{
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"# Introduction to programming for Geoscientists through Python\n",
"### [Gerard Gorman](http://www.imperial.ac.uk/people/g.gorman), [Nicolas Barral](http://www.imperial.ac.uk/people/n.barral)\n",
"\n",
"# Lecture 5: Array computing and curve plotting"
]
},
{
"cell_type": "markdown",
"metadata": {
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"slide_type": "slide"
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"source": [
"Learning objectives: \n",
"\n",
"* Learn how to compute using numpy arrays, *i.e.* vectorise code.\n",
"* Learn how to generate 2D graphs."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Vectors and arrays\n",
"\n",
"You have known **vectors** since high school mathematics, *e.g.*, point $(x,y)$ in the plane, point $(x,y,z)$ in space. In general, we can describe a vector $v$ as an $n$-tuple of numbers: $v=(v_0,\\ldots,v_{n-1})$. One way to store vectors in Python is by using *lists*: $v_i$ is stored as *v[i]*."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
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"source": [
"**Arrays** are a generalization of vectors where we can have multiple indices: $A_{i,j}$, $A_{i,j,k}$. In Python code this is represented as a nested list (see previous lecture), accessed as *A[i][j]*, *A[i][j][k]*.\n",
"\n",
"Example: table of numbers, one index for the row, one for the column\n",
"$$\n",
"\\left\\lbrack\\begin{array}{cccc}\n",
"0 & 12 & -1 & 5q\n",
"-1 & -1 & -1 & 0\\cr\n",
"11 & 5 & 5 & -2\n",
"\\end{array}\\right\\rbrack\n",
"\\hspace{1cm}\n",
"A =\n",
"\\left\\lbrack\\begin{array}{ccc}\n",
"A_{0,0} & \\cdots & A_{0,n-1}\\cr\n",
"\\vdots & \\ddots & \\vdots\\cr\n",
"A_{m-1,0} & \\cdots & A_{m-1,n-1}\n",
"\\end{array}\\right\\rbrack\n",
"$$\n",
"The number of indices in an array is the *rank* or *number of dimensions*. Using these terms, a vector can be described as a one-dimensional array, or rank 1 array."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
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},
"source": [
"In practice, we use [Numerical Python (*NumPy*)](http://www.numpy.org/) arrays instead of lists to represent mathematical arrays because it is **much** faster for large arrays.\n",
"\n",
"Let's consider an example where we store $(x,y)$ points along a curve in Python lists and numpy arrays:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true,
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"slide_type": "subslide"
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},
"outputs": [],
"source": [
"# Sample function\n",
"def f(x):\n",
" return x**3\n",
"\n",
"# Generate n points in [0,1]\n",
"n = 5\n",
"dx = 1.0/(n-1) # x spacing\n",
"\n",
"xlist = [i*dx for i in range(n)] # Python lists\n",
"ylist = [f(x) for x in xlist]\n",
"\n",
"# Turn these Python lists into Numerical Python (NumPy) arrays:\n",
"from numpy import *\n",
"x2 = array(xlist)\n",
"y2 = array(ylist)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Instead of first making lists with $x$ and $y = f (x)$ data, and then turning lists into arrays, we can make NumPy arrays\n",
"directly:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"n = 5 # number of points\n",
"x2 = linspace(0, 1, n) # generates n points between 0 and 1\n",
"y2 = zeros(n) # n zeros (float data type by default)\n",
"for i in range(n): \n",
" y2[i] = f(x2[i])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"List comprehensions create lists, not arrays, but we can do:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"y2 = array([f(xi) for xi in x2]) # list -> array"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### When and where to use NumPy arrays\n",
"\n",
"* Python lists can hold any sequence of any Python objects, however, NumPy arrays can only hold objects of the same type.\n",
"* Arrays are most efficient when the elements are of basic number types (*float*, *int*, *complex*).\n",
"* In that case, arrays are stored efficiently in the computer's memory and we can compute very efficiently with the array elements.\n",
"* Mathematical operations on whole arrays can be done without loops in Python. For example,"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": [
"x = linspace(0, 2, 10001)\n",
"y = zeros(10001)\n",
"for i in range(len(x)):\n",
" y[i] = sin(x[i])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"can be coded as"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"y = sin(x)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"In the latter case the loop over all elements is now performed in a very efficient C function.\n",
"\n",
"Operations on whole arrays, instead of using Python *for*-loops, is called vectorization and is a very **convenient**, **efficient** and therefore important programming technique to master."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Let's consider a simple vectorisation example: a loop to compute $x$ coordinates (*x2*) and $y=f(x)$ coordinates (*y2*) along a function curve:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"x2 = linspace(0, 1, n)\n",
"y2 = zeros(n)\n",
"for i in range(n):\n",
" y2[i] = f(x2[i])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"This computation can be replaced by:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"x2 = linspace(0, 1, n)\n",
"y2 = f(x2)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"The advantage of this approach is:\n",
"\n",
"* There is no need to allocate space for y2 (via the NumPy *zeros* function).\n",
"* There is no need for a loop.\n",
"* It is *much* faster."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## How vectorised functions work\n",
"Consider the function"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"def f(x):\n",
" return x**3"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"$f(x)$ is intended for a number $x$, *i.e.* a *scalar*. So what happens when we call *f(x2)*, where *x2* is an NumPy array? **The function simply evaluates $x^3$ for an array x**. NumPy supports arithmetic operations on arrays, which correspond to the equivalent operations on each element, *e.g.*:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"text/plain": [
"array([ 0.00000000e+00, 6.66600003e-05, 1.33306669e-04, ...,\n",
" 2.45252956e-01, 2.45252960e-01, 2.45252961e-01])"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x**3 # x[i]**3 forr all i\n",
"cos(x) # cos(x[i]) for all i\n",
"x**3 + x*cos(x) # x[i]**3 + x[i]*cos(x[i]) for all i\n",
"x/3*exp(-x*0.5) # x[i]/3*exp(-x[i]*0.5) for all i "
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"In each of these cases a highly optimised C function is actually called to evaluate the expression. In this example, the *cos* function called for an *array* is imported from *numpy* rather than from the *math* module which only acts on scalars.\n",
"\n",
"Notes:\n",
"\n",
"* Functions that can operate on arrays are called **vectorized functions**.\n",
"* Vectorization is the process of turning a non-vectorized expression/algorithm into a vectorized expression/algorithm.\n",
"* Mathematical functions in Python automatically work for both scalar and array (vector) arguments, *i.e.* no vectorization is needed by the programmer.\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"### Watch out for references Vs. copies of arrays!\n",
"Consider this code:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"42.0\n"
]
}
],
"source": [
"a=x\n",
"a[-1] = 42\n",
"print(x[-1])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Notice what happened here - we changed a value in *a* but the corresponding value in *x* was also changed! This is because *a* refers to the same array as *x*. If you really want a seperate copy of *x* then we have to make an explicit copy:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [],
"source": [
"a = x.copy()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.1: Fill lists and arrays with function values\n",
"A function with many applications in science is defined as:\n",
"$h(x) = \\frac{1}{\\sqrt{2\\pi}}\\exp(-0.5x^2)$\n",
"\n",
"* Fill two lists *xlist* and *hlist* with *x* and *h(x)* values for uniformly spaced *x* coordinates in [−4, 4]. You may adapt the first example in the lecture 4 notes.\n",
"\n",
"* Fill two arrays *x* and *y* with *x* and *h(x)* values, respectively, where *h(x)* is defined above. Let the *x* values be uniformly spaced in [−4, 4]. Use list comprehensions to create the *x* and *y* arrays.\n",
"\n",
"* Vectorize the code by creating the *x* values using the *linspace* function and by evaluating *h(x)* for an array argument."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.2: Apply a function to a vector\n",
"Given a vector $v = (2, 3, −1)$ and a function $f(x) = x^3 + xe^x + 1$, apply $f$ to each element in $v$. Then calculate $f(v)$ as $v^3 + ve^v + 1$ using vector computing rules. Show that the two results are equal."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.3: Simulate by hand a vectorized expression\n",
"Suppose *x* and *t* are two arrays of the same length, and given the vectorized expression,\n",
"\n",
"```python\n",
"y = cos(sin(x)) + exp(1/t)\n",
"```\n",
"\n",
"assuming *x* holds two elements, 0 and 2, and *t* holds the elements 1 and 1.5, calculate by hand (using a calculator) the *y* array.\n",
"\n",
"Write a program that mimics the series of computations you did by hand (use explicit loops, but at the end you can use NumPy functionality to check the results)."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Generalised array indexing\n",
"We can select a slice of an array using *a[start:stop:inc]*, where the slice *start:stop:inc* implies a set of indices starting from *start*, up to *stop* in increments of *inc*. In fact, any integer list or array can be used to indicate a set of indices:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1. 2. 3. 4. 5. 6. 7. 8.]\n"
]
}
],
"source": [
"a = linspace(1, 8, 8)\n",
"print(a)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1. 10. 3. 4. 5. 6. 10. 10.]\n"
]
}
],
"source": [
"a[[1,6,7]] = 10 # i.e. set the elements with indicies 1,6, and 7 in the list to 10.\n",
"print(a)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1. 10. -2. 4. 5. -2. 10. 10.]\n"
]
}
],
"source": [
"a[range(2,8,3)] = -2 # same as a[2:8:3] = -2\n",
"print(a)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Even boolean expressions can also be used to select part of an array(!)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[-2. -2.]\n"
]
}
],
"source": [
"print(a[a < 0]) # pick out all negative elements"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 1. 10. 10. 4. 5. 10. 10. 10.]\n"
]
}
],
"source": [
"a[a < 0] = a.max() # if a[i]<0, set a[i]=10\n",
"print(a)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.4: Demonstrate array slicing\n",
"Create an array *w* with values 0, 0.1, 0.2, ..., 3. Write out *w[:]*, *w[:-2]*, *w[::5]*, *w[2:-2:6]*. Convince yourself in each case that you understand which elements of the array are printed."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Plotting curves - the basics\n",
"First of all, a little house keeping. There are quite a few ways of plotting graphs etc. in Python. Currently the best way is using [PyLab](http://wiki.scipy.org/PyLab). If you are someone who likes the details then please go to the [PyLab](http://wiki.scipy.org/PyLab) website and read. Secondly, because we are doing this within IPython NoteBook, and we do not want additional windows popping up all over the place, we execute this next line:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Populating the interactive namespace from numpy and matplotlib\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/home/nbuser/anaconda3_431/lib/python3.6/site-packages/IPython/core/magics/pylab.py:160: UserWarning: pylab import has clobbered these variables: ['f']\n",
"`%matplotlib` prevents importing * from pylab and numpy\n",
" \"\\n`%matplotlib` prevents importing * from pylab and numpy\"\n"
]
}
],
"source": [
"%pylab inline"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Now, onwards and upwards...\n",
"\n",
"A curve $y = f(x)$ stored in the 1D NumPy arrays *x* and *y* can easily be plotted:"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"image/png": 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tKyJzRSRDRDKKiopc/Nbu6eNt+dTVGz1qR3UbEeGxy0cS6Gfn7nc246hv/gu3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\nSinlQ/4/iboA1k3RIWkAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"t = linspace(0, 3, 51)\n",
"y = t**2*exp(-t**2)\n",
"plot(t, y)\n",
"show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Plots also should have **labels** on the axis, a **title**, and sometimes a specific extent of the axis (perhaps you wish to easily compare two graphs side-by-side):"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def f(t):\n",
" return t**2*exp(-t**2)\n",
"\n",
"t = linspace(0, 3, 51) # Generates 51 points between 0 and 3\n",
"y = f(t)\n",
"plot(t, y)\n",
"\n",
"xlabel('t')\n",
"ylabel('y')\n",
"legend(('t^2*exp(-t^2)',))\n",
"axis([0, 3, -0.05, 0.6]) # specify the extent of the axes [tmin, tmax, ymin, ymax]\n",
"\n",
"title('My second PyLab graph')\n",
"show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.5: Plot a formula\n",
"* Make a plot of the function $y(t) = v_0t − 0.5gt^2$ for $v_0 = 10$, $g = 9.81$, and $t \\in [0, 2v_0/g]$. The label on the *x* axis should be 'time (s)' and the label on the *y* axis should be 'height (m)'.\n",
"* Extend the program such that the minimum and maximum *x* and *y* values are computed, and use the extreme values to specify the extent of the *x* and *y* axes. Add some space above the heighest curve."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.6: Plot another formula\n",
"The function\n",
"$f(x, t) = \\exp(-(x - 3t)^2)\\sin(3\\pi(x - t))$\n",
"\n",
"describes, for a fixed value of *t*, a wave localized in space. Make a program that visualizes this function as a function of *x* on the interval [−4, 4] when *t* = 0."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Multiple curves in one plot\n",
"We can also plot several curves in one plot:"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"data": {
"image/png": 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InW+xWzzDOzbgx/v6EOTrxc2z/+CrTYftVrbb6fMgFOXCn7OtjkRzUjopaJfk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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def f1(t):\n",
" return t**2*exp(-t**2)\n",
"\n",
"def f2(t):\n",
" return t**2*f1(t)\n",
"\n",
"t = linspace(0, 3, 51)\n",
"y1 = f1(t)\n",
"y2 = f2(t)\n",
"\n",
"# Matlab-style syntax:\n",
"plots = plot(t, y1, t, y2)\n",
"legend(plots, ('t^4*exp(-t^2)', 't^4*exp(-t^2)'), loc='best')\n",
"xlabel('t')\n",
"ylabel('y')\n",
"title('Plotting two curves in the same plot')\n",
"show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"When plotting multiple curves in the same plot, PyLab usually does a good job in making sure that the different lines actually look different. However, sometimes you need to take action yourself (*e.g.* if you need to print your graph out in black&white). To do this we can add an extra argument to the plot command where we specify what we want - *e.g.* \"r-\" means a *red line*, while \"bo\" means *blue circles*:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"data": {
"image/png": 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ZqtnZ9gv+u9+pvvGGLdodcCLxF52vvhbDcG3C4QLZmVlzV7V2xJdftrSoXbu6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\nrkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot(t, y1, 'r-', t, y2, 'bo')\n",
"show()"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"For further examples check out the [PyLab website](http://scipy.org/PyLab)."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.7: Plot a formula for several parameters\n",
" - Make a program in which you define a set of $v_0$ values and plot the corresponding curves $y(t) = v_0t − 0.5gt^2$ in the same figure (set $g = 9.81$). Let $t \\in [0, 2v_0/g$] for each curve, which implies that you need a different vector of $t$ coordinates for each curve.\n",
" - Make a similar program where the values of $v_0$ are given by the user through an iPython widget as a series of numbers separated by a comma."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## 2D arrays\n",
"When we have a table of numbers,\n",
"\n",
"$$\n",
"\\left\\lbrack\\begin{array}{cccc}\n",
"0 & 12 & -1 & 5\\cr\n",
"-1 & -1 & -1 & 0\\cr\n",
"11 & 5 & 5 & -2\n",
"\\end{array}\\right\\rbrack\n",
"$$\n",
"\n",
"(*i.e.* a *matrix*) it is natural to use a two-dimensional array $A_{i,j}$ with one index for the rows and one for the columns:\n",
"\n",
"$$\n",
"A = \n",
"\\left\\lbrack\\begin{array}{ccc}\n",
"A_{0,0} & \\cdots & A_{0,n-1}\\cr\n",
"\\vdots & \\ddots & \\vdots\\cr\n",
"A_{m-1,0} & \\cdots & A_{m-1,n-1}\n",
"\\end{array}\\right\\rbrack\n",
"$$\n",
"\n",
"Let's recreate this array using NumPy:"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 0. 12. -1. 5.]\n",
" [ -1. -1. -1. 0.]\n",
" [ 11. 5. 5. -2.]]\n"
]
}
],
"source": [
"A = zeros((3,4))\n",
"A[0,0] = 0\n",
"A[1,0] = -1\n",
"A[2,0] = 11\n",
"\n",
"A[0,1] = 12\n",
"A[1,1] = -1\n",
"A[2,1] = 5\n",
"\n",
"A[0,2] = -1\n",
"A[1,2] = -1\n",
"A[2,2] = 5\n",
"\n",
"# we can also use the same syntax that we used for nested lists\n",
"\n",
"A[0][3] = 5\n",
"A[1][3] = 0\n",
"A[2][3] = -2\n",
"\n",
"print(A)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Next let's convert a nested list from a previous example into a 2D array:"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[0, 32.0], [10, 50.0], [20, 68.0], [30, 86.0], [40, 104.0], [50, 122.0], [60, 140.0], [70, 158.0], [80, 176.0], [90, 194.0], [100, 212.0]]\n"
]
}
],
"source": [
"Cdegrees = range(0, 101, 10)\n",
"Fdegrees = [9./5*C + 32 for C in Cdegrees]\n",
"table = [[C, F] for C, F in zip(Cdegrees, Fdegrees)]\n",
"print(table)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 0. 32.]\n",
" [ 10. 50.]\n",
" [ 20. 68.]\n",
" [ 30. 86.]\n",
" [ 40. 104.]\n",
" [ 50. 122.]\n",
" [ 60. 140.]\n",
" [ 70. 158.]\n",
" [ 80. 176.]\n",
" [ 90. 194.]\n",
" [ 100. 212.]]\n"
]
}
],
"source": [
"# Convert this into a NumPy array:\n",
"table2 = array(table)\n",
"print(table2)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"To see the number of elements in each dimension:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(11, 2)\n"
]
}
],
"source": [
"print(table2.shape)"
]
},
{
"cell_type": "markdown",
"metadata": {
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"source": [
"*i.e.* 11 rows and 2 columns."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Let's write a loop over all array elements of A:"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"table2[0,0] = 0\n",
"table2[0,1] = 32\n",
"table2[1,0] = 10\n",
"table2[1,1] = 50\n",
"table2[2,0] = 20\n",
"table2[2,1] = 68\n",
"table2[3,0] = 30\n",
"table2[3,1] = 86\n",
"table2[4,0] = 40\n",
"table2[4,1] = 104\n",
"table2[5,0] = 50\n",
"table2[5,1] = 122\n",
"table2[6,0] = 60\n",
"table2[6,1] = 140\n",
"table2[7,0] = 70\n",
"table2[7,1] = 158\n",
"table2[8,0] = 80\n",
"table2[8,1] = 176\n",
"table2[9,0] = 90\n",
"table2[9,1] = 194\n",
"table2[10,0] = 100\n",
"table2[10,1] = 212\n"
]
}
],
"source": [
"for i in range(table2.shape[0]):\n",
" for j in range(table2.shape[1]):\n",
" print('table2[%d,%d] = %g' % (i, j, table2[i,j]))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Alternatively:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"index (0, 0) has value 0\n",
"index (0, 1) has value 32\n",
"index (1, 0) has value 10\n",
"index (1, 1) has value 50\n",
"index (2, 0) has value 20\n",
"index (2, 1) has value 68\n",
"index (3, 0) has value 30\n",
"index (3, 1) has value 86\n",
"index (4, 0) has value 40\n",
"index (4, 1) has value 104\n",
"index (5, 0) has value 50\n",
"index (5, 1) has value 122\n",
"index (6, 0) has value 60\n",
"index (6, 1) has value 140\n",
"index (7, 0) has value 70\n",
"index (7, 1) has value 158\n",
"index (8, 0) has value 80\n",
"index (8, 1) has value 176\n",
"index (9, 0) has value 90\n",
"index (9, 1) has value 194\n",
"index (10, 0) has value 100\n",
"index (10, 1) has value 212\n"
]
}
],
"source": [
"for index_tuple, value in ndenumerate(table2):\n",
" print('index %s has value %g' % (index_tuple, table2[index_tuple]))"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
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},
"source": [
"We can also extract slices from multi-dimensional arrays as before. For example, extract the second column:"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[ 32. 50. 68. 86. 104. 122. 140. 158. 176. 194. 212.]\n"
]
}
],
"source": [
"print(table2[:, 1]) # 2nd column (index 1)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "subslide"
}
},
"source": [
"Play with this more complicated example:"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 1. 2. 3. 4. 5. 6.]\n",
" [ 7. 8. 9. 10. 11. 12.]\n",
" [ 13. 14. 15. 16. 17. 18.]\n",
" [ 19. 20. 21. 22. 23. 24.]\n",
" [ 25. 26. 27. 28. 29. 30.]]\n"
]
}
],
"source": [
"t = linspace(1, 30, 30).reshape(5, 6)\n",
"print(t)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[ 9. 10. 11. 12.]\n",
" [ 21. 22. 23. 24.]]\n"
]
}
],
"source": [
"print(t[1:-1:2, 2:])"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.8: Implement matrix-vector multiplication\n",
"A matrix $\\mathbf{A}$ and a vector $\\mathbf{b}$, represented in Python as a 2D array and a 1D array respectively, are given by:\n",
"\n",
"$$\n",
"\\mathbf{A} = \\left\\lbrack\\begin{array}{ccc}\n",
"0 & 12 & -1\\cr\n",
"-1 & -1 & -1\\cr\n",
"11 & 5 & 5\n",
"\\end{array}\\right\\rbrack\n",
"$$\n",
"\n",
"$$\n",
"\\mathbf{b} = \\left\\lbrack\\begin{array}{c}\n",
"-2\\cr\n",
"1\\cr\n",
"7\n",
"\\end{array}\\right\\rbrack\n",
"$$\n",
"\n",
"Multiplying a matrix by a vector results in another vector $\\mathbf{c}$, whose components are defined by the general rule\n",
"\n",
"$$\\mathbf{c}_i = \\sum_j\\mathbf{A}_{i,j}\\mathbf{b}_j$$\n",
"\n",
"Define $\\mathbf{A}$ and $\\mathbf{b}$ as NumPy arrays, and multiply them together using the above rule."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true,
"slideshow": {
"slide_type": "subslide"
}
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"## Exercise 5.9: Plot meteorological data\n",
"The file data/precipitation.csv contains monthly precipitation data for South-East England between 1916 and 2016 provided by the [Met Office](https://www.metoffice.gov.uk/hadobs/hadukp/data/download.html). Each row represents a year, and each column a month.\n",
"To open the data file, use the command below:\n",
"\n",
"```python\n",
"import numpy as np\n",
"data = np.loadtxt(fname='data/precipitation.csv', delimiter=',')\n",
"```\n",
"\n",
" - Open the data file, and print its content. Observe that it is a multi-dimensional Numpy array. Print its shape. How many years are represented ?\n",
" \n",
" - Write a function that converts a length in mm into a length in inches. Use this function to convert the precipitation data into inches in a vectorized way.\n",
" \n",
" - With a `for` loop, print (in a formatted way) for each month, the year where the monthly precipitation was the highest. A line like:
\n",
"`The maximal precipitation for January is 181.9 mm/month and was reached in 2014.`
\n",
"is expected.
\n",
"Hint: you should use the [max](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.amax.html) and [argmax](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.argmax.html) functions provided by Numpy.\n",
"\n",
" - Plot the [average](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.mean.html), [maximal](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.amax.html) and [minimal](https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.amin.html) monthly precipitation with respect to the year. Don't forget to decorate the plot properly (add a plot title, legends and axis labels).\n",
" \n",
" - Plot the same quantities, but only for the last twenty years.\n",
" \n",
" - Plot the same quantities, but only for years ending in 6."
]
},
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