MTH 453/553 Numerical Calculus III (Numerical Linear Algebra)

Instructor Information

Dr. Jeffrey Ovall

Office: Fariborz Maseeh Hall (FMH), 464R

Office Hours: TR 2-3pm, or by special appointment

Course Overview

This course provides the necessary background for understanding and applying computational algorithms for the key problems of linear algebra: solving linear systems, least-squares problems and eigenvalue problems.

Prerequisites: MTH 452 or equivalent

Course Meeting Times: TR 11am-12:15pm

Course Meeting Location: Fariborz Maseeh Hall (FMH), 419

Course Reference Number (CRN): 61468 (453), 61475 (553)

Syllabus

Learning Outcomes

By the end of this course, students will be able to:

Work fluently with matrix and vector operations, including block structures and triangular systems
Understand the fundamental theorems of linear algebra that support key algorithms
Compute and apply triangular decompositions (e.g. LU and Cholesky) for solving linear systems
Analyze the computational cost of solving linear systems by direct methods
Compute and interpret vector and matrix norms, and condition numbers
Understand how perturbations affect solutions of linear systems (error analysis)
Understand the fundamental theory behind least-squares problems
Understand orthogonal decompositions (e.g. QR and SVD) and apply them to solve least-squares problems
Understand the fundamental theory behind eigenvalue problems
Understand iterative methods (e.g. power method and subspace iteration) and apply them to solve eigenvalue problems

Textbook and Topics

We will be using the textbook Numerical Mathematics, by Jeffrey S. Ovall(opens in new tab) . Known typos can be found at my website for the book (opens in new tab); there are at least three in the chapters we will cover.

Topics: We will cover much of Chapters 8-11, as outlined below.

First-order differential equations (Chapter 8)
- Basic Definitions, Notation and Results
- Matrix and Vector Products Algorithmically
- Submatrices and Block Matrices, Permutation Matrices
- Triangular Matrices
- Fundamental Theorems of Linear Algebra
Direct Methods for Solving Linear Systems (Chapter 9)
- LU Decomposition
- Cholesky and LDLT Decompositions
- Vector and Matrix Norms
- Condition Number, Perturbation Results for Linear Systems
Least-Squares Problems (Chapter 10)
- Least-Squares Problems
- QR-Decomposition
- Singular Value Decomposition
- Rank Estimation, Reduced Rank Matrix Approximation
Eigenvalue Problems (Chapter 11)
- Similarity, Diagonalizability and Triangularizability
- Further Theoretical Results
- The Power Method
- Subspace Iteration and FEAST

If time permits, we may cover other topics such as: Francis's QR Algorithm for computing eigenvalues (Section 11.5), or material from Chapter 12 on iterative methods for linear systems.

Course Grade

Your course grade will be assigned based on the percentage earned of 400 possible points: 25 points for each of four assignments, 100 points for the midterm exam, 200 points for the final exam. Scores on each assignment and exam may be adjusted (in your favor) in order to achieve a fair distribution of grades. Students registered at the 500-level will have some additional problems on assignments and exams.

Scores on each assignment and exam may be adjusted (in your favor) in order to achieve a fair distribution of grades. At the end of the quarter each student's points will be added and their percentage of the total points will be calculated. You are guaranteed that, if your percentage meets the traditional university standards, then you will get at least the following grade:

A: 90-100%
B: 80-89%
C: 70-79%
D: 60-69%
F: 0-59%

In assigning final grades, plusses/minuses will be used. If you have chosen P/NP grading, a grade of P will be assigned if you would have gotten at least a C under the grading scheme described above, and a grade of NP otherwise. If you are concerned about your performance at any point in the term, come talk to me---make an appointment if necessary.

Extra Credit: Do not expect extra credit or make-up work.

Instructions for Assignments: Assigned problems and due dates for each assignment are given in the Weekly Schedule section. If I decide to change the problem set for a given assignment, the class will be notified by e-mail, and the relevant information will be updated in the Weekly Schedule.

Your assignments should be clearly written, and well-organized. If your penmanship is poor, consider using LaTeX or a word processor.
I will select only a subset of problems from each assignment for grading, but solutions to all assigned problems will be made available after the due date.
You are welcome to (even encouraged to) discuss the assignments with each other, but your write-ups should reflect your own work and understanding.
Problems required for those registered at the 500-level are indicated with asterisks. Optional problems, which will not be graded, are given in parentheses.
Assignments are to be turned in before midnight on the Friday of the week in which they appear in the Weekly Schedule.
E-mail your solutions and supporting code/documents to me at jovall@pdx.edu, and CC yourself on the e-mail, so that we both have a record of when you submitted it.
Your solutions (not counting supporting code) should consist of a single PDF of modest size (< 10MB) file name of the form LastName.pdf (e.g. Ovall.pdf).

Exam Dates: Because the dates of exams are given well in advance, make/adjust your travel plans accordingly. Only in exceptional cases (left to my discretion, but including observance of religious holidays) will an exam be given on an alternate date. Unless a missed exam is due to a properly documented sickness or family emergency, an alternate exam date will be given only if we have a written agreement to do so, made at least one week before the originally-scheduled exam date.

Midterm: Thursday, April 30 (Week 5)
Final: Thursday, June 11, 11:10am-1:00pm

Weekly Schedule

The topics listed below may need to be adjusted as the term progresses. Assignments are due before midnight on the Friday of the week in which they appear in the calendar below. Recall that problems listed in parentheses are optional for everyone and problems with an asterisk are required of those registered for MTH 553.

Weekly Topics and Assignments
Week	Tuesday	Thursday	Assignment
1: Mar 31, Apr 2	Basic definitions, notation and results (8.1)	Matrix/vector products algorithmically (8.2)
2: Apr 7, Apr 9	Matrix/vector products algorithmically (8.2) Submatrices and Block Matrices (8.3)	Submatrices and Block Matrices (8.3) Triangular Matrices (8.4)
3: Apr 14, Apr 16	Triangular Matrices (8.4)	Fundamental Theorems of Linear Algebra (8.5)	Section 8.1: 1, 4, 6, 7, 8, 13 Section 8.2:* 1, 2, 5, 6, 8* Section 8.3: 1, 2, (5), 6, 7, (8) Section 8.4:* 1, 3, 5, (6)
4: Apr 21, Apr 23	LU Decomposition (9.1)	Cholesky and LDLT Decompositions (9.2)
5: Apr 28, Apr 30	Vector and Matrix Norms (9.3)	In-class Midterm	Section 8.5: 1, 3, (6), 7, 8* Section 9.1: 2, 3, 7* Section 9.2: 1, 4, (7), (8), 9
6: May 5, May 7	Condition Number, Perturbation Results (9.4)	Least Squares Problems (10.1)
7: May 12, May 14	QR Decomposition (10.2)	Singular Value Decomposition (10.3)	Section 9.3: 1, 2, 4, 8, (9) Section 9.4: 1, (2), 3, (4), (5), 6 Section 10.1: 1, (2), 8, 11*
8: May 19, May 21	Eigenvalue "basics" (11.1) Spectral Theorem (11.1)	Courant-Fischer Theorem (11.2) Geršgorin Disk Theorem (11.2) Power Method (11.3)
9: May 26, May 28	Power Method (11.3)	Subspace iteration and FEAST (11.3)	Section 10.2: 1, 2, 3, (10) Section 10.3: 1, 3, (4), 5, 6, 7 Section 11.1:* 1, 2, 5, 12*
10: Jun 2, Jun 4	Subspace iteration and FEAST (11.3)	Optional Topic Review for final exam

MTH 453/553 Numerical Calculus III, Spring 2026