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fabiogaluppo/Fun with Generating Functions and SymPy.ipynb

Last active May 29, 2019 01:45
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Fun with Generating Functions and SymPy.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"from sympy import *\n",
"from functools import lru_cache"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"x = Symbol('x')"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def f(func):\n",
"\treturn func"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/(2*x + 1)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df0 = f(1/(2*x + 1))\n",
"df0"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"def d(func, variable = Symbol('x'), n = 1):\n",
"\treturn diff(func, variable, n)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1/(2*x + 1),\n",
" -2/(2*x + 1)**2,\n",
" 8/(2*x + 1)**3,\n",
" -48/(2*x + 1)**4,\n",
" 384/(2*x + 1)**5)"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df1 = d(df0)\n",
"df2 = d(df1)\n",
"df3 = d(df2)\n",
"df4 = d(df3)\n",
"df0, df1, df2, df3, df4"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"def to_python_function(func, *variables):\n",
"\tif len(variables) == 0:\n",
"\t\treturn lambdify(Symbol('x'), func)\n",
"\treturn lambdify(tuple(variables), func)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.0, -2.0, 8.0, -48.0, 384.0)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DF0 = to_python_function(df0)\n",
"DF1 = to_python_function(df1)\n",
"DF2 = to_python_function(df2)\n",
"DF3 = to_python_function(df3)\n",
"DF4 = to_python_function(df4)\n",
"DF0(0), DF1(0), DF2(0), DF3(0), DF4(0)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"t*(-v0 + v1) + v0"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v0, v1, t = Symbol('v0'), Symbol('v1'), Symbol('t')\n",
"lerp = f(v0 + t * (v1 - v0))\n",
"lerp"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.5, 0.6, 16.37)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"LERP = to_python_function(lerp, v0, v1, t)\n",
"LERP(0, 1, 0.5), LERP(0, 1, 0.6), LERP(12, 23.5, 0.38)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{t, v0, v1}"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lerp.free_symbols"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"def get_variables(func):\n",
"\tfree_symbols = list(func.free_symbols)\n",
"\tfree_symbols.sort(key = lambda s: str(s))\n",
"\treturn tuple(free_symbols)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(t, v0, v1)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"get_variables(lerp)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"# https://www.mathsisfun.com/algebra/completing-square.html\n",
"def completing_the_square(expr):\n",
"\tvs = get_variables(expr)\n",
"\tif (len(vs) != 1):\n",
"\t\traise Exception('expr must have just 1 variable')\n",
"\tpoly = Poly(expr)\n",
"\tif (degree(poly) != 2):\n",
"\t\traise Exception('expr must be degree 2')\n",
"\ta, b, c = poly.all_coeffs()\n",
"\tif (a != 1):\n",
"\t\traise Exception('coefficient of degree 2 must be equal to 1')\n",
"\td = b / 2\n",
"\te = c - d**2\n",
"\tvariable = vs[0]\n",
"\treturn (variable + d)**2 + e"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 3)**2 - 2"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"completing_the_square(x**2 + 6*x + 7)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**2 + 6*x + 7"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expand(completing_the_square(x**2 + 6*x + 7))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 6)**2 - 28"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"completing_the_square(x**2 + 12*x + 8)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**2 + 12*x + 8"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expand(completing_the_square(x**2 + 12*x + 8))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"# https://www.codecogs.com/latex/eqneditor.php"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"\\sum_{k=0}^{n}\\frac{f^{(k)}(x)}{k!}x^{k}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"\\sum_{k=0}^{n}\\frac{f^{(k)}(x)}{k!}x^{k}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"def compute_derivatives(func, n):\n",
"\tvs = get_variables(func)\n",
"\tif (len(vs) != 1):\n",
"\t\traise Exception('func must have just 1 variable')\n",
"\tvariable = vs[0]\n",
"\tf = func\n",
"\tdfs = list()\n",
"\tdfs.append(f)\n",
"\tfor i in range(1, n + 1):\n",
"\t\tdfs.append(d(dfs[i - 1], variable))\n",
"\tDFS = [to_python_function(df) for df in dfs]\n",
"\treturn dfs, DFS"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1/(2*x + 1), -2/(2*x + 1)**2, 8/(2*x + 1)**3, -48/(2*x + 1)**4]"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dfs, DFS = compute_derivatives(1/(2*x + 1), 3)\n",
"dfs"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.0, -2.0, 8.0, -48.0)"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DF0, DF1, DF2, DF3 = DFS\n",
"DF0(0), DF1(0), DF2(0), DF3(0)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1/(2*y + 1), -2/(2*y + 1)**2, 8/(2*y + 1)**3, -48/(2*y + 1)**4]"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dfs, _ = compute_derivatives(1/(2*Symbol('y') + 1), 3)\n",
"dfs"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [],
"source": [
"def compute_dividends(func, n):\n",
"\tf = func\n",
"\t_, DFS = compute_derivatives(f, n)\n",
"\treturn np.array([DF(0) for DF in DFS])"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"n!= \n",
"\\begin{cases}\n",
"1 & \\text{ if } x=0 \\, \\vee \\, x=1 \\\\ \n",
"2 & \\text{ if } x=2 \\\\\n",
"n(n-1)! & \\text{ }\n",
"\\end{cases}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"n!= \n",
"\\begin{cases}\n",
"1 & \\text{ if } n=0 \\, \\vee \\, n=1 \\\\ \n",
"2 & \\text{ if } n=2 \\\\\n",
"n(n-1)! & \\text{ }\n",
"\\end{cases}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [],
"source": [
"@lru_cache(maxsize = 100)\n",
"def factorial(n):\n",
"\tif n == 0: return 1\n",
"\telif n == 1: return 1\n",
"\telif n == 2: return 2\n",
"\telse: return n * factorial(n - 1)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., -2., 8., -48., 384.])"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_dividends(1/(2*x + 1), 4)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3628800, 24, 1405006117752879898543142606244511569936384000000000, 5040)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"factorial(10), factorial(4), factorial(42), factorial(7)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [],
"source": [
"def compute_divisors(n):\n",
"\treturn np.array([factorial(i) for i in range(n + 1)])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1, 1, 2, 6, 24])"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_divisors(4)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [],
"source": [
"def compute_Taylor_and_MacLaurin_series(func, n):\n",
"\tdef compute_coefficients(func, n):\n",
"\t\treturn compute_dividends(func, n) / compute_divisors(n)\n",
"\treturn compute_coefficients(func, n)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., -2., 4., -8., 16.])"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(1/(2*x + 1), 4)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., -1., 1., -1., 1., -1.])"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(1/(1 + x), 5)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1., 1., 1., 1., 1., 1.])"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(1/(1 - x), 5)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0., 0., 1., 3., 6., 10., 15., 21., 28.])"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(x**2/(1 - x)**3, 8)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0. , 1. , 0.5 , 0.33, 0.25, 0.2 , 0.17, 0.14, 0.12])"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.round(compute_Taylor_and_MacLaurin_series(ln(1/(1 - x)), 8), 2)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [],
"source": [
"def compute_partial_derivatives(func, n = 1):\n",
"\tf = func\n",
"\tvs = get_variables(f)\n",
"\tdvs = [d(f, v, n) for v in vs]\n",
"\tDVS = [to_python_function(dv, *get_variables(dv)) for dv in dvs]\n",
"\treturn list(vs), dvs, DVS"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[t, v0, v1]"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"vs, dvs, DVS = compute_partial_derivatives(lerp)\n",
"vs"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[-v0 + v1, -t + 1, t]"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dvs"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1, -1, 2)"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"DV0, DV1, DV2 = DVS[0], DVS[1], DVS[2]\n",
"DV0(1, 2), DV1(2), DV2(2)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [],
"source": [
"@lru_cache(maxsize = 100)\n",
"def comb(n, k):\n",
"\tif (n < k):\n",
"\t\traise Exception('k must be lesser than n')\n",
"\treturn factorial(n) / (factorial(n - k) * factorial(k))"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"\\binom{3}{2}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"\\binom{3}{2}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.0"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"comb(3, 2)"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"\\binom{5}{2}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"\\binom{5}{2}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10.0"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"comb(5, 2)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
"# https://www.mathsisfun.com/pascals-triangle.html\n",
"def pascal_triangle(n):\n",
"\treturn [comb(n, k) for k in range(n + 1)]"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0], 1.0)"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(0), sum(pascal_triangle(0))"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0, 1.0], 2.0)"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(1), sum(pascal_triangle(1))"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0, 2.0, 1.0], 4.0)"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(2), sum(pascal_triangle(2))"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0, 3.0, 3.0, 1.0], 8.0)"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(3), sum(pascal_triangle(3))"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0, 4.0, 6.0, 4.0, 1.0], 16.0)"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(4), sum(pascal_triangle(4))"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([1.0, 5.0, 10.0, 10.0, 5.0, 1.0], 32.0)"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle(5), sum(pascal_triangle(5))"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [],
"source": [
"def pascal_triangle_symbolic(n):\n",
"\tp = pascal_triangle(n)\n",
"\tx = Symbol('x')\n",
"\treturn sum([p[i] * x**i for i in range(len(p))])"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"\\binom{4}{0}+\\binom{4}{1}x+\\binom{4}{2}x^{2}+\\binom{4}{3}x^{3}+\\binom{4}{4}x^{4}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"\\binom{4}{0}+\\binom{4}{1}x+\\binom{4}{2}x^{2}+\\binom{4}{3}x^{3}+\\binom{4}{4}x^{4}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0*x**4 + 4.0*x**3 + 6.0*x**2 + 4.0*x + 1.0"
]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pascal_triangle_symbolic(4)"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [],
"source": [
"def counting_with_gf(expr, *exprs):\n",
"\tdef check_expression(expr, variable):\n",
"\t\tvs = get_variables(expr)\n",
"\t\tif (len(vs) != 1):\n",
"\t\t\traise Exception('expression must have just 1 variable')\n",
"\t\tif (vs[0] != variable):\n",
"\t\t\traise Exception('expression must have x as variable')\n",
"\t\treturn expr\n",
"\tx = Symbol('x')\n",
"\tacc = check_expression(expr, x)\n",
"\tfor expr in exprs:\n",
"\t\tacc = acc * check_expression(expr, x)\n",
"\treturn expand(acc), Poly(acc, x).coeffs()"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 1)*(x**2 + x + 1)"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(x**0 + x**1 + x**2) * (x**0 + x**1)"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**3 + 2*x**2 + 2*x + 1"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expand((x**0 + x**1 + x**2) * (x**0 + x**1))"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1, 2, 2, 1]"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Poly((x**0 + x**1 + x**2) * (x**0 + x**1), x).coeffs()"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(x + 1, [1, 1])"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"counting_with_gf(x**0 + x**1)"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**3 + 2*x**2 + 2*x + 1"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf, cs = counting_with_gf((x**0 + x**1 + x**2), (x**0 + x**1))\n",
"gf"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1, 2, 2, 1]"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"cs"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import matplotlib.pyplot as plt\n",
"interval = np.round(np.arange(-0.9, 0.8, 0.025), 3)\n",
"func1 = lambda x: 1/(1 - x)\n",
"points1 = np.round(np.fromiter((func1(x) for x in interval), interval.dtype), 3)\n",
"plt.plot(points1, label = \"1/(1 - x)\")\n",
"func2 = lambda x: 1 + x + x**2 + x**3 + x**4 + x**5 + x**6 + x**7 + x**8 \n",
"points2 = np.round(np.fromiter((func2(x) for x in interval), interval.dtype), 3)\n",
"plt.plot(points2, label = \"1 + x + x**2 + ... + x**8\")\n",
"plt.legend()\n",
"plt.grid(axis = 'y', linestyle = '-')\n",
"plt.gca().axes.get_xaxis().set_visible(False)"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [],
"source": [
"def gf_builder(expr, *actions):\n",
"\tlhs, rhs = 1, expr\n",
"\tfor action in actions:\n",
"\t\tlhs = lhs + action\n",
"\t\trhs = rhs + expand(action * expr)\n",
"\treturn rhs.removeO() / lhs"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 + x + x**2 + x**3 + O(x**4)"
]
},
"execution_count": 66,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"S = 1 + x + x**2 + x**3 + O(x**4)\n",
"S"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x + x**2 + x**3 + x**4 + O(x**5)"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expand(x*S)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 68,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(S - expand(x*S)).removeO()"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
" S = 1+x+x^{2}+x^{3}+x^{4}+... \\\\\n",
"- xS = 0+x+x^{2}+x^{3}+x^{4}+x^{5}+... \\\\\n",
"----------------- \\\\\n",
"(1-x)S = 1 \\\\\n",
"\\\\\n",
"S=\\frac{1}{1-x}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
" S = 1+x+x^{2}+x^{3}+x^{4}+... \\\\\n",
"- xS = 0+x+x^{2}+x^{3}+x^{4}+x^{5}+... \\\\\n",
"----------------- \\\\\n",
"(1-x)S = 1 \\\\\n",
"\\\\\n",
"S=\\frac{1}{1-x}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/(-x + 1)"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf_builder(1 + x + x**2 + x**3 + O(x**4), -x)"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1., 1., 1., 1.])"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(gf_builder(S, -x), 3)"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
" S = 1-x+x^{2}-x^{3}+x^{4}+... \\\\\n",
"+ xS = 0+x-x^{2}+x^{3}-x^{4}+x^{5}+... \\\\\n",
"----------------- \\\\\n",
"(1+x)S = 1 \\\\\n",
"\\\\\n",
"S=\\frac{1}{1+x}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
" S = 1-x+x^{2}-x^{3}+x^{4}+... \\\\\n",
"+ xS = 0+x-x^{2}+x^{3}-x^{4}+x^{5}+... \\\\\n",
"----------------- \\\\\n",
"(1+x)S = 1 \\\\\n",
"\\\\\n",
"S=\\frac{1}{1+x}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/(x + 1)"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf_builder(1 - x + x**2 - x**3 + O(x**4), x)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
" S = 2+x+2x^{2}+2x^{3}+2x^{4}+... \\\\\n",
"- xS = 0+2x+2x^{2}+2x^{3}+2x^{4}+2x^{5}+... \\\\\n",
"-------------------- \\\\\n",
"(1-x)S = 2 \\\\\n",
"\\\\\n",
"S=\\frac{2}{1-x}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
" S = 2+x+2x^{2}+2x^{3}+2x^{4}+... \\\\\n",
"- xS = 0+2x+2x^{2}+2x^{3}+2x^{4}+2x^{5}+... \\\\\n",
"-------------------- \\\\\n",
"(1-x)S = 2 \\\\\n",
"\\\\\n",
"S=\\frac{2}{1-x}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2/(-x + 1)"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf_builder(2 + 2*x + 2*x**2 + 2*x**3 + O(x**4), -x)"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
" S = 3+9x+27x^{2}+81x^{3}+... \\\\\n",
"- 3xS = 0+9x+27x^{2}+81x^{3}+... \\\\\n",
"----------------- \\\\\n",
"(1-3x)S = 3 \\\\\n",
"\\\\\n",
"S=\\frac{3}{1-3x}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
" S = 3+9x+27x^{2}+81x^{3}+... \\\\\n",
"- 3xS = 0+9x+27x^{2}+81x^{3}+... \\\\\n",
"----------------- \\\\\n",
"(1-3x)S = 3 \\\\\n",
"\\\\\n",
"S=\\frac{3}{1-3x}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3/(-3*x + 1)"
]
},
"execution_count": 77,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf_builder(3 + 9*x + 27*x**2 + 81*x**3 + O(x**4), -3*x)"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
" S = 1+3x+5x^{2}+7x^{3}+9x^{4}+... \\\\\n",
"- xS = 0+x+3x^{2}+5x^{3}+7x^{4}+9x^{5}+... \\\\\n",
"-------------------- \\\\\n",
"(1-x)S = 1+2x+2x^{2}+2x^{3}+2x^{4}+2x^{5}+... \\\\\n",
"\\\\\n",
"S=\\frac{1+\\frac{2x}{1-x}}{1-x}=\\frac{1}{1-x}+\\frac{2x}{(1-x)^{2}}=\\frac{1+x}{(1-x)^{2}}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
" S = 1+3x+5x^{2}+7x^{3}+9x^{4}+... \\\\\n",
"- xS = 0+x+3x^{2}+5x^{3}+7x^{4}+9x^{5}+... \\\\\n",
"-------------------- \\\\\n",
"(1-x)S = 1+2x+2x^{2}+2x^{3}+2x^{4}+2x^{5}+... \\\\\n",
"\\\\\n",
"S=\\frac{1+\\frac{2x}{1-x}}{1-x}=\\frac{1}{1-x}+\\frac{2x}{(1-x)^{2}}=\\frac{1+x}{(1-x)^{2}}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(2*x**4 + 2*x**3 + 2*x**2 + 2*x + 1)/(-x + 1)"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gf_builder(1 + 3*x + 5*x**2 + 7*x**3 + 9*x**4 + O(x**5), -x)"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1., 3., 5., 7., 9.])"
]
},
"execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series((2*x**4 + 2*x**3 + 2*x**2 + 2*x + 1)/(-x + 1), 4)"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([1., 3., 5., 7., 9.])"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series((1 + x)/(1 - x)**2, 4)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., 3., 5., 7., 9., 11., 13., 15., 17., 19., 21.])"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series((1 + x)/(1 - x)**2, 10)"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{align}\n",
"A(n) = \\begin{cases}\n",
"1 & \\text{ if } x=0 \\\\ \n",
"3 & \\text{ if } x=1 \\\\\n",
"3A(n-1)- 2A(n-2) & \\text{ }\n",
"\\end{cases}\n",
"\\end{align}"
],
"text/plain": [
"<IPython.core.display.Latex object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%latex\n",
"\\begin{align}\n",
"A(n) = \\begin{cases}\n",
"1 & \\text{ if } n=0 \\\\ \n",
"3 & \\text{ if } n=1 \\\\\n",
"3A(n-1)- 2A(n-2) & \\text{ }\n",
"\\end{cases}\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [],
"source": [
"@lru_cache(maxsize = 100)\n",
"def A(n):\n",
"\tif n == 0: return 1\n",
"\telif n == 1: return 3\n",
"\telse: return 3 * A(n - 1) - 2 * A(n - 2)"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1, 3, 7, 15, 31, 63, 127, 255, 511)"
]
},
"execution_count": 85,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A(0), A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1/(2*x**2 - 3*x + 1)"
]
},
"execution_count": 86,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gfA = gf_builder(1 + 3*x + 7*x**2 + 15*x**3 + 31*x**4 + 63*x**5 + O(x**6), -3*x, 2*x**2)\n",
"gfA"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., 3., 7., 15., 31., 63., 127., 255., 511.])"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compute_Taylor_and_MacLaurin_series(gfA, 8)"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 + x + x**2 + x**3 + x**4 + x**5 + O(x**6)"
]
},
"execution_count": 88,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(1/(1-x)).truncate()"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 - 2*x + 4*x**2 - 8*x**3 + 16*x**4 - 32*x**5 + O(x**6)"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(1/(2*x+1)).truncate()"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 - x + x**2 - x**3 + x**4 - x**5 + O(x**6)"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(1/(1+x)).truncate()"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**2 + 3*x**3 + 6*x**4 + 10*x**5 + O(x**6)"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(x**2/(1-x)**3).truncate()"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"x**2 + 3*x**3 + 6*x**4 + 10*x**5 + 15*x**6 + 21*x**7 + 28*x**8 + O(x**9)"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(x**2/(1-x)**3).truncate(9)"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[28, 21, 15, 10, 6, 3, 1, 0, 0]"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Poly(fps(x**2/(1-x)**3).truncate(9).removeO()).all_coeffs()"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[0, 0, 1, 3, 6, 10, 15, 21, 28]"
]
},
"execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Poly(fps(x**2/(1-x)**3).truncate(9).removeO()).all_coeffs()[::-1]"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 1., 1., 2., 3., 5., 8., 13., 21., 34., 55., 89.])"
]
},
"execution_count": 95,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# http://discrete.openmathbooks.org/dmoi2/section-27.html\n",
"# https://aofa.cs.princeton.edu/30gf/\n",
"compute_Taylor_and_MacLaurin_series(1/(1-x-x**2), 10)"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Poly(fps(1/(1-x-x**2)).truncate(11).removeO()).all_coeffs()[::-1]"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 + x + x**2 + x**3 + x**4 + x**5 + x**6 + x**7 + x**8 + x**9 + x**10 + O(x**11)"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fps(1/(1-x-x**2)).truncate(11)"
]
}
],
"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": "3.6.5"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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