EGR 103/Concept List/S22

Lecture 1 - 1/7 - Course Introduction

• Main class page: EGR 103L.
• See information on PDF of slide show on Sakai in Resources section
• Sakai page: Sakai 103L page; grades, surveys and tests, some assignment submissions
• Pundit page: EGR 103; reference lists
• Ed page: 103L page; message board for questions

Lecture 2 - 1/10- Programs and Programming

• Almost all languages have input, output, math, conditional execution (decisions), and repetition (loops)
• Seven steps of programming The Seven Steps Poster
  • Watch video on Developing an Algorithm
  • Watch video on A Seven Step Approach to Solving Programming Problems
• Consider how to decide if a number is a prime number - code is available in the Box drive for the class under Lectures / Lec 03
• To play with Python:
  • Install it on your machine or a public machine: Download
• Quick tour of Python
  • Editing window, variable explorer, and console
  • Main numerical types: whole numbers (int) and numbers with decimals (float)
  • + - * // (rounded division) and % (remainder / modula) produce in if both sides are an int, float if either or both are floats
  • / (regular division) and // (rounded division) produces float with ints or floats
  • ** to do powers

Lecture 3 - 1/14 - Indexing and "Number" Types

• Demonstrated how cells work in Spyder editor
• Slices allow us to extract information from a collection or change information in mutable collections
  • a[0] is the element in a at the start
  • a[3] is the element in a three away from the start
  • a[-1] is the last element of a
  • a[-2] is the second-to-last element of a
  • a[:] is all the elements in a because what is really happening is:
    • a[start:until] where start is the first index and until is just *past* the last index;
    • a[3:7] will return a[3] through a[6] in a 4-element array
    • a[start:until:increment] will skip indices by increment instead of 1
    • To go backwards, a[start:until:-increment] will start at an index and then go backwards until getting at or just past until.
  • For 2-D arrays, you can index items with either separate row and column indices or indices separated by commas:
    • a[2][3] is the same as a[2, 3]
    • Only works for arrays!
• Getting values from a user:
  • VAR = input("prompt: ") will ask the user for a value and stores whatever they type as a string
  • NUM = int(VAR) will convert the item in VAR to an integer if it looks like an integer; error otherwise
  • NUM = float(VAR) will convert the item in VAR to a float if it looks like a float; error otherwise
• Finished prime number checker - code is available in the Box drive for the class under Lectures / Lec 03
  • Looked at for loops for running code multiple times
  • Looked at if...else trees for making decisions
  • Created a variable to track whether we thought the number was prime
• Python doesn't know everything to start with; may need to import things
  • import MODULE means using MODULE.function() to run
  • import MODULE as NAME means using NAME.function() to run
  • from MODULE import FUNCTION means using FUNCTION() to run
  • from MODULE import * means bringing in everything from the module under their own names - dangerous!
• Python is a "typed" language - variables have types. We will use several types:
  • Focus of the day: int, float, and array
  • Focus a little later: string, list, tuple
  • Focus later: dictionary, set
  • Focus way later: map, filter, zip
• int: integers; Python 3 can store these perfectly
• float: floating point numbers - "numbers with decimal points" - Python sometimes has problems storing floating point items exactly
• array
  • Requires numpy, usually with import numpy as np
  • Organizational unit for storing rectangular arrays of numbers
  • Generally create with np.array(LIST) where depth of nested LIST is dimensionality of array
    • np.array([1, 2, 3]) is a 1-dimensional array with 3 elements
    • np.array([[1, 2, 3], [4, 5, 6]]) is a 2-dimension array with 2 rows and 3 columns
• Math with "Number" types works the way you expect
  • ** * / // % + -
  • With arrays, * and / work element by element; *matrix* multiplication is a different character (specifically, @)
• Relational operators can compare "Number" Types and work the way you expect with True or False as an answer
  • < <= == >= > !=
  • With arrays, either same size or one is a single value; result will be an array of True and False the same size as the array

Lecture 4 - 1/21 - Other Types, Formatted Printing

• Lists are set off with [ ] and entries can be any valid type (including other lists!); entries can be of different types from other entries; list items can be changed and mutable items within lists can be changed. Lists can be "grown" by using += with the list.
• Tuples are indicated by commas without square brackets (and are usually shown with parentheses - which are required if trying to make a tuple an entry in a tuple or a list); tuple items cannot be changed but mutable items within tuples can be
• Strings are set off with " " or ' ' and contain characters; string items cannot be changed
• For lists, tuples, and strings:
  • Using + concatenates the two collections
  • Using * with them makes creates a collection with the original repeated that many times
  • Using += will create a new item with something appended to the old item; the "something" needs to be the same type (list, tuple, or string); this may seem to break the "can't be changed" rule but really a += b is a = a + b which creates a new a.
• Creating formatted strings using {} and .format() (format strings, standard format specifiers) -- focus was on using s for string and e or f for numerical types, minimumwidth.precision, and possibly a + in front to force printing + for positive numbers.
  • Using {} by themselves will substitute items in order from the format() function into the string that gets created
  • Putting a number in the {} will tell format which thing to get
  • Format specification comes after a : in the {}; if you do not specify a location index, you still have to put a colon in the {}
  • {:s} means string and {:Xs} where X is an integer means reserve at least that much space for a left-formatted string
  • {:f} means floating point (default 6 digits after decimal point) and {:X.Yf} reserves at least X spaces (including + or - and the . if it is there) with Y digits after the decimal point for t right-justified number
  • {:e} means floating point (default 6 digits after decimal point) and {:X.Ye} reserves at least X spaces (including + or - and the . if it is there and the letter e and the + or - after the e and the two or three digit number after that) with Y digits after the decimal point for t right-justified number
• Aside - Format Specification Mini-Language has all the possibilities; we will cover some but not all of these in later classes
• You can enter numbers in scientific notation with a number followed by the letter 3 and then a number or negative number for the power of 10; for example, x = 6.02e23 or e = -1.6e-19
  • float can convert scientific notation as well:

float("1e-5")

• To read more:
  • Note! Many of the tutorials below use Python 2 so instead of print(thing) it shows print thing
  • Lists at tutorialspoint
  • Tuples at tutorialspoint

Lecture 5 - 1/24 - Functions

• Defined functions can be multiple lines of code and have multiple outputs.

•  def FNAME(local1, local2, ...):
       CODE
       return THING1, THING2, ...
  

  • Four different types of input parameters - we mainly talked about the first two kinds:
    • Required (listed first)
    • Named with defaults (second)
  • We will cover the other kinds in more detail later
    • Additional positional arguments ("*args") (third)
      • Function will create a tuple containing these items in order
    • Additional keyword arguments ("**kwargs") (last)
      • Function will create a dictionary of keyword and value pairs
  • Function ends when indentation stops or when the function hits a return statement
  • Return returns single item as an item of that type; if there are multiple items returned, they are stored and returned in a tuple
  • If there is a left side to the function call, it either needs to be a single variable name or a tuple with as many entries as the number of items returned

Lecture 6 - 1/28 - Relational Operators, Decisions, and Loops

• <= < == >= > != work with many types; just be careful about interpreting
• not can reverse while and and or can combine logical expressions
• Basics of decisions using if...elif...else
  • Must have logic after if
  • Can have as many elif with logic after
  • Can have an else without logic at the end
  • Flow is solely dependent on indentation!
  • Branches can contain other trees for follow-up questions
• If you have an array of logical True and False values, use ARRAY.all() to determine if all the entries are true or ARRAY.any() to determine if one or more are true.
• Basics of while loops and for loops.
• Using a list to keep track of counts.
• Characters in strings have "numerical" values based on the ASCII table (https://www.asciitable.com/)
  • Numbers are earlier than lower case letters; lower case letters are earlier than upper case letters
  • Strings are sorted character by character; if one string is shorter than another, it is considered less
    • "Hello" < "Hi" since the "e" comes before the "i"
    • "Zebra" < "apple" since the upper case "Z" is before the lower case "a"
    • "go" < "gone" since the first two characters match and then the word is done
• To get the numerical value of a single character, use ord("A") or replace the A with the character you want
• To get the character a number represents, use chr(NUM)
• To apply either ord or chr to multiple items, use a map; to see the results, make a list out of the map
• Trinket

Lecture 7 - 1/31 - More on Loops; Dictionaries

• break will break out of a loop early, while continue will cause a loop to go to its next iteration immediately
• Using the enumerate type to provide a collection of indices and values to a loop
• Dictionaries are collections of key : value pairs set off with { }; keys can be any immutable type (int, float, string, tuple) and must be unique; values can be any type and do not need to be unique
  • d[KEY] = VALUE will add the VALUE to the dictionary with a key of KEY
  • d[KEY] will return the value associated with the KEY if there is one OR will give an error if there is not
  • d.get(KEY) will return the value associated with KEY if there is one or None if there is not
  • d.get(KEY, DEFAULT) will return the value associated with KEY if there is one or default if there is not
  • Dictionary at tutorialspoint
• Translation demo with Morse code and NATO phonetic alphabet

Lecture 8 - 2/4 - Iterative Methods

• Taylor series fundamentals
• Maclaurin series approximation for exponential uses Chapra 4.2 to compute terms in an infinite sum.

\( y=e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!} \)

so

\( \begin{align} y_{init}&=1\\ y_{new}&=y_{old}+\frac{x^n}{n!} \end{align} \)

• Newton Method for finding square roots uses Chapra 4.2 to iteratively solve using a mathematical map. To find \(y\) where \(y=\sqrt{x}\): \( \begin{align} y_{init}&=1\\ y_{new}&=\frac{y_{old}+\frac{x}{y_{old}}}{2} \end{align} \)
• See Python version of Fig. 4.2 and modified version of 4.2 in the Resources section of Sakai page under Chapra Pythonified

Lecture 9 - 2/7 - Binary

• Different number systems convey information in different ways.
  • Roman Numerals
  • Chinese Numbers
  • Binary Numbers
    • We went through how to convert between decimal and binary
• Floats (specifically double precision floats) are stored with a sign bit, 52 fractional bits, and 11 exponent bits. The exponent bits form a code:
  • 0 (or 00000000000): the number is either 0 or a denormal
  • 2047 (or 11111111111): the number is either infinite or not-a-number
  • Others: the power of 2 for scientific notation is 2**(code-1023)
    • The largest number is thus just *under* 2**1024 (ends up being (2-2**-52)**1024\(\approx 1.798\times 10^{308}\).
    • The smallest normal number (full precision) is 2**(-1022)\(\approx 2.225\times 10^{-308}\).
    • The smallest denormal number (only one significant binary digit) is 2**(-1022)/2**53 or 5e-324.
  • When adding or subtracting, Python can only operate on the common significant digits - meaning the smaller number will lose precision.
  • (1+1e-16)-1=0 and (1+1e-15)-1=1.1102230246251565e-15
  • Avoid intermediate calculations that cause problems: if x=1.7e308,
    • (x+x)/x is inf
    • x/x + x/x is 2.0
  • $$e^x=\lim_{n\rightarrow \infty}\left(1+\frac{x}{n}\right)^n$$

# Exponential Demo

<syntaxhighlightlang=python> import numpy as np import matplotlib.pyplot as plt

def exp_calc(x, n):

   return (1 + x/n)**n

if __name__ == "__main__":

   n = np.logspace(0, 17, 1000)
   y = exp_calc(1, n)
   fig, ax = plt.subplots(num=1, clear=True)
   ax.semilogx(n, y)
   fig.savefig('ExpDemoPlot1.png')
   
   # Focus on right part
   n = np.logspace(13, 16, 1000)
   y = exp_calc(1, n)
   fig, ax = plt.subplots(num=2, clear=True)
   ax.semilogx(n, y)
   fig.savefig('ExpDemoPlot2.png')

</syntaxhighlight>

• 

  estimates for calculating $$e$$ with $$n$$ between 1 and $$1*10^{17}$$

• 

  $$n$$ between $$10^{13}$$ and $$10^{16}$$ showing region when roundoff causes problems

Lecture 10 - 2/11 - Monte Carlo Methods

• From Wikipedia: Monte Carlo method
• Wrote program to flip one coin, flip many coins, and keep track of flips
  • one_coin, many_coins, too_many_coins in Box folder for Lec 10
• Wrote program to roll on die, roll many dice, and keep track of rolls; also wrote a program to see how often "n of a kind" happens
  • one_die, many_dice, same in Box folder for Lec 10
• Discussed Random Walks and went through code that will graph results - documented version of walker code: