From 4239e488f6fae9869782ffbac9f39747b58afdda Mon Sep 17 00:00:00 2001 From: Brett Morris Date: Wed, 6 Jun 2018 15:56:55 -0700 Subject: [PATCH] adding sympy --- radspots/sympy.ipynb | 171 +++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 171 insertions(+) create mode 100644 radspots/sympy.ipynb diff --git a/radspots/sympy.ipynb b/radspots/sympy.ipynb new file mode 100644 index 0000000..70ad509 --- /dev/null +++ b/radspots/sympy.ipynb @@ -0,0 +1,171 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "IPython console for SymPy 0.7.6.1 (Python 3.5.4-64-bit) (ground types: python)\n" + ] + } + ], + "source": [ + "import sympy\n", + "\n", + "from sympy import symbols, init_session, sqrt\n", + "init_session(quiet=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "tau0, p0, b, P, a, Gamma, delta, sigma, sigma_tau_0 = symbols('tau0 p0 b P a Gamma delta sigma, sigma_tau_0', \n", + " positive=True, real=True) " + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [], + "source": [ + "T = 2 * tau0 * sqrt(1 - b**2)\n", + "tau = 2 * tau0 * p0 / sqrt(1 - b**2)\n", + "theta = tau/T\n", + "Q = sqrt(Gamma * T) * delta/sigma\n", + "sigma_tau = T * sqrt(6 * theta) / Q\n", + "sigma_T = T * sqrt(2 * theta) / Q" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ p_0 = \\frac{\\tau T}{4 \\tau_0^2}$$ " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ \\sigma_{p_0}^2 = \\left(\\frac{\\partial p_0}{\\partial \\tau}\\right)^2 \\sigma_\\tau^2 + \\left(\\frac{\\partial p_0}{\\partial \\tau}\\frac{\\partial p_0}{\\partial T}\\right) \\mathrm{cov}(\\tau,T)+\\left(\\frac{\\partial p_0}{\\partial T}\\right)^2 \\sigma_T^2 +\\left(\\frac{\\partial p_0}{\\partial \\tau_0}\\right)^2 \\sigma_{\\tau_0}^2 $$\n", + "\n", + "\n", + "$$ \\sigma_{p_0}^2 = \\left(\\frac{T \\sigma_\\tau}{4 \\tau_0^2} \\right)^2 + \n", + " \\left(\\frac{\\tau T }{16 \\tau_0^4} b \\theta^2 T^2\\right) + \n", + " \\left(\\frac{\\tau \\sigma_T}{4 \\tau_0^2} \\right)^2 + \n", + " \\left(\\frac{\\tau T }{2 \\tau_0^3} \\sigma_{\\tau_0}\\right)^2 $$\n" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [], + "source": [ + "sigma_p0_sq = ((T * sigma_tau / 4 / tau0**2)**2 + \n", + " (tau * T / 16 / tau0**4) * b * theta**2 * T**2 + \n", + " (tau * sigma_T / 4 / tau0**2)**2 + \n", + " (tau * T / 2 / tau0**3 * sigma_tau_0)**2)" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/latex": [ + "$$\\frac{\\sqrt{p_{0}}}{\\sqrt{\\Gamma} \\delta \\tau_{0}} \\sqrt{- \\frac{1}{\\left(- b^{2} + 1\\right)^{\\frac{5}{2}}} \\left(- \\Gamma b \\delta^{2} p_{0}^{2} \\tau_{0}^{2} \\left(- b^{2} + 1\\right)^{\\frac{3}{2}} - 4 \\Gamma \\delta^{2} p_{0} \\sigma_{\\tau 0}^{2} \\left(- b^{2} + 1\\right)^{\\frac{5}{2}} + p_{0}^{2} \\sigma^{2} \\tau_{0} \\left(b^{2} - 1\\right) - 3 \\sigma^{2} \\tau_{0} \\left(- b^{2} + 1\\right)^{3}\\right)}$$" + ], + "text/plain": [ + " _______________________________________________________________\n", + " ╱ ⎛ 3/2 \n", + " ╱ ⎜ 2 2 2 ⎛ 2 ⎞ 2 2 ⎛ 2 ⎞\n", + " ____ ╱ -⎝- Γ⋅b⋅δ ⋅p₀ ⋅τ₀ ⋅⎝- b + 1⎠ - 4⋅Γ⋅δ ⋅p₀⋅σ_τ_0 ⋅⎝- b + 1⎠\n", + "╲╱ p₀ ⋅ ╱ ──────────────────────────────────────────────────────────────\n", + " ╱ 5/2 \n", + " ╱ ⎛ 2 ⎞ \n", + " ╲╱ ⎝- b + 1⎠ \n", + "──────────────────────────────────────────────────────────────────────────────\n", + " ___ \n", + " ╲╱ Γ ⋅δ⋅τ₀ \n", + "\n", + "_________________________________________________\n", + "5/2 3⎞ \n", + " 2 2 ⎛ 2 ⎞ 2 ⎛ 2 ⎞ ⎟ \n", + " + p₀ ⋅σ ⋅τ₀⋅⎝b - 1⎠ - 3⋅σ ⋅τ₀⋅⎝- b + 1⎠ ⎠ \n", + "──────────────────────────────────────────────── \n", + " \n", + " \n", + " \n", + "─────────────────────────────────────────────────\n", + " \n", + " " + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "sqrt(sigma_p0_sq).simplify()" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "sqrt(p0)*sqrt(-(-Gamma*b*delta**2*p0**2*tau0**2*(-b**2 + 1)**(3/2) - 4*Gamma*delta**2*p0*sigma_tau_0**2*(-b**2 + 1)**(5/2) + p0**2*sigma**2*tau0*(b**2 - 1) - 3*sigma**2*tau0*(-b**2 + 1)**3)/(-b**2 + 1)**(5/2))/(sqrt(Gamma)*delta*tau0)\n" + ] + } + ], + "source": [ + "print(sqrt(sigma_p0_sq).simplify())" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "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": "3.5.4" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}