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
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Link to our Canvas Class
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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.
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Daily Assignments - Reading - Handouts
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DAY 1 - Tuesday 8/29
PRE-CLASS:
CLASS TIME:Welcome to Numerical Analysis - Jupyter-Notebooks
AFTER CLASS:
Prep for Day 2 and
EXTRAS:
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.
There are lots of resources to get help with Python programming online. Here is one:
Extras about Jupyter Notebooks
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DAY 2 - Thursday 8/31
PRE-CLASS:Video: Introduction to Python in Jupyter Notebooks
CLASS TIME:
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 AssignedProgramming in Python
AFTER CLASS:
Prep for Day 3
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DAY 3 - Tuesday 9/5
PRE-CLASS:Video: Taylor Polynomials Intro
CLASS TIME:
Video: Taylor Polynomials Notebook Demo
Jupyter Notebook: Taylor Polynomials
HW3 AssignedWork on Taylor Polynomials Homework
AFTER CLASS:
Prep for Day 4
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DAY 4 - Thursday 9/7
PRE-CLASS:Video: Error in Taylor Polynomials Intro
CLASS TIME:
Video: Taylor Polynomials Example
Jupyter Notebook: Error Bound
HW4 AssignedWork on Error in Taylor Polynomials Homework
AFTER CLASS:
Prep for Day 5
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DAY 5 - Tuesday 9/12
PRE-CLASS:Video: Intro to Computational Error
CLASS TIME:
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 AssignedWork on General Computational Error Homework
AFTER CLASS:
Prep for Day 6Real Life Computation Errors - Article
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DAY 6 - Thursday 9/14
PRE-CLASS:Video: Bisection Method
CLASS TIME:
Video: Jupyter Bisection Method Example
Jupyter Notebook: Bisection Method - Code Outline
Jupyter Notebook: Bisection Method - Solution
HW6 AssignedWork on Bisection Method Homework
AFTER CLASS:
Prep for Day 7
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DAY 7 - Tuesday 9/19
PRE-CLASS:Video: Newtons Method
CLASS TIME:
Video: Jupyter Newtons Method Example
Video: Fun Application of Newtons Method
Jupyter Notebook: Newtons Method
HW7 AssignedWork on Newtons Method Homework
AFTER CLASS:
Prep for Day 8
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DAY 8 - Thursday 9/21
PRE-CLASS:Video: Introduction to Fixed Point Iteration
CLASS TIME:
Worksheet: Fixed Point Iteration
Jupyter Notebook: Fixed Point Iteration
Video: Worksheet Solutions
Video: Contraction Mapping Theorem
HW8 AssignedWork on Fixed Point Iteration and Contraction Mapping HW
AFTER CLASS:
Prep for Day 9
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DAY 9 - Tuesday 9/26
PRE-CLASS:Video: Aitken Error and Method
CLASS TIME:
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 AssignedWork on Aitken Homework and Applied Examples
AFTER CLASS:
Prep for Day 10
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DAY 10 - Thursday 9/28
PRE-CLASS:Video: Logistic Map - Analysis When I say HW12 I mean HW9!
CLASS TIME:
Video: Jupyter Walkthrough - Logistic Map
Video: Unstable Problems - Perturbation Analysis
Jupyter Notebooks: Logistic Map
HW10 AssignedWork on Perturbation Theory Problems - AND - Past problems you struggled with
AFTER CLASS:
Prep for Day 11
Video: Extra - Newton Fractal
Jupyter Notebook: Newtons Fractal
Extra - 2D Newton Fractals - Website Link
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DAY 11 - Tuesday 10/3
PRE-CLASS:Video: Polynomial Interpolation
CLASS TIME:
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 AssignedWork on Polynomial Interpolation
AFTER CLASS:
Prep for Day 12
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DAY 12 - Thursday 10/5
PRE-CLASS:Video: Jupyter Notebook - Interpolation
CLASS TIME:
Video: Error in Polynomial Interpolation
Jupyter Notebook - Error in Interpolation
HW12 AssignedExplore error in Polynomial Interpolation:
AFTER CLASS:
Prep for Day 13 - on Tuesday 10/17
EXTRAS:
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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
CLASS TIME:
Jupyter Notebook - Homework 13
HW13 Assigned
Photo of Sofia - Homework 13
Sofia Data .mat - Homework 13
Python Logo - Homework 13
HW13 AssignedWork on the image repair code!
AFTER CLASS:
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?Prep for Day 14
EXTRAS:
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DAY 14 - Thursday 10/19
PRE-CLASS:Video: Intro to Spline Interpolation and Cubic Spline Formula
CLASS TIME:
Video: Example and Psuedo Code for Spline Interpolation
Jupyter Notebook - Spline Interpolation
HW14 AssignedExplore error in Polynomial Interpolation:
AFTER CLASS:
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?Prep for Day 15
EXTRAS:
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DAY 15 - Tuesday 10/24
PRE-CLASS:Video: Intro to Numerical Integration
CLASS TIME:
Video: Simpsons Rule
Jupyter Notebook - Numerical Integration - Trapezoid Rule
Jupyter Notebook - Numerical Integration - Simpsons Rule
HW15 AssignedNumerical Integration:
AFTER CLASS:
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?Prep for Day 16
EXTRAS:
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DAY 16 - Thursday 10/26
PRE-CLASS:Video: Error in Trapezoid Rule
CLASS TIME:
Video: Asymptotic Error and Error in Simpsons Rule
HW16 AssignedExplore error in Numerical Integration:
AFTER CLASS:
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.Prep for Day 17
EXTRAS:
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DAY 17 - Tuesday 10/31
PRE-CLASS:Video: Introduction to Numerical Differentiation
CLASS TIME:
Video: Central Difference Formula
Video: Deriving Higher Order Derivatives
HW17 AssignedExplore Numerical Differentiation:
AFTER CLASS:
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?Prep for Day 18
EXTRAS:
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DAY 18 - Thursday 11/2
PRE-CLASS:Video: Ordinary Differential Equations
CLASS TIME:
Video: Existence and Uniqueness
Video: Stability and Introduction to Direction Fields
Video: Jupyter Notebook Direction Fields
Jupyter Notebook - Direction Fields
HW18 AssignedGet an understanding of Ordinary Differential Equations
AFTER CLASS:
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?Prep for Day 19
EXTRAS:
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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.
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DAY 19 - Thursday 11/9
PRE-CLASS:Video: Euler's Method
CLASS TIME:
Video: Error in Eulers Method
HW19 AssignedEulers Method
AFTER CLASS:
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?Prep for Day 20
EXTRAS:
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DAY 20 - Tuesday 11/14
PRE-CLASS:Video: Convergence and Stability
CLASS TIME:
Video: Implicit Methods
HW20 AssignedEulers Method
AFTER CLASS:
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?Prep for Day 21
EXTRAS:
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DAY 21 - Thursday 11/16
PRE-CLASS:Video: Higher Order Methods ODE - Taylor
CLASS TIME:
Video: Runge Kutta
HW21 AssignedExplore Higher Order Methods for ODEs
AFTER CLASS:
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?Thanksgiving Break Next Week - Watch videos for a review of linear systems and start homework for Gaussian Elimination.
EXTRAS:
Homework 20 and 21 are due after Thanksgiving on 11/26.
- Review of Linear Systems - Thanksgiving Week
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DAY 22 - Thursday 11/28
PRE-CLASS:Video: Intro Gaussian Elimination
CLASS TIME:
Video: Psuedo Code
Video: Code Walkthrough
Gaussian Elimination - Code
HW22 Assigned - OptionalGaussian Elimination
AFTER CLASS:
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?
Prep for Day 23
EXTRAS:
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DAY 23 - Thursday 11/30
PRE-CLASS:Video: Intro Iterative Methods for Systems
CLASS TIME:
Video: Error in Iterative Methods for Systems
Gauss Seidel - Code
HW23 Assigned - OptionalGauss Seidel
AFTER CLASS:
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?
Start working on the Final Exam.
EXTRAS:We will hand out the final reflection in class.
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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
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
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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
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!
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