483 lines
11 KiB
483 lines
11 KiB
{
|
|
"cells": [
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# $\\text{sinc}(x)$ and Its Derivatives\n",
|
|
"\\begin{align}\n",
|
|
"\\text{sinc}(x) &= \\begin{cases} \n",
|
|
" \\frac{x^4}{120}-\\frac{x^2}{6}+1 & |x| \\leq \\epsilon \\\\\n",
|
|
" \\frac{\\sin(x)}{x} & \\text{otherwise}\n",
|
|
" \\end{cases}\\\\\n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 1,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"import sys\n",
|
|
"\n",
|
|
"import numpy\n",
|
|
"from sympy import *\n",
|
|
"from sympy.plotting import plot\n",
|
|
"\n",
|
|
"x = Symbol(\"x\")\n",
|
|
"x0 = Symbol(\"x_0\")"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 2,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def taylor_series(f, n=5):\n",
|
|
" dfs = [f]\n",
|
|
" for i in range(n):\n",
|
|
" dfs.append(dfs[i].diff())\n",
|
|
" return sum([\n",
|
|
" dfs[i].subs({\"x\": x0}) / factorial(i) * (x - x0)**i\n",
|
|
" for i in range(len(dfs))\n",
|
|
" ])"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 3,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{\\sin{\\left(x \\right)}}{x}$"
|
|
],
|
|
"text/plain": [
|
|
"sin(x)/x"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"sinc = sin(x) / x\n",
|
|
"display(sinc)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 4,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{x^{4}}{120} - \\frac{x^{2}}{6} + 1$"
|
|
],
|
|
"text/plain": [
|
|
"x**4/120 - x**2/6 + 1"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"sinc_ts = taylor_series(sinc)\n",
|
|
"display(limit(sinc_ts, x0, 0))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 5,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"name": "stdout",
|
|
"output_type": "stream",
|
|
"text": [
|
|
"0.9999998333333333\n",
|
|
"0.999999833333325\n",
|
|
"2.2204460492503130808472633361816406250000000000000e-16\n",
|
|
"1.4901161193847656250000000000000000e-8\n",
|
|
"1.220703125000000000000000e-4\n"
|
|
]
|
|
}
|
|
],
|
|
"source": [
|
|
"x1 = 1e-3\n",
|
|
"print(1.0 - x1**2 / 6)\n",
|
|
"print(1.0 - x1**2 * (1 / 6 + x1**2 / 120))\n",
|
|
"x1 = sys.float_info.epsilon\n",
|
|
"print(N(x1, 50))\n",
|
|
"print(f\"{N(sqrt(x1), 35):e}\")\n",
|
|
"print(f\"{N(sqrt(sqrt(x1)), 25):e}\")"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## $\\frac{d}{dx} \\text{sinc}(x)$\n",
|
|
"\\begin{align}\n",
|
|
"\\text{sinc}'(x) &= \\begin{cases} \n",
|
|
" 0 & x = 0 \\\\\n",
|
|
" \\frac{x\\cos(x) - \\sin(x)}{x^2} & \\text{otherwise}\n",
|
|
" \\end{cases}\\\\\n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 6,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)}}{x^{2}}$"
|
|
],
|
|
"text/plain": [
|
|
"(x*cos(x) - sin(x))/x**2"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"dsinc = simplify(sinc.diff(x))\n",
|
|
"display(dsinc)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 7,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle - \\frac{x^{5}}{840} + \\frac{x^{3}}{30} - \\frac{x}{3}$"
|
|
],
|
|
"text/plain": [
|
|
"-x**5/840 + x**3/30 - x/3"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"dsinc_ts = taylor_series(dsinc)\n",
|
|
"display(limit(dsinc_ts, x0, 0))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## $\\frac{d^2}{dx^2} \\text{sinc}(x)$\n",
|
|
"\\begin{align}\n",
|
|
"\\text{sinc}''(x) &= \\begin{cases} \n",
|
|
" \\frac{-1}{3} & x = 0 \\\\\n",
|
|
" \\frac{-x^2\\sin(x) - 2x\\cos(x) + 2\\sin(x)}{x^3} & \\text{otherwise}\n",
|
|
" \\end{cases}\\\\\n",
|
|
"\\end{align}"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 8,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{- x^{2} \\sin{\\left(x \\right)} - 2 x \\cos{\\left(x \\right)} + 2 \\sin{\\left(x \\right)}}{x^{3}}$"
|
|
],
|
|
"text/plain": [
|
|
"(-x**2*sin(x) - 2*x*cos(x) + 2*sin(x))/x**3"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"ddsinc = simplify(dsinc.diff(x))\n",
|
|
"display(ddsinc)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 9,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle - \\frac{x^{4}}{168} + \\frac{x^{2}}{10} - \\frac{1}{3}$"
|
|
],
|
|
"text/plain": [
|
|
"-x**4/168 + x**2/10 - 1/3"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"ddsinc_ts = taylor_series(ddsinc)\n",
|
|
"display(limit(ddsinc_ts, x0, 0))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Derivatives of $sinc(\\|x\\|)$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 10,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"def norm(x):\n",
|
|
" return sqrt(sum(x**2))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 11,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{\\sin{\\left(\\sqrt{x^{2} + y^{2} + z^{2}} \\right)}}{\\sqrt{x^{2} + y^{2} + z^{2}}}$"
|
|
],
|
|
"text/plain": [
|
|
"sin(sqrt(x**2 + y**2 + z**2))/sqrt(x**2 + y**2 + z**2)"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"X = numpy.array(symbols(\"x y z\"))\n",
|
|
"sinc_normx = sinc.subs({\"x\": norm(X)})\n",
|
|
"display(sinc_normx)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Gradient of $\\text{sinc}(\\|x\\|)$\n",
|
|
"\\begin{align}\n",
|
|
"\\nabla \\text{sinc}(\\|x\\|) &= \\frac{\\text{sinc}'(\\|x\\|)}{\\|x\\|}x \\\\\n",
|
|
"\\frac{\\text{sinc}'(x)}{x} &= \\begin{cases}\n",
|
|
" -\\frac{x^4}{840} + \\frac{x^2}{30} - \\frac{1}{3}& |x| \\leq \\epsilon\\\\\n",
|
|
" \\frac{x\\cos(x)-\\sin(x)}{x^3} & \\text{otherwise}\n",
|
|
" \\end{cases}\n",
|
|
"\\end{align}\n",
|
|
"where $\\epsilon = 10^{-4}$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 12,
|
|
"metadata": {
|
|
"scrolled": true
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"dsinc_normx = numpy.array(sinc_normx.diff(X))\n",
|
|
"my_dsinc_normx = dsinc.subs({\"x\": norm(X)}) / norm(X) * X\n",
|
|
"for i in range(3):\n",
|
|
" assert (simplify((dsinc_normx - my_dsinc_normx)[i]) == 0)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 13,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{x \\cos{\\left(x \\right)} - \\sin{\\left(x \\right)}}{x^{3}}$"
|
|
],
|
|
"text/plain": [
|
|
"(x*cos(x) - sin(x))/x**3"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"dsinc_x = dsinc / x\n",
|
|
"display(dsinc_x)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 14,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle - \\frac{x^{4}}{840} + \\frac{x^{2}}{30} - \\frac{1}{3}$"
|
|
],
|
|
"text/plain": [
|
|
"-x**4/840 + x**2/30 - 1/3"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"dsinc_x_ts = taylor_series(dsinc_x)\n",
|
|
"display(limit(dsinc_x_ts, x0, 0))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Hessian of $\\text{sinc}(\\|x\\|)$\n",
|
|
"\\begin{align}\n",
|
|
"\\nabla^2 \\text{sinc}(\\|x\\|) &= \\left(\\frac{\\text{sinc}''(\\|x\\|)}{\\|x\\|^2} - \\frac{\\text{sinc}'(\\|x\\|)}{\\|x\\|^3}\\right)xx^T + \\frac{\\text{sinc}'(\\|x\\|)}{\\|x\\|}I \\\\\n",
|
|
"\\frac{\\text{sinc}''(\\|x\\|)}{\\|x\\|^2} - \\frac{\\text{sinc}'(\\|x\\|)}{\\|x\\|^3} &= \\begin{cases} \n",
|
|
" \\frac{x^{4}}{7560}-\\frac{x^{2}}{210}+\\frac{1}{15} & |x| \\leq \\epsilon \\\\\n",
|
|
" \\frac{-x^{2} \\sin (x)-3 x \\cos (x)+3 \\sin (x)}{x^{5}} & \\text{otherwise}\n",
|
|
" \\end{cases}\n",
|
|
"\\end{align}\n",
|
|
"where $\\epsilon = 0.1$."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 15,
|
|
"metadata": {},
|
|
"outputs": [],
|
|
"source": [
|
|
"ddsinc_normx = numpy.array(Matrix(Matrix(dsinc_normx).diff(X)))\n",
|
|
"my_ddsinc_normx = ((ddsinc.subs({\"x\": norm(X)}) / norm(X)**2 -\n",
|
|
" dsinc.subs({\"x\": norm(X)}) / norm(X)**3) *\n",
|
|
" X.reshape(3, 1) @ X.reshape(1, 3) +\n",
|
|
" dsinc.subs({\"x\": norm(X)}) / norm(X) * numpy.eye(3))\n",
|
|
"for i in range(3):\n",
|
|
" for j in range(3):\n",
|
|
" assert (simplify(ddsinc_normx[i, j][0] - my_ddsinc_normx[i, j]) == 0)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 16,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{- x^{2} \\sin{\\left(x \\right)} - 3 x \\cos{\\left(x \\right)} + 3 \\sin{\\left(x \\right)}}{x^{5}}$"
|
|
],
|
|
"text/plain": [
|
|
"(-x**2*sin(x) - 3*x*cos(x) + 3*sin(x))/x**5"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"ddsinc_x = simplify(ddsinc.subs({\"x\": x}) / x**2 - dsinc.subs({\"x\": x}) / x**3)\n",
|
|
"display(ddsinc_x)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": 17,
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"data": {
|
|
"text/latex": [
|
|
"$\\displaystyle \\frac{x^{4}}{7560} - \\frac{x^{2}}{210} + \\frac{1}{15}$"
|
|
],
|
|
"text/plain": [
|
|
"x**4/7560 - x**2/210 + 1/15"
|
|
]
|
|
},
|
|
"metadata": {},
|
|
"output_type": "display_data"
|
|
}
|
|
],
|
|
"source": [
|
|
"ddsinc_x_ts = taylor_series(ddsinc_x)\n",
|
|
"display(limit(ddsinc_x_ts, x0, 0.0))"
|
|
]
|
|
}
|
|
],
|
|
"metadata": {
|
|
"kernelspec": {
|
|
"display_name": "Python 3",
|
|
"language": "python",
|
|
"name": "python3"
|
|
},
|
|
"language_info": {
|
|
"codemirror_mode": {
|
|
"name": "ipython",
|
|
"version": 3
|
|
},
|
|
"file_extension": ".py",
|
|
"mimetype": "text/x-python",
|
|
"name": "python",
|
|
"nbconvert_exporter": "python",
|
|
"pygments_lexer": "ipython3",
|
|
"version": "3.9.5"
|
|
},
|
|
"varInspector": {
|
|
"cols": {
|
|
"lenName": 16,
|
|
"lenType": 16,
|
|
"lenVar": 40
|
|
},
|
|
"kernels_config": {
|
|
"python": {
|
|
"delete_cmd_postfix": "",
|
|
"delete_cmd_prefix": "del ",
|
|
"library": "var_list.py",
|
|
"varRefreshCmd": "print(var_dic_list())"
|
|
},
|
|
"r": {
|
|
"delete_cmd_postfix": ") ",
|
|
"delete_cmd_prefix": "rm(",
|
|
"library": "var_list.r",
|
|
"varRefreshCmd": "cat(var_dic_list()) "
|
|
}
|
|
},
|
|
"types_to_exclude": [
|
|
"module",
|
|
"function",
|
|
"builtin_function_or_method",
|
|
"instance",
|
|
"_Feature"
|
|
],
|
|
"window_display": false
|
|
}
|
|
},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|
|
|