18.335J/6.337J: Introduction to Numerical Methods

This is the repository of course materials for the 18.335J/6.337J course at MIT, taught by Prof. Steven G. Johnson, in Spring 2021.

Syllabus

Lectures: Monday/Wednesday/Friday 3–4pm (via Zoom videoconference). Lecture videos and handwritten notes will be posted online. Office Hours: Thursday 4–5pm (via Zoom videoconference). Zoom and video access requires MIT Touchstone authentication.

Topics: Advanced introduction to numerical linear algebra and related numerical methods. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating-point standard, sparse and structured matrices, and linear algebra software. Other topics may include memory hierarchies and the impact of caches on algorithms, nonlinear optimization, numerical integration, FFTs, and sensitivity analysis. Problem sets will involve use of Julia, a Matlab-like environment (little or no prior experience required; you will learn as you go).

Launch a Julia environment in the cloud:

Prerequisites: Understanding of linear algebra (18.06, 18.700, or equivalents). 18.335 is a graduate-level subject, however, so much more mathematical maturity, ability to deal with abstractions and proofs, and general exposure to mathematics is assumed than for 18.06!

Textbook: The primary textbook for the course is Numerical Linear Algebra by Trefethen and Bau. (Readable online with MIT certificates.)

Other Reading: Previous terms can be found in branches of the 18335 git repository. The course notes from 18.335 in much earlier terms can be found on OpenCourseWare. For a review of iterative methods, the online books Templates for the Solution of Linear Systems (Barrett et al.) and Templates for the Solution of Algebraic Eigenvalue Problems are useful surveys.

Grading: 40% problem sets (about six, ~biweekly). 20% take-home mid-term exam (posted Thursday Apr. 15 and due Friday Apr. 16), 40% final project (one-page proposal due Friday March 26, project due Thursday May 20).

• Psets will be submitted electronically via Canvas. Submit a good-quality PDF scan of any handwritten solutions and also a PDF printout of a Julia notebook of your computational solutions.

• Piazza discussion board

TA/grader: Mo Chen.

Collaboration policy: Talk to anyone you want to and read anything you want to, with three exceptions: First, you may not refer to homework solutions from the previous terms in which I taught 18.335. Second, make a solid effort to solve a problem on your own before discussing it with classmates or googling. Third, no matter whom you talk to or what you read, write up the solution on your own, without having their answer in front of you.

• You can use psetpartners.mit.edu to help you find classmates to chat with.

Final Projects: The final project will be an 8–15 page paper reviewing some interesting numerical algorithm not covered in the course. See the 18.335 final-projects page for more information, including topics from past semesters.

Lecture Summaries and Handouts

Lecture 1 (Feb 3)

• video
• handwritten notes for this and subsequent lectures
• pset 1 and accompanying notebook, due Monday, March 1.
• Newton's method for square roots and accompanying notebook.

Brief overview of the huge field of numerical methods, and outline of the small portion that this course will cover. Key new concerns in numerical analysis, which don't appear in more abstract mathematics, are (i) performance (traditionally, arithmetic counts, but now memory access often dominates) and (ii) accuracy (both floating-point roundoff errors and also convergence of intrinsic approximations in the algorithms).

As a starting example, considered the convergence of Newton's method (as applied to square roots); see the handout and Julia notebook above.

Further reading: Googling "Newton's method" will find lots of references; as usual, the Wikipedia article on Newton's method is a reasonable starting point. Beware that the terminology for the convergence order (linear, quadratic, etc.) is somewhat different in this context from the terminology for discretization schemes (first-order, second-order, etc.); see e.g. the linked Wikipedia article. Homer Reid's notes on machine arithmetic for 18.330 are an excellent introduction that covers several applications and algorithms for root-finding. For numerical computation in 18.335, we will be using the Julia language: see this information on Julia at MIT.

Lecture 2 (Feb 19)

• video
• notes on floating-point (18.335 Fall 2007; also slides)
• Julia floating-point notebook
• some floating-point myths

New topic: Floating-point arithmetic

The basic issue is that, for computer arithmetic to be fast, it has to be done in hardware, operating on numbers stored in a fixed, finite number of digits (bits). As a consequence, only a finite subset of the real numbers can be represented, and the question becomes which subset to store, how arithmetic on this subset is defined, and how to analyze the errors compared to theoretical exact arithmetic on real numbers.

In floating-point arithmetic, we store both an integer coefficient and an exponent in some base: essentially, scientific notation. This allows large dynamic range and fixed relative accuracy: if fl(x) is the closest floating-point number to any real x, then |fl(x)-x| < ε|x| where ε is the machine precision. This makes error analysis much easier and makes algorithms mostly insensitive to overall scaling or units, but has the disadvantage that it requires specialized floating-point hardware to be fast. Nowadays, all general-purpose computers, and even many little computers like your cell phones, have floating-point units.

Overview of floating-point representations, focusing on the IEEE 754 standard (see also handout from previous lecture). The key point is that the nearest floating-point number to x, denoted fl(x), has the property of uniform relative precision (for |x| and 1/|x| < than some range, ≈10³⁰⁰ for double precision) that |fl(x)−_x_| ≤ εmachine|x|, where εmachine is the relative "machine precision" (about 10⁻¹⁶ for double precision). There are also a few special values: ±Inf (e.g. for overflow), NaN, and ±0 (e.g. for underflow).

Went through some simple examples in Julia (see notebook above), illustrating basic syntax and a few interesting tidbits. In particular, we looked at two examples of catastrophic cancellation and how it can sometimes be avoided by rearranging a calculation.

Further reading: Trefethen, lecture 13. What Every Computer Scientist Should Know About Floating Point Arithmetic (David Goldberg, ACM 1991). William Kahan, How Java's floating-point hurts everyone everywhere (2004): contains a nice discussion of floating-point myths and misconceptions. A brief but useful summary can be found in this Julia-focused floating-point overview by Prof. John Gibson.

Julia tutorial (Feb 19: 5pm) — optional

• video
• tutorial notes online
• Julia cheat-sheet
• Julia intro slides

On Friday, 19 February, at 5pm via Zoom, I will give an (attendance-optional!) Julia tutorial, introducing the Julia programming language and environment that we will use this term. Please see the tutorial notes online.

Please try to install Julia and the IJulia interface first via the abovementioned tutorial notes. Several people will be at the tutorial session to help answer installation questions. Alternatively, you can use Julia online at without installing anything (although running things on your own machine is usually faster and eliminates timeout frustrations).

Lecture 3 (Feb 22)

• video
• notes on the accuracy and stability of floating-point summation

Continued discussion from Julia floating-point notebook of last lecture, starting with catastrophic cancellation and moving on to error accumulation in summing many floating-point numbers.

Began a more rigorous analysis of summation, accuracy, and stability (see notes).

Further reading: See the further reading from the previous lecture. Trefethen, lectures 14, 15, and 3. See also the Wikipedia article on asymptotic ("big O") notation; note that for expressions like O(ε) we are looking in the limit of small arguments rather than of large arguments (as in complexity theory), but otherwise the ideas are the same. A classic paper on the accuracy of summation is Higham (1993), "The accuracy of floating point summation".

Lecture 4 (Feb 24)

• video
• notes on the equivalence of norms

Continuing notes from last time, noted that the "forwards" error of summation depends on a ratio called the "condition number" that we will generalize later in the course, and in fact the forwards relative error can be arbitrarily large for inputs that sum to nearly zero. This doesn't mean that the algorithm is "bad", however — in fact, any fixed-precision summation algorithm will have this problem.

A better way to evaluate accuracy of algorithms is given by the notion of numerical stability, most commonly by the concept of backwards stability which we now define. We can then straightforwardly prove that the naive summation algorithm is, in fact, backwards stable (see notes).

When quantifying errors, a central concept is a norm, and we saw in our proof of backwards stability of summation that it is useful to be able to choose different norms in different circumstances. Defined norms (as in lecture 3 of Trefethen): for a vector space V, a norm takes any v∈V and gives you a real number ‖v‖ satisfying three properties:

• Positive definite: ‖v‖≥0, and =0 if and only if v=0
• Scaling: ‖αv‖ = |α|⋅‖v‖ for any scalar α.
• Triangle inequality: ‖v+w‖≤‖v‖+‖w‖

There are many norms, for many different vector spaces. Gave examples of norms of column vectors: Lₚ norms (usually p = 1, 2, or ∞) and weighted L₂ norms. A (complete) normed vector space is called a Banach space, and these error concepts generalize to f(x) when f and x are in any Banach spaces (including scalars, column vectors, matrices, …though infinite-dimensional Banach spaces are trickier).

Equivalence of norms. Described fact that any two norms are equivalent up to a constant bound, and showed that this means that stability in one norm implies stability in all norms. See notes handout for a proof.

Further reading: Trefethen, lectures 14, 15, and 3. If you don't immediately recognize that A'A has nonnegative real eigenvalues (it is positive semidefinite), now is a good time to review your linear algebra; see also Trefethen lecture 24.

Lecture 5 (Feb 26)

• video

Especially important in numerical analysis are functions where the inputs and/or outputs are matrices, and for these cases we need matrix norms. The most important matrix norms are those that are related to matrix operations. Started with the Frobenius norm. Related the Frobenius norm ‖A‖F to the square root of the sum of eigenvalues of A'A, which are called the singular values σ²; we will do much more on singular values later, but for now noted that they equal the squared eigenvalues of A if A'A (Hermitian). Also defined the induced matrix norm, and bounded it below by the maximum eigenvalue magnitude of A (if A is square). For the L₂ induced norm, related it (without proof for now) to the maximum singular value. A useful property of the induced norm is ‖AB‖≤‖A‖⋅‖B‖. Multiplication by a unitary matrix Q (Q' = Q⁻¹) preserves both the Frobenius and L₂ induced norms.

Relate backwards error to forwards error, and backwards stability to forwards error (or "accuracy" as the book calls it). Show that, in the limit of high precision, the forwards error can be bounded by the backwards error multiplied by a quantity κ, the relative condition number. The nice thing about κ is that it involves only exact linear algebra and calculus, and is completely separate from considerations of floating-point roundoff. Showed that, for differentiable functions, κ can be written in terms of the induced norm of the Jacobian matrix.

Calculated condition number for square root, summation, and matrix-vector multiplication, as well as solving Ax=b, similar to the book. Defined the condition number of a matrix: for f(x)=Ax, the condition number is ‖A‖⋅‖x‖/‖Ax‖, which is bounded above by κ(A)=‖A‖⋅‖A⁻¹‖.

Further reading: Trefethen, lectures 12, 14, 15, 24. See any linear-algebra textbook for a review of eigenvalue problems, especially Hermitian/real-symmetric ones. See also these notes from 18.06 for a basic overview.

Lecture 6 (Mar 1)

• video
• pset 1 solutions
• pset 2: due Friday March 12 at 3pm.

Related matrix L₂ norm to eigenvalues of B=AᵀA (or Āᵀ=A^* for complex A). B is obviously Hermitian (Bᵀ=B), and with a little more work showed that it is positive semidefinite: xᵀBx≥0 for any x. Reviewed and re-derived properties of eigenvalues and eigenvectors of Hermitian and positive-semidefinite matrices. Hermitian means that the eigenvalues are real, the eigenvectors are orthogonal (or can be chosen orthogonal). Also, a Hermitian matrix must be diagonalizable (I skipped the proof for this; we will prove it later in a more general setting). Positive semidefinite means that the eigenvalues are nonnegative.

Proved that, for a Hermitian matrix B, the Rayleigh quotient R(x)=xᵀBx/xᵀx is bounded above and below by the largest and smallest eigenvalues of B (the "min–max theorem"). Hence showed that the L₂ induced norm of A is the square root of the largest eigenvalue of B=AᵀA. Similarly, showed that the L₂ induced norm of A⁻¹, or more generally the supremum of ‖x‖/‖Ax‖, is equal to the square root of the inverse of the smallest eigenvalue of AᵀA

Understanding norms and condition numbers of matrices therefore reduces to understanding the eigenvalues of AᵀA (or AAᵀ). However, looking at it this way is unsatisfactory for several reasons. First, we would like to solve one eigenproblem, not two. Second, working with things like AᵀA explicitly is often bad numerically, because it squares the condition number [showed that κ(AᵀA)=κ(A)²] and hence exacerbates roundoff errors. Third, we would really like to get some better understanding of A itself. All of these concerns are addressed by the singular value decomposition or SVD.

Explicitly constructed SVD (both "thin" and thick/unitary) in terms of eigenvectors/eigenvalues of AᵀA and AAᵀ. Recall from above that we related the singular values to induced L₂ norm and condition number.

Talked a little about the SVD and low-rank approximations (more on this in homework), e.g. graphically illustrated via image compression, or principal component analysis (PCA), e.g. illustrated with this nice demo of human locomotion analysis.

Further reading: Trefethen, lectures 4, 5, 11.

Lecture 7 (Mar 3)

• video

Finished SVD topics from last time.

• least-squares IJulia notebook

Introduced least-squares problems and went through some examples (notebook).

Further reading: Trefethen, lectures 7, 8, 18, 19

Lecture 8 (Mar 5)

• video

Finished least-squares notebook. Reviewed some basic consequences of SVD: κ(A) from σ max/min ratio, κ(Q)=1, κ(AᵀA)=κ(A)². Key consequence for least-squares: normal equations formulation squares the condition number—not a good idea if we can avoid it.

Introduced the alternative of QR factorization (finding an orthonormal basis for the column space of the matrix). Explained why, if we can do it accurately, this will give a good way to solve least-squares problems.

• Per Persson's 2006 18.335 Gram-Schmidt slides
• Gram-Schmidt IJulia notebook

Gave the simple, but unstable, construction of the Gram-Schmidt algorithm, to find a QR factorization.

Discussed loss of orthogonality in classical Gram-Schmidt, using a simple example (see slides), especially in the case where the matrix has nearly dependent columns to begin with. Showed modified Gram-Schmidt and argued how it (mostly) fixes the problem. (For more numerical examples, see the notebook.)

Further reading: Trefethen, lectures 7, 8, 18, 19. The Wikipedia Gram-Schmidt article is also nice. It turns out that modified GS is backwards stable in the sense that the product QR is close to A, i.e. the function f(A) = QᵀR is backwards stable in MGS; this is why solving systems with Q,R (appropriately used as discussed in Trefethen lecture 19) is an accurate approximation to solving them with A. For a review of the literature on backwards-stability proofs of MGS, see e.g. this 2006 paper by Paige et al. [SIAM J. Matrix Anal. Appl. 28, pp. 264-284].

Lecture 9 (Mar 9: Tuesday is an "MIT Monday")

• video
• Householder QR notes from Per Persson.

Floating-point operation (flop) count for classical or modified Gram–Schmidt.

Re-interpret Gram-Schmidt in matrix form as Q = AR₁R₂..., i.e. as multiplying A on the right by a sequence of upper-triangular matrices to get Q. The problem is that these matrices R may be very badly conditioned, leading to an inaccurate Q and loss of orthogonality. Instead of multiplying A on the right by R's to get Q, however, we can instead multiply A on the left by Q's to get R. In homework (pset 2), you will show that any algorithm consisting of multiplying by a sequence of Q's is backwards stable! This leads us to the Householder QR algorithm.

Introduced Householder QR, emphasized the inherent stability properties of multiplying by a sequence of unitary matrices (as shown in pset 2). Show how we can convert a matrix to upper-triangular form (superficially similar to Gaussian elimination) via unitary Householder reflectors.

Further reading: Trefethen, lectures 7, 8, 10, 16.

Lecture 10 (Mar 10)

• video

Finished Householder QR derivation from last time, including the detail that one has a choice of Householder reflectors...we choose the sign to avoid taking differences of nearly-equal vectors. Emphasized that we don't need to explicitly compute Q if we instead store the Householder reflector vectors, and can still compute Qx and Qᵀx quickly.

Operation count for Gram-Schmidt (2mn²) vs. Householder (2mn² - 2n³/3). Noted that associativity matters in matrix products: there is a big difference in computational cost between v(vᵀX) and (vvᵀ)X! Evidently, Householder is at least as accurate as modified GS while being slightly faster. But does fewer operations really mean it is faster?

• performance experiments with matrix multiplication (one-page or full-size versions)

Counting arithmetic operation counts is no longer enough. Illustrate this with some performance experiments on a much simpler problem, matrix multiplication (see handouts). This leads us to analyze memory-access efficiency and caches and points the way to restructuring many algorithms.

Further reading: Trefethen, lectures 7, 8, 10, 16.

Lecture 11 (Mar 12)

• video
• pset 2 solutions and IJulia notebook
• pset 3: due Friday, March 26 at 3pm
• IJulia matrix-multiplication notebook
• ideal-cache terminology

Outline of the memory hierarchy: CPU, registers, L1/L2 cache, main memory, and presented simple 2-level ideal-cache model that we can analyze to get the basic ideas.

Analyzed cache complexity of simple row-column matrix multiply, showed that it asymptotically gets no benefit from the cache. Presented blocked algorithm, and showed that it achieves optimal Θ(n³/√Z) cache complexity.

Discussed some practical difficulties of the blocked matrix multiplication: algorithm depends on cache-size Z, and multi-level memories require multi-level blocking. Discussed how these ideas are applied to the design of modern linear-algebra libraries (LAPACK) by building them out of block operations (performed by an optimized BLAS).

Further reading: Wikipedia has a reasonable introduction to memory locality that you might find useful. The optimized matrix multiplication shown on the handouts is called ATLAS, and you can find out more about it on the ATLAS web page. The "Cache-oblivious algorithms" paper (1999) describes the ideal cache model and analysis for various algorithms. Notes on the switch from LINPACK to LAPACK/BLAS in Matlab. The MIT course 6.172 has two lecture videos (first and second) on cache-efficient algorithms, including a discussion of matrix multiplication.

Lecture 12 (Mar 15)

• video
• experiments with cache-oblivious matrix-multiplication (handout or full-size slides) … see Julia notebook above for more recent results.

Introduced the concept of optimal cache-oblivious algorithms. Discussed cache-oblivious matrix multiplication in theory and in practice (see handout, Julia notebook, and Frigo et. al paper above).

Discussion of spatial locality and cache lines, with examples of dot products and matrix additions (both of which are "level 1 BLAS" operations with no temporal locality), and the critical implications for loop ordering when traversing multidimensional arrays.

Further reading: Frigo et al. paper from previous lecture. A typical free-software BLAS these days is OpenBLAS. See Register Allocation in Kernel Generators (talk by M. Frigo, 2007) on the difficulty of optimizing for the last level of cache (the registers) in matrix multiplication (compared to FFTs), and why a simple cache-oblivious algorithm is no longer enough. See e.g. the Wikipedia article on row-major and column-major order and the resulting Julia performance tip.

Lecture 13 (Mar 17)

• video

Review of Gaussian elimination. Reviewed the fact (from 18.06) that this givs an A=LU factorization, and that we then solve Ax=b by solving Ly=b (forward substitution: doing the same steps to b that we did to A during elimination to get y) and then solving Ux=y (backsubstitution). Emphasized that you should almost never compute A⁻¹ explicitly. It is just as cheap to keep L and U around, since triangular solves are essentially the same cost as a matrix-vector multiplication. Computing A⁻¹ is usually a mistake: you can't do anything with A⁻¹ that you couldn't do with L and U, and you are wasting both computations and accuracy in computing A⁻¹. A⁻¹ is useful in abstract manipulations, but whenever you see "x=A⁻¹b" you should interpret it for computational purposes as solving Ax=b by LU or some other method.

In Julia, x = A \ b solves Ax=b by a specialized method depending on the type of A, and F = factorization(A) followed by x = F \ b stores the factorization (e.g. LU) for re-use on subsequent right-hand sides.

Further reading: Trefethen, lectures 20–22.

Lecture 14 (Mar 19)

• video

Showed with an example that Gaussian elimination (LU) without row swaps ("pivoting") is numerically unstable. (In principle, we could instead solve Ax=b with Householder QR, but we'd rather not: QR flop count 2m³ - 2m³/3 = 4m³/3 is double that of LU.)

Introduced partial pivoting, and pointed out (omitting bookkeeping details) that this can be expressed as a PA=LU factorization where P is a permutation. Discussed backwards stability of LU, and mentioned example where U matrix grows exponentially fast with m to point out that the backwards stability result is practically useless here, and that the (indisputable) practicality of Gaussian elimination is more a result of the types of matrices that arise in practice.

Discussed Cholesky factorization, which is Gaussian elimation for the special case of Hermitian positive-definite matrices, where we can save a factor of two in time and memory. More generally, if the matrix A has a special form, one can sometimes take advantage of this to have a more efficient Ax=b solver, for example: Hermitian positive-definite (Cholesky), tridiagonal or banded (linear-time solvers), lower/upper triangular (forward/backsubstitution), generic sparse (A is mostly zero: sparse-direct and iterative solvers, to be discussed later; typically only worthwhile when the matrix is much bigger than 1000×1000).

Further reading: Trefethen, lectures 20–23. See all of the special cases of LAPACK's linear-equation solvers.

Lecture 15 (March 24 — NO CLASS MONDAY)

New topic: eigenproblems. Reviewed the usual formulation of eigenproblems and the characteristic polynomial, mentioned extensions to generalized eigenproblems and SVDs. Discussed diagonalization, defective matrices, and the generalization ot the Schur factorization.

Further reading: Trefethen, lecture 24, 25. See this Wilkinson polynomial Julia notebook for some experiments with polynomial roots in Julia, as well as this more recent 18.06 notebook.