{
"cells": [
{
"cell_type": "markdown",
"id": "conditional-vermont",
"metadata": {},
"source": [
"#### `==` vs `is`"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "serial-chamber",
"metadata": {},
"outputs": [],
"source": [
"a = 1.0\n",
"b = 1.0\n",
"print(a == b)\n",
"print(a is b)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "nominated-completion",
"metadata": {},
"outputs": [],
"source": [
"a = [1, 2, 3]\n",
"b = [1, 2, 3]\n",
"print(a == b)\n",
"print(a is b)"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "vietnamese-needle",
"metadata": {},
"outputs": [],
"source": [
"a = [1, 2, 3]\n",
"b = a\n",
"print(a == b)\n",
"print(a is b)\n",
"b[0] = 42\n",
"print(a)"
]
},
{
"cell_type": "markdown",
"id": "random-dispute",
"metadata": {},
"source": [
"#### numpy copy vs view"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "higher-contamination",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "continental-panel",
"metadata": {},
"outputs": [],
"source": [
"a = np.arange(10)\n",
"b = a[5] #copy numeric type\n",
"b = 42\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "bronze-heather",
"metadata": {},
"outputs": [],
"source": [
"a = np.arange(10)\n",
"b = a #alias\n",
"b[5] = 42\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "rational-thursday",
"metadata": {},
"outputs": [],
"source": [
"a = np.arange(10)\n",
"b = a[5:10] #view\n",
"b[0] = 42\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "arranged-hello",
"metadata": {},
"outputs": [],
"source": [
"a = np.arange(10)\n",
"b = np.array(a[5:10]) #copy array view\n",
"b[0] = 42\n",
"a"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "finite-copyright",
"metadata": {},
"outputs": [],
"source": [
"a = np.arange(10)\n",
"b = a[[5, 6, 7, 8, 9]] #copy by advanced indexing\n",
"b[0] = 42\n",
"a"
]
},
{
"cell_type": "markdown",
"id": "endangered-throw",
"metadata": {},
"source": [
"### Give an equivalent python function for the following mathematical expressions\n",
"Vector and matrix variables are represented by numpy arrays.\n",
"Run the assert cells to check your solution."
]
},
{
"cell_type": "markdown",
"id": "dependent-crime",
"metadata": {},
"source": [
"$f(\\mathbf{x}, \\mathbf{y}) = \\sum \\limits_n x_n y_n$\n",
"$\\qquad \\mathbf{x}, \\mathbf{y} \\in \\mathbb{R}^N$\n",
"$\\qquad 0 \\leq n \\lt N$"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "forbidden-ireland",
"metadata": {},
"outputs": [],
"source": [
"# solution using a loop\n",
"def f(x, y):\n",
" s = 0\n",
" for n in range(x.shape[0]):\n",
" s += x[n] * y[n]\n",
" return s"
]
},
{
"cell_type": "markdown",
"id": "sixth-works",
"metadata": {},
"source": [
"Which is equivalent to the dot product:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "needed-haiti",
"metadata": {},
"outputs": [],
"source": [
"# solution using the numpy dot product (more efficient and concise)\n",
"def f(x, y):\n",
" return x @ y"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "referenced-siemens",
"metadata": {},
"outputs": [],
"source": [
"assert f(np.array([1, 2]), np.array([3, 4])) == 11"
]
},
{
"cell_type": "markdown",
"id": "rising-quantum",
"metadata": {},
"source": [
"$g(\\mathbf{X}, \\mathbf{y})_m = \\sum \\limits_n X_{m, n} y_n$\n",
"$\\qquad \\mathbf{X} \\in \\mathbb{R}^{MxN}$\n",
"$\\quad \\mathbf{y} \\in \\mathbb{R}^{N}$\n",
"$\\quad \\mathbf{g}(\\mathbf{X}, \\mathbf{y}) \\in \\mathbb{R}^{M}$\n",
"$\\qquad 0 \\leq m \\lt M \\quad 0 \\leq n \\lt N$"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "racial-retail",
"metadata": {},
"outputs": [],
"source": [
"# solution using for loops\n",
"def g(X, y):\n",
" s = np.zeros(X.shape[0])\n",
" for m in range(X.shape[0]):\n",
" for n in range(X.shape[1]):\n",
" s[m] += X[m, n] * y[n]\n",
" return s"
]
},
{
"cell_type": "markdown",
"id": "increasing-brief",
"metadata": {},
"source": [
"Which is again equivalent to the dot product:"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "reliable-confusion",
"metadata": {},
"outputs": [],
"source": [
"# solution using the numpy dot product (more efficient and concise)\n",
"def g(X, y):\n",
" return X @ y"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "impressive-policy",
"metadata": {},
"outputs": [],
"source": [
"X = np.array([[0, 2, 4],\n",
" [1, 3, 5]])\n",
"assert (g(X, np.array([1, 2, 3])) == np.array([16, 22])).all()"
]
},
{
"cell_type": "markdown",
"id": "thrown-dimension",
"metadata": {},
"source": [
"$\\Phi(\\mathbf x, N)_{m, n} = x_m^n$\n",
"$\\qquad \\mathbf x \\in \\mathbb R^M$\n",
"$\\quad \\mathbf\\Phi(\\mathbf x, N) \\in \\mathbb R^{MxN}$\n",
"$\\qquad 0 \\leq m \\lt M$\n",
"$\\qquad 0 \\leq n \\lt N$"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "qualified-polish",
"metadata": {},
"outputs": [],
"source": [
"# solution using for loops\n",
"def phi(x, N):\n",
" phi = np.zeros((x.shape[0], N))\n",
" for m in range(phi.shape[0]):\n",
" for n in range(phi.shape[1]):\n",
" phi[m, n] = x[m]**n\n",
" return phi"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "monthly-lingerie",
"metadata": {},
"outputs": [],
"source": [
"# solution using numpy broadcasting (more efficient and concise) \n",
"def phi(x, N):\n",
" return np.power(x[:, np.newaxis], np.arange(N))"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "religious-quarter",
"metadata": {},
"outputs": [],
"source": [
"assert (phi(np.array([1, 2, 3]), 3) == np.array([[1, 1, 1],\n",
" [1, 2, 4],\n",
" [1, 3, 9]])).all()"
]
}
],
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"display_name": "Python 3",
"language": "python",
"name": "python3"
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"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
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"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
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