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‎1.Integration.ipynb

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‎2.Differentiation.ipynb ‎2a.DifferentialEQ-Ordinary.ipynb

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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Numerical Methods: "
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"1. Integration\n",
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"4. Differential Equations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Integration [source](https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"from scipy.integrate import quad, dblquad, tplquad"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"$ y = mx +c$"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"integral value = 6.499999999999999 , absolute error = 7.216449660063516e-14\n"
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]
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}
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],
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"source": [
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"def f(x):\n",
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" return 5*x + 4\n",
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"\n",
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"x_lower = 0 # the lower limit of x\n",
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"x_upper = 1 # the upper limit of x\n",
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"## Differentiation\n",
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"\n",
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"val, abserr = quad(f, x_lower, x_upper)\n",
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"print(\"integral value =\", val, \", absolute error =\", abserr)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"* Bessel function"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"from scipy.special import jn, yn, jn_zeros, yn_zeros"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.7366751370811073 9.389126882496403e-13\n"
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]
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}
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],
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"source": [
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"def integrand(x, n):\n",
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"\n",
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" \"\"\"\n",
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" Bessel function of first kind and order n.\n",
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" \"\"\"\n",
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" return jn(n, x)\n",
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"\n",
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"x_lower = 0 # the lower limit of x\n",
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"x_upper = 10 # the upper limit of x\n",
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"val, abserr = quad(integrand, x_lower, x_upper, args=(3,))\n",
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"\n",
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"print(val, abserr)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"* Gaussian function"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"numerical = 1.7724538509027912 4.6261378229003154e-14\n",
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"analytical = 1.7724538509055159\n"
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]
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}
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],
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"source": [
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"val, abserr = quad(lambda x: np.exp(-x ** 2), -5.0, 5.0)\n",
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"print(\"numerical =\", val, abserr)\n",
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"analytical = np.sqrt(np.pi)\n",
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"print(\"analytical =\", analytical)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.7853981633974476 1.3753098510218528e-08\n"
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]
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}
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],
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"source": [
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"def integrand(x, y):\n",
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" return np.exp(-x**2-y**2)\n",
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"\n",
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"x_lower = 0\n",
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"x_upper = 10\n",
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"y_lower = 0\n",
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"y_upper = 10\n",
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"\n",
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"val, abserr = dblquad(integrand, x_lower, x_upper,\\\n",
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" lambda x : y_lower, lambda x: y_upper)\n",
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"print(val, abserr)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Ordinary Differential Equations [Source](https://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html)"
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]
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},
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"$\\large{Ax = b}$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"-------"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.1"
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"version": "3.7.3"
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}
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},
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"nbformat": 4,

‎2b.DifferentialEQ-Partial.ipynb

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‎3.Interpolation.ipynb

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{
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"cells": [],
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"metadata": {},
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

‎4.Optimization.ipynb

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