Joanna Bieri

Applied Math Professor

Department of Economics

University of Redlands

Numerical Analysis
Fall 2022

Office Location:
Duke Hall #209
Phone: 909-748-8630
E-Mail: joanna_bieri@redlands.edu
(Email or Teams are my preferred contact methods)

Programming Lab:
Wednesday 6-8pm Duke 206

Office Hours:
Make an Appointment:
Book an appointment on Calendly

Link to our Canvas Class
This is where grades will be posted and announcements will be made.

Link to the Book

Link to Google Colab
This is where you can collaborate on a cloud based Jupyter Notebook. Sign up for a free account. You should still have Anaconda and Jupyter Notebooks on your computer. It runs much faster off the cloud!



• Important Course Documents
  
  Course Syllabus
  Schedule of Topics
  NOTE: as the semester progresses we may change up the schedule a bit to suit our class pace and interests. The most recent schedule will be posted here.
  
• Daily Assignments - Reading - Handouts
  
  Each day I will post the lecture videos, homework, additional reading, and other information. Make sure to check here for each day of class.
  • DAY 1 - Tuesday 8/29
    
    
    PRE-CLASS:
    CLASS TIME:

    Welcome to Numerical Analysis - Jupyter-Notebooks
    
    

    AFTER CLASS:

    Prep for Day 2 and
    HW1 Assigned
    Download Anaconda and get a Jupyter Notebook running!
    Video: Introduction to Jupyter Notebooks Ignore the part at the end where I say to post to your groups... instead bring your comments and questions to class.
    

    EXTRAS:

    There are lots of resources to get help with Python programming online. Here is one:
    Extras about Jupyter Notebooks
    

    
    
  • DAY 2 - Thursday 8/31
    
    
    PRE-CLASS:

    Video: Introduction to Python in Jupyter Notebooks
    Jupyter Notebook: Introduction to Python Programming Ignore the part at the end where I say to post to your groups... instead bring your comments and questions to class.
    
    HW2 Assigned

    CLASS TIME:

    Programming in Python

    AFTER CLASS:

    
    Prep for Day 3

    EXTRAS:

    
    
  • DAY 3 - Tuesday 9/5
    
    
    PRE-CLASS:

    Video: Taylor Polynomials Intro
    Video: Taylor Polynomials Notebook Demo
    Jupyter Notebook: Taylor Polynomials
    
    HW3 Assigned

    CLASS TIME:

    Work on Taylor Polynomials Homework

    AFTER CLASS:

    
    Prep for Day 4

    EXTRAS:

    
    
  • DAY 4 - Thursday 9/7
    
    
    PRE-CLASS:

    Video: Error in Taylor Polynomials Intro
    Video: Taylor Polynomials Example
    Jupyter Notebook: Error Bound
    
    HW4 Assigned

    CLASS TIME:

    Work on Error in Taylor Polynomials Homework

    AFTER CLASS:

    
    Prep for Day 5
    

    EXTRAS:

    
    
  • DAY 5 - Tuesday 9/12
    
    
    PRE-CLASS:

    Video: Intro to Computational Error
    Video: Notebook Example - Precision Errors
    Video: Intro to Rounding and Chopping Error
    Video: Notebook Example - Rounding/Chopping Errors
    Video: Intro to Error Propagation, Function Eval, and Relative Error
    Video: Notebook Example - Error Propagation, Function Eval
    Jupyter Notebook: Computational Error
    HW5 Assigned

    CLASS TIME:

    Work on General Computational Error Homework

    AFTER CLASS:

    
    Prep for Day 6

    EXTRAS:

    Real Life Computation Errors - Article

    
    
  • DAY 6 - Thursday 9/14
    
    PRE-CLASS:

    Video: Bisection Method
    Video: Jupyter Bisection Method Example
    Jupyter Notebook: Bisection Method - Code Outline
    Jupyter Notebook: Bisection Method - Solution
    
    HW6 Assigned

    CLASS TIME:

    Work on Bisection Method Homework

    AFTER CLASS:

    
    Prep for Day 7
    

    EXTRAS:

    
    
  • DAY 7 - Tuesday 9/19
    
    PRE-CLASS:

    Video: Newtons Method
    Video: Jupyter Newtons Method Example
    Video: Fun Application of Newtons Method
    Jupyter Notebook: Newtons Method
    
    HW7 Assigned

    CLASS TIME:

    Work on Newtons Method Homework

    AFTER CLASS:

    
    Prep for Day 8

    EXTRAS:

    
    
  • DAY 8 - Thursday 9/21
    
    
    PRE-CLASS:

    Video: Introduction to Fixed Point Iteration
    Worksheet: Fixed Point Iteration
    Jupyter Notebook: Fixed Point Iteration
    Video: Worksheet Solutions
    Video: Contraction Mapping Theorem
    
    HW8 Assigned

    CLASS TIME:

    Work on Fixed Point Iteration and Contraction Mapping HW

    AFTER CLASS:

    
    Prep for Day 9
    

    EXTRAS:

    
    
  • DAY 9 - Tuesday 9/26
    
    
    PRE-CLASS:

    Video: Aitken Error and Method
    Video: Ill Behaving Problems
    Video: Jupyter Walkthrough + Applied Example
    Jupyter Notebook: Aitkin Code - Outline - You fill in the blanks
    Jupyter Notebook: Aitkin Code - Solution
    Jupyter Notebook: The Problem with Multiple Roots
    Jupyter Notebook: Applied Rootfinding Example
    
    HW9 Assigned

    CLASS TIME:

    Work on Aitken Homework and Applied Examples

    AFTER CLASS:

    
    Prep for Day 10

    EXTRAS:

    
    
  • DAY 10 - Thursday 9/28
    
    
    PRE-CLASS:

    Video: Logistic Map - Analysis When I say HW12 I mean HW9!
    Video: Jupyter Walkthrough - Logistic Map
    Video: Unstable Problems - Perturbation Analysis
    Jupyter Notebooks: Logistic Map
    
    HW10 Assigned

    CLASS TIME:

    Work on Perturbation Theory Problems - AND - Past problems you struggled with

    AFTER CLASS:

    
    Prep for Day 11
    

    EXTRAS:

    Video: Extra - Newton Fractal
    Jupyter Notebook: Newtons Fractal
    Extra - 2D Newton Fractals - Website Link
    

    
    
  • DAY 11 - Tuesday 10/3
    
    
    PRE-CLASS:

    Video: Polynomial Interpolation
    Video: Jupyter Notebook - Interpolation
    Video: Divided Differences
    Worksheet - Example for Pseudo Code See Polynomial Interpolation Lecture Notes for Solutions.
    
    Try to write your own code using the worksheet and pseudo code you created. See how far you can get without looking at my code.
    Jupyter Notebook: Polynomial Interpolation
    Jupyter Notebook: Divided Difference
    
    HW11 Assigned

    CLASS TIME:

    Work on Polynomial Interpolation

    AFTER CLASS:

    
    Prep for Day 12

    EXTRAS:

    
    
  • DAY 12 - Thursday 10/5
    
    
    PRE-CLASS:

    Video: Jupyter Notebook - Interpolation
    Video: Error in Polynomial Interpolation
    Jupyter Notebook - Error in Interpolation
    HW12 Assigned

    CLASS TIME:

    Explore error in Polynomial Interpolation:
    

    AFTER CLASS:

    Prep for Day 13 - on Tuesday 10/17
    

    EXTRAS:

    
    
  • Fall Break
    
    University Fall Break is Monday 10/9 and Tuesday 10/10. Our class will not meet on Thursday 10/12. This means that you have a full week away from Numerical Analysis. Please take the time to rest your minds and get caught up if you need to.
  • DAY 13 - Tuesday 10/17
    
    
    PRE-CLASS:

    Video: Image Repair and Interpolation
    Jupyter Notebook - Homework 13
    HW13 Assigned
    
    Photo of Sofia - Homework 13
    Sofia Data .mat - Homework 13
    Python Logo - Homework 13
    HW13 Assigned

    CLASS TIME:

    Work on the image repair code!
    
    Can you imagine other things you might do iteratively to an image file? Blur? Increase Contrast? Change the colors? What would the code look like for these processes?

    AFTER CLASS:

    Prep for Day 14

    EXTRAS:

    
    
  • DAY 14 - Thursday 10/19
    
    
    PRE-CLASS:

    Video: Intro to Spline Interpolation and Cubic Spline Formula
    Video: Example and Psuedo Code for Spline Interpolation
    Jupyter Notebook - Spline Interpolation
    HW14 Assigned

    CLASS TIME:

    Explore error in Polynomial Interpolation:
    
    Questions:
    Do you have any questions about the homework? Did everyone finish?
    Were you able to write your own code for spline interpolation? What were the hardest parts of this? Where are you getting stuck?
    What is the difference between Linear and Cubic Spline interpolation?
    What are the properties of a Cubic Spline? There are three!
    What are some differences between Polynomial Interpolation and Spline Interpolation?
    Is the piecewise cubic spline function continuous everywhere between a and b?

    AFTER CLASS:

    Prep for Day 15
    
    

    EXTRAS:

    
    
  • DAY 15 - Tuesday 10/24
    
    
    PRE-CLASS:

    Video: Intro to Numerical Integration
    Video: Simpsons Rule
    Jupyter Notebook - Numerical Integration - Trapezoid Rule
    Jupyter Notebook - Numerical Integration - Simpsons Rule
    HW15 Assigned

    CLASS TIME:

    Numerical Integration:
    
    Questions:
    Do you have any questions about the homework? Did everyone finish?
    Were you able to write your own code for Trapezoid and Simpsons Rule? What were the hardest parts of this? Where are you getting stuck?
    Were you able to find the formula and write code for Midpoint Rule?
    Does your code get the same results as the book when doing the same functions?
    Can you brainstorm some ways to get even better numerical integration techniques? Did anyone look at problem 10?

    AFTER CLASS:

    Prep for Day 16
    

    EXTRAS:

    
    
  • DAY 16 - Thursday 10/26
    
    
    PRE-CLASS:

    Video: Error in Trapezoid Rule
    Video: Asymptotic Error and Error in Simpsons Rule
    HW16 Assigned

    CLASS TIME:

    Explore error in Numerical Integration:
    
    Questions:
    Do you have any questions about the homework? Did everyone finish?
    Were you able to write your own code for spline interpolation? What were the hardest parts of this? Where are you getting stuck?
    Were you able to demonstrate h^2 and h^4 error ratios in your numerical code?
    Does the idea of asymptotic error correction make sense?
    There are lots of other numerical integration techniques. You could look up some other scipy functions and see what methods those functions use for numerical integration.

    AFTER CLASS:

    Prep for Day 17
    

    EXTRAS:

    
    
  • DAY 17 - Tuesday 10/31
    
    
    PRE-CLASS:

    Video: Introduction to Numerical Differentiation
    Video: Central Difference Formula
    Video: Deriving Higher Order Derivatives
    HW17 Assigned

    CLASS TIME:

    Explore Numerical Differentiation:
    
    Questions:
    Do you have any questions about the homework? Did everyone finish?
    Were you able to write your own code for numerical differentiation? What were the hardest parts of this? Where are you getting stuck?
    Were you able to derive the two approximations given in the homework? If not, where are you stuck?
    What other questions do you have about numerical differentiation?

    AFTER CLASS:

    Prep for Day 18
    

    EXTRAS:

    
    
  • DAY 18 - Thursday 11/2
    
    
    PRE-CLASS:

    Video: Ordinary Differential Equations
    Video: Existence and Uniqueness
    Video: Stability and Introduction to Direction Fields
    Video: Jupyter Notebook Direction Fields
    Jupyter Notebook - Direction Fields
    HW18 Assigned

    CLASS TIME:

    Get an understanding of Ordinary Differential Equations
    
    Questions:
    What does it mean for a solution to exist and be unique? How do you test this?
    What does it mean for an ODE to be unstable or ill-conditioned? What types of problems might this cause?
    What is a direction field? What is it useful for?

    AFTER CLASS:

    Prep for Day 19

    EXTRAS:

    
    
  • Catch Up Day - Tuesday 11/7
    We will still have class time where you can work together to get homework done and have questions answered, but there is no new homework assigned. Take this time to get caught up.
  • DAY 19 - Thursday 11/9
    
    
    PRE-CLASS:

    Video: Euler's Method
    Video: Error in Eulers Method
    HW19 Assigned

    CLASS TIME:

    Eulers Method
    
    Questions:
    Did you get your code written and working?
    What are the three ways to think about deriving Eulers method?
    How do we use Richardsons error approximation?
    What are you doing for Halloween?

    AFTER CLASS:

    Prep for Day 20
    

    EXTRAS:

    
    
  • DAY 20 - Tuesday 11/14
    
    
    PRE-CLASS:

    Video: Convergence and Stability
    Video: Implicit Methods
    HW20 Assigned

    CLASS TIME:

    Eulers Method
    
    Questions:
    Did you get your code written and working?
    What is the difference between stability of ODE and stability of the method?
    What does it mean for a method to converge?
    What does in mean when an ode numerical method is implicit?
    What are the strengths and weaknesses of Eulers Method? Backward Eulers method?

    AFTER CLASS:

    Prep for Day 21

    EXTRAS:

    
    
  • DAY 21 - Thursday 11/16
    
    PRE-CLASS:

    Video: Higher Order Methods ODE - Taylor
    Video: Runge Kutta
    HW21 Assigned

    CLASS TIME:

    Explore Higher Order Methods for ODEs
    
    Were you able to alter your Euler code so you can use Taylor and Runge-Kutta methods?
    What are the pro's and con's of Taylor Methods?
    How did we derive the second order Runge-Kutta method?
    Would Runge-Kutta still work if we chose a different gamma?

    AFTER CLASS:

    Thanksgiving Break Next Week - Watch videos for a review of linear systems and start homework for Gaussian Elimination.
    Homework 20 and 21 are due after Thanksgiving on 11/26.
    

    EXTRAS:

    
    
  • Review of Linear Systems - Thanksgiving Week
    
    
    Review Materials

    Video: Intro Linear Systems
    Video: Python - Numpy - Matrix Arithmetic
    Jupyter Notebook - Matrix Arithmetic

    
    
  • DAY 22 - Thursday 11/28
    
    
    PRE-CLASS:

    Video: Intro Gaussian Elimination
    Video: Psuedo Code
    Video: Code Walkthrough
    Gaussian Elimination - Code
    HW22 Assigned - Optional

    CLASS TIME:

    Gaussian Elimination
    
    What are the basic steps of Gaussian elimination?
    Why do we do pivoting? What are the benefits of this?
    What is an upper triangular matrix system and why are they easy to solve?
    

    AFTER CLASS:

    Prep for Day 23

    EXTRAS:

    
    
  • DAY 23 - Thursday 11/30
    
    
    PRE-CLASS:

    Video: Intro Iterative Methods for Systems
    Video: Error in Iterative Methods for Systems
    Gauss Seidel - Code
    HW23 Assigned - Optional

    CLASS TIME:

    Gauss Seidel
    
    
    What is the difference between Jacobi and Gauss Seidel Iteration?
    Why do we care about convergence?
    Are iterative methods always better or worse than Gaussian Elimination?
    

    AFTER CLASS:

    Start working on the Final Exam.

    EXTRAS:

    We will hand out the final reflection in class.
    

  • Final Reflection
    
    
    Self Reflection - Due Wednesday 12/6 by 11:59pm
    Final submissions to your homework assignments are due Wednesday at 11:59pm too.
    
    Final Exam - Optional
• Homework Solutions - Review
  
  HW 2 - Introduction to Python SOLN
  HW 3 - Taylor Polynomials SOLN
  HW 4 - Error in Taylor Polynomials SOLN
  HW 5 - Error in Numerical Methods SOLN
  HW 6 - Bisection Method SOLN
  HW 7 - Newtons Method SOLN
  HW 8 - Fixed Point Iterations SOLN
  HW 9 - Aitken Iterations and Ill-Behaving Problems SOLN
  HW 10 - Logistic Map and Ill-Behaving Problems SOLN
  HW 11 - Polynomial Interpolation SOLN
  HW 12 - Error in Polynomial Interpolation SOLN
  HW 13 - Image Repair SOLN
  HW 14 - Spline Interpolation SOLN
  HW 15 - Numerical Integration SOLN
  HW 16 - Error in Numerical Integration SOLN
  HW 17 - Numerical Differentiation SOLN
  HW 18 - Intro to ODES SOLN
  HW 19 - Euler's Method SOLN
  HW 20 - Backward Euler SOLN
  HW 21 - Taylor and Runge-Kutta SOLN
  
  
  
• Lecture Notes - Important Links
  
  USE AT YOUR OWN RISK - There could be typos that are corrected in class, so please use common sense and your own notes to compare to what you see here:
  Download Anaconda
  Introduction to Jupyter Notebooks
  Typesetting using LaTeX - Can be used in Markdown
  Intro and Review of Taylor Series
  Error in Taylor Polynomials
  A General Discussion of Computational Error
  Bisection Method
  Newtons Method
  Fixed Point Iteration - Contraction Mapping Theorem
  Aitken Extrapolation - Multiple Root Issues
  Stability Issues in Root Finding Methods
  Polynomial Interpolation - Lagrange - Newton Divided Difference
  Polynomial Interpolation - Error in Interpolation
  Natural Cubic Spline
  Numerical Integration
  Error in Numerical Integration
  Numerical Differentiation
  Error Analysis - Richardson's and Aitken's Integration
  Introduction to Ordinary Differential Equations
  Euler's Methods
  Introduction to Implicit Methods
  Taylor and Runge-Kutta Methods
  Adams-Bashforth and Adams-Moulton Methods
  Review of Matrix Algebra and Linear Systems
  Gaussian Elimination and Partial Pivoting
  Jacobi Iteration
  
  
  




I catch all the buggies in my code!