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tutorial.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Setup"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [],
"source": [
"#%pip install --upgrade symbolica\n",
"#%pip install --upgrade ipywidgets\n",
"from symbolica import *\n",
"from IPython.display import Markdown as md, display as dp\n",
"from ipywidgets import interact, interactive, fixed, interact_manual\n",
"import ipywidgets as widgets\n",
"\n",
"def display(self) -> None:\n",
" dp(md(self.to_latex().replace('gamma', '\\\\gamma').replace('mu', '\\\\mu')))\n",
"\n",
"def display_graph(self: Graph) -> None:\n",
" dp(md('```mermaid\\n' + self.to_mermaid() + '\\n```'))\n",
"\n",
"\n",
"set_license_key('gcolab-demo-key-do-not-use-elsewhere')"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<style>\n",
".cell-output-ipywidget-background {\n",
" background-color: transparent !important;\n",
"}\n",
":root {\n",
" --jp-widgets-color: var(--vscode-editor-foreground);\n",
" --jp-widgets-font-size: var(--vscode-editor-font-size);\n",
"}\n",
"</style>\n"
],
"text/plain": [
"<IPython.core.display.HTML object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"%%html\n",
"<style>\n",
".cell-output-ipywidget-background {\n",
" background-color: transparent !important;\n",
"}\n",
":root {\n",
" --jp-widgets-color: var(--vscode-editor-foreground);\n",
" --jp-widgets-font-size: var(--vscode-editor-font-size);\n",
"}\n",
"</style>"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Expressions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Parse expressions with `E`:"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"f(x,y)^2\u001b[33m+\u001b[0m(x+1)*(y+1)\n"
]
}
],
"source": [
"e = E('(x+1)(y+1)+f(x,y)^2')\n",
"print(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Build expressions by defining symbols with `S` and combining them:"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$2 x+x^{2}+\\left(y+1\\right)^{4}+1$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x, f, y = S('x', 'f', 'y')\n",
"e = ((1+x)**2 + (1+y)**4).expand(x)\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Print modes:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x\u001b[33m+\u001b[0mf(x,1)²\n"
]
}
],
"source": [
"print(e.format())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Parse numbers with `N`:"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1/10*x*f(x,1)\n"
]
}
],
"source": [
"e = N(0.1, 0.01) * x * f(x,1)\n",
"print(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Loosely precision-tracking floats:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1.23456\n",
"1.23400\n",
"5.61e-4\n"
]
}
],
"source": [
"print(N('1.234563`6'))\n",
"print(N('1.234`6'))\n",
"print(N('1.23456`6') - N('1.234`6'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Symbols have a namespace:"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x\u001b[33m+\u001b[0mx\n",
"x\u001b[33m+\u001b[0m\u001b[2;3mtrace\u001b[0m\u001b[2m::\u001b[0mx\n"
]
}
],
"source": [
"e = E('trace::x') + x\n",
"print(e)\n",
"print(e.format(show_namespaces=True))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Assign attributes to symbols:"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$3 dot\\!\\left(x,x\\right)+3 dot\\!\\left(x,y\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x, y = S('x', 'y')\n",
"dot = S('dot',is_symmetric=True, is_linear=True)\n",
"e = dot(x+y,3*x)\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Operations on expressions"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(x+1)*(2*x+x^2+1)^-1\u001b[33m+\u001b[0m1\n"
]
},
{
"data": {
"text/markdown": [
"$$\\frac{1}{x+1}+1$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"$$\\frac{1}{x+1} \\left(x+2\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y=S('y')\n",
"e = 1+(1+x)/((1+x)**2).expand()\n",
"print(e)\n",
"display(e.cancel())\n",
"display(e.factor())"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x = (-c+c^2)*f(c)^-1 , y = \u001b[33m-\u001b[0mc^2\n"
]
}
],
"source": [
"c = S('c')\n",
"x_r, y_r = Expression.solve_linear_system([f(c)*x + y + c, y + c**2], [x, y])\n",
"print('x =', x_r, ', y =', y_r)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x\u001b[33m+\u001b[0m1\n"
]
}
],
"source": [
"e = E('x+1')\n",
"e.save('test') # save in compressed binary format\n",
"e2 = Expression.load('test')\n",
"print(e2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Canonize tensors:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"g(mu(2),mu(3))*fc(k,mu(3),mu(4),mu(2),k,mu(4))\u001b[33m-\u001b[0mg(mu(3),mu(4))*fc(k,mu(2),k,mu(3),mu(2),mu(4))\n",
"0\n"
]
}
],
"source": [
"g = S('g', is_symmetric=True)\n",
"fc = S('fc', is_cyclesymmetric=True)\n",
"mu, k = S('mu', 'k')\n",
"\n",
"e = g(mu(2), mu(3))*fc(mu(4), mu(2), k, mu(4), k, mu(3))-g(mu(4),mu(3))*fc(k,mu(2),k,mu(3),mu(2),mu(4))\n",
"print(e)\n",
"print(e.canonize_tensors([(mu(1), 0), (mu(2), 0), (mu(3), 0), (mu(4), 0)]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Series expansion"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1\u001b[33m+\u001b[0mx\u001b[33m+\u001b[0m3/2*x^2\u001b[33m+\u001b[0m2*x^3\u001b[33m+\u001b[0m65/24*x^4+𝒪(x^5)\n"
]
}
],
"source": [
"s = E('sin(x)/(1-x)').series(x, 0, 5).exp()\n",
"print(s)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "e1e1bb6978134c6896133dd735b729be",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"interactive(children=(IntSlider(value=4, description='depth', max=20, min=1), Output()), _dom_classes=('widget…"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"@interact(depth=(1,20))\n",
"def update_series(depth=4):\n",
" e = E('sin(x)/(1-x)').series(S('x'), 0, depth).exp()\n",
" return md(e.to_latex())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Matrices"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$\\begin{pmatrix}\\frac{2-3t+2t^{2}}{-3+2t} & \\frac{-7+4t}{-3+2t} \\\\ \\frac{-21+12t}{-6+4t} & \\frac{-24+17t}{-6+4t}\\end{pmatrix}$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"t = S('t')\n",
"\n",
"a = Matrix.from_nested([[t, 2], [3, 4]])\n",
"display(a.inv() + a)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$\\begin{pmatrix}6+t \\\\ 15\\end{pmatrix}$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"b = Matrix.vec([1,3])\n",
"display(a*b)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$\\begin{pmatrix}\\frac{-1}{-3+2t} \\\\ \\frac{-3+3t}{-6+4t}\\end{pmatrix}$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display(a.solve(b))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Polynomials"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Any expression can be turned into a (rational) polynomial:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x\n",
"y\n",
"z^-1\n",
"f(w)\n"
]
}
],
"source": [
"a = E('x+ x* y + f(w)*x /z + 5').to_polynomial()\n",
"\n",
"for v in a.get_var_list():\n",
" print(v)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Pattern matching"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$x+f\\!\\left(x^{2},x,f\\!\\left(x\\right)\\right)+2$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/markdown": [
"$$y^{2}+f\\!\\left(y^{4},y^{2},f\\!\\left(x\\right)\\right)+2$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"e = f(x**2, x, f(x))+x+2\n",
"display(e)\n",
"display(e.replace(x, y**2, level_range=(0,1)))"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$x f\\!\\left(5,y,f\\!\\left(5,y\\right)\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y = S('y')\n",
"e = x*f(x, y, f(x, y))\n",
"e = e.replace(x, 5, level_range=(1,2))\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Wildcards can match a single atom in the subexpression:"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$x f\\!\\left(x,y^{2},f\\!\\left(x,2,y^{2}\\right)\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x_ = S('x_')\n",
"e = x*f(x, y, f(x, 2, y))\n",
"e = e.replace(x_, x_**2, x_.req(lambda x: x == y))\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A wildcard can be restricted:"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$3628800$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x_, fact = S('x_', 'fact')\n",
"\n",
"e = fact(10).replace(fact(x_), x_*fact(x_ - 1), x_ > 1, repeat=True).replace(fact(1), 1)\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"No match here:"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$f\\!\\left(x y z\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y_ , z= S('y_', 'z')\n",
"e = f(x*y*z)\n",
"e = e.replace(f(x_*x), 1)\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A single `_` means it matches a single atom completely. `__` means it can match a non-empty subset and `___` means match empty subsets as well."
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"f(0,1,2,3,4,100)\n",
"f(0,1,2,3,100,5)\n",
"f(0,1,2,100,4,5)\n",
"f(0,1,100,3,4,5)\n",
"f(0,100,2,3,4,5)\n"
]
}
],
"source": [
"x__, x___, y_, y__, y___ = S(\n",
" 'x__', 'x___', 'y_', 'y__', 'y___')\n",
"e = f(*[i for i in range(6)])\n",
"\n",
"for r in e.replace_iter(f(x__, x_, y___), f(x__, 100, y___)):\n",
" print(r)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Match without replacement:"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"5\n",
"4\n",
"3\n",
"2\n",
"1\n",
"0\n"
]
}
],
"source": [
"for r in e.match(f(x___, x_, y___)):\n",
" print(r[x_])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lists are wrapped in the `arg` function:"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\u001b[35marg\u001b[0m(1,2,3,4)\n"
]
},
{
"data": {
"text/plain": [
"f(1,2,3,4)"
]
},
"execution_count": 96,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"e = f(1,2,3,4)\n",
"a = e.replace(f(x__), x__)\n",
"print(a)\n",
"\n",
"f(a)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The right-hand side can also be a function that takes the match dictionary as arguments:"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$k\\!\\left(\\mu1\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"k, x_ = S('k', 'x_')\n",
"\n",
"r = (k(1)).replace(k(x_), lambda m: E(\"k(mu{})\".format(m[x_])))\n",
"display(r)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Examples\n",
"- Check if a graph is connected by shrinking edges"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"vs(2,2,4,4,5,5)\n"
]
}
],
"source": [
"vs = S('vs', is_symmetric=True)\n",
"x_, l1___, l2___, r1___, r2___ = S('x_', 'l1___', 'l2___', 'r1___', 'r2___')\n",
"g = vs(0, 1, 4)*vs(1, 2, 5)*vs(0, 2, 3)*vs(3, 4, 5) # Mercedes topology\n",
"\n",
"g = g.replace(vs(l1___,x_,r1___)*vs(l2___,x_,r2___),\n",
" vs(l1___,r1___,l2___,r2___), repeat=True)\n",
"print(g)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"- Write code that applies the gamma chain relation:\n",
"$$\n",
"\\gamma(\\mu_1,\\ldots,\\nu,\\nu,\\ldots\\mu_n) = D \\gamma(\\mu_1,\\ldots,\\mu_n)\\\\\n",
"\\gamma(\\mu_1,\\ldots,\\nu_2,\\nu_1,\\ldots\\mu_n) = 2g(\\nu_1,\\nu_2) \\gamma(\\mu_1,\\ldots,\\mu_n) - \\gamma(\\mu_1,\\ldots,\\nu_1,\\nu_2,\\ldots\\mu_n) ; \\nu_2 > \\nu_1\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$-D \\gamma\\!\\left(\\mu_0,\\mu_0,\\mu_1,\\mu_2,\\mu_3\\right)+2 D g\\!\\left(\\mu_0,\\mu_1\\right) \\gamma\\!\\left(\\mu_0,\\mu_2,\\mu_3\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"mu = S(*['mu_' + str(i) for i in range(10)])\n",
"gamma, D = S('gamma', 'D')\n",
"g = S('g', is_symmetric=True)\n",
"\n",
"e = gamma(mu[0],mu[1],mu[0], mu[2], mu[4], mu[4], mu[3])\n",
"\n",
"e = e.replace(gamma(x___, x_, x_, y___), D*gamma(x___,y___), repeat=True)\n",
"e = e.replace(gamma(x___, x_, y_, y___), 2*g(x_,y_)*gamma(x___, y___) - gamma(x___, y_, x_, y___), x_.req_cmp_gt(y_, True), repeat=True).expand()\n",
"\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Graph generation"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Symmetry factor: 1/1\n"
]
},
{
"data": {
"text/markdown": [
"```mermaid\n",
"graph TD;\n",
" 0[\"0\"];\n",
" 1[\"0\"];\n",
" 2[\"1\"];\n",
" 3[\"2\"];\n",
" 0 -->|\"q\"| 1;\n",
" 0 ---|\"gl\"| 3;\n",
" 1 -->|\"q\"| 0;\n",
" 1 ---|\"gl\"| 2;\n",
"\n",
"```"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Symmetry factor: 1/2\n"
]
},
{
"data": {
"text/markdown": [
"```mermaid\n",
"graph TD;\n",
" 0[\"0\"];\n",
" 1[\"0\"];\n",
" 2[\"1\"];\n",
" 3[\"2\"];\n",
" 0 ---|\"gl\"| 1;\n",
" 0 ---|\"gl\"| 1;\n",
" 0 ---|\"gl\"| 3;\n",
" 1 ---|\"gl\"| 2;\n",
"\n",
"```"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"g, q = S('gl', 'q')\n",
"\n",
"external= [(1, (None, g)), (2, (None, g))]\n",
"vertices = [((None, g), (None, g), (None, g), (None, g)),\n",
" ((None, g), (None, g), (None, g)),\n",
" ((None, g), (True, q), (False, q))]\n",
"\n",
"graphs = graphs = Graph.generate(external, vertices, max_loops=1, max_bridges=0)\n",
"\n",
"for graph, sym in graphs.items():\n",
" print(f\"Symmetry factor: 1/{sym}\")\n",
" display_graph(graph)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Expression evaluation"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One-time evaluation"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"10.0\n"
]
}
],
"source": [
"print((x+5).evaluate({x: 5.}, {}))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Evaluate nested expressions"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\\begin{align*}\n",
"e &= x + \\pi + \\cos(x) + f(g(x+1),2x)\\\\\n",
"f(y,z) &=y^2 + z^2 y^2\\\\\n",
"g(y) &=x+y+5\n",
"\\end{align*}"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[25864.42651932832], [46990.103027429504]]\n"
]
}
],
"source": [
"x, y, z, pi, f, g = S(\n",
" 'x', 'y', 'z', 'pi', 'f', 'g')\n",
"\n",
"e1 = E(\"x + pi + cos(x) + f(g(x+1),x*2)\")\n",
"fd = E(\"y^2 + z^2*y^2\")\n",
"gd = E(\"x + y + 5\")\n",
"\n",
"ev = e1.evaluator({pi: N(22)/7},\n",
" {(f, \"f\", (y, z)): fd,\n",
" (g, \"g\", (y, )): gd},\n",
" [x])\n",
"res = ev.evaluate([[5.], [6.]])\n",
"print(res)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Evaluating a large expression"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {},
"outputs": [],
"source": [
"f13 = E(open('f13.txt').read())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"One-time evaluation:"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"16.4 ms ± 130 μs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
]
}
],
"source": [
"vars = [\"alpha\", \"amuq\", \"ammu\", \"xcp1\", \"e1245\", \"xcp4\", \"e3e2\", \"e1234\", \"e2345\", \"e1235\",\n",
" \"e1345\", \"amel2\", \"e2e1\", \"e5e2\", \"e4e2\", \"e3e1\", \"e4e1\", \"e5e1\", \"ammu2\", \"amuq2\", \"e5e3\",\n",
" \"e4e3\", \"x5\", \"x6\", \"x1\", \"x3\", \"x4\", \"xcp3\", \"xcp2\"]\n",
"\n",
"%timeit r = f13.evaluate({S(x): float(i) for i, x in enumerate(vars)}, {})"
]
},
{
"cell_type": "code",
"execution_count": 101,
"metadata": {},
"outputs": [],
"source": [
"evaluator = f13.evaluator({}, {}, [S(x) for x in vars], iterations=1)"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"89.9 μs ± 1.14 μs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
]
}
],
"source": [
"%timeit evaluator.evaluate([[float(i) for i in range(len(vars))]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compile instantly with custom inline assembly"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {},
"outputs": [],
"source": [
"compiled_evaluator = evaluator.compile('f13', 'f13.cpp', 'f13.so', )"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"19.4 μs ± 36.6 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)\n"
]
}
],
"source": [
"%timeit compiled_evaluator.evaluate([[float(i) for i in range(len(vars))]])"
]
},
{
"cell_type": "code",
"execution_count": 105,
"metadata": {},
"outputs": [
{
"data": {
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" ('mul', ('temp', 75), [('param', 5), ('param', 17)]),\n",
" ('mul', ('temp', 76), [('param', 0), ('param', 3)]),\n",
" ('mul', ('temp', 77), [('param', 0), ('param', 19)]),\n",
" ('mul', ('temp', 78), [('param', 0), ('param', 5)]),\n",
" ('mul', ('temp', 79), [('param', 9), ('param', 19)]),\n",
" ('mul', ('temp', 80), [('param', 19), ('param', 27)]),\n",
" ('mul', ('temp', 81), [('param', 5), ('param', 14)]),\n",
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" ('mul', ('temp', 117), [('const', 18), ('temp', 115)]),\n",
" ('mul', ('temp', 118), [('const', 8), ('temp', 115)]),\n",
" ('mul', ('temp', 119), [('param', 14), ('temp', 115)]),\n",
" ('mul', ('temp', 120), [('const', 39), ('temp', 115)]),\n",
" ('mul', ('temp', 121), [('const', 16), ('temp', 115)]),\n",
" ('mul', ('temp', 122), [('const', 31), ('temp', 115)]),\n",
" ('mul', ('temp', 123), [('const', 17), ('temp', 115)]),\n",
" ('mul', ('temp', 124), [('param', 13), ('param', 16)]),\n",
" ('mul', ('temp', 125), [('param', 14), ('temp', 124)]),\n",
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" ('mul', ('temp', 127), [('param', 0), ('param', 9)]),\n",
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" ('mul', ('temp', 129), [('const', 3), ('temp', 127)]),\n",
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" ('mul', ('temp', 143), [('param', 13), ('temp', 141)]),\n",
" ('mul', ('temp', 144), [('param', 19), ('temp', 141)]),\n",
" ('mul', ('temp', 145), [('param', 17), ('temp', 141)]),\n",
" ('mul', ('temp', 146), [('param', 14), ('param', 20)]),\n",
" ('mul', ('temp', 147), [('const', 43), ('temp', 146)]),\n",
" ('mul', ('temp', 148), [('param', 9), ('temp', 146)]),\n",
" ('mul', ('temp', 149), [('param', 0), ('temp', 146)]),\n",
" ('mul', ('temp', 150), [('param', 0), ('param', 4)]),\n",
" ('mul', ('temp', 151), [('const', 6), ('temp', 150)]),\n",
" ('mul', ('temp', 152), [('param', 18), ('temp', 150)]),\n",
" ('mul', ('temp', 153), [('param', 7), ('param', 18)]),\n",
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" ('mul', ('temp', 158), [('param', 20), ('param', 21)]),\n",
" ('mul', ('temp', 159), [('param', 9), ('temp', 158)]),\n",
" ('mul', ('temp', 160), [('param', 14), ('param', 21)]),\n",
" ('mul', ('temp', 161), [('const', 15), ('temp', 160)]),\n",
" ('mul', ('temp', 162), [('param', 7), ('param', 21)]),\n",
" ('mul', ('temp', 163), [('param', 7), ('param', 14)]),\n",
" ('mul', ('temp', 164), [('const', 12), ('temp', 163)]),\n",
" ('mul', ('temp', 165), [('const', 26), ('temp', 163)]),\n",
" ('mul', ('temp', 166), [('const', 16), ('temp', 163)]),\n",
" ('mul', ('temp', 167), [('const', 17), ('temp', 163)]),\n",
" ('mul', ('temp', 168), [('param', 3), ('param', 16)]),\n",
" ('mul', ('temp', 169), [('param', 5), ('param', 16)]),\n",
" ('mul', ('temp', 170), [('param', 0), ('param', 10)]),\n",
" ('mul', ('temp', 171), [('param', 3), ('param', 13)]),\n",
" ('mul', ('temp', 172), [('const', 15), ('temp', 171)]),\n",
" ('mul', ('temp', 173), [('param', 5), ('param', 13)]),\n",
" ('mul', ('temp', 174), [('const', 15), ('temp', 173)]),\n",
" ('mul', ('temp', 175), [('param', 0), ('param', 8)]),\n",
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" ('mul', ('temp', 177), [('const', 44), ('temp', 176)]),\n",
" ('mul', ('temp', 178), [('param', 20), ('param', 27)]),\n",
" ('mul', ('temp', 179), [('const', 44), ('temp', 178)]),\n",
" ('mul', ('temp', 180), [('const', 26), ('param', 9)]),\n",
" ('add', ('temp', 181), [('temp', 180), ('temp', 20)]),\n",
" ('add', ('temp', 182), [('temp', 180), ('temp', 87)]),\n",
" ('add', ('temp', 183), [('temp', 95), ('temp', 180)]),\n",
" ('add', ('temp', 184), [('temp', 180), ('temp', 94)]),\n",
" ('add', ('temp', 185), [('temp', 180), ('temp', 85)]),\n",
" ('mul', ('temp', 186), [('param', 18), ('temp', 180)]),\n",
" ('mul', ('temp', 187), [('const', 12), ('param', 9)]),\n",
" ('add', ('temp', 188), [('temp', 187), ('temp', 95)]),\n",
" ('add', ('temp', 189), [('temp', 187), ('temp', 85)]),\n",
" ('mul', ('temp', 190), [('param', 18), ('temp', 187)]),\n",
" ('mul', ('temp', 191), [('param', 14), ('param', 19)]),\n",
" ('mul', ('temp', 192), [('const', 8), ('temp', 191)]),\n",
" ('mul', ('temp', 193), [('const', 39), ('temp', 191)]),\n",
" ('mul', ('temp', 194), [('param', 0), ('param', 16)]),\n",
" ('mul', ('temp', 195), [('const', 39), ('param', 27)]),\n",
" ('mul', ('temp', 196), [('param', 9), ('temp', 195)]),\n",
" ('mul', ('temp', 197), [('const', 18), ('param', 0)]),\n",
" ('mul', ('temp', 198), [('const', 42), ('param', 27)]),\n",
" ('mul', ('temp', 199), [('param', 14), ('temp', 198)]),\n",
" ('mul', ('temp', 200), [('param', 19), ('param', 20)]),\n",
" ('mul', ('temp', 201), [('param', 4), ('param', 21)]),\n",
" ('mul', ('temp', 202), [('param', 18), ('param', 20)]),\n",
" ('mul', ('temp', 203), [('const', 22), ('param', 28)]),\n",
" ('mul', ('temp', 204), [('param', 14), ('temp', 203)]),\n",
" ('mul', ('temp', 205), [('param', 19), ('param', 21)]),\n",
" ('mul', ('temp', 206), [('param', 19), ('param', 28)]),\n",
" ('mul', ('temp', 207), [('param', 4), ('param', 19)]),\n",
" ('mul', ('temp', 208), [('const', 119), ('param', 28)]),\n",
" ('mul', ('temp', 209), [('param', 9), ('temp', 208)]),\n",
" ('mul', ('temp', 210), [('const', 25), ('param', 9)]),\n",
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" -12672,\n",
" 12864,\n",
" -5856,\n",
" 14976,\n",
" -17856,\n",
" -8928,\n",
" 14784,\n",
" 17664,\n",
" -7200,\n",
" -4704,\n",
" 10272,\n",
" -20736,\n",
" -10944,\n",
" 7584,\n",
" 14016,\n",
" -9888,\n",
" -17280,\n",
" 12096,\n",
" -12480,\n",
" 6048,\n",
" 10656,\n",
" -6240,\n",
" 7008,\n",
" -16896,\n",
" 15936,\n",
" 21312,\n",
" 24000,\n",
" -7776,\n",
" -8352,\n",
" 18816,\n",
" -26112,\n",
" -9120,\n",
" -7968,\n",
" -12096,\n",
" -13824,\n",
" -13248,\n",
" 5280,\n",
" -14208,\n",
" 11328,\n",
" 20160,\n",
" -15744,\n",
" -11424,\n",
" -11712,\n",
" -5472,\n",
" 15360,\n",
" -8160,\n",
" 16320,\n",
" -22272,\n",
" -16320,\n",
" -11808,\n",
" -14400,\n",
" 13824,\n",
" -15168,\n",
" 8736,\n",
" 20736,\n",
" -13344,\n",
" -10272,\n",
" -12864])"
]
},
"execution_count": 105,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"evaluator.get_instructions()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Transformers"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Transformers are like functions that execute internally and take an expression and yield another expression."
]
},
{
"cell_type": "code",
"execution_count": 109,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2*x\u001b[33m+\u001b[0mx^2\u001b[33m+\u001b[0m1\n"
]
}
],
"source": [
"t = Transformer().expand().collect(x)\n",
"e = E('(1+x)^2')\n",
"r = t(e)\n",
"print(r)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Transformers can be bound to an expression:"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2*x+x^2+1"
]
},
"execution_count": 110,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = ((x+1)**2).transform().expand()\n",
"t.execute()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The following code tries to expand `x_`, but it does not work:"
]
},
{
"cell_type": "code",
"execution_count": 112,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$f\\!\\left(2 x+x^{2}+1\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"x, f, x_ = S('x', 'f', 'x_')\n",
"e = f((x+1)**2)\n",
"e = e.replace(f(x_), f(x_.transform().expand()))\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Expansion is executed after substituting the matched value of `x_`, since `replace` calls the transformer when it matched"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Hello from Python!\n"
]
},
{
"data": {
"text/markdown": [
"$$\\left(x+1\\right)^{4}$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def some_function(x):\n",
" print('Hello from Python!')\n",
" return x**2\n",
"\n",
"\n",
"e = f((x+1)**2)\n",
"e = e.replace(f(x_), x_.transform().map(some_function))\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Transformers can be used to create a computational graph: a series of instructions to execute"
]
},
{
"cell_type": "code",
"execution_count": 113,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$34 f\\!\\left(0\\right)+55 f\\!\\left(1\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"T = Transformer() # short-hand\n",
"\n",
"e = f(10)\n",
"tr = T.repeat(\n",
" T.expand(),\n",
" T.replace(f(x_), f(x_ - 1) + f(x_ - 2), x_ > 1),\n",
")\n",
"\n",
"display(tr(e))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Transformers support branching:"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y\n",
"2*y\u001b[33m+\u001b[0m4\n"
]
}
],
"source": [
"t = T.map_terms(T.if_changed(T.replace(x, y), T.print()))\n",
"print(t(x + y + 4))"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"4\n",
"4\n"
]
},
{
"data": {
"text/plain": [
"4"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = T.map_terms(T.repeat(\n",
" T.replace(y, 4),\n",
" T.if_changed(T.replace(x, y),\n",
" T.break_chain()),\n",
" T.print() # print of y is never reached\n",
"))\n",
"t(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Custom normalization"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Create a function that is linear in the variable $n$"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [],
"source": [
"n, x_, y, y_ = S('n', 'x_', 'y', 'y_')\n",
"dot2_ = S('dotp_', is_symmetric=True, is_linear=True)\n",
"dot2 = S('dot2', is_symmetric=True, is_linear=True, custom_normalization=Transformer().replace(dot2_(n*x_,y_), n*dot2_(x_, y_)))"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n*dot2(x,x)\u001b[33m+\u001b[0mn^2*dot2(x,y)\n"
]
}
],
"source": [
"print(E('dot2(n*x,x + n*y)'))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Streams"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Terms can be streamed to and from disk automatically:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"2*y\n",
"normalized 2*y\n"
]
}
],
"source": [
"x, y = S('x', 'y')\n",
"#e = ((x+1)**20).expand()\n",
"stream = TermStreamer(n_cores=4)\n",
"stream.push(x)\n",
"stream.push(y)\n",
"stream = stream.map(Transformer().replace(x, y))\n",
"\n",
"stream.normalize()\n",
"stream.map(Transformer().print())\n",
"\n",
"e = stream.to_expression()\n",
"print('normalized', e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"-----"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercises"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"I(n) = I(n-1) \\frac{2 + (4-2 \\epsilon) - 2 n}{2(n - 1) m^2}\\,; n \\geq 1\n",
"$$\n"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$\\frac{1}{72} ep m^{-18} topo\\!\\left(1\\right)+\\frac{223}{10080} ep^{2} m^{-18} topo\\!\\left(1\\right)+\\frac{1}{5670} ep^{3} m^{-18} topo\\!\\left(1\\right)-\\frac{101}{5760} ep^{4} m^{-18} topo\\!\\left(1\\right)-\\frac{229}{17280} ep^{5} m^{-18} topo\\!\\left(1\\right)-\\frac{13}{2880} ep^{6} m^{-18} topo\\!\\left(1\\right)-\\frac{7}{8640} ep^{7} m^{-18} topo\\!\\left(1\\right)-\\frac{1}{13440} ep^{8} m^{-18} topo\\!\\left(1\\right)-\\frac{1}{362880} ep^{9} m^{-18} topo\\!\\left(1\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ep, m, n_ = S('ep', 'm', 'n_')\n",
"topo = S('topo')\n",
"T = Transformer()\n",
"\n",
"e = E(\"topo(10)\")\n",
"e = e.replace(topo(n_),\n",
" topo(n_ - 1) * (2 + (4-2*ep) - 2 * n_) /\n",
" (2*(n_ - 1)) / m**2,\n",
" n_.req_gt(1), repeat=True).expand()\n",
"display(e)"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$[\\frac{5040ep+8028ep^{2}+64ep^{3}-6363ep^{4}-4809ep^{5}-1638ep^{6}-294ep^{7}-27ep^{8}-ep^{9}}{362880}] m^{-18} topo\\!\\left(1\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"e = E(\"topo(10)\")\n",
"e = e.replace(topo(n_),\n",
" topo(n_ - 1) * Expression.COEFF(2 + (4-2*ep) - 2 * n_) /\n",
" (2*(n_ - 1)) / m**2,\n",
" n_.req_gt(1), repeat=True)\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"1. Write code that applies the trace rule\n",
"$$Tr(\\mu_1,\\ldots,\\mu_n) = \\sum_{k=2}^n (-1)^k g(\\mu_1,\\mu_k) Tr(\\mu_2,\\ldots,\\mu_{k-1},\\mu_{k+1},\\ldots \\mu_n)$$"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/markdown": [
"$$f\\!\\left(p_1,p_2,p_3,p_4,p_5,p_6\\right)$$"
],
"text/plain": [
"<IPython.core.display.Markdown object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from symbolica import Expression\n",
"\n",
"p = S(*['p_' + str(i + 1) for i in range(10)])\n",
"p_ = S(*['p_' + str(i + 1) + '_' for i in range(10)])\n",
"g = S('g', is_symmetric=True)\n",
"\n",
"e = f(*p[:6])\n",
"display(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here is the FORM code that does the trace:\n",
"\n",
"```\n",
"#-\n",
"off statistics;\n",
"V p,p0,p1,...,p20;\n",
"CF f;\n",
"CF d(s);\n",
"\n",
"L F = f(p1,...,p10);\n",
"\n",
"#do i = 10,1,-2\n",
" id f(p1?,...,p`i'?) = \n",
" #if `i' > 2\n",
" + d(p1, p2) * f(p3,...,p`i')\n",
" #else\n",
" + d(p1,p2)\n",
" #endif\n",
" \n",
" #do k = 3,{`i'-1},1\n",
" + (-1)^(`k'+1) * d(p1, p`k') * f(p2,...,p{`k'-1},p{`k'+ 1},...,p`i')\n",
" #enddo\n",
"\n",
" #if `i' > 2\n",
" + (-1)^(`i') * d(p1, p`i') * f(p2,...,p{`i'-1})\n",
" #endif\n",
" ;\n",
" .sort:trace-`i';\n",
"#enddo\n",
"\n",
"id f = 4;\n",
"id f(?a) = 0;\n",
"\n",
"Print +s;\n",
".end\n",
"```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"2. Apply the gluon Feynman rule"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"((-d(nu2,mom(0))+d(nu2,mom(4)))*d(mu4,mu5)-(-d(mu4,mom(0))+d(mu4,mom(4)))*d(nu2,mu5)-d(nu2,mu4)*d(mu5,mom(0))-d(nu2,mu4)*d(mu5,mom(4))+d(nu2,mu5)*d(mu4,mom(0))+d(mu4,mu5)*d(nu2,mom(4)))*((d(nu1,mom(0))-d(nu1,mom(1)))*d(mu1,mu8)-(d(mu8,mom(0))-d(mu8,mom(1)))*d(mu1,nu1)-d(mu1,nu1)*d(mu8,mom(0))-d(mu1,mu8)*d(nu1,mom(1))+d(nu1,mu8)*d(mu1,mom(0))+d(nu1,mu8)*d(mu1,mom(1)))*(-(-d(mu5,mom(0))+d(mu5,mom(3)))*d(mu6,mu11)+(-d(mu5,mom(3))+d(mu5,mom(4)))*d(mu6,mu11)-(-d(mu6,mom(3))+d(mu6,mom(4)))*d(mu5,mu11)+(-d(mu11,mom(0))+d(mu11,mom(3)))*d(mu5,mu6)+(d(mu6,mom(0))-d(mu6,mom(4)))*d(mu5,mu11)-(d(mu11,mom(0))-d(mu11,mom(4)))*d(mu5,mu6))*(-(-d(mu6,mom(0))+d(mu6,mom(2)))*d(mu7,mu10)+(-d(mu6,mom(2))+d(mu6,mom(3)))*d(mu7,mu10)-(-d(mu7,mom(2))+d(mu7,mom(3)))*d(mu6,mu10)+(-d(mu10,mom(0))+d(mu10,mom(2)))*d(mu6,mu7)+(d(mu7,mom(0))-d(mu7,mom(3)))*d(mu6,mu10)-(d(mu10,mom(0))-d(mu10,mom(3)))*d(mu6,mu7))*(-(-d(mu7,mom(0))+d(mu7,mom(1)))*d(mu8,mu9)+(-d(mu7,mom(1))+d(mu7,mom(2)))*d(mu8,mu9)-(-d(mu8,mom(1))+d(mu8,mom(2)))*d(mu7,mu9)+(-d(mu9,mom(0))+d(mu9,mom(1)))*d(mu7,mu8)+(d(mu8,mom(0))-d(mu8,mom(2)))*d(mu7,mu9)-(d(mu9,mom(0))-d(mu9,mom(2)))*d(mu7,mu8))*(-(d(mu1,mom(1))-d(mu1,mom(2)))*d(mu2,mu9)+(d(mu2,mom(1))-d(mu2,mom(2)))*d(mu1,mu9)-d(mu1,mu2)*d(mu9,mom(1))-d(mu1,mu2)*d(mu9,mom(2))+d(mu1,mu9)*d(mu2,mom(1))+d(mu2,mu9)*d(mu1,mom(2)))*(-(d(mu2,mom(2))-d(mu2,mom(3)))*d(mu3,mu10)+(d(mu3,mom(2))-d(mu3,mom(3)))*d(mu2,mu10)-d(mu2,mu3)*d(mu10,mom(2))-d(mu2,mu3)*d(mu10,mom(3))+d(mu2,mu10)*d(mu3,mom(2))+d(mu3,mu10)*d(mu2,mom(3)))*(-(d(mu3,mom(3))-d(mu3,mom(4)))*d(mu4,mu11)+(d(mu4,mom(3))-d(mu4,mom(4)))*d(mu3,mu11)-d(mu3,mu4)*d(mu11,mom(3))-d(mu3,mu4)*d(mu11,mom(4))+d(mu3,mu11)*d(mu4,mom(3))+d(mu4,mu11)*d(mu3,mom(4)))\n"
]
}
],
"source": [
"k = S('mom', is_symmetric=True)\n",
"vx, coll = S('vx', 'coll')\n",
"nu = S(*['nu%d' % i for i in range(3)])\n",
"mu = S(*['mu%d' % i for i in range(12)])\n",
"d = S('d', is_symmetric=True, is_linear=True)\n",
"v = S('v', is_linear=True)\n",
"k_ = S(*['k%d_' % i for i in range(5)])\n",
"mu_ = S(*['mu%d_' % i for i in range(5)])\n",
"\n",
"x_, y_, x__, x___, y___ = S('x_', 'y_', 'x__', 'x___', 'y___')\n",
"\n",
"# k(0) is external momentum\n",
"r = vx(-k(0), k(0)-k(1), k(1), nu[1], mu[1], mu[8]) *\\\n",
" vx(-k(1), k(2), k(1)-k(2), mu[1], mu[2], mu[9]) *\\\n",
" vx(-k(2), k(3), k(2)-k(3), mu[2], mu[3], mu[10]) *\\\n",
" vx(-k(3), k(4), k(3)-k(4), mu[3], mu[4], mu[11]) *\\\n",
" vx(-k(4), k(0), k(4)-k(0), mu[4], nu[2], mu[5]) *\\\n",
" vx(-k(4)+k(0), -k(3)+k(4), k(3)-k(0), mu[5], mu[11], mu[6]) *\\\n",
" vx(-k(3)+k(0), -k(2)+k(3), k(2)-k(0), mu[6], mu[10], mu[7]) *\\\n",
" vx(-k(2)+k(0), -k(1)+k(2), k(1)-k(0), mu[7], mu[9], mu[8])\n",
"\n",
"r2 = r.replace(vx(k_[1], k_[2], k_[3], mu_[1], mu_[2], mu_[3]),\n",
" (- d(mu_[1], mu_[3]) * d(k_[1], mu_[2])\n",
" + d(mu_[1], mu_[2]) * d(k_[1], mu_[3])\n",
" + d(mu_[2], mu_[3]) * d(k_[2], mu_[1])\n",
" - d(mu_[1], mu_[2]) * d(k_[2], mu_[3])\n",
" - d(mu_[2], mu_[3]) * d(k_[3], mu_[1])\n",
" + d(mu_[1], mu_[3]) * d(k_[3], mu_[2])\n",
" )\n",
" )\n",
"\n",
"print(r2.format(terms_on_new_line=True))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solutions"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [],
"source": [
"\n",
"T = Transformer()\n",
"i1, i2 = S('i1', 'i2')\n",
"p = S(*['p' + str(i + 1) for i in range(10)])\n",
"p_ = S(*['p' + str(i + 1) + '_' for i in range(10)])\n",
"x_, y_, x___, y___, z___, i1_, i2_ = S(\n",
" 'x_', 'y_', 'x___', 'y___', 'z___', 'i1_', 'i2_')\n",
"\n",
"gamma = S('γ')\n",
"\n",
"\n",
"# Canonicalize a gamma string\n",
"def gamma_canonicalize() -> Transformer:\n",
" return T.repeat(\n",
" T.expand(),\n",
" T.replace(gamma(i1_, x___, x_, x_, y___, i2_),\n",
" d(x_, x_)*gamma(i1_, x___, y___, i2_)),\n",
" T.replace(gamma(i1_, x___, x_, y_, z___, i2_), -gamma(i1_, x___, y_, x_, z___, i2_)+2*g(y_, x_)*gamma(i1_, x___, z___, i2_),\n",
" x_.req_cmp_gt(y_, True))\n",
" )"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Canonicalization of γ(i1,p1,p2,p4,p5,p4,p6,i2)\n",
"\t 2*g(p4,p5)*γ(i1,p1,p2,p4,p6,i2)\u001b[33m-\u001b[0md(p4,p4)*γ(i1,p1,p2,p5,p6,i2)\n",
"Trace of γ(p1,p2,p3,p3,p4,p5,p6,p7)\n",
"\t 4*d(p1,p2)*d(p3,p3)*d(p4,p5)*d(p6,p7)\n",
"\u001b[33m-\u001b[0m4*d(p1,p2)*d(p3,p3)*d(p4,p6)*d(p5,p7)\n",
"\u001b[33m+\u001b[0m4*d(p1,p2)*d(p3,p3)*d(p4,p7)*d(p5,p6)\n",
"\u001b[33m-\u001b[0m4*d(p1,p4)*d(p2,p5)*d(p3,p3)*d(p6,p7)\n",
"\u001b[33m+\u001b[0m4*d(p1,p4)*d(p2,p6)*d(p3,p3)*d(p5,p7)\n",
"\u001b[33m-\u001b[0m4*d(p1,p4)*d(p2,p7)*d(p3,p3)*d(p5,p6)\n",
"\u001b[33m+\u001b[0m4*d(p1,p5)*d(p2,p4)*d(p3,p3)*d(p6,p7)\n",
"\u001b[33m-\u001b[0m4*d(p1,p5)*d(p2,p6)*d(p3,p3)*d(p4,p7)\n",
"\u001b[33m+\u001b[0m4*d(p1,p5)*d(p2,p7)*d(p3,p3)*d(p4,p6)\n",
"\u001b[33m-\u001b[0m4*d(p1,p6)*d(p2,p4)*d(p3,p3)*d(p5,p7)\n",
"\u001b[33m+\u001b[0m4*d(p1,p6)*d(p2,p5)*d(p3,p3)*d(p4,p7)\n",
"\u001b[33m-\u001b[0m4*d(p1,p6)*d(p2,p7)*d(p3,p3)*d(p4,p5)\n",
"\u001b[33m+\u001b[0m4*d(p1,p7)*d(p2,p4)*d(p3,p3)*d(p5,p6)\n",
"\u001b[33m-\u001b[0m4*d(p1,p7)*d(p2,p5)*d(p3,p3)*d(p4,p6)\n",
"\u001b[33m+\u001b[0m4*d(p1,p7)*d(p2,p6)*d(p3,p3)*d(p4,p5)\n",
"4*d(p1,p2)*d(p3,p4)\u001b[33m-\u001b[0m4*d(p1,p3)*d(p2,p4)\u001b[33m+\u001b[0m4*d(p1,p4)*d(p2,p3)\n"
]
}
],
"source": [
"\n",
"# Perform the trace of a closed gamma string,\n",
"# working from the chain length `max_len` back to 0\n",
"def gamma_trace(max_len: int) -> Transformer:\n",
" trace = T\n",
" for l in range(max_len, 0, -1):\n",
" if l % 2 == 1:\n",
" trace = trace.replace(gamma(*p_[:l]), 0) # odd trace is 0\n",
" continue\n",
"\n",
" trace = trace.chain(\n",
" T.replace(gamma(x___, x_, x_, y___),\n",
" d(x_, x_)*gamma(x___, y___)),\n",
" T.replace(gamma(\n",
" *p_[:l]), sum((-1)**(k+1) * d(p_[0], p_[k]) * gamma(*p_[1:k], *p_[k+1:l]) for k in range(1, l))).expand())\n",
" return trace.replace(gamma(), 4)\n",
"\n",
"\n",
"e = gamma(i1, p[0], p[1], p[3], p[4], p[3], p[5], i2)\n",
"print('Canonicalization of', e)\n",
"e = e.transform().chain(gamma_canonicalize()).execute()\n",
"print('\\t', e)\n",
"\n",
"\n",
"e = gamma(p[0], p[1], p[2], p[2], p[3], p[4], p[5], p[6])\n",
"print('Trace of', e)\n",
"e = e.transform().chain(gamma_trace(10)).execute()\n",
"print('\\t', e.format(terms_on_new_line=True))\n",
"\n",
"e = gamma(p[0], p[1], p[2], p[3]).transform().chain(gamma_trace(10)).execute()\n",
"print(e)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Gluon exercise"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Terms 9\n",
"Terms 72\n",
"Terms 560\n",
"Terms 2654\n",
"Terms 3541\n",
"Terms 6044\n",
"Terms 10516\n",
"Terms 9652\n"
]
},
{
"data": {
"text/plain": [
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]
},
"execution_count": 56,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def gluon(r: Expression) -> Expression:\n",
" while True:\n",
" r2 = r.replace(vx(k_[1], k_[2], k_[3], mu_[1], mu_[2], mu_[3])*x___,\n",
" (- d(mu_[1], mu_[3]) * d(k_[1], mu_[2])\n",
" + d(mu_[1], mu_[2]) * d(k_[1], mu_[3])\n",
" + d(mu_[2], mu_[3]) * d(k_[2], mu_[1])\n",
" - d(mu_[1], mu_[2]) * d(k_[2], mu_[3])\n",
" - d(mu_[2], mu_[3]) * d(k_[3], mu_[1])\n",
" + d(mu_[1], mu_[3]) * d(k_[3], mu_[2])\n",
" )*x___\n",
" ).expand()\n",
"\n",
" if r == r2:\n",
" return r\n",
"\n",
" r2 = r2.replace(d(mu_[0], mu_[1])*d(mu_[0], mu_[2]),\n",
" d(mu_[1], mu_[2]), ~mu_[0].req_type(AtomType.Fn), level_range=(0, 0), repeat=True)\n",
"\n",
" r2 = r2.replace(d(mu_[0], mu_[0]), 4, ~mu_[\n",
" 0].req_type(AtomType.Fn), level_range=(0, 0), repeat=True)\n",
"\n",
" r2 = r2.replace(d(k(x_), k(y_)), k(x_, y_),\n",
" level_range=(0, 0), repeat=True)\n",
" r2 = r2.replace(d(mu_[0], k(y_))**2, k(y_, y_), ~mu_[0].req_type(AtomType.Fn),\n",
" level_range=(0, 0), repeat=True)\n",
" r2 = r2.replace(d(mu_[0], mu_[1])**2, 4, ~mu_[0].req_type(AtomType.Fn) & ~mu_[1].req_type(AtomType.Fn), # 4 instead of D\n",
" level_range=(0, 0), repeat=True)\n",
" print('Terms', len(r2))\n",
"\n",
" r = r2\n",
"\n",
"gluon(r)\n",
"#print('\\t', gluon(r).format(terms_on_new_line=True))"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 1 │ 769.00 B │\n",
"\tOut │ 9 │ 6.03KB │ ⧗ 139.75µs\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 9 │ 6.03KB │\n",
"\tOut │ 72 │ 41.30KB │ ⧗ 3.47ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 72 │ 41.30KB │\n",
"\tOut │ 560 │ 266.76KB │ ⧗ 14.05ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 560 │ 266.76KB │\n",
"\tOut │ 2654 │ 1.08MB │ ⧗ 64.69ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 2654 │ 1.08MB │\n",
"\tOut │ 3541 │ 1.16MB │ ⧗ 273.99ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 3541 │ 1.16MB │\n",
"\tOut │ 6044 │ 1.50MB │ ⧗ 340.87ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 6044 │ 1.50MB │\n",
"\tOut │ 10516 │ 1.76MB │ ⧗ 502.35ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 10516 │ 1.76MB │\n",
"\tOut │ 9652 │ 851.32KB │ ⧗ 573.17ms\n",
"Stats for \u001b[1mgluon\u001b[0m:\n",
"\tIn │ 9652 │ 851.32KB │\n",
"\tOut │ 9652 │ 851.32KB │ ⧗ 94.00ms\n"
]
}
],
"source": [
"def gluon_transformer():\n",
" return Transformer().repeat(\n",
" Transformer().stats('gluon',\n",
" Transformer().replace(\n",
" vx(k_[1], k_[2], k_[3], mu_[\n",
" 1], mu_[2], mu_[3])*x__,\n",
" (- d(mu_[1], mu_[3]) * d(k_[1], mu_[2])\n",
" + d(mu_[1], mu_[2]) *\n",
" d(k_[1], mu_[3])\n",
" + d(mu_[2], mu_[3]) * d(k_[2], mu_[1])\n",
" - d(mu_[1], mu_[2]) * d(k_[2], mu_[3])\n",
" - d(mu_[2], mu_[3]) * d(k_[3], mu_[1])\n",
" + d(mu_[1], mu_[3]) * d(k_[3], mu_[2]))*x__,\n",
" level_range=(0, 0)\n",
" ).expand().\n",
" repeat(Transformer().replace(\n",
" d(mu_[0], mu_[1]) *\n",
" d(mu_[0], mu_[2]),\n",
" d(mu_[1], mu_[2]), ~mu_[\n",
" 0].req_type(AtomType.Fn),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(k(x_), k(y_)),\n",
" k(x_, y_),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(mu_[0], k(y_))**2,\n",
" k(y_, y_),\n",
" ~mu_[0].req_type(\n",
" AtomType.Fn),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(mu_[0], mu_[1])**2,\n",
" 4,\n",
" ~mu_[0].req_type(AtomType.Fn) & ~mu_[\n",
" 1].req_type(AtomType.Fn),\n",
" level_range=(0, 0)\n",
" ))).check_interrupt()\n",
" )\n",
"\n",
"\n",
"e = r.transform().chain(gluon_transformer()).execute()"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [],
"source": [
"def gluon_stream(r):\n",
" s = TermStreamer('gluon', n_cores=1)\n",
" s.push(r)\n",
"\n",
" for _ in range(10):\n",
" s = s.map(Transformer().replace(\n",
" vx(k_[1], k_[2], k_[3], mu_[\n",
" 1], mu_[2], mu_[3])*x__,\n",
" (- d(mu_[1], mu_[3]) * d(k_[1], mu_[2])\n",
" + d(mu_[1], mu_[2]) * d(k_[1], mu_[3])\n",
" + d(mu_[2], mu_[3]) * d(k_[2], mu_[1])\n",
" - d(mu_[1], mu_[2]) * d(k_[2], mu_[3])\n",
" - d(mu_[2], mu_[3]) * d(k_[3], mu_[1])\n",
" + d(mu_[1], mu_[3]) * d(k_[3], mu_[2])\n",
" )\n",
" * x__,\n",
" level_range=(0, 0)\n",
" ).expand())\n",
"\n",
" # s.normalize()\n",
"\n",
" s = s.map(Transformer().repeat(Transformer().replace(\n",
" d(mu_[0], mu_[1]) *\n",
" d(mu_[0], mu_[2]),\n",
" d(mu_[1], mu_[2]), ~mu_[\n",
" 0].req_type(AtomType.Fn),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(k(x_), k(y_)),\n",
" k(x_, y_),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(mu_[0], k(y_))**2,\n",
" k(y_, y_),\n",
" ~mu_[0].req_type(\n",
" AtomType.Fn),\n",
" level_range=(0, 0)\n",
" ).replace(\n",
" d(mu_[0], mu_[1])**2,\n",
" 4,\n",
" ~mu_[0].req_type(AtomType.Fn) & ~mu_[\n",
" 1].req_type(AtomType.Fn),\n",
" level_range=(0, 0)\n",
" )))\n",
"\n",
" s.normalize()\n",
"\n",
" r = s.to_expression()\n",
" print(r.format(terms_on_new_line=True))\n",
" print(len(r))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Trace4"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In 4 dimensions:\n",
"$$\\gamma(\\mu_1,\\mu_2,\\mu_3,\\ldots) = \\epsilon^{\\mu_1 \\mu_2 \\mu_3 \\nu} \\gamma_5 \\gamma(\\nu,\\ldots) + \\gamma(\\mu_1,\\ldots) g(\\mu_2,\\mu_3) - \\gamma(\\mu_2,\\ldots) g(\\mu_1,\\mu_3) + \\gamma(\\mu_3,\\ldots) g(\\mu_1,\\mu_2)$$\n",
"\n",
"- Repeatedly applying this rule (with unique dummy index $\\nu$) yields an expression in terms of Levi-Civita symbols and metrics\n",
"- The Levi-Civita symbol is fully antisymmetric: in 4D we can only have 4 unique indices. Thus input with many dummy indices will have many canceling combinations"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"ename": "TypeError",
"evalue": "Symbol python::gamma redefined with new attributes.",
"output_type": "error",
"traceback": [
"\u001b[31m---------------------------------------------------------------------------\u001b[39m",
"\u001b[31mTypeError\u001b[39m Traceback (most recent call last)",
"\u001b[36mCell\u001b[39m\u001b[36m \u001b[39m\u001b[32mIn[59]\u001b[39m\u001b[32m, line 15\u001b[39m\n\u001b[32m 13\u001b[39m levi, sign = S(\u001b[33m'\u001b[39m\u001b[33mlevi\u001b[39m\u001b[33m'\u001b[39m, \u001b[33m'\u001b[39m\u001b[33msign\u001b[39m\u001b[33m'\u001b[39m, is_antisymmetric=\u001b[38;5;28;01mTrue\u001b[39;00m)\n\u001b[32m 14\u001b[39m d = S(\u001b[33m'\u001b[39m\u001b[33mds\u001b[39m\u001b[33m'\u001b[39m, is_symmetric=\u001b[38;5;28;01mTrue\u001b[39;00m)\n\u001b[32m---> \u001b[39m\u001b[32m15\u001b[39m gamma = \u001b[43mS\u001b[49m\u001b[43m(\u001b[49m\u001b[33;43m'\u001b[39;49m\u001b[33;43mgamma\u001b[39;49m\u001b[33;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mis_cyclesymmetric\u001b[49m\u001b[43m=\u001b[49m\u001b[38;5;28;43;01mTrue\u001b[39;49;00m\u001b[43m)\u001b[49m\n\u001b[32m 16\u001b[39m x___, y___, D = S(\u001b[33m'\u001b[39m\u001b[33mx___\u001b[39m\u001b[33m'\u001b[39m, \u001b[33m'\u001b[39m\u001b[33my___\u001b[39m\u001b[33m'\u001b[39m, \u001b[33m'\u001b[39m\u001b[33mD\u001b[39m\u001b[33m'\u001b[39m)\n\u001b[32m 19\u001b[39m \u001b[38;5;28;01mdef\u001b[39;00m\u001b[38;5;250m \u001b[39m\u001b[34mlevi_contract\u001b[39m(l: \u001b[38;5;28mlist\u001b[39m[Expression], r: \u001b[38;5;28mlist\u001b[39m[Expression]) -> Expression:\n",
"\u001b[31mTypeError\u001b[39m: Symbol python::gamma redefined with new attributes."
]
}
],
"source": [
"import random\n",
"from itertools import permutations\n",
"import time\n",
"\n",
"T = Transformer()\n",
"mu = S('mu', 'nu')\n",
"i = S(*['mu<{}>'.format(i) for i in range(21)])\n",
"\n",
"mu = S(*['mu<{}>'.format(i) for i in range(21)])\n",
"nu = S(*['nu<{}>'.format(i) for i in range(21)]) # dummy\n",
"muw = S(*['mu{}_'.format(i) for i in range(21)])\n",
"g5 = S('g5')\n",
"levi, sign = S('levi', 'sign', is_antisymmetric=True)\n",
"d = S('ds', is_symmetric=True)\n",
"gamma = S('gamma', is_cyclesymmetric=True)\n",
"x___, y___, D = S('x___', 'y___', 'D')\n",
"\n",
"\n",
"def levi_contract(l: list[Expression], r: list[Expression]) -> Expression:\n",
" a = [(tuple(zip(l, perm)), sign(*perm).replace(sign(x___), 1)) for perm in permutations(r)]\n",
"\n",
" res = N(0)\n",
" for (c, sig) in a:\n",
" s = sig\n",
" for b in c:\n",
" s = s * d(b[0], b[1])\n",
" res = res + s\n",
"\n",
" return res\n",
"\n",
"\n",
"K = 16\n",
"e = 4*gamma(*mu[:K])\n",
"\n",
"t = time.time()\n",
"\n",
"for i in range(K):\n",
" e = e.replace(\n",
" gamma(muw[1], muw[2], muw[3], x___),\n",
" levi(muw[1], muw[2], muw[3], nu[i])*g5*gamma(nu[i], x___) +\n",
" gamma(muw[1], x___)*d(muw[2], muw[3]) - gamma(muw[2], x___)*d(muw[1], muw[3]) + gamma(muw[3], x___)*d(muw[1], muw[2])).expand()\n",
" e = e.replace(g5**2, 1)\n",
"print('yo', time.time() - t)\n",
"\n",
"e = e.replace(g5*gamma(x___), gamma(g5, x___))\n",
"e = e.replace(gamma(muw[0], muw[1], muw[2]), 0)\n",
"e = e.replace(gamma(muw[0], muw[1]), d(muw[0], muw[1]), repeat=True)\n",
"\n",
"print('yo1', time.time() - t)\n",
"\n",
"# TODO: break and continue statement in loops?, IfMatch, Jump?\n",
"t = T.repeat(\n",
" T.repeat(T.replace(\n",
" d(muw[0], muw[1])*d(muw[0], muw[2]), d(muw[1], muw[2]))),\n",
" T.replace(d(muw[0], muw[0]), 4),\n",
" T.repeat(T.replace(\n",
" levi(muw[0], x___)*d(muw[0], muw[1]), sign(muw[0], x___)*levi(muw[1], x___,))),\n",
" T.replace(levi(muw[0], muw[1], muw[6], muw[7])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[6], muw[7])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1]], [muw[4], muw[5]])),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[7])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[7])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2]], [muw[4], muw[5], muw[6]])),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[3])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[3])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2], muw[3]], [muw[4], muw[5], muw[6], muw[7]])),\n",
" T.replace(sign(x___), 1),\n",
" T.expand()\n",
")\n",
"\n",
"e = t(e)\n",
"print(len(e))\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"\n",
"t = T.repeat(\n",
" T.repeat(T.replace(\n",
" d(muw[0], muw[1])*d(muw[0], muw[2]), d(muw[1], muw[2])).replace(\n",
" d(muw[0], muw[1])**2,4)),\n",
" T.replace(d(muw[0], muw[0]), 4),\n",
" T.repeat(T.replace(\n",
" levi(muw[0], x___)*d(muw[0], muw[1]), sign(muw[0], x___)*levi(muw[1], x___))),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[3])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[3])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2], muw[3]], [muw[4], muw[5], muw[6], muw[7]])),\n",
" T.replace(sign(x___), 1),\n",
" T.expand(),\n",
")\n",
"\n",
"e2 = t(e2)\n",
"print(len(e2))\n",
"\n",
"# print(e.format(terms_on_new_line=True))\n",
"# print(e2.format(terms_on_new_line=True))\n",
"\n",
"# evaluate\n",
"vals = [[random.uniform(0, 1) for j in range(4)]\n",
" for i in range(K)]\n",
"\n",
"\n",
"for x in range(K):\n",
" for y in range(x + 1, K):\n",
" e = e.replace(d(mu[x], mu[y]), sum(\n",
" [vals[x][i] * vals[y][i] for i in range(4)]))\n",
" e2 = e2.replace(d(mu[x], mu[y]), sum(\n",
" [vals[x][i] * vals[y][i] for i in range(4)]))\n",
"\n",
"print(e)\n",
"print(e2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Old"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import copy\n",
"import random\n",
"\n",
"T = Transformer()\n",
"mu = S(*['mu<{}>'.format(i) for i in range(21)])\n",
"nu = S(*['nu<{}>'.format(i) for i in range(21)]) # dummy\n",
"muw = S(*['mu{}_'.format(i) for i in range(21)])\n",
"g5 = S('g5')\n",
"levi, sign = S('levi', 'sign', is_antisymmetric=True)\n",
"d = S('d', is_symmetric=True)\n",
"gamma = S('gamma', is_cyclesymmetric=True)\n",
"x___, y___, D = S('x___', 'y___', 'D')\n",
"\n",
"\n",
"def levi_contract(l: list[Expression], r: list[Expression]) -> Expression:\n",
" from itertools import permutations\n",
" a = [(tuple(zip(l, perm)), sign(*perm)) for perm in permutations(r)]\n",
"\n",
" res = N(0)\n",
" for t in a:\n",
" s = t[1]\n",
" for b in t[0]:\n",
" s = s * d(b[0], b[1])\n",
" res = res + s\n",
"\n",
" return res.replace(sign(x___), 1)\n",
"\n",
"\n",
"K = 4\n",
"e = 4*gamma(*mu[:K])\n",
"\n",
"\n",
"for i in range(K):\n",
" e = e.replace(\n",
" gamma(muw[1], muw[2], muw[3], x___),\n",
" levi(muw[1], muw[2], muw[3], nu[i])*g5*gamma(nu[i], x___) +\n",
" gamma(muw[1], x___)*d(muw[2], muw[3]) - gamma(muw[2], x___)*d(muw[1], muw[3]) + gamma(muw[3], x___)*d(muw[1], muw[2])).expand()\n",
" e = e.replace(g5**2, 1, repeat=True)\n",
"\n",
"e = e.replace(g5*gamma(x___), gamma(g5, x___))\n",
"e = e.replace(gamma(muw[0], muw[1], muw[2]), 0, repeat=True)\n",
"e = e.replace(gamma(muw[0], muw[1]), d(muw[0], muw[1]), repeat=True)\n",
"\n",
"\n",
"e2 = copy.deepcopy(e)\n",
"\n",
"# TODO: break and continue statement in loops?, IfMatch, Jump?\n",
"t = T.repeat(\n",
" T.repeat(T.replace(\n",
" d(muw[0], muw[1])*d(muw[0], muw[2]), d(muw[1], muw[2]))),\n",
" T.replace(d(muw[0], muw[0]), 4),\n",
" T.repeat(T.replace(\n",
" levi(muw[0], x___)*d(muw[0], muw[1]), sign(muw[0], x___)*levi(muw[1], x___,))),\n",
" T.replace(levi(muw[0], muw[1], muw[6], muw[7])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[6], muw[7])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1]], [muw[4], muw[5]])),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[7])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[7])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2]], [muw[4], muw[5], muw[6]])),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[3])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[3])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2], muw[3]], [muw[4], muw[5], muw[6], muw[7]])),\n",
" T.replace(sign(x___), 1),\n",
" T.expand()\n",
")\n",
"\n",
"e = t(e)\n",
"print(len(e))\n",
"\n",
"t = T.repeat(\n",
" T.repeat(T.replace(\n",
" d(muw[0], muw[1])*d(muw[0], muw[2]), d(muw[1], muw[2]))),\n",
" T.replace(d(muw[0], muw[0]), 4),\n",
" T.repeat(T.replace(\n",
" levi(muw[0], x___)*d(muw[0], muw[1]), sign(muw[0], x___)*levi(muw[1], x___))),\n",
" T.replace(levi(muw[0], muw[1], muw[2], muw[3])*levi(muw[4], muw[5], muw[6], muw[7]),\n",
" sign(muw[0], muw[1], muw[2], muw[3])*sign(muw[4], muw[5], muw[6], muw[7]) *\n",
" levi_contract([muw[0], muw[1], muw[2], muw[3]], [muw[4], muw[5], muw[6], muw[7]])),\n",
" T.replace(sign(x___), 1),\n",
" T.expand(),\n",
")\n",
"\n",
"e2 = t(e2)\n",
"print(len(e2))\n",
"\n",
"# print(e.format(terms_on_new_line=True))\n",
"# print(e2.format(terms_on_new_line=True))\n",
"\n",
"# evaluate\n",
"vals = [[random.uniform(0, 1) for j in range(4)]\n",
" for i in range(K)]\n",
"\n",
"\n",
"for x in range(K):\n",
" for y in range(x + 1, K):\n",
" e = e.replace(d(mu[x], mu[y]), sum(\n",
" [vals[x][i] * vals[y][i] for i in range(4)]))\n",
" e2 = e2.replace(d(mu[x], mu[y]), sum(\n",
" [vals[x][i] * vals[y][i] for i in range(4)]))\n",
"\n",
"print(e)\n",
"print(e2)\n"
]
}
],
"metadata": {
"kernelspec": {
"display_name": ".venv",
"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": "3.13.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
@benruijl
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benruijl commented May 15, 2024 •
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Also available here: https://colab.research.google.com/drive/1Dh0I3UmJyu6CYzJQQfF2cs8K-xmoTE_B?usp=sharing

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