Rick Muller, Sandia National Laboratories

version 0.62, Updated Dec 15, 2016 by Ryan Smith, Cal State East Bay

Using Python 3.5.2 | Anaconda 4.1.1

This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

**If skipping to other sections, it's a good idea to run this first:**

```
In [1]:
```import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline

- 0. Preliminary
- 1. Python Overview
- 1.1 Using Python as a Calculator
- 1.2 Strings
- 1.3 Lists
- 1.4 Iteration, Indentation, and Blocks
- 1.5 Slicing
- 1.6 Booleans and Truth Testing
- 1.7 Code Example: The Fibonacci Sequence
- 1.8 Functions
- 1.9 Recursion and Factorials
- 1.10 Two More Data Structures: Tuples and Dictionaries
- 1.11 Plotting with Matplotlib
- 1.12 Conclusion of the Python Overview

- 2. Numpy and Scipy
- 2.1 Making vectors and matrices
- 2.2 Linspace, matrix functions, and plotting
- 2.3 Matrix operations
- 2.4 Matrix Solvers
- 2.5 Example: Finite Differences
- 2.6 One-Dimensional Harmonic Oscillator using Finite Difference
- 2.7 Special Functions
- 2.8 Least squares fitting
- 2.9 Monte Carlo, random numbers, and computing $\pi$
- 2.10 Numerical Integration
- 2.11 Fast Fourier Transform and Signal Processing

- 3. Intermediate Python
- 4. Speeding Python: Timeit, Profiling, Cython, SWIG, and PyPy
- 5. References
- 6. Acknowledgements

Python is the programming language of choice for many scientists to a large degree because it offers a great deal of power to analyze and model scientific data with relatively little overhead in terms of learning, installation or development time. It is a language you can pick up in a weekend, and use for the rest of your life.

The Python Tutorial is a great place to start getting a feel for the language. To complement this material, I taught a Python Short Course years ago to a group of computational chemists during a time that I was worried the field was moving too much in the direction of using canned software rather than developing one's own methods. I wanted to focus on what working scientists needed to be more productive: parsing output of other programs, building simple models, experimenting with object oriented programming, extending the language with C, and simple GUIs.

I'm trying to do something very similar here, to cut to the chase and focus on what scientists need. In the last year or so, the IPython Project has put together a notebook interface that I have found incredibly valuable. A large number of people have released very good IPython Notebooks that I have taken a huge amount of pleasure reading through. Some ones that I particularly like include:

- Rob Johansson's excellent notebooks, including Scientific Computing with Python and Computational Quantum Physics with QuTiP lectures;
- XKCD style graphs in matplotlib;
- A collection of Notebooks for using IPython effectively
- A gallery of interesting IPython Notebooks

I find IPython notebooks an easy way both to get important work done in my everyday job, as well as to communicate what I've done, how I've done it, and why it matters to my coworkers. I find myself endlessly sweeping the IPython subreddit hoping someone will post a new notebook. In the interest of putting more notebooks out into the wild for other people to use and enjoy, I thought I would try to recreate some of what I was trying to get across in the original Python Short Course, updated by 15 years of Python, Numpy, Scipy, Matplotlib, and IPython development, as well as my own experience in using Python almost every day of this time.

IPython notebooks are now called Jupyter notebooks.

There are two branches of current releases in Python: the older-syntax Python 2, and the newer-syntax Python 3. This schizophrenia is largely intentional: when it became clear that some non-backwards-compatible changes to the language were necessary, the Python dev-team decided to go through a five-year (or so) transition, during which the new language features would be introduced and the old language was still actively maintained, to make such a transition as easy as possible. We're now (2016) past the halfway point, and people are moving to python 3.

These notes are written with Python 3 in mind.

If you are new to python, try installing Anaconda Python 3.5 (supported by Continuum) and you will automatically have all libraries installed with your distribution. These notes assume you have a Python distribution that includes:

- Python version 3;
- Numpy, the core numerical extensions for linear algebra and multidimensional arrays;
- Scipy, additional libraries for scientific programming;
- Matplotlib, excellent plotting and graphing libraries;
- IPython, with the additional libraries required for the notebook interface.

Here are some other options for various ways to run python:

- Continuum supports a bundled, multiplatform Python package called Anaconda
- Entought Python Distribution, also known as EPD. You can either purchase a license to use EPD, or there is also a free version that you can download and install.
**Linux**Most distributions have an installation manager. Redhat has yum, Ubuntu has apt-get. To my knowledge, all of these packages should be available through those installers.**Mac**I use Macports, which has up-to-date versions of all of these packages.**Windows**The PythonXY package has everything you need: install the package, then go to Start > PythonXY > Command Prompts > IPython notebook server.**Cloud**This notebook is currently running on the IPython notebook viewer, which allows the notebook to be viewed but not interactively.

This is a quick introduction to Python. There are lots of other places to learn the language more thoroughly. I have collected a list of useful links, including ones to other learning resources, at the end of this notebook. If you want a little more depth, Python Tutorial is a great place to start, as is Zed Shaw's Learn Python the Hard Way.

The lessons that follow make use of the IPython notebooks. There's a good introduction to notebooks in the IPython notebook documentation that even has a nice video on how to use the notebooks. You should probably also flip through the IPython tutorial in your copious free time.

Briefly, notebooks have code cells (that are generally followed by result cells) and text cells. The text cells are the stuff that you're reading now. The code cells start with "In []:" with some number generally in the brackets. If you put your cursor in the code cell and hit Shift-Enter, the code will run in the Python interpreter and the result will print out in the output cell. You can then change things around and see whether you understand what's going on. If you need to know more, see the IPython notebook documentation or the IPython tutorial.

Many of the things I used to use a calculator for, I now use Python for:

```
In [2]:
```2+2

```
Out[2]:
```

```
In [3]:
```(50-5*6)/4

```
Out[3]:
```

There are some gotchas compared to using a normal calculator.

```
In [4]:
```7/33

```
Out[4]:
```

There used to be gotchas in division in python 2, like C or Fortran integer division, where division truncates the remainder and returns an integer. In version 3, Python returns a floating point number. If for some reason you are using Python 2, you can fix this by importing the module from the future features:

`from __future__ import division`

In the last few lines, we have sped by a lot of things that we should stop for a moment and explore a little more fully. We've seen, however briefly, two different data types: **integers**, also known as *whole numbers* to the non-programming world, and **floating point numbers**, also known (incorrectly) as *decimal numbers* to the rest of the world.

We've also seen the first instance of an **import** statement. Python has a huge number of libraries included with the distribution. To keep things simple, most of these variables and functions are not accessible from a normal Python interactive session. Instead, you have to import the name. For example, there is a **math** module containing many useful functions. To access, say, the square root function, you can either first import the sqrt function from the math library:

```
In [5]:
```from math import sqrt
sqrt(81)

```
Out[5]:
```

or you can simply import the math library itself

```
In [6]:
```import math
math.sqrt(81)

```
Out[6]:
```

You can define variables using the equals (=) sign:

```
In [7]:
```width = 20
length = 30
area = length*width
area

```
Out[7]:
```

If you try to access a variable that you haven't yet defined, you get an error:

```
In [9]:
``````
volume
```

```
```

and you need to define it:

```
In [10]:
```depth = 10
volume = area*depth
volume

```
Out[10]:
```

You can name a variable *almost* anything you want. It needs to start with an alphabetical character or "_", can contain alphanumeric charcters plus underscores ("_"). Certain words, however, are reserved for the language:

```
and, as, assert, break, class, continue, def, del, elif, else, except,
exec, finally, for, from, global, if, import, in, is, lambda, not, or,
pass, print, raise, return, try, while, with, yield
```

Trying to define a variable using one of these will result in a syntax error:

```
In [11]:
```return = 0

```
```

```
In [14]:
``````
'Hello, World!'
```

```
Out[14]:
```

or double quotes

```
In [15]:
``````
"Hello, World!"
```

```
Out[15]:
```

But not both at the same time, unless you want one of the symbols to be part of the string.

```
In [16]:
``````
"He's a Rebel"
```

```
Out[16]:
```

```
In [17]:
``````
'She asked, "How are you today?"'
```

```
Out[17]:
```

```
In [18]:
```greeting = "Hello, World!"

The **print** statement is often used for printing character strings:

```
In [19]:
```print(greeting)

```
```

But it can also print data types other than strings:

```
In [20]:
```print("The area is ",area)

```
```

You can use the + operator to concatenate strings together:

```
In [21]:
```statement = "Hello," + "World!"
print(statement)

```
```

Don't forget the space between the strings, if you want one there.

```
In [22]:
```statement = "Hello, " + "World!"
print(statement)

```
```

You can use + to concatenate multiple strings in a single statement:

```
In [23]:
```print( "This " + "is " + "a " + "longer " + "statement.")

```
```

```
In [24]:
```days_of_the_week = ["Sunday","Monday","Tuesday","Wednesday","Thursday","Friday","Saturday"]

You can access members of the list using the **index** of that item:

```
In [25]:
```days_of_the_week[2]

```
Out[25]:
```

*n*th element from the end of the list, you can use a negative index. For example, the -1 element of a list is the last element:

```
In [26]:
```days_of_the_week[-1]

```
Out[26]:
```

You can add additional items to the list using the .append() command:

```
In [27]:
```languages = ["Fortran","C","C++"]
languages.append("Python")
print(languages)

```
```

The **range()** command is a convenient way to make sequential lists of numbers:

```
In [28]:
```list(range(10))

```
Out[28]:
```

```
In [29]:
```list(range(2,8))

```
Out[29]:
```

*step* of 1 between elements. You can also give a fixed step size via a third command:

```
In [30]:
```evens = list(range(0,20,2))
evens

```
Out[30]:
```

```
In [31]:
```evens[3]

```
Out[31]:
```

Lists do not have to hold the same data type. For example,

```
In [32]:
```["Today",7,99.3,""]

```
Out[32]:
```

However, it's good (but not essential) to use lists for similar objects that are somehow logically connected. If you want to group different data types together into a composite data object, it's best to use **tuples**, which we will learn about below.

You can find out how long a list is using the **len()** command:

```
In [33]:
```help(len)

```
```

```
In [34]:
```len(evens)

```
Out[34]:
```

```
In [35]:
```for day in days_of_the_week:
print(day)

```
```

This code snippet goes through each element of the list called **days_of_the_week** and assigns it to the variable **day**. It then executes everything in the indented block (in this case only one line of code, the print statement) using those variable assignments. When the program has gone through every element of the list, it exists the block.

(Almost) every programming language defines blocks of code in some way. In Fortran, one uses END statements (ENDDO, ENDIF, etc.) to define code blocks. In C, C++, and Perl, one uses curly braces {} to define these blocks.

Python uses a colon (":"), followed by indentation level to define code blocks. Everything at a higher level of indentation is taken to be in the same block. In the above example the block was only a single line, but we could have had longer blocks as well:

```
In [36]:
```for day in days_of_the_week:
statement = "Today is " + day
print(statement)

```
```

**range()** command is particularly useful with the **for** statement to execute loops of a specified length:

```
In [37]:
```for i in range(20):
print("The square of ",i," is ",i*i)

```
```

```
In [38]:
```for letter in "Sunday":
print(letter)

```
```

*slicing* operation, which you can also use on any sequence. We already know that we can use *indexing* to get the first element of a list:

```
In [39]:
```days_of_the_week[0]

```
Out[39]:
```

If we want the list containing the first two elements of a list, we can do this via

```
In [40]:
```days_of_the_week[0:2]

```
Out[40]:
```

or simply

```
In [41]:
```days_of_the_week[:2]

```
Out[41]:
```

If we want the last items of the list, we can do this with negative slicing:

```
In [42]:
```days_of_the_week[-2:]

```
Out[42]:
```

which is somewhat logically consistent with negative indices accessing the last elements of the list.

You can do:

```
In [43]:
```workdays = days_of_the_week[1:5]
print(workdays)

```
```

Since strings are sequences, you can also do this to them:

```
In [44]:
```day = "Saturday"
abbreviation = day[:3]
print(abbreviation)

```
```

**range()** function specifies the step):

```
In [45]:
```numbers = list(range(0,21))
evens = numbers[6::2]
evens

```
Out[45]:
```

We have now learned a few data types. We have integers and floating point numbers, strings, and lists to contain them. We have also learned about lists, a container that can hold any data type. We have learned to print things out, and to iterate over items in lists. We will now learn about **boolean** variables that can be either True or False.

We invariably need some concept of *conditions* in programming to control branching behavior, to allow a program to react differently to different situations. If it's Monday, I'll go to work, but if it's Sunday, I'll sleep in. To do this in Python, we use a combination of **boolean** variables, which evaluate to either True or False, and **if** statements, that control branching based on boolean values.

For example:

```
In [46]:
```if day == "Sunday":
print("Sleep in")
else:
print("Go to work")

```
```

(Quick quiz: why did the snippet print "Go to work" here? What is the variable "day" set to?)

Let's take the snippet apart to see what happened. First, note the statement

```
In [47]:
```day == "Sunday"

```
Out[47]:
```

*equality testing*. If the two items are equal, it returns True, otherwise it returns False. In this case, it is comparing two variables, the string "Sunday", and whatever is stored in the variable "day", which, in this case, is the other string "Saturday". Since the two strings are not equal to each other, the truth test has the false value.

The if statement that contains the truth test is followed by a code block (a colon followed by an indented block of code). If the boolean is true, it executes the code in that block. Since it is false in the above example, we don't see that code executed.

The first block of code is followed by an **else** statement, which is executed if nothing else in the above if statement is true. Since the value was false, this code is executed, which is why we see "Go to work".

Try setting the day equal to "Sunday" and then running the above if/else statement. Did it work as you thought it would?

You can compare any data types in Python:

```
In [48]:
```1 == 2

```
Out[48]:
```

```
In [49]:
```50 == 2*25

```
Out[49]:
```

```
In [50]:
```3 < 3.14159

```
Out[50]:
```

```
In [51]:
```1 == 1.0

```
Out[51]:
```

```
In [52]:
```1 != 0

```
Out[52]:
```

```
In [53]:
```1 <= 2

```
Out[53]:
```

```
In [54]:
```1 >= 1

```
Out[54]:
```

We see a few other boolean operators here, all of which which should be self-explanatory. Less than, equality, non-equality, and so on.

Particularly interesting is the 1 == 1.0 test, which is true, since even though the two objects are different data types (integer and floating point number), they have the same *value*. There is another boolean operator **is**, that tests whether two objects are the same object:

```
In [55]:
```1 is 1.0

```
Out[55]:
```

Why is 1 not the same as 1.0? Different data type. You can check the data type:

```
In [56]:
```type(1)

```
Out[56]:
```

```
In [57]:
```type(1.0)

```
Out[57]:
```

We can do boolean tests on lists as well:

```
In [58]:
```[1,2,3] == [1,2,4]

```
Out[58]:
```

```
In [59]:
```hours = 5
0 < hours < 24

```
Out[59]:
```

If statements can have **elif** parts ("else if"), in addition to if/else parts. For example:

```
In [60]:
```if day == "Sunday":
print ("Sleep in")
elif day == "Saturday":
print ("Do chores")
else:
print ("Go to work")

```
```

Of course we can combine if statements with for loops, to make a snippet that is almost interesting:

```
In [61]:
```for day in days_of_the_week:
statement = "On " + day + ":"
print (statement)
if day == "Sunday":
print (" Sleep in")
elif day == "Saturday":
print (" Do chores")
else:
print (" Go to work")

```
```

**bool()** function.

```
In [62]:
```bool(1)

```
Out[62]:
```

```
In [63]:
```bool(0)

```
Out[63]:
```

```
In [64]:
```bool(["This "," is "," a "," list"])

```
Out[64]:
```

The Fibonacci sequence is a sequence in math that starts with 0 and 1, and then each successive entry is the sum of the previous two. Thus, the sequence goes 0,1,1,2,3,5,8,13,21,34,55,89,...

A very common exercise in programming books is to compute the Fibonacci sequence up to some number **n**. First I'll show the code, then I'll discuss what it is doing.

```
In [65]:
```n = 10
sequence = [0,1]
for i in range(2,n): # This is going to be a problem if we ever set n <= 2!
sequence.append(sequence[i-1]+sequence[i-2])
print (sequence)

```
```

Let's go through this line by line. First, we define the variable **n**, and set it to the integer 20. **n** is the length of the sequence we're going to form, and should probably have a better variable name. We then create a variable called **sequence**, and initialize it to the list with the integers 0 and 1 in it, the first two elements of the Fibonacci sequence. We have to create these elements "by hand", since the iterative part of the sequence requires two previous elements.

We then have a for loop over the list of integers from 2 (the next element of the list) to **n** (the length of the sequence). After the colon, we see a hash tag "#", and then a **comment** that if we had set **n** to some number less than 2 we would have a problem. Comments in Python start with #, and are good ways to make notes to yourself or to a user of your code explaining why you did what you did. Better than the comment here would be to test to make sure the value of **n** is valid, and to complain if it isn't; we'll try this later.

In the body of the loop, we append to the list an integer equal to the sum of the two previous elements of the list.

After exiting the loop (ending the indentation) we then print out the whole list. That's it!

```
In [66]:
```def fibonacci(sequence_length):
"Return the Fibonacci sequence of length *sequence_length*"
sequence = [0,1]
if sequence_length < 1:
print("Fibonacci sequence only defined for length 1 or greater")
return
if 0 < sequence_length < 3:
return sequence[:sequence_length]
for i in range(2,sequence_length):
sequence.append(sequence[i-1]+sequence[i-2])
return sequence

We can now call **fibonacci()** for different sequence_lengths:

```
In [67]:
```fibonacci(2)

```
Out[67]:
```

```
In [68]:
```fibonacci(12)

```
Out[68]:
```

**docstring**, and is a special kind of comment that is often available to people using the function through the python command line:

```
In [69]:
```help(fibonacci)

```
```

If you define a docstring for all of your functions, it makes it easier for other people to use them, since they can get help on the arguments and return values of the function.

Next, note that rather than putting a comment in about what input values lead to errors, we have some testing of these values, followed by a warning if the value is invalid, and some conditional code to handle special cases.

Functions can also call themselves, something that is often called *recursion*. We're going to experiment with recursion by computing the factorial function. The factorial is defined for a positive integer **n** as

First, note that we don't need to write a function at all, since this is a function built into the standard math library. Let's use the help function to find out about it:

```
In [70]:
```from math import factorial
help(factorial)

```
```

This is clearly what we want.

```
In [71]:
```factorial(20)

```
Out[71]:
```

However, if we did want to write a function ourselves, we could do recursively by noting that

$$ n! = n(n-1)!$$The program then looks something like:

```
In [72]:
```def fact(n):
if n <= 0:
return 1
return n*fact(n-1)

```
In [73]:
```fact(20)

```
Out[73]:
```

Recursion can be very elegant, and can lead to very simple programs.

Before we end the Python overview, I wanted to touch on two more data structures that are very useful (and thus very common) in Python programs.

A **tuple** is a sequence object like a list or a string. It's constructed by grouping a sequence of objects together with commas, either without brackets, or with parentheses:

```
In [74]:
```t = (1,2,'hi',9.0)
t

```
Out[74]:
```

Tuples are like lists, in that you can access the elements using indices:

```
In [75]:
```t[1]

```
Out[75]:
```

However, tuples are *immutable*, you can't append to them or change the elements of them:

```
In [76]:
```t.append(7)

```
```

```
In [77]:
```t[1]=77

```
```

```
In [78]:
```('Bob',0.0,21.0)

```
Out[78]:
```

```
In [79]:
```positions = [
('Bob',0.0,21.0),
('Cat',2.5,13.1),
('Dog',33.0,1.2)
]

```
In [80]:
```def minmax(objects):
minx = 1e20 # These are set to really big numbers
miny = 1e20
for obj in objects:
name,x,y = obj
if x < minx:
minx = x
if y < miny:
miny = y
return minx,miny
x,y = minmax(positions)
print(x,y)

```
```

Here we did two things with tuples you haven't seen before. First, we unpacked an object into a set of named variables using *tuple assignment*:

```
>>> name,x,y = obj
```

We also returned multiple values (minx,miny), which were then assigned to two other variables (x,y), again by tuple assignment. This makes what would have been complicated code in C++ rather simple.

Tuple assignment is also a convenient way to swap variables:

```
In [81]:
```x,y = 1,2
y,x = x,y
x,y

```
Out[81]:
```

**Dictionaries** are an object called "mappings" or "associative arrays" in other languages. Whereas a list associates an integer index with a set of objects:

```
In [82]:
```mylist = [1,2,9,21]

*key*, and the corresponding dictionary entry is the *value*. A dictionary can use (almost) anything as the key. Whereas lists are formed with square brackets [], dictionaries use curly brackets {}:

```
In [83]:
```ages = {"Rick": 46, "Bob": 86, "Fred": 21}
print("Rick's age is ",ages["Rick"])

```
```

There's also a convenient way to create dictionaries without having to quote the keys.

```
In [84]:
```dict(Rick=46,Bob=86,Fred=20)

```
Out[84]:
```

The **len()** command works on both tuples and dictionaries:

```
In [85]:
```len(t)

```
Out[85]:
```

```
In [86]:
```len(ages)

```
Out[86]:
```

We can generally understand trends in data by using a plotting program to chart it. Python has a wonderful plotting library called Matplotlib. The Jupyter notebook interface we are using for these notes has that functionality built in.

First off, it is important to import the library. We did this at the very beginning of this whole jupyter notebook, but here it is in case you've jumped straight here without running the first code line:

```
In [87]:
```import matplotlib.pyplot as plt
%matplotlib inline

The **%matplotlib inline** command makes it so plots are within this notebook. To plot to a separate window, use instead:

`%matplotlib qt`

```
In [88]:
```fibs = fibonacci(10)

Next lets generate the factorials.

```
In [89]:
```facts = []
for i in range(10):
facts.append(factorial(i))

Now we use the Matplotlib function **plot** to compare the two.

```
In [90]:
```plt.plot(facts,'-ob',label="factorial")
plt.plot(fibs,'-dg',label="Fibonacci")
plt.xlabel("n")
plt.legend()

```
Out[90]:
```

The factorial function grows much faster. In fact, you can't even see the Fibonacci sequence. It's not entirely surprising: a function where we multiply by n each iteration is bound to grow faster than one where we add (roughly) n each iteration.

Let's plot these on a semilog plot so we can see them both a little more clearly:

```
In [91]:
```plt.semilogy(facts,label="factorial")
plt.semilogy(fibs,label="Fibonacci")
plt.xlabel("n")
plt.legend()

```
Out[91]:
```

There is, of course, much more to the language than we've covered here. I've tried to keep this brief enough so that you can jump in and start using Python to simplify your life and work. My own experience in learning new things is that the information doesn't "stick" unless you try and use it for something in real life.

You will no doubt need to learn more as you go. I've listed several other good references, including the Python Tutorial and Learn Python the Hard Way. Additionally, now is a good time to start familiarizing yourself with the Python Documentation, and, in particular, the Python Language Reference.

Tim Peters, one of the earliest and most prolific Python contributors, wrote the "Zen of Python", which can be accessed via the "import this" command:

```
In [92]:
```import this

```
```

No matter how experienced a programmer you are, these are words to meditate on.

Numpy contains core routines for doing fast vector, matrix, and linear algebra-type operations in Python. Scipy contains additional routines for optimization, special functions, and so on. Both contain modules written in C and Fortran so that they're as fast as possible. Together, they give Python roughly the same capability that the Matlab program offers. (In fact, if you're an experienced Matlab user, there a guide to Numpy for Matlab users just for you.)

First off, it is important to import the library. Again, we did this at the very beginning of this whole jupyter notebook, but here it is in case you've jumped straight here without running the first code line:

```
In [93]:
```import numpy as np

```
In [94]:
```np.array([1,2,3,4,5,6])

```
Out[94]:
```

**array** that gives the numeric type. There are a number of types listed here that your matrix can be. Some of these are aliased to single character codes. The most common ones are 'd' (double precision floating point number), 'D' (double precision complex number), and 'i' (int32). Thus,

```
In [95]:
```np.array([1,2,3,4,5,6],'d')

```
Out[95]:
```

```
In [96]:
```np.array([1,2,3,4,5,6],'D')

```
Out[96]:
```

```
In [97]:
```np.array([1,2,3,4,5,6],'i')

```
Out[97]:
```

To build matrices, you can either use the array command with lists of lists:

```
In [98]:
```np.array([[0,1],[1,0]],'d')

```
Out[98]:
```

**zeros** command:

```
In [99]:
```np.zeros((3,3),'d')

```
Out[99]:
```

```
In [100]:
```np.zeros(3,'d')

```
Out[100]:
```

```
In [101]:
```np.zeros((1,3),'d')

```
Out[101]:
```

or column vectors:

```
In [102]:
```np.zeros((3,1),'d')

```
Out[102]:
```

There's also an **identity** command that behaves as you'd expect:

```
In [103]:
```np.identity(4,'d')

```
Out[103]:
```

as well as a **ones** command.

```
In [104]:
```np.linspace(0,1)

```
Out[104]:
```

```
In [105]:
```np.linspace(0,1,11)

```
Out[105]:
```

**linspace** is an easy way to make coordinates for plotting. Functions in the numpy library (all of which are imported into IPython notebook) can act on an entire vector (or even a matrix) of points at once. Thus,

```
In [106]:
```x = np.linspace(0,2*np.pi)
np.sin(x)

```
Out[106]:
```

In conjunction with **matplotlib**, this is a nice way to plot things:

```
In [107]:
```plt.plot(x,np.sin(x))
plt.show()

```
```

```
In [108]:
```0.125*np.identity(3,'d')

```
Out[108]:
```

as well as when you add two matrices together. (However, the matrices have to be the same shape.)

```
In [109]:
```np.identity(2,'d') + np.array([[1,1],[1,2]])

```
Out[109]:
```

```
In [110]:
```np.identity(2)*np.ones((2,2))

```
Out[110]:
```

To get matrix multiplication, you need the **dot** command:

```
In [111]:
```np.dot(np.identity(2),np.ones((2,2)))

```
Out[111]:
```

**dot** can also do dot products (duh!):

```
In [112]:
```v = np.array([3,4],'d')
np.sqrt(np.dot(v,v))

```
Out[112]:
```

as well as matrix-vector products.

**determinant**, **inverse**, and **transpose** functions that act as you would suppose. Transpose can be abbreviated with ".T" at the end of a matrix object:

```
In [113]:
```m = np.array([[1,2],[3,4]])
m.T

```
Out[113]:
```

**diag()** function that takes a list or a vector and puts it along the diagonal of a square matrix.

```
In [114]:
```np.diag([1,2,3,4,5])

```
Out[114]:
```

We'll find this useful later on.

```
In [115]:
```A = np.array([[1,1,1],[0,2,5],[2,5,-1]])
b = np.array([6,-4,27])
np.linalg.solve(A,b)

```
Out[115]:
```

There are a number of routines to compute eigenvalues and eigenvectors

**eigvals**returns the eigenvalues of a matrix**eigvalsh**returns the eigenvalues of a Hermitian matrix**eig**returns the eigenvalues and eigenvectors of a matrix**eigh**returns the eigenvalues and eigenvectors of a Hermitian matrix.

```
In [116]:
```A = np.array([[13,-4],[-4,7]],'d')
np.linalg.eigvalsh(A)

```
Out[116]:
```

```
In [117]:
```np.linalg.eigh(A)

```
Out[117]:
```

Now that we have these tools in our toolbox, we can start to do some cool stuff with it. Many of the equations we want to solve in Physics involve differential equations. We want to be able to compute the derivative of functions:

$$ y' = \frac{y(x+h)-y(x)}{h} $$by *discretizing* the function $y(x)$ on an evenly spaced set of points $x_0, x_1, \dots, x_n$, yielding $y_0, y_1, \dots, y_n$. Using the discretization, we can approximate the derivative by

We can write a derivative function in Python via

```
In [118]:
```def nderiv(y,x):
"Finite difference derivative of the function f"
n = len(y)
d = np.zeros(n,'d') # assume double
# Use centered differences for the interior points, one-sided differences for the ends
for i in range(1,n-1):
d[i] = (y[i+1]-y[i])/(x[i+1]-x[i])
d[0] = (y[1]-y[0])/(x[1]-x[0])
d[n-1] = (y[n-1]-y[n-2])/(x[n-1]-x[n-2])
return d

Let's see whether this works for our sin example from above:

```
In [119]:
```x = np.linspace(0,2*np.pi)
dsin = nderiv(np.sin(x),x)
plt.plot(x,dsin,label='numerical')
plt.plot(x,np.cos(x),label='analytical')
plt.title("Comparison of numerical and analytical derivatives of sin(x)")
plt.legend()

```
Out[119]:
```

Pretty close!

Now that we've convinced ourselves that finite differences aren't a terrible approximation, let's see if we can use this to solve the one-dimensional harmonic oscillator.

We want to solve the time-independent Schrodinger equation

$$ -\frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2} + V(x)\psi(x) = E\psi(x)$$for $\psi(x)$ when $V(x)=\frac{1}{2}m\omega^2x^2$ is the harmonic oscillator potential. We're going to use the standard trick to transform the differential equation into a matrix equation by multiplying both sides by $\psi^*(x)$ and integrating over $x$. This yields

$$ -\frac{\hbar}{2m}\int\psi(x)\frac{\partial^2}{\partial x^2}\psi(x)dx + \int\psi(x)V(x)\psi(x)dx = E$$We will again use the finite difference approximation. The finite difference formula for the second derivative is

$$ y'' = \frac{y_{i+1}-2y_i+y_{i-1}}{x_{i+1}-x_{i-1}} $$We can think of the first term in the Schrodinger equation as the overlap of the wave function $\psi(x)$ with the second derivative of the wave function $\frac{\partial^2}{\partial x^2}\psi(x)$. Given the above expression for the second derivative, we can see if we take the overlap of the states $y_1,\dots,y_n$ with the second derivative, we will only have three points where the overlap is nonzero, at $y_{i-1}$, $y_i$, and $y_{i+1}$. In matrix form, this leads to the tridiagonal Laplacian matrix, which has -2's along the diagonals, and 1's along the diagonals above and below the main diagonal.

The second term turns leads to a diagonal matrix with $V(x_i)$ on the diagonal elements. Putting all of these pieces together, we get:

```
In [120]:
```def Laplacian(x):
h = x[1]-x[0] # assume uniformly spaced points
n = len(x)
M = -2*np.identity(n,'d')
for i in range(1,n):
M[i,i-1] = M[i-1,i] = 1
return M/h**2

```
In [121]:
```x = np.linspace(-3,3)
m = 1.0
ohm = 1.0
T = (-0.5/m)*Laplacian(x)
V = 0.5*(ohm**2)*(x**2)
H = T + np.diag(V)
E,U = np.linalg.eigh(H)
h = x[1]-x[0]
# Plot the Harmonic potential
plt.plot(x,V,color='k')
for i in range(4):
# For each of the first few solutions, plot the energy level:
plt.axhline(y=E[i],color='k',ls=":")
# as well as the eigenfunction, displaced by the energy level so they don't
# all pile up on each other:
plt.plot(x,-U[:,i]/np.sqrt(h)+E[i])
plt.title("Eigenfunctions of the Quantum Harmonic Oscillator")
plt.xlabel("Displacement (bohr)")
plt.ylabel("Energy (hartree)")

```
Out[121]:
```

We've made a couple of hacks here to get the orbitals the way we want them. First, I inserted a -1 factor before the wave functions, to fix the phase of the lowest state. The phase (sign) of a quantum wave function doesn't hold any information, only the square of the wave function does, so this doesn't really change anything.

But the eigenfunctions as we generate them aren't properly normalized. The reason is that finite difference isn't a real basis in the quantum mechanical sense. It's a basis of Dirac δ functions at each point; we interpret the space betwen the points as being "filled" by the wave function, but the finite difference basis only has the solution being at the points themselves. We can fix this by dividing the eigenfunctions of our finite difference Hamiltonian by the square root of the spacing, and this gives properly normalized functions.

The solutions to the Harmonic Oscillator are supposed to be Hermite polynomials. The Wikipedia page has the HO states given by

$$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right) H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$$Let's see whether they look like those. There are some special functions in the Numpy library, and some more in Scipy. Hermite Polynomials are in Numpy:

```
In [122]:
```from numpy.polynomial.hermite import Hermite
def ho_evec(x,n,m,ohm):
vec = [0]*9
vec[n] = 1
Hn = Hermite(vec)
return (1/np.sqrt(2**n*factorial(n)))*pow(m*ohm/np.pi,0.25)*np.exp(-0.5*m*ohm*x**2)*Hn(x*np.sqrt(m*ohm))

Let's compare the first function to our solution.

```
In [123]:
```plt.plot(x,ho_evec(x,0,1,1),label="Analytic")
plt.plot(x,-U[:,0]/np.sqrt(h),label="Numeric")
plt.xlabel('x (bohr)')
plt.ylabel(r'$\psi(x)$')
plt.title("Comparison of numeric and analytic solutions to the Harmonic Oscillator")
plt.legend()

```
Out[123]:
```

The agreement is almost exact.

**subplot** command to put multiple comparisons in different panes on a single plot (run **%matplotlib qt** on a separate line first to plot in a separate window):

```
In [124]:
```phase_correction = [-1,1,1,-1,-1,1]
for i in range(6):
plt.subplot(2,3,i+1)
plt.plot(x,ho_evec(x,i,1,1),label="Analytic")
plt.plot(x,phase_correction[i]*U[:,i]/np.sqrt(h),label="Numeric")

```
```

Other than phase errors (which I've corrected with a little hack: can you find it?), the agreement is pretty good, although it gets worse the higher in energy we get, in part because we used only 50 points.

The Scipy module has many more special functions:

```
In [125]:
```from scipy.special import airy,jn,eval_chebyt,eval_legendre
plt.subplot(2,2,1)
x = np.linspace(-1,1)
Ai,Aip,Bi,Bip = airy(x)
plt.plot(x,Ai)
plt.plot(x,Aip)
plt.plot(x,Bi)
plt.plot(x,Bip)
plt.title("Airy functions")
plt.subplot(2,2,2)
x = np.linspace(0,10)
for i in range(4):
plt.plot(x,jn(i,x))
plt.title("Bessel functions")
plt.subplot(2,2,3)
x = np.linspace(-1,1)
for i in range(6):
plt.plot(x,eval_chebyt(i,x))
plt.title("Chebyshev polynomials of the first kind")
plt.subplot(2,2,4)
x = np.linspace(-1,1)
for i in range(6):
plt.plot(x,eval_legendre(i,x))
plt.title("Legendre polynomials")
# plt.tight_layout()
plt.show()

```
```

```
In [126]:
```raw_data = """\
3.1905781584582433,0.028208609537968457
4.346895074946466,0.007160804747670053
5.374732334047101,0.0046962988461934805
8.201284796573875,0.0004614473299618756
10.899357601713055,0.00005038370219939726
16.295503211991434,4.377451812785309e-7
21.82012847965739,3.0799922117601088e-9
32.48394004282656,1.524776208284536e-13
43.53319057815846,5.5012073588707224e-18"""

```
In [127]:
```data = []
for line in raw_data.splitlines():
words = line.split(',')
data.append(list(map(float,words)))
data = np.array(data)

```
In [128]:
```plt.title("Raw Data")
plt.xlabel("Distance")
plt.plot(data[:,0],data[:,1],'bo')

```
Out[128]:
```

Since we expect the data to have an exponential decay, we can plot it using a semi-log plot.

```
In [129]:
```plt.title("Raw Data")
plt.xlabel("Distance")
plt.semilogy(data[:,0],data[:,1],'bo')

```
Out[129]:
```

For a pure exponential decay like this, we can fit the log of the data to a straight line. The above plot suggests this is a good approximation. Given a function $$ y = Ae^{ax} $$ $$ \log(y) = ax + \log(A) $$ Thus, if we fit the log of the data versus x, we should get a straight line with slope $a$, and an intercept that gives the constant $A$.

There's a numpy function called **polyfit** that will fit data to a polynomial form. We'll use this to fit to a straight line (a polynomial of order 1)

```
In [130]:
```params = np.polyfit(data[:,0],np.log(data[:,1]),1)
a = params[0] # the coefficient of x**1
logA = params[1] # the coefficient of x**0
# plot if curious:
# plt.plot(data[:,0],np.log(data[:,1]),'bo')
# plt.plot(data[:,0],data[:,0]*a+logA,'r')

Let's see whether this curve fits the data.

```
In [131]:
```x = np.linspace(1,45)
plt.title("Raw Data")
plt.xlabel("Distance")
plt.semilogy(data[:,0],data[:,1],'bo',label='data')
plt.semilogy(x,np.exp(logA)*np.exp(a*x),'r-',label='fit')
plt.legend()

```
Out[131]:
```

```
In [132]:
```gauss_data = """\
-0.9902286902286903,1.4065274110372852e-19
-0.7566104566104566,2.2504438576596563e-18
-0.5117810117810118,1.9459459459459454
-0.31887271887271884,10.621621621621626
-0.250997150997151,15.891891891891893
-0.1463309463309464,23.756756756756754
-0.07267267267267263,28.135135135135133
-0.04426734426734419,29.02702702702703
-0.0015939015939017698,29.675675675675677
0.04689304689304685,29.10810810810811
0.0840994840994842,27.324324324324326
0.1700546700546699,22.216216216216214
0.370878570878571,7.540540540540545
0.5338338338338338,1.621621621621618
0.722014322014322,0.08108108108108068
0.9926849926849926,-0.08108108108108646"""
gdata = []
for line in gauss_data.splitlines():
words = line.split(',')
gdata.append(list(map(float,words)))
gdata = np.array(gdata)
plt.plot(gdata[:,0],gdata[:,1],'bo')

```
Out[132]:
```

This data looks more Gaussian than exponential. If we wanted to, we could use **polyfit** for this as well, but let's use the **curve_fit** function from Scipy, which can fit to arbitrary functions. You can learn more using help(curve_fit).

First define a general Gaussian function to fit to.

```
In [133]:
```def gauss(x,A,a): return A*np.exp(a*x**2)

Now fit to it using **curve_fit**:

```
In [134]:
```from scipy.optimize import curve_fit
params,conv = curve_fit(gauss,gdata[:,0],gdata[:,1])
x = np.linspace(-1,1)
plt.plot(gdata[:,0],gdata[:,1],'bo')
A,a = params
plt.plot(x,gauss(x,A,a),'r-')

```
Out[134]:
```

**curve_fit** routine we just used is built on top of a very good general **minimization** capability in Scipy. You can learn more at the scipy documentation pages.

Many methods in scientific computing rely on Monte Carlo integration, where a sequence of (pseudo) random numbers are used to approximate the integral of a function. Python has good random number generators in the standard library. The **random()** function from the numpy library gives pseudorandom numbers uniformly distributed between 0 and 1:

```
In [135]:
```# from random import random
rands = []
for i in range(100):
rands.append(np.random.random())

Or, more elegantly:

```
In [136]:
```rands = np.random.rand(100)
plt.plot(rands,'o')

```
Out[136]:
```

**np.random.random()** uses the Mersenne Twister algorithm, which is a highly regarded pseudorandom number generator. There are also functions to generate random integers, to randomly shuffle a list, and functions to pick random numbers from a particular distribution, like the normal distribution:

```
In [137]:
```mu, sigma = 0, 0.1 # mean and standard deviation
s=np.random.normal(mu, sigma,1000)

**numpy.random.normal**:

```
In [138]:
```count, bins, ignored = plt.hist(s, 30, normed=True)
plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
linewidth=2, color='r')

```
Out[138]:
```

```
In [139]:
```npts = 5000
xs = 2*np.random.rand(npts)-1
ys = 2*np.random.rand(npts)-1
r = xs**2+ys**2
ninside = (r<1).sum()
plt.figure(figsize=(6,6)) # make the figure square
plt.title("Approximation to pi = %f" % (4*ninside/float(npts)))
plt.plot(xs[r<1],ys[r<1],'b.')
plt.plot(xs[r>1],ys[r>1],'r.')
plt.figure(figsize=(8,6)) # change the figsize back to standard size for the rest of the notebook

```
Out[139]:
```

The idea behind the program is that the ratio of the area of the unit circle to the square that inscribes it is $\pi/4$, so by counting the fraction of the random points in the square that are inside the circle, we get increasingly good estimates to $\pi$.

The above code uses some higher level Numpy tricks to compute the radius of each point in a single line, to count how many radii are below one in a single line, and to filter the x,y points based on their radii. To be honest, I rarely write code like this: I find some of these Numpy tricks a little too cute to remember them, and I'm more likely to use a list comprehension (see below) to filter the points I want, since I can remember that.

As methods of computing $\pi$ go, this is among the worst. A much better method is to use Leibniz's expansion of arctan(1):

$$\frac{\pi}{4} = \sum_k \frac{(-1)^k}{2*k+1}$$```
In [140]:
```n = 100
total = 0
for k in range(n):
total += pow(-1,k)/(2*k+1.0)
print(4*total)

```
```

**decimal** module, if you're interested.

Integration can be hard, and sometimes it's easier to work out a definite integral using an approximation. For example, suppose we wanted to figure out the integral:

$$\int_0^\infty\exp(-x)dx$$(It turns out that this is equal to 1, as you can work out easily with a pencil :) )

```
In [141]:
```def f(x): return np.exp(-x)
x = np.linspace(0,10)
plt.plot(x,np.exp(-x))

```
Out[141]:
```

**quad** (since sometimes numerical integration is called *quadrature*), that we can use for this:

```
In [142]:
```from scipy.integrate import quad
quad(f,0,np.inf)

```
Out[142]:
```

The first number in the tuple is the result, the second number is an estimate of the absolute error in the result.

There are also 2d and 3d numerical integrators in Scipy. See the docs for more information.

```
In [143]:
```from scipy.fftpack import fft,fftfreq
npts = 4000
nplot = int(npts/10)
t = np.linspace(0,120,npts)
def sig(t): return 50*np.sin(2*np.pi*2.0*t) + 20*np.sin(2*np.pi*5.0*t) + 10*np.sin(2*np.pi*8.0*t) + 2*np.random.rand(npts)
Vsignal = sig(t)
FFT = abs(fft(Vsignal))
freqs = fftfreq(npts, t[1]-t[0])
FFT_plot = FFT[0:int(len(freqs)/2)]
freqs_plot = freqs[0:int(len(freqs)/2)]
plt.subplot(211)
plt.plot(t[:nplot], Vsignal[:nplot])
plt.xlabel ('time (s)')
plt.ylabel ('voltage\nmeasured (V)')
plt.subplot(212)
plt.semilogy(freqs_plot,FFT_plot**2,'-')
plt.xlabel ('frequency (Hz)')
plt.ylabel ('power\nspectrum (a.u.)')
plt.ylim([1e-1,np.max(FFT_plot**2)])
plt.tight_layout()

```
```

As more and more of our day-to-day work is being done on and through computers, we increasingly have output that one program writes, often in a text file, that we need to analyze in one way or another, and potentially feed that output into another file.

Suppose we have the following output in CSV (comma separated values) format, a format that originally came from Microsoft Excel, and is increasingly used as a data interchange format in big data applications. How would we parse that?

```
In [144]:
```csv = """\
1, -6095.12544083, 0.03686, 1391.5
2, -6095.25762870, 0.00732, 10468.0
3, -6095.26325979, 0.00233, 11963.5
4, -6095.26428124, 0.00109, 13331.9
5, -6095.26463203, 0.00057, 14710.8
6, -6095.26477615, 0.00043, 20211.1
7, -6095.26482624, 0.00015, 21726.1
8, -6095.26483584, 0.00021, 24890.5
9, -6095.26484405, 0.00005, 26448.7
10, -6095.26484599, 0.00003, 27258.1
11, -6095.26484676, 0.00003, 28155.3
12, -6095.26484693, 0.00002, 28981.7
13, -6095.26484693, 0.00002, 28981.7"""
csv

```
Out[144]:
```

**splitlines()**, we see that a list is created where line gets separated into a string:

```
In [145]:
```lines = csv.splitlines()
lines

```
Out[145]:
```

Splitting is a big concept in text processing. We used **splitlines()** here, and next we'll use the more general **.split(",")** function below to split each line into comma-delimited words.

We now want to do three things:

- Skip over the lines that don't carry any information
- Break apart each line that does carry information and grab the pieces we want
- Turn the resulting data into something that we can plot.

To break apart each line, we will use **.split(",")**. Let's see what it does to one of the lines:

```
In [146]:
```lines[4].split(",")

```
Out[146]:
```

What does **split()** do?

```
In [147]:
```help("".split)

```
```

```
In [148]:
```for line in lines:
# do something with each line
words = line.split(",")

We need to add these results at each step to a list:

```
In [149]:
```data = []
for line in csv.splitlines()[2:]:
words = line.split(',')
data.append(list(map(float,words)))
data = np.array(data)
data

```
Out[149]:
```

**for** loop to iterate over each line. However, we skipped the first two (the lines[2:] only takes the lines starting from index 2), since lines[0] contained the title information, and lines[1] contained underscores. Similarly, **[:5]** instead would take the first five lines.
We pass the comma string ",'"into the split function, so that it breaks to a new word every time it sees a comma. Next, to simplify things a bit, we're using the **map()** command to repeatedly apply a single function (**float()**) to a list, and to return the output as a list. Finally, we turn the list of lists into a numpy arrray structure.

```
In [150]:
```plt.plot(data[:,0],data[:,1],'-o')
plt.xlabel('step')
plt.ylabel('Energy (hartrees)')
plt.title('Convergence of NWChem geometry optimization for Si cluster\n')

```
Out[150]:
```

```
In [151]:
```energies = data[:,1]
minE = np.min(energies)
energies_eV = 27.211*(energies-minE)
plt.plot(data[:,0],energies_eV,'-o')
plt.xlabel('step')
plt.ylabel('Energy (eV)')
plt.title('Convergence of NWChem geometry optimization for Si cluster')

```
Out[151]:
```

```
In [152]:
```filename= 'DS0004.csv'
data = np.genfromtxt(filename,delimiter=',',skip_header=17 )
x_values = data[:,0]
y_values = data[:,1]

```
In [153]:
```plt.plot(x_values, y_values)

```
Out[153]:
```

**genfromtxt()**. For larger data sets, the library `pandas`

might be helpful.

Strings are a big deal in most modern languages, and hopefully the previous sections helped underscore how versatile Python's string processing techniques are. We will continue this topic in this section.

We can print out lines in Python using the print command.

```
In [154]:
```print("I have 3 errands to run")

```
```

```
In [155]:
``````
"I have 3 errands to run"
```

```
Out[155]:
```

**print** even converts some arguments to strings for us:

```
In [156]:
```a,b,c = 1,2,3
print("The variables are ",1,2,3)

```
```

As versatile as this is, you typically need more freedom over the data you print out. For example, what if we want to print a bunch of data to exactly 4 decimal places? We can do this using formatted strings.

Formatted strings share a syntax with the C **printf** statement. We make a string that has some funny *format characters* in it, and then pass a bunch of variables into the string that fill out those characters in different ways.

For example,

```
In [157]:
```print("Pi as a decimal = %d" % np.pi)
print("Pi as a float = %f" % np.pi)
print("Pi with 4 decimal places = %.4f" % np.pi)
print("Pi with overall fixed length of 10 spaces, with 6 decimal places = %10.6f" % np.pi)
print("Pi as in exponential format = %e" % np.pi)

```
```

We use a percent sign in two different ways here. First, the format character itself starts with a percent sign. %d or %i are for integers, %f is for floats, %e is for numbers in exponential formats. All of the numbers can take number immediately after the percent that specifies the total spaces used to print the number. Formats with a decimal can take an additional number after a dot . to specify the number of decimal places to print.

The other use of the percent sign is after the string, to pipe a set of variables in. You can pass in multiple variables (if your formatting string supports it) by putting a tuple after the percent. Thus,

```
In [158]:
```print("The variables specified earlier are %d, %d, and %d" % (a,b,c))

```
```

This is a simple formatting structure that will satisfy most of your string formatting needs. More information on different format symbols is available in the string formatting part of the standard docs.

It's worth noting that more complicated string formatting methods are in development, but I prefer this system due to its simplicity and its similarity to C formatting strings.

Recall we discussed multiline strings. We can put format characters in these as well, and fill them with the percent sign as before.

```
In [159]:
```form_letter = """\
%s
Dear %s,
We regret to inform you that your product did not
ship today due to %s.
We hope to remedy this as soon as possible.
From,
Your Supplier
"""
print(form_letter % ("July 1, 2016","Valued Customer Bob","alien attack"))

```
```

```
In [160]:
```form_letter = """\
%(date)s
Dear %(customer)s,
We regret to inform you that your product did not
ship today due to %(lame_excuse)s.
We hope to remedy this as soon as possible.
From,
Your Supplier
"""
print(form_letter % {"date" : "July 1, 2016","customer":"Valued Customer Bob","lame_excuse":"alien attack"})

```
```

By providing a little bit more information, you're less likely to make mistakes, like referring to your customer as "alien attack".

As a scientist, you're less likely to be sending bulk mailings to a bunch of customers. But these are great methods for generating and submitting lots of similar runs, say scanning a bunch of different structures to find the optimal configuration for something.

For example, you can use the following template for NWChem input files:

```
In [161]:
```nwchem_format = """
start %(jobname)s
title "%(thetitle)s"
charge %(charge)d
geometry units angstroms print xyz autosym
%(geometry)s
end
basis
* library 6-31G**
end
dft
xc %(dft_functional)s
mult %(multiplicity)d
end
task dft %(jobtype)s
"""

```
In [162]:
```oxygen_xy_coords = [(0,0),(0,0.1),(0.1,0),(0.1,0.1)]
charge = 0
multiplicity = 1
dft_functional = "b3lyp"
jobtype = "optimize"
geometry_template = """\
O %f %f 0.0
H 0.0 1.0 0.0
H 1.0 0.0 0.0"""
for i,xy in enumerate(oxygen_xy_coords):
thetitle = "Water run #%d" % i
jobname = "h2o-%d" % i
geometry = geometry_template % xy
print("---------")
print(nwchem_format % dict(thetitle=thetitle,charge=charge,jobname=jobname,jobtype=jobtype,
geometry=geometry,dft_functional=dft_functional,multiplicity=multiplicity))

```
```

This is a very bad geometry for a water molecule, and it would be silly to run so many geometry optimizations of structures that are guaranteed to converge to the same single geometry, but you get the idea of how you can run vast numbers of simulations with a technique like this.

We used the **enumerate** function to loop over both the indices and the items of a sequence, which is valuable when you want a clean way of getting both. **enumerate** is roughly equivalent to:

```
In [163]:
```def my_enumerate(seq):
l = []
for i in range(len(seq)):
l.append((i,seq[i]))
return l
my_enumerate(oxygen_xy_coords)

```
Out[163]:
```

**generators** (see below) so that it doesn't have to create a big list, which makes it faster for really long sequenes.

```
In [164]:
```np.linspace(0,1)

```
Out[164]:
```

or it can take three arguments, for the starting point, the ending point, and the number of points:

```
In [165]:
```np.linspace(0,1,5)

```
Out[165]:
```

You can also pass in keywords to exclude the endpoint:

```
In [166]:
```np.linspace(0,1,5,endpoint=False)

```
Out[166]:
```

Right now, we only know how to specify functions that have a fixed number of arguments. We'll learn how to do the more general cases here.

If we're defining a simple version of linspace, we would start with:

```
In [167]:
```def my_linspace(start,end):
npoints = 50
v = []
d = (end-start)/float(npoints-1)
for i in range(npoints):
v.append(start + i*d)
return v
my_linspace(0,1)

```
Out[167]:
```

We can add an optional argument by specifying a default value in the argument list:

```
In [168]:
```def my_linspace(start,end,npoints = 50):
v = []
d = (end-start)/float(npoints-1)
for i in range(npoints):
v.append(start + i*d)
return v

This gives exactly the same result if we don't specify anything:

```
In [169]:
```my_linspace(0,1)

```
Out[169]:
```

But also let's us override the default value with a third argument:

```
In [170]:
```my_linspace(0,1,5)

```
Out[170]:
```

```
In [171]:
```def my_linspace(start,end,npoints=50,**kwargs):
endpoint = kwargs.get('endpoint',True)
v = []
if endpoint:
d = (end-start)/float(npoints-1)
else:
d = (end-start)/float(npoints)
for i in range(npoints):
v.append(start + i*d)
return v
my_linspace(0,1,5,endpoint=False)

```
Out[171]:
```

What the keyword argument construction does is to take any additional keyword arguments (i.e. arguments specified by name, like "endpoint=False"), and stick them into a dictionary called "kwargs" (you can call it anything you like, but it has to be preceded by two stars). You can then grab items out of the dictionary using the **get** command, which also lets you specify a default value. I realize it takes a little getting used to, but it is a common construction in Python code, and you should be able to recognize it.

There's an analogous *args that dumps any additional arguments into a list called "args". Think about the **range** function: it can take one (the endpoint), two (starting and ending points), or three (starting, ending, and step) arguments. How would we define this?

```
In [172]:
```def my_range(*args):
start = 0
step = 1
if len(args) == 1:
end = args[0]
elif len(args) == 2:
start,end = args
elif len(args) == 3:
start,end,step = args
else:
raise Exception("Unable to parse arguments")
v = []
value = start
while True:
v.append(value)
value += step
if value > end: break
return v

**break** statement, that allows us to exit a for loop if some conditions are met, and an exception statement, that causes the interpreter to exit with an error message. For example:

```
In [173]:
```my_range()

```
```

```
In [174]:
```evens1 = [2*i for i in range(10)]
print(evens1)

```
```

You can also put some boolean testing into the construct:

```
In [175]:
```odds = [i for i in range(20) if i%2==1]
odds

```
Out[175]:
```

**iterators** are a way of making virtual sequence objects. Consider if we had the nested loop structure:

```
for i in range(1000000):
for j in range(1000000):
```

Inside the main loop, we make a list of 1,000,000 integers, just to loop over them one at a time. We don't need any of the additional things that a lists gives us, like slicing or random access, we just need to go through the numbers one at a time. And we're making 1,000,000 of them.

**iterators** are a way around this. For example, the **xrange** function is the iterator version of range. This simply makes a counter that is looped through in sequence, so that the analogous loop structure would look like:

```
for i in xrange(1000000):
for j in xrange(1000000):
```

Even though we've only added two characters, we've dramatically sped up the code, because we're not making 1,000,000 big lists.

We can define our own iterators using the **yield** statement:

```
In [176]:
```def evens_below(n):
for i in range(n):
if i%2 == 0:
yield i
return
for i in evens_below(9):
print(i)

```
```

We can always turn an iterator into a list using the **list** command:

```
In [177]:
```list(evens_below(9))

```
Out[177]:
```

There's a special syntax called a **generator expression** that looks a lot like a list comprehension:

```
In [178]:
```evens_gen = (i for i in range(9) if i%2==0)
for i in evens_gen:
print(i)

```
```

A factory function is a function that returns a function. They have the fancy name *lexical closure*, which makes you sound really intelligent in front of your CS friends. But, despite the arcane names, factory functions can play a very practical role.

Suppose you want the Gaussian function centered at 0.5, with height 99 and width 1.0. You could write a general function.

```
In [179]:
```def gauss(x,A,a,x0):
return A*np.exp(-a*(x-x0)**2)

```
In [180]:
```def gauss_maker(A,a,x0):
def f(x):
return A*np.exp(-a*(x-x0)**2)
return f

```
In [181]:
```x = np.linspace(0,1)
g = gauss_maker(99.0,20,0.5)
plt.plot(x,g(x))

```
Out[181]:
```

Everything in Python is an object, including functions. This means that functions can be returned by other functions. (They can also be passed into other functions, which is also useful, but a topic for another discussion.) In the **gauss_maker** example, the *g* function that is output "remembers" the A, a, x0 values it was constructed with, since they're all stored in the local memory space (this is what the *lexical closure* really refers to) of that function.

Factories are one of the more important of the Software Design Patterns, which are a set of guidelines to follow to make high-quality, portable, readable, stable software. It's beyond the scope of the current work to go more into either factories or design patterns, but I thought I would mention them for people interested in software design.

*Serialization* refers to the process of outputting data (and occasionally functions) to a database or a regular file, for the purpose of using it later on. In the very early days of programming languages, this was normally done in regular text files. Python is excellent at text processing, and you probably already know enough to get started with this.

When accessing large amounts of data became important, people developed database software based around the Structured Query Language (SQL) standard. I'm not going to cover SQL here, but, if you're interested, I recommend using the sqlite3 module in the Python standard library.

As data interchange became important, the eXtensible Markup Language (XML) has emerged. XML makes data formats that are easy to write parsers for, greatly simplifying the ambiguity that sometimes arises in the process. Again, I'm not going to cover XML here, but if you're interested in learning more, look into Element Trees, now part of the Python standard library.

Python has a very general serialization format called **pickle** that can turn any Python object, even a function or a class, into a representation that can be written to a file and read in later. But, again, I'm not going to talk about this, since I rarely use it myself. Again, the standard library documentation for pickle is the place to go.

What I am going to talk about is a relatively recent format call JavaScript Object Notation (JSON) that has become very popular over the past few years. There's a module in the standard library for encoding and decoding JSON formats. The reason I like JSON so much is that it looks almost like Python, so that, unlike the other options, you can look at your data and edit it, use it in another program, etc.

Here's a little example:

```
In [182]:
```# Data in a json format:
json_data = """\
{
"a": [1,2,3],
"b": [4,5,6],
"greeting" : "Hello"
}"""
import json
loaded_json=json.loads(json_data)
loaded_json

```
Out[182]:
```

Your data sits in something that looks like a Python dictionary, and in a single line of code, you can load it into a Python dictionary for use later.

In the same way, you can, with a single line of code, put a bunch of variables into a dictionary, and then output to a file using json:

```
In [183]:
```json.dumps({"a":[1,2,3],"b":[9,10,11],"greeting":"Hola"})

```
Out[183]:
```

Functional programming is a very broad subject. The idea is to have a series of functions, each of which generates a new data structure from an input, without changing the input structure at all. By not modifying the input structure (something that is called not having *side effects*), many guarantees can be made about how independent the processes are, which can help parallelization and guarantees of program accuracy. There is a Python Functional Programming HOWTO in the standard docs that goes into more details on functional programming. I just wanted to touch on a few of the most important ideas here.

There is an **operator** module that has function versions of most of the Python operators. For example:

```
In [184]:
```from operator import add, mul
add(1,2)

```
Out[184]:
```

```
In [185]:
```mul(3,4)

```
Out[185]:
```

These are useful building blocks for functional programming.

**lambda** operator allows us to build *anonymous functions*, which are simply functions that aren't defined by a normal **def** statement with a name. For example, a function that doubles the input is:

```
In [186]:
```def doubler(x): return 2*x
doubler(17)

```
Out[186]:
```

We could also write this as:

```
In [187]:
```lambda x: 2*x

```
Out[187]:
```

And assign it to a function separately:

```
In [188]:
```another_doubler = lambda x: 2*x
another_doubler(19)

```
Out[188]:
```

**lambda** is particularly convenient (as we'll see below) in passing simple functions as arguments to other functions.

**map** is a way to repeatedly apply a function to a list:

```
In [189]:
```list(map(float,'1 2 3 4 5'.split()))

```
Out[189]:
```

**reduce** is a way to repeatedly apply a function to the first two items of the list. There already is a **sum** function in Python that is a reduction:

```
In [190]:
```sum([1,2,3,4,5])

```
Out[190]:
```

We can use **reduce** to define an analogous **prod** function:

```
In [191]:
```from functools import reduce
def prod(l): return reduce(mul,l)
prod([1,2,3,4,5])

```
Out[191]:
```

We've seen a lot of examples of **objects** in Python. We create a string object with quote marks:

```
In [192]:
```mystring = "Hi there"

and we have a bunch of methods we can use on the object:

```
In [193]:
```mystring.split()

```
Out[193]:
```

```
In [194]:
```mystring.startswith('Hi')

```
Out[194]:
```

```
In [195]:
```len(mystring)

```
Out[195]:
```

Object oriented programming simply gives you the tools to define objects and methods for yourself. It's useful anytime you want to keep some data (like the characters in the string) tightly coupled to the functions that act on the data (length, split, startswith, etc.).

As an example, we're going to bundle the functions we did to make the 1d harmonic oscillator eigenfunctions with arbitrary potentials, so we can pass in a function defining that potential, some additional specifications, and get out something that can plot the orbitals, as well as do other things with them, if desired.

```
In [196]:
```class Schrod1d:
"""\
Schrod1d: Solver for the one-dimensional Schrodinger equation.
"""
def __init__(self,V,start=0,end=1,npts=50,**kwargs):
m = kwargs.get('m',1.0)
self.x = np.linspace(start,end,npts)
self.Vx = V(self.x)
self.H = (-0.5/m)*self.laplacian() + np.diag(self.Vx)
return
def plot(self,*args,**kwargs):
titlestring = kwargs.get('titlestring',"Eigenfunctions of the 1d Potential")
xstring = kwargs.get('xstring',"Displacement (bohr)")
ystring = kwargs.get('ystring',"Energy (hartree)")
if not args:
args = [3]
x = self.x
E,U = np.linalg.eigh(self.H)
h = x[1]-x[0]
# Plot the Potential
plt.plot(x,self.Vx,color='k')
for i in range(*args):
# For each of the first few solutions, plot the energy level:
plt.axhline(y=E[i],color='k',ls=":")
# as well as the eigenfunction, displaced by the energy level so they don't
# all pile up on each other:
plt.plot(x,U[:,i]/np.sqrt(h)+E[i])
plt.title(titlestring)
plt.xlabel(xstring)
plt.ylabel(ystring)
return
def laplacian(self):
x = self.x
h = x[1]-x[0] # assume uniformly spaced points
n = len(x)
M = -2*np.identity(n,'d')
for i in range(1,n):
M[i,i-1] = M[i-1,i] = 1
return M/h**2

The ** init()** function specifies what operations go on when the object is created. The

For example, to do an infinite square well potential, we have a function that is 0 everywhere. We don't have to specify the barriers, since we'll only define the potential in the well, which means that it can't be defined anywhere else.

```
In [197]:
```square_well = Schrod1d(lambda x: 0*x,m=10)
square_well.plot(4,titlestring="Square Well Potential")

```
```

We can similarly redefine the Harmonic Oscillator potential.

```
In [198]:
```ho = Schrod1d(lambda x: x**2,start=-3,end=3)
ho.plot(6,titlestring="Harmonic Oscillator")

```
```

Let's define a finite well potential:

```
In [199]:
```def finite_well(x,V_left=1,V_well=0,V_right=1,d_left=10,d_well=10,d_right=10):
V = np.zeros(x.size,'d')
for i in range(x.size):
if x[i] < d_left:
V[i] = V_left
elif x[i] > (d_left+d_well):
V[i] = V_right
else:
V[i] = V_well
return V
fw = Schrod1d(finite_well,start=0,end=30,npts=100)
fw.plot()

```
```

A triangular well:

```
In [200]:
```def triangular(x,F=30): return F*x
tw = Schrod1d(triangular,m=10)
tw.plot()

```
```

Or we can combine the two, making something like a semiconductor quantum well with a top gate:

```
In [201]:
```def tri_finite(x): return finite_well(x)+triangular(x,F=0.025)
tfw = Schrod1d(tri_finite,start=0,end=30,npts=100)
tfw.plot()

```
```

The first rule of speeding up your code is not to do it at all. As Donald Knuth said:

"We should forget about small efficiencies, say about 97% of the time: premature optimization is the root of all evil."

The second rule of speeding up your code is to only do it if you really think you need to do it. Python has two tools to help with this process: a timing program called **timeit**, and a very good code profiler. We will discuss both of these tools in this section, as well as techniques to use to speed up your code once you know it's too slow.