"## Exact arithmatic with rounded values\n",
"\n",
"Floating point arithmatic is therefore exact arithmatic with rounding.\n",
"\n",
"We define $\\oplus, \\ominus, \\otimes, \\oslash$ as the floating point arithmatic versions of $+, -, \\times, /$.\n",
"\n",
"Our relative accuracy is therefore\n",
"\n",
"$$ \\frac{\\lvert \\left( x \\otimes y \\right) - \\left( x \\times y \\right) \\rvert}{\\lvert x \\times y \\rvert} \\leq \\epsilon_\\text{machine} $$"
]
},
{
"cell_type": "markdown",
"id": "a29861b2-1ef5-4ba9-aee1-95aef6df6602",
"metadata": {},
"source": [
"## Composition\n",
"\n",
"I would hope that this is true\n",
"\n",
"$$ \\frac{\\lvert \\left( \\left( x \\otimes y \\right) \\otimes z \\right) - \\left( \\left( x \\times y \\right) \\times z \\right) \\rvert}{\\lvert \\left( x \\times y \\right) \\times z \\rvert} \\leq^? \\epsilon_\\text{machine} $$"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "8ff05b82-ac34-4a03-8e90-a82d6b0bdb9c",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Let's test it with addition\n",
"f1(x; y = 1, z = -1) = (x + y) + z;\n",
"a = 1e-15\n",
"plot(x -> relativeerror(f1, x -> x, x), xlims=(-a, a), label=\"relative_error((x + 1) - 1)\")"
]
},
{
"cell_type": "markdown",
"id": "782dc76f-dc75-475b-a600-62050cab0770",
"metadata": {},
"source": [
"## Who to blame?\n",
"\n",
"1) $\\text{tmp} = fl \\left( x + 1 \\right)$\n",
"\n",
"2) $fl \\left( \\text{tmp} - 1 \\right)$"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "34f39644-525a-414c-89d2-1991236c97d2",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"big(1.0e-15) = 1.000000000000000077705399876661079238307185601195015145492561714490875601768494e-15\n",
"BigFloat(\"1e-15\") = 1.000000000000000000000000000000000000000000000000000000000000000000000000000003e-15\n"
]
}
],
"source": [
"# Note this 'gotcha' with big floats\n",
"@show big(1e-15); # First float, then bigFloat\n",
"@show BigFloat(\"1e-15\"); # bigFloat from the start"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "38b72f52-74c7-466c-ae12-50cf784b459b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"relative_error = abs(tmp - tmp_big) / abs(tmp_big) = 1.102230246251563524952071662733800140614440125894379682676737388538642032894338e-16\n"
]
}
],
"source": [
"# 1) tmp = fl(x + 1)\n",
"tmp = 1e-15 + 1\n",
"tmp_big = BigFloat(1e-15) + 1\n",
"@show relative_error = abs(tmp - tmp_big) / abs(tmp_big);"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "28abd7d0-94d1-45c8-9c55-63210528253a",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"relative_error = abs(result - result_big) / abs(result_big) = 0.0\n"
]
}
],
"source": [
"# 2) fl(tmp - 1)\n",
"result = tmp - 1\n",
"result_big = big(tmp) - 1\n",
"@show relative_error = abs(result - result_big) / abs(result_big);"
]
},
{
"cell_type": "markdown",
"id": "58153d2a-9396-42c6-ae20-9cd5f735221b",
"metadata": {},
"source": [
"So, which part is to blame?\n",
"\n",
"Step 1 has the rounding. Step 2 is exact.\n",
"\n",
"But Step 2 represents *subtractive cancellation* and is the loss of accuracy in the final result."
]
},
{
"cell_type": "markdown",
"id": "89aabcfa-4875-48cb-b4b9-e0755a12fe38",
"metadata": {},
"source": [
"## Conditioning\n",
"\n",
"So which functions cause small errors to grow?\n",
"\n",
"Consider a function $f : X \\rightarrow Y$.\n",
"The **condition number** is defined as\n",
"\n",
"$$ \\hat{\\kappa} = \\lim_{\\delta \\rightarrow 0} \\max_{\\lvert \\delta x \\rvert < \\delta} \\frac{f \\left( x + \\delta x \\right) - f \\left( x \\right)}{\\lvert \\delta x \\rvert} = \\max_{\\delta x} \\frac{\\lvert \\delta f \\rvert}{\\lvert \\delta x \\rvert} $$\n",
"\n",
"If $f$ is differentiable, then $\\hat{\\kappa} = \\lvert f' \\left( x \\right) \\lvert$."
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"Floating point numbers offer relative accuracy, so we define the **relative condition number** as\n",
"\n",
"$$ \\kappa = \\max_{\\delta x} \\frac{\\lvert \\delta f \\rvert / \\lvert f \\rvert}{\\lvert \\delta x \\rvert / \\lvert x \\rvert} = \\max_{\\delta x} \\frac{\\lvert \\delta f \\rvert / \\lvert \\delta x \\rvert}{\\lvert f \\rvert / \\vert x \\rvert} $$\n",
"\n",
"If $f$ is differentiable, then $\\kappa = \\left\\lvert f' \\left( x \\right) \\right\\rvert \\frac{\\lvert x \\rvert}{\\lvert f \\rvert}$."
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