This set is due Thursday, May 8, at the beginning of lecture. (There will be another homework set, due the scheduled day of the final exam, Thursday May 15, at 11am, so I recommend you try to complete this set earlier than the scheduled deadline.)

You can work on your own, or in groups of up to three members. In case you cannot find anybody to work with, and do not know how to program, let me know as soon as possible, and we will find an alternative. As usual, you can still collaborate with others not in your group, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc.

1. Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)

2. Write a program that, given a square matrix (diagonalizable and) with real entries, computes approximations to its eigenvalues using the -algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the (tolerance) error within which the approximations will be found. Apply your method to a symmetric matrix, and check the number of iterations the process requires, as a function of the tolerance error.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. To help verify that your algorithm is proceeding correctly, at each step of the iteration have your program indicate clearly what the matrices and are, and what the new (output) matrix is.

Please make the algorithm as explicit as possible. Meaning: Do not use shortcuts already built into the software; most CASs already have functions that perform the Gram-Schmidt process to a given set of vectors, or functions that give the decomposition of a matrix. Instead, I want you to program these subroutines as well.

The programming language you use is up to you. Maple, Mathlab, Sage are standard choices, but if you prefer a different language, it should be fine. Let me know, just in case.

3. Do the same, but now for Francis’s algorithm. Apply it to the same matrix. (Here there are more matrices and some vectors the algorithm may want to display along the way. For instance, whenever a matrix is put into upper Heissenberg form, indicate what the reflectors used along the way are.)

43.614000-116.202000

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One Response to 403 – HW 3 – Computing eigenvalues

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

Ignas: It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here. […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]

Given any field automorphism of $\mathbb C$, the rational numbers are fixed. In fact, any number that is explicitly definable in $\mathbb C$ (in the first order language of fields) is fixed. (Actually, this means that we can only ensure that the rationals are fixed, I expand on this below.) Any construction of a wild automorphism uses the axiom of choice. Se […]

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The question immediately reminded me of this. Here is an argument following the same basic idea at the beginning of that argument: First, consider $B=\{x\in A\mid A\cap(-\infty,x]$ is countable$\}$, and note that $B$ itself is countable: The point is that if $x\in B$ then $A\cap(-\infty,x]\subseteq B$. Now, if $B\ne\emptyset$, let $t=\sup B$, fix an increasi […]

[…] 15. Reflectors. Francis’s algorithm (conclusion). Homework 3, due May […]