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"# Python Arrays "
]
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"source": [
"### Import Libraries"
]
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"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline"
]
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"source": [
"## Array basics were covered earlier\n",
"- Review those if needed\n",
"- [See this page](https://www.w3schools.com/python/python_lists.asp) for more information on built-in Python Lists.\n",
"\n",
"## Numpy arrays\n",
"\n",
"- Python lists are nice, but for numerical work, numpy arrays are much better.\n",
"- 90% of the work we will do in science, math, and engineering, is best done using numpy arrays.\n",
"- All of the above regarding Python lists is the same, **except**\n",
" - Each element type needs to be the same.\n",
" - The way we create arrays is different.\n",
"\n",
"### Some good links:\n",
"- [Numpy for Matlab users](https://docs.scipy.org/doc/numpy-1.15.0/user/numpy-for-matlab-users.html)\n",
" - [This too](http://mathesaurus.sourceforge.net/matlab-numpy.html)\n",
"- [Tentative numpy tutorial](https://scipy.github.io/old-wiki/pages/Tentative_NumPy_Tutorial)\n",
"- [Numpy quickstart tutorial](https://numpy.org/doc/stable/user/quickstart.html)\n",
"- [Linear algebra](http://docs.scipy.org/doc/numpy/reference/routines.linalg.html)\n",
"- [Cookbook with good recipes](https://scipy-cookbook.readthedocs.io)"
]
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"source": [
"### Length of a list\n",
"`len` function"
]
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"source": [
"a = np.array([1.1, 2.2, 3.3, 4.4])"
]
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"source": [
"## Creating arrays\n",
"\n",
"- `array` (we've already seen this)\n",
"- `linspace`, `arange`\n",
"- `empty`, `zeros`, `ones`, `full`\n"
]
},
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"outputs": [],
"source": [
"# linspace"
]
},
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"outputs": [],
"source": [
"# arange\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
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},
"outputs": [],
"source": [
"# empty"
]
},
{
"cell_type": "code",
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"outputs": [],
"source": [
"# zeros"
]
},
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"outputs": [],
"source": [
"# ones"
]
},
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"cell_type": "code",
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"metadata": {},
"outputs": [],
"source": [
"# full"
]
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"source": [
"### Access elements\n",
"- use square brackets with index number\n",
"- array indexing **starts at zero**\n",
"- for 2D arrays (matrix) `x[i,j]` is row `i`, column `j`\n",
"- **can access backwards**0\n",
" - `x[-1]` is the last element\n",
" - `x[-2]` is the second to the last element, etc.\n",
" - See the figure below. If we wrap the array into a circle. Then element 0 is at the top. We index by +1 in moving clockwise, and index by -1 in moving counter-clockwise.\n",
" \n",
"
"
]
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"source": [
"x = np.arange(3,21,3) # x is [3 6 9 12 15 18]\n",
"print(x)\n",
"\n",
"# Access elements with index values\n",
"\n",
"# or use an \"index array\"\n"
]
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"### Array slices\n",
"- Can access multiple elements using syntax: `a[s:e]`\n",
" - `s` is the start (optional)\n",
" - `e` is the end (optional)\n",
" - (stops short of the end)"
]
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"x = np.arange(0,11,1.5) \n",
"\n",
" # get all elements\n",
" # get indices 2 to the end\n",
" # get elements start to one before 2nd to last (stops short)\n",
" # get indices 2 to 4 (stops short of 5)"
]
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"source": [
"### Example\n",
"\n",
"We have an array of times $t$, and we want the array of differences $dt$ between times.\n",
"- How big is $t$?\n",
"- How big will $dt$ be?\n",
"- Create $dt$ using a loop.\n",
"- Create $dt$ using slice-indexing.\n",
" - Note, for arrays `a` and `b`, `a-b` is done element-wise.\n",
"\n",
"
"
]
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"source": [
"t = np.array([ \n",
" 0.01136646, 0.01655847, 0.02548247, 0.07698612, 0.17964401,\n",
" 0.19813924, 0.22419994, 0.28622096, 0.28884997, 0.3627257 ,\n",
" 0.42555072, 0.44965767, 0.47257237, 0.59335946, 0.73842352,\n",
" 0.85900369, 0.88851226, 0.89153229, 0.89355469, 0.97705246,\n",
" ])\n",
"\n",
"# loop\n",
"\n",
"# slice\n"
]
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"source": [
"### Array properties\n",
"\n",
"- `size` and `shape`"
]
},
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"source": [
"x = np.array([[1,2,3,4],[7,8,9,10]])\n",
"print(x)\n",
"\n",
"np.shape(x) # rows x columns"
]
},
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"source": [
"np.size(x) # total number of elements"
]
},
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"source": [
"len(x) # length of first dimension (rows)"
]
},
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"source": [
"x = np.array([1,2,3,4])\n",
"len(x)"
]
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"source": [
"### Arithmetic"
]
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"outputs": [],
"source": [
"a = np.arange(9)\n",
"a"
]
},
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"source": [
"#------- THIS\n",
"\n",
"b = 0.1 * a # multiplies each element of a by 0.1 and assigns whole resulting array to b\n",
"\n",
"#------- NOT THIS\n",
"\n",
"b = np.empty(len(a))\n",
"for i in range(len(b)):\n",
" b[i] = 0.1*a[i]\n",
" \n",
"#-------\n",
"\n",
"print(b)"
]
},
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"source": [
"a+b # adds element-wise"
]
},
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"cell_type": "code",
"execution_count": null,
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"source": [
"a**2 # squares element-wise"
]
},
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"source": [
"### BE CAREFUL about assignment"
]
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"source": [
"x = np.array([1,2,3,4]) \n",
"y = x # note, y is now just a reference to x.\n",
"y[3] = 44 # if we change an element of y, it will change in x too!\n",
"print(x) # this is not usually what we want\n",
"print(y)"
]
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"source": [
"x = np.array([1,2,3,4])\n",
"y = x.copy() # now x and y are two different objects\n",
"y[3] = 44 # could also do y = np.array(x)\n",
"print(x)\n",
"print(y)\n"
]
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"source": [
"## Basic Plots"
]
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"source": [
"x = np.linspace(0,20,1000)\n",
"y = np.exp(-x/5)*np.sin(x)\n",
"\n",
"plt.plot(x,y);"
]
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"source": [
"### Problem 1\n",
"\n",
"* Make an array of $x$ from 0 to 5 with 1000 points.\n",
"* create the following arrays\n",
" * $2x$\n",
" * $x^2 -3x + 4$\n",
" * $\\sin(5x)$\n",
"* Plot each array versus $x$ on a single plot."
]
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"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n"
]
},
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"source": [
"### Problem 2\n",
"\n",
"* Ideal gas law\n",
"$$\\rho = \\frac{MP}{RT}$$\n",
"* M=29 kg/kmol\n",
"* R=8314 J/kmol*K\n",
"* P=101325 Pa\n",
"* **Find $\\rho$ for $T=300,400,\\ldots1000$ K**\n",
" * Store $\\rho$ and $T$ as arrays\n",
" * Plot $\\rho$ versus $T$"
]
},
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"source": []
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"source": [
"### Problem 3\n",
"\n",
"* Redo Problem 2 by writing a function for $\\rho(T)$ as you would have in a previous lecture.\n",
" * Call the function with an array of $T$ and store the result as an array of $\\rho$.\n",
" * Plot $\\rho$ versus $T$."
]
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"source": [
"### Problem 4\n",
"\n",
"- You have an array of $x$ points $x=0.0, 0.1, \\ldots, 50$.\n",
"- Create $x$\n",
"- Create $y=\\sin(x)$\n",
"- At interior points, use a central difference approximation to find the derivative \n",
"$$\\left(\\frac{dy}{dx}\\right)_i \\approx \\frac{y_{i+1}-y_{i-1}}{2\\Delta x},$$\n",
"where $\\Delta x = x_1-x_0$.\n",
"- At edge point use a one-sided difference approximation,\n",
"$$\\left(\\frac{dy}{dx}\\right)_0 \\approx \\frac{y_{1}-y_{0}}{\\Delta x},$$\n",
"$$\\left(\\frac{dy}{dx}\\right)_{n-1} \\approx \\frac{y_{n-1}-y_{n-2}}{\\Delta x}.$$\n",
"- Plot $dy/dx$ versus $x$ , and $\\cos(x)$ versus $x$."
]
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"source": [
"# y: * (* * * * * *) * # boundary, (interior), boundary\n",
"# 0 1 2 3 4 5 6 7\n",
"# n-2 n-1\n"
]
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"source": [
"### Problem 5\n",
"- Solve the rate equation below to $t=5$ s using the Explicit Euler method $y_{i+1} = y_i + \\Delta tf(y_i,t_i).$\n",
"\n",
"$$\\frac{dy}{dt} = -ky,$$\n",
"$$y_0=1,$$\n",
"$$k=2.$$\n",
"\n",
"- Use $\\Delta t = 0.5$.\n",
"- Store $t$ and $y$ in arrays.\n",
"- Plot $y$ versus $t$, and $y_0e^{-kt}$ versus $t$ on the same plot."
]
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"source": [
"### Problem 6\n",
"- The Haaland equation relates the friction factor $f$ in turbulent pipe flow to the Reynolds number $Re$.\n",
"$$\\frac{1}{\\sqrt{f}} = -1.8\\log_{10}\\left[\\left(\\frac{\\epsilon/D}{3.7}\\right)^{1.11}+ \\frac{6.9}{Re}\\right].$$\n",
"\n",
"- Write a function to compute $f(Re, \\epsilon/D)$.\n",
"- Create an array of 100 points that is uniformly spaced on a log scale, for $1000\\le Re\\le 1\\times 10^8$\n",
"- Plot $f$ versus $\\log_{10}(Re)$. \n",
"- Use $\\epsilon/D=0.001$."
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