Welcome to this course

In this course we will learn some of the most common numerical techniques and algorithms used to efficiently solve problems expressed using large matrices. We focus on detailed understanding about the performance of these methods when they are applied to large-scale systems and study topics such as convergence, accuracy and efficiency.

The lectures and exercises of this course are taught jointly with SF2524 and SF3580. The course SF3580 is for PhD students only.

Programming language in this course.

SF2524: Matlab or Julia
SF3580 (PhD students): Julia

This course is offered in study period 2 starting. Before the course starts, you are welcome to try Quiz: Background. If you can't access, see FAQ. Don't worry if you don't do so well on the background quiz, you can do it as many times as you like.

Why matrix computations?

The most common computational approaches to scientific problems involve, in some way, a subproblem containing a matrix. The efficiency and robustness of the computational method used for this matrix problem is often crucial for the efficiency and robustness of the entire approach. After an accurate discretization, a partial differential equation will be approximated by a problem involving a large matrix. Google's page-rank algorithm involves the computation with a huge transition matrix representing links between web pages. In order to achieve good efficiency and robustness for these approaches, you need detailed understanding of matrix computations.

Matrix computations is important in machine learning and AI: http://www.fast.ai/2017/07/17/num-lin-alg/.

In other words: Accept reality that the matrix is everywhere. Take the red pill and listen to

a youtube seminar by David Gleich

or

the enthusiastic TED-talk about the beauty of matrices by Margot Gerritsen (Stanford)

What do I need to do?

The examination of the course consists of correctly completing

The course has mandatory homeworks, and each homework can give you bonus points to the exam. If you in addition complete the course training tasks you can get additional bonus. See further description on the page for Homework & bonus points rules.

How do I learn what all this?

The learning activities of the course consist of lectures, homeworks, canvas-quizzes and wiki-work. The course material can be found under modules.

Prerequisites

A basic course in Numerical Methods, for instance SF1544 (or equivalent).
Recommended but not strictly necessary: A continuation course in numerical analysis, for instance SF2520 applied numerical methods, which can be read in parallel.

This course is elective for all master's programmes at KTH. The course is conditionally elective in the second year of the Master's programme in applied and computational mathematics, and mandatory for the Computational mathematics (beräkningsmatematik) track. It is conditionally elective for the Master's programme in Computer simulation for science and engineering (COSSE, TDTNM) as well as the the Nordic N5TeAM Master's Programme, Applied and Engineering Mathematics (TITMM). This course is suitable for students taking the programme Computer science and engineering (CDATE) if you have taken the course SF1626 Calculus in Several Variable (flervariabelanalys).

This course is a subset of the PhD-level course SF3580 - Numerical linear algebra.

Lecture plan:

The course consists of blocks:

Block 1: Large sparse eigenvalue problems
- Topics: Power method, Rayleigh quotient iteration, Arnoldi's method, Repeated Gram-Schmidt, Modified Gram-Schmidt, convergence characterization for Arnoldi's method for eigenvalue problems
Block 2: Large sparse linear systems
- Topics: GMRES, CG, CGN, BiCG, convergence of Krylov methods, preconditioning
Block 3: Dense eigenvalue algorithms (QR-method)
- Topics: QR-method, Hessenberg reduction, implicit QR-method
Block 4: Functions of matrices
- Topics: Taylor definition, Jordan definition, Contour integral definition, Schur-Parlett method, scaling-and-squaring for the matrix exponential, methods for the matrix square root, methods for the sign function
Block 5: (only for PhD students taking SF3580) Matrix equations

Lecture slides can be found here (or here).

Week	Contents
1	Lecture 1: Intro + RQ + PM: block1.pdf: §1.1
	Lecture 2: Conv. theory PM + Inv. it + RQI + AM intro. block1.pdf: §1.1, §1.3
	Lecture 2.5 (video 6 min): Rayleigh-Ritz (RR) and Bauer-Fike (BF)
	Lecture 3: CGS, MGS, RGS, AG +AM. block1.pdf: §1.2, §1.3
2	Lecture 3.5 (video 5 min): Convergence of AM preparation
	Lecture 4: Convergence AM. §1.4
	Lecture 4.5: (video 7 min) Breakdown
	Lecture 5: Linsys Intro + GMRES. block2.pdf: §2.1.1
3	Lecture 6: GMRES conv. disks. block2.pdf: §2.1.2
	Lecture 7: CG + CG convergence. block2.pdf: §2.2
	Lecture 8: CGNE + BiCG. block2.pdf: §2.3
4	Lecture 9: Precond GMRES, intro QR-algorithm. block3.pdf: §3.1-3.2
	Lecture 9.5 (video 5 min): Precond CG part 1 also in Quiz HW2
	Lecture 9.8 (video 6 min): Precond CG part 2 also in Quiz HW2
	Lecture 10: QR-method. Improvement 1: phase 1. block3.pdf: §3.2.1
5	Lecture 11: QR-method. Improvement 2: phase 2. Improvement 2. block3.pdf: §3.2.2
	Lecture 12: QR-method shifts + conv: block3.pdf §3.3
6	Lecture 13: Matrix functions intro. block4.pdf: §4.1
	Lecture 14: Matrix exponention + Mat sqrt + Mat sign. block4.pdf: §4.2
7	Lecture 15: Krylov for matexp integrators. block4.pdf: §4.3

See modules for more information.