EGR 103/Concept List Fall 2019

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This page will be used to keep track of the commands and major concepts for each lecture in EGR 103.

Lecture 1 - Introduction

  • Class web page: EGR 103L; assignments, contact info, readings, etc - see slides on Errata/Notes page
  • Sakai page: Sakai 103L page; grades, surveys and tests, some assignment submissions
  • CampusWire page: CampusWire 103L page; message board for questions - you need to be in the class and have the access code to subscribe.

Lecture 2 - Programs and Programming

Lecture 3 - "Number" Types

  • Python is a "typed" language - variables have types
  • We will use eight types:
    • Focus of the day: int, float, and array
    • Focus a little later: string, list, tuple
    • Focus later: dictionary, set
  • int: integers; Python can store these perfectly
  • float: floating point numbers - "numbers with decimal points" - Python sometimes has problems
  • array
    • Requires numpy, usually with import numpy as np
    • Organizational unit for storing rectangular arrays of numbers
  • Math with "Number" types works the way you expect
    • ** * / // % + -
  • 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 an array or put information into an array
  • a[0] is the element in a at the start
  • a[3] is the element in a three away from the start
  • 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 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 - Other Types and Functions

  • 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
  • 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)
  • 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
  • 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
    • Dictionary at tutorialspoint
  • Defined functions can be multiple lines of code and have multiple outputs.
    • Four different types of input parameters:
      • 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 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 5 - Format, Logic, Decisions, and Loops

Lecture 6 - String Things and Loops

  • ord to get numerical value of each character
  • chr to get character based on integer
  • map(fun, sequence) to apply a function to each item in a sequence
  • Basics of while loops
  • Basics of for loops
  • List comprehensions
    • [FUNCTION for VAR in SEQUENCE if LOGIC]
      • The FUNCTION should return a single thing (though that thing can be a list, tuple, etc)
      • The "if LOGIC" part is optional
      • [k for k in range(3)] creates [0, 1, 2]
      • [k**2 for k in range (5, 8)] creates [25, 36, 49]
      • [k for k in 'hello' if k<'i'] creates ['h', 'e']
      • [(k,k**2) for k in range(11) if k%3==2] creates [(2, 4), (5, 25), (8, 64)]
    • Wait - that's the simplified version...here:
  • 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 7 - Applications

# tpir.py from class:

import numpy as np
import time

def create_price(low=100, high=1500):
    return np.random.randint(low, high+1)
    
def get_guess():
    guess = int(input('Guess: '))
    return guess
    
def check_guess(actual, guess):
    if actual > guess:
        print('Higher!')
    elif actual < guess:
        print('Lower!')

    
if __name__ == '__main__':
    #print(create_price(0, 100))
    the_price = create_price()
    the_guess = get_guess()
    start_time = time.clock()
    #print(the_guess)
    while the_price != the_guess and (time.clock() < start_time+30):
        check_guess(the_price, the_guess)
        the_guess = get_guess()
    
    if the_price==the_guess:    
        print('You win!!!!!!!')
    else:
        print('LOOOOOOOOOOOOOOOSER')

# nato_trans.py from class:

fread = open('NATO.dat', 'r')

d = {}

for puppies in fread:
    #print(puppies) $ if you want to see the whole line
    
    #key = puppies[0]
    #value = puppies[:-1]
    #d[key] = value
    
    d[puppies[0]] = puppies[:-1]

fread.close()

hamster = input('Word: ').upper()

for kittens in hamster:
    #print(d[letter], end=' ')
    print(d.get(kittens, 'XXX'), end=' ')
    
'''
In class - one question was "in cases where there is not a code, can it
return the original value instead of XXX" -- yes:
    print(d.get(kittens, kittens))
'''
  • Data file we used:
# NATO.dat from class:

Alfa
Bravo
Charlie
Delta
Echo
Foxtrot
Golf
Hotel
India
Juliett
Kilo
Lima
Mike
November
Oscar
Papa
Quebec
Romeo
Sierra
Tango
Uniform
Victor
Whiskey
X-ray
Yankee
Zulu

Lecture 8 - Taylor Series and Iterative Solutions

  • 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 - Binary and Floating Point Numbers

  • Different number systems convey information in different ways.
    • Roman Numerals
    • Chinese Numbers
    • Ndebe Igbo Numbers
    • Binary Numbers
      • We went through how to convert between decimal and binary
    • Kibibytes et al
  • "One billion dollars!" may not mean the same thing to different people: Long and Short Scales
  • 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
  • In cases where mathematical formulas have limits to infinity, you have to pick numbers large enough to properly calculate values but not so large as to cause errors in computing:
    • $$e^x=\lim_{n\rightarrow \infty}\left(1+\frac{x}{n}\right)^n$$
# 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 - Monte Carlo Methods

  • See walk1 in Resources section of Sakai

Lecture 11 - Style, Code Formatters, Docstrings, and More Walking

  • Discussion of PEP and PEP8 in particular
  • Autostylers include black, autopep8, and yapf -- we will mainly use black
    • To get the package:
      • On Windows start an Anaconda Prompt (Start->Anaconda3->Anaconda Prompt) or on macOS open a terminal and change to the \users\name\Anaconda3 folder
      • pip install black should install the code
    • To use that package:
      • Change to the directory where you files lives. On Windows, to change drives, type the driver letter and a colon by itself on a line, then use cd and a path to change directories; on macOS, type cd /Volumes/NetID where NetID is your NetID to change into your mounted drive.
      • Type black FILE.py and note that this will actually change the file - be sure to save any changes you made to the file before running black
      • As noted in class, black automatically assumes 88 characters in a line; to get it to use the standard 80, use the -l 80 adverb, e.g. black FILE.py -l 80
  • Docstrings
    • We will be using the numpy style at docstring guide
    • Generally need a one-line summary, summary paragraph (if needed), a list of parameters, and a list of returns
    • Specific formatting chosen to allow Spyder's built in help tab to format file in a pleasing way
  • More walking
    • We went through the walk_1 code again and then decided on three different ways we could expand it and looked at how that might impact the code:
    • Choose from more integers than just 1 and -1 for the step: very minor impact on code
    • Choose from a selection of floating point values: minor impact other than a bit of documentation since ints and floats operate in similar ways
    • Walk in 2D rather than along a line: major impact in terms of needing to return x and y value for the step, store x and y value for the location, plot things differently
    • All codes from today will be on Sakai in Resources folder

Lecture 12 - Arrays and Matrix Representation in Python

  • 1-D and 2-D Arrays
    • Python does mathematical operations differently for 1 and 2-D arrays
  • Matrix multiplication (by hand)
  • Matrix multiplication (using @ in Python)
  • To multiply matrices A and B ($$C=A\times B$$ in math or C=A@B in Python) using matrix multiplication, the number of columns of A must match the number of rows of B; the results will have the same number of rows as A and the same number of columns as B. Order is important
  • Setting up linear algebra equations
  • Determinants of matrices and the meaning when the determinant is 0
    • Shortcuts for determinants of 1x1, 2x2 and 3x3 matrices (see class notes for processes)
$$\begin{align*} \mbox{det}([a])&=a\\ \mbox{det}\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)&=ad-bc\\ \mbox{det}\left(\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\right)&=aei+bfg+cdh-afh-bdi-ceg\\ \end{align*}$$

Lecture 13 - Linear Algebra and Solutions

  • Inverses of matrices:
    • Generally, $$\mbox{inv}(A)=\frac{\mbox{cof}(A)^T}{\mbox{det}(A)}$$ there the superscript T means transpose...
      • And $$\mbox{det}(A)=\sum_{i\mbox{ or }j=0}^{N-1}a_{ij}(-1)^{i+j}M_{ij}$$ for some $$j$$ or $$i$$...
        • And $$M_{ij}$$ is a minor of $$A$$, specifically the determinant of the matrix that remains if you remove the $$i$$th row and $$j$$th column or, if $$A$$ is a 1x1 matrix, 1
          • And $$\mbox{cof(A)}$$ is a matrix where the $$i,j$$ entry $$c_{ij}=(-1)^{i+j}M_{ij}$$
    • Good news - for this class, you need to know how to calculate inverses of 1x1 and 2x2 matrices only:
$$ \begin{align} \mbox{inv}([a])&=\frac{1}{a}\\ \mbox{inv}\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)&=\frac{\begin{bmatrix}d &-b\\-c &a\end{bmatrix}}{ad-bc} \end{align}$$
  • Converting equations to a matrix system:
    • For a certain circuit, conservation equations learned in upper level classes will yield the following two equations:
$$\begin{align} \frac{v_1-v_s}{R1}+\frac{v_1}{R_2}+\frac{v_1-v_2}{R_3}&=0\\ \frac{v_2-v_1}{R_3}+\frac{v_2}{R_4}=0 \end{align}$$
  • Assuming $$v_s$$ and the $$R_k$$ values are known, to write this as a matrix equation, you need to get $$v_1$$ and $$v_2$$ on the left and everything else on the right:
$$\begin{align} \left(\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}\right)v_1+\left(-\frac{1}{R_3}\right)v_2&=\frac{v_s}{R_1}\\ \left(-\frac{1}{R_3}\right)v_1+\left(\frac{1}{R_3}+\frac{1}{R_4}\right)v_2&=0 \end{align}$$
  • Now you can write this as a matrix equation:

$$ \newcommand{\hmatch}{\vphantom{\frac{1_s}{R_1}}} \begin{align} \begin{bmatrix} \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3} & -\frac{1}{R_3} \\ -\frac{1}{R_3} & \frac{1}{R_3}+\frac{1}{R_4} \end{bmatrix} \begin{bmatrix} \hmatch v_1 \\ \hmatch v_2 \end{bmatrix} &= \begin{bmatrix} \frac{v_s}{R_1} \\ 0 \end{bmatrix} \end{align}$$

Lecture 14 - Solution Sweeps, Norms, and Condition Numbers

  • See Python:Linear_Algebra#Sweeping_a_Parameter for example code on solving a system of equations when one parameter (either in the coefficient matrix or in the forcing vector or potentially both)
  • Chapra 11.2.1 for norms
  • Chapra 1.2.2 for condition numbers
    • np.linalg.cond() in Python
    • Note: base-10 logarithm of condition number gives number of digits of precision possibly lost due to system geometry and scaling (top of p. 295 in Chapra)

Lecture 15

  • Test Review

Lecture 16

  • Test



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