Course Announcements

4/3/25: 1. Papers on best polynomial and rational approximation have been added to the group of papers on orthogonal polynomials.; 2. A paper on the current popular research topic of mixed and variable floating point arithmetic has been added to the group of papers posted early in the semester on floating point arithmetic and algorithms.
4/3/25: 1. Papers on parallel time-stepping and applications have been posted.; 2. A paper on Gauss-Kronod quadrature has been posted.; 3. A paper on changing stepsizes in linear mutlistep codes has been posted.
4/1/25: 1. Sets 29 and 30 of the class notes have been posted.; 2. The due date of Graded Homework 4 is now April 14.
3/31/25: 1. The last material on Exam 2 is Gauss Quadrature, i.e., Set 25 and associated homework and study problems.; 2. Set 28 of the class notes has been posted.
3/26/25: 1. Minor clarifications have been made to Set 25.; 2. Information on Exam 2 has been posted on the class webpage.
3/24/25: 1. The solutions to the written problems of Graded Homework 3 have been posted.; 2. The derivations of the boundary conditions and the tabulation of the functions for the spline usage problem in the programming part of Graded Homework 3 have been posted.
3/20/25: 1. In Set 22, the slides discussing the economization of the monomial form using successive approximations to x^k have been modified to clarify the procedure and the equivalence to the change-of-basis approach. Read it before the lecture on Friday and prepare any questions for the beginning of class.
3/19/25: 1. In Set 21, slide 68 has been modified for clarification and references added. Pointers to proofs have also been added for the recurrence statments and the convergence of the truncated GFS.; 2. Study Question Set 3 has been posted along with related material in a classic paper by Gautschi and a copy of relevant pages from the Quateroni et al. textbook with Matlab Gaussian Quadrature codes.
3/17/25: 1. A typo has been fixed on slide 9 of Set 21.; 2. Sets 25, 26, and 27 of the class notes have been posted.
3/6/25: 1. Exam 1 has been graded and will be returned at the end of class today.; 2. An anonymous grade list for Exam 1 has been posted.
3/6/25: 1. The beginning of the Hilbert space notes in Set 21 has been replaced with a version that has more complete proofs in some cases and references to literature for others.
3/5/25: 1. Sets 21, 22, 23, and 24 of the class notes have been posted.
3/4/25: 1. Solution comments and Matlab codes posted for Graded Homework 2 Program
3/2/25: 1. Sets 19 and 20 have been posted.
2/28/25: 1. Graded Homework 3 has been posted.; 2. Solutions for Exam 1 have been posted.; 3. Quiz 3 grades have been posted.
2/26/25: 1. A paper by Xiang on the convergence of interpolation strategies has been posted. This paper has a good set of pointers to the literature on this subject. You are strongly encouraged to read it at some point but are not required so to do for this class. Note this paper is also referenced in the convergence of quadrature algorithms that we will discuss this semester.
2/19/25: 1. A typo on slide 22 of Set 14 has been fixed and the set reposted.; 2. The solutions to Quiz 3 have been posted.
2/17/25: 1. Sets 17 and 18 on splines have been posted.; 2. Several papers on splines have been posted. These are not required reading rather they are for your reference and later study.
2/14/25: 1. Quiz 3 will be given on Wednesday 2/19/25.
2/12/25: 1. Information about Exam 1 has been posted (follow the Exam link on the course page) and sent as a Canvas announcement.
2/10/25: 1. The solutions for Graded Homework 1 have been posted.; 2. Sets 15 and 16 of the class notes have been posted. Note that Set 15 has the local change of variables for using a reference interval, e.g., [-1,1], for a generic interval [a,b], needed for using Cauchy points in Graded Homework 2.; 3. Papers by Gautschi, Wilkinson, and, Austin and Xu have been added to the polynomial interpolation papers list.
2/7/25: 1. Graded Homework 2 has been posted.
2/5/25: 1. The solutions for Quiz 2 have been posted.
2/1/25: 1. Study Questions Set 2 and associated solutions have been posted.
1/31/25: 1. Graded Homework 1 has been posted.; 2. Quiz 2 will be given on Monday.
1/29/25: 1. Minor typos have been fixed in Theorem 7.1, its proof and on slide 34 of Set 7.; 2. Based on today's lecture, clarifying details of the construction of the key result in the recursive form of the divided differences have been added to Set 7.
1/28/25: 1. Your quiz grades have been posted in Canvas.; 2. You may pick up your graded quiz during office hours or at an arranged appointment.; 3. The solutions have been posted and you should consult them before picking up your quiz if you have questions.; 4. Lectures on Wednesday and Friday this week will concern Sets 7 and 8. Note that Set 8 begins the algorithm discussions and is related to the assigned reading of the paper of Berrut and Trefethen. Understanding the complexity implications of Barycentric forms 1 and 2 is a central goal of the interpolation portion of the semester and requires reading the set as preparation.; 5. Set 9 on implementations of the Newton form will be covered in detail early next week and you are expected to have read the material and prepared for the lectures.; 6. The material on conditioning, error and stability in Sets 11, 12 and 13 will be summarized in two lectures that concentrate on the key points. Therefore, it is critical that you prepare for these lectures. This is also related to the assigned reading of the paper by Higham.; 7. Set 14 on the Hermite-Birkhoff interpolation problem has been posted.
1/21/25: 1. The solutions to Quiz 1 have been posted.
1/16/25: 1.The extra set of notes on the integer encodings had a typo for the 2's complement table on page 17. It was a latex error that used the 1's complement table. This has been fixed. The summary table on page 20 had the correct 2's complement encoding.
1/15/25: 1. A typo in the Lemma 4.2 (standard model of arithmetic with gradual underflow) in Set 4 has been fixed.; 2. Sets 11 through 13 have been posted. The cover fundamental material on the conditioning of polynomials, stability of interpolation, and convergence/divergence of interpolating polynomials. You are required to be familiar with these sets before the assoiated lectures.; 3. Several papers related to interpolation have been posted. You are required to read two and two others are recommended. The others are for your reference.
1/13/25: 1. The solutions for Study Question Set 1 have been posted.; 2. Sets 6 through 10 have been posted. These form the fundamentals of the theory and computation of the basic polynomial interpolation problem. Set 10 is extra and will not be discussed in a lecture. Anyone interested in discussing it should set up a meeting outside class. The lectures concerning the others will assume that you have read and are familiar with these sets.
1/8/25: 1. Sets 4 and 5 of the class notes have been posted.
1/7/25: 1. Extra Notes Set 1 of the class notes has been posted.; 2. Study Questions Set 1 has been posted.
1/6/25: 1. Set 3 of the class notes has been posted.
1/5/25: 1. The class syllabus has been posted.; 2. Sets 0 and 1 have been posted. The are similar to those from FCM 1 Fall 2024 with the appropriate updates to course content.; 3. A description of the format for submitting program solutions has been posted.; 4. Example program solutions have been posted.; 5. Set 2 of the class notes has been posted.

Course Announcements

4/3/25

1. Papers on best polynomial and rational approximation have been added to the group of papers on orthogonal polynomials.

2. A paper on the current popular research topic of mixed and variable floating point arithmetic has been added to the group of papers posted early in the semester on floating point arithmetic and algorithms.

4/3/25

1. Papers on parallel time-stepping and applications have been posted.

2. A paper on Gauss-Kronod quadrature has been posted.

3. A paper on changing stepsizes in linear mutlistep codes has been posted.

4/1/25

1. Sets 29 and 30 of the class notes have been posted.

2. The due date of Graded Homework 4 is now April 14.

3/31/25

1. The last material on Exam 2 is Gauss Quadrature, i.e., Set 25 and associated homework and study problems.

2. Set 28 of the class notes has been posted.

3/26/25

1. Minor clarifications have been made to Set 25.

2. Information on Exam 2 has been posted on the class webpage.

3. Graded Homework 4 has been posted.

3/24/25

1. The solutions to the written problems of Graded Homework 3 have been posted.

2. The derivations of the boundary conditions and the tabulation of the functions for the spline usage problem in the programming part of Graded Homework 3 have been posted.

3/20/25

3/19/25

1. In Set 21, slide 68 has been modified for clarification and references added. Pointers to proofs have also been added for the recurrence statments and the convergence of the truncated GFS.

2. Study Question Set 3 has been posted along with related material in a classic paper by Gautschi and a copy of relevant pages from the Quateroni et al. textbook with Matlab Gaussian Quadrature codes.

3/17/25

1. A typo has been fixed on slide 9 of Set 21.

2. Sets 25, 26, and 27 of the class notes have been posted.

3/6/25

1. Exam 1 has been graded and will be returned at the end of class today.

2. An anonymous grade list for Exam 1 has been posted.

3/6/25

1. The beginning of the Hilbert space notes in Set 21 has been replaced with a version that has more complete proofs in some cases and references to literature for others.

3/5/25

1. Sets 21, 22, 23, and 24 of the class notes have been posted.

3/4/25

1. Solution comments and Matlab codes posted for Graded Homework 2 Program

3/2/25

1. Sets 19 and 20 have been posted.

2/28/25

1. Graded Homework 3 has been posted.

2. Solutions for Exam 1 have been posted.

3. Quiz 3 grades have been posted.

2/26/25

2/19/25

1. A typo on slide 22 of Set 14 has been fixed and the set reposted.

2. The solutions to Quiz 3 have been posted.

2/17/25

1. Sets 17 and 18 on splines have been posted.

2. Several papers on splines have been posted. These are not required reading rather they are for your reference and later study.

2/14/25

1. Quiz 3 will be given on Wednesday 2/19/25.

2/12/25

1. Information about Exam 1 has been posted (follow the Exam link on the course page) and sent as a Canvas announcement.

2/10/25

1. The solutions for Graded Homework 1 have been posted.

2. Sets 15 and 16 of the class notes have been posted. Note that Set 15 has the local change of variables for using a reference interval, e.g., [-1,1], for a generic interval [a,b], needed for using Cauchy points in Graded Homework 2.

3. Papers by Gautschi, Wilkinson, and, Austin and Xu have been added to the polynomial interpolation papers list.

2/7/25

1. Graded Homework 2 has been posted.

2/5/25

1. The solutions for Quiz 2 have been posted.

2/1/25

1. Study Questions Set 2 and associated solutions have been posted.

1/31/25

1. Graded Homework 1 has been posted.

2. Quiz 2 will be given on Monday.

1/29/25

1. Minor typos have been fixed in Theorem 7.1, its proof and on slide 34 of Set 7.

2. Based on today's lecture, clarifying details of the construction of the key result in the recursive form of the divided differences have been added to Set 7.

1/28/25

1. Your quiz grades have been posted in Canvas.

2. You may pick up your graded quiz during office hours or at an arranged appointment.

3. The solutions have been posted and you should consult them before picking up your quiz if you have questions.

5. Set 9 on implementations of the Newton form will be covered in detail early next week and you are expected to have read the material and prepared for the lectures.

6. The material on conditioning, error and stability in Sets 11, 12 and 13 will be summarized in two lectures that concentrate on the key points. Therefore, it is critical that you prepare for these lectures. This is also related to the assigned reading of the paper by Higham.

7. Set 14 on the Hermite-Birkhoff interpolation problem has been posted.

1/21/25

1. The solutions to Quiz 1 have been posted.

1/16/25