EGR 103/Concept List/F22

Lecture 1 - 8/29 - Course Introduction

• Main class page: EGR 103L
  • Includes links to Sakai, Pundit, and Ed pages
• Sakai page: Sakai 103L page; grades, surveys and tests, some assignment submissions; first day slideshow in Resources section

Lecture 2 - 8/27 - Programs and Programming

• Almost all languages have input, output, math, conditional execution (decisions), and repetition (loops)
• Seven steps of programming The Seven Steps Poster. Also, for Monday's class:
  • Watch video on Developing an Algorithm
  • Watch video on A Seven Step Approach to Solving Programming Problems
• Problem: Consider how to decide if a number is a prime number
  • Some "shortcuts" for specific factors but need to have a generalized approach
  • See if number is evenly divisible by any integer between 2 and the square root of the number - but how do we ask the computer to do that?
• Quick tour of Python
  • Console (with history tab), variable explorer (with other tabs), and editing window
  • Main numerical types: whole numbers (int) and numbers with decimals (float)
  • Can use % (called "mod") to get "remainder"
    • If both items are integers, result is an integer; if either is a float, result is a float
  • Relational operators: < <= == >= > !=
    • Result is is either True or False
• Comments in code:
  • If there is a #, Python ignores everything remaining in that line after the #
  • If there are """ or , Python ignores everything until the closing """ or
  • If you use # %% in Spyder, the editing window will set up a cell and light up the cell your cursor is in. Cells have no impact on how the code runs, just how the code appears in the window

Lecture 3 - 9/5 - "Number" Types

• Python is a "typed" language
  • Focus of the day: int, float, and array
    • 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
  • Focus a little later: string, list, tuple
  • Focus later: dictionary, set
  • Focus way later: map, filter, zip
• Basic operations and types
  • + - * // (rounded division) and % (remainder / modulo) produce int 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
  • VAR = input("prompt: ") will ask the user for a value and stores whatever they type as a string (broken in some versions of Spyder!)
  • NUM = int(VAR)
    • If VAR is an int or a float, it will return an int rounded towards 0
    • If VAR is a string, it will return an int only if the string looks exactly like an integer
  • NUM = float(VAR)
    • If VAR is an int or a float, it will return a float with the same value
    • If VAR is a string, it will return a float if the string looks like a float, including scientific notation such as float("1.23e4")
• Arrays
  • 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
  • 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
• 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!

Lecture 4 - 9/9 - Other Types

• 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 or l.append().
• 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.
• 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" is True since the "e" comes before the "i"
    • "Zebra" < "apple" is True since the upper case "Z" is before the lower case "a"
    • "go" < "gone" is True 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

• 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
• 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")

Lecture 5 - 9/12 - Functions

• Defined functions can be multiple lines of code and have multiple outputs.
• The function can see everything in main, but main cannot see things created in the function.
  • Best bet is to pretend the function cannot see things in main - pass everything in that you need to see!

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

  • Four different types of input parameters - we only really talked about the first three kinds:
    • Required (listed first)
    • Named with defaults (second)
    • 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
• Dictionaries
  • Object where the index (called the key) can be any immutable (integer, float, string, or tuple); the value can be anything.

Lecture 6 - 9/16 - Loops and Decisions

• The Price is Right!
• Logic
  • <= < == >= > != work with many types; just be careful about interpreting
  • not can reverse while and and or can combine logical expressions; most expressions can be written in two ways (while/if TRUE or while/if not FALSE)
• 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
• Basics of loops using while
  • Must have logic; gets evaluated at start and not again until branch ends
  • Useful when you do not know how many times a loop will run (input validation example)
  • break in a loop can break out early
  • continue in a loop goes back to the top early
• Basics of loops using for
  • for VAR in ITERABLE
    • VAR will take on each item in ITERABLE one at a time; ITERABLE can be a string, list, tuple, array, or range
      • range(N) creates something similar to [0, 1, 2, 3, 4, ..., N-1]
      • range(M, N) creates something similar to [M, M+1, M+2, ..., N-1]
      • range only creates integers

Lecture 7 - 9/19 - Loops and Accounting

• Looked at looping through letters in a phrase
• Logic: ITEM in THING will be true if ITEM is a subunit of THING
  • "a" in "subway" would be True
  • "way" in "subway" would be True
  • "as" in "subway" would be False
• Many ways to keep track of items
  • Counter variable
  • List with different indices to track different items
• Many ways to evaluate items
  • For loop with an if tree
  • For loop with another for loop

Lecture 8 - 9/23 - Dictionaries

• 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
  • Dictionary at tutorialspoint
• Storing values in a dictionary
  • Different loops
  • zip
• Translation demo with Morse code and NATO phonetic alphabet
  • Loading lines of text from a file
  • Splitting strings with split
• Don't copy code from a PDF!
  • waffle versus waffle versus waffle
    • waffle versus waffle versus waffle
  • In the first version, the f-f-l are three separate characters, in the second version it is ff-l and in the third, it is ffl; if you try to copy/paste these into Spyder, the words are 6, 5, or 4 characters, respectively!

Lecture 9 - 9/26 - Random Numbers and Logical Masks

• np.random.randint(low, high, size)
  • low defaults to 0
  • number is an integer [low, high)
  • if size is 2 dimensions or more, must be in a tuple
• np.random.uniform(low, high, size)
  • low defaults to 0, high defaults to 1
  • number is a float [low, high) with a uniform distribution over range
  • if size is 2 dimensions or more, must be in a tuple
• np.random.normal(loc, scale, size)
  • loc (average) defaults to 0, scale (spread) defaults to 1
  • number can be anything; concentrated around loc depending on how big scale is
  • if size is 2 dimensions or more, must be in a tuple
• Basics of Python:Logical Masks

Lecture 10 - 9/30 - 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 11 - 10/3 - 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

• Want to see Amharic?

list(map(chr, range(4608, 4992)))

• Want to see the Greek alphabet?

for k in range(913,913+25):
    print(chr(k), chr(k+32))

Lecture 12 - 9/27 - Monte Carlo Methods

• From Wikipedia: Monte Carlo method
• Several demonstrations in class (coins, dice, darts)