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EGR 103/Concept List Spring 2020

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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
  • Pundit page: EGR 103; reference lists
  • CampusWire page: CampusWire 103L page; message board for questions - you need to be in the class and have the access code 6393 to subscribe.

Lecture 2 - Programs and Programming


Lecture 3 - "Number" Types

  • To play with Python:
    • Install it on your machine or a public machine: Download
  • Quick tour of Python
    • Editing window, variable explorer, and console
    • Run icon (F5)
  • You are not expected to remember any of the specifics about how Python stores things or works with them yet!
  • Python is a "typed" language - variables have types
  • We will use eight types:
    • Focus of the day: int, float, and array
    • Basics today, 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 storing floating point items exactly
  • 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
    • ** * / // % + -
  • 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 string contains an immutable collection of characters
    • Using + with strings concatenates strings
    • Using * with strings makes a string with the original repeated
  • A tuple contains an immutable collection of other types
    • Using + with tuples concatenates tuples
    • Using * with tuples makes a tuple with the original repeated
  • A list contains an immutable collection of other types
    • Using + with lists concatenates lists
    • Using * with lists makes a list with the original repeated

Lecture 4 - More on Types

  • 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
  • More advanced slices:
  • 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 5 - Printing and Decisions

Lecture 6 - Decisions

  • Rolling dice
# tpir.py from class:

def roll(num=1, sides=6):
    print(num, sides)
    dice = np.random.randint(1, sides+1, num)
    print(dice)
    return dice
  • Checking rolls:
# tpir.py from class:

import numpy as np

def eval_hand(dice):
    sorted_dice = dice*1
    sorted_dice.sort()
    if sorted_dice[0] == sorted_dice[-1]:
        hand_type = 3
        tie_value = [sorted_dice[0]]
    elif (sorted_dice[0] == sorted_dice[1]-1 and
          sorted_dice[0] == sorted_dice[2]-2):
        hand_type = 2
        tie_value = [sorted_dice[2]]
    elif (sorted_dice[1] == sorted_dice[0] or
          sorted_dice[1] == sorted_dice[2]):
        hand_type = 1
        tie_value = [sorted_dice[1]]
        if sorted_dice[1] == sorted_dice[0]:
            tie_value += [sorted_dice[2]]
        else:
            tie_value += [sorted_dice[0]]
    else:
        hand_type = 0
        tie_value = sorted_dice[::-1]
    return hand_value, tie_value

Lecture 7 - Loops

# tpir.py from class:

import numpy as np

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__':
    the_price = create_price()
    the_guess = get_guess()
    while the_price != the_guess:
        check_guess(the_price, the_guess)
        the_guess = get_guess()
    
    if the_price==the_guess:    
        print('You win!!!!!!!')
    else:
        print('LOOOOOOOOOOOOOOOSER')
  • Getting temperatures:
# get_temps.py from class:

T = int(input('Temp: '))
Tlist = []
while T>=0:
    Tlist += [T]
    print(Tlist)

    T = int(input('Temp: '))

Lecture 8 - 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 - Dictionaries and Loading

Lecture 10 - Monte Carlo Methods

  • From Wikipedia: Monte Carlo method
  • See file in "Programs From Class" folder under Resources on Sakai

Lecture 11 - Binary and Floating Point Numbers

  • Different number systems convey information in different ways.
    • Roman Numerals
    • Chinese 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

Lecture 12 - Matrix Operations

  • 1D arrays are neither rows nor columns - they are 1D!
  • 2D arrays have...two dimensions, one of which might be 1
  • Dot products as heart of matrix multiplication
    • Inner dimensions must match; outer dimensions equal dimensions of result
  • Reformatting linear algebra expressions as matrix equations
  • Calculating determinants and inverses
    • 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*} $
  • 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} $


Lecture 13 - Linear Algebra

  • 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)
  • 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 - Linear Algebra with Parameter Sweeps

Lecture 15 - Review

EGR 103/Spring 2020/Test 1

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