Posts 61 - 70 of 101

結果: Matrices

Apologies to Gram-Schmidt

This is a follow-up to my previous follow-up, posted several days ago. A very careful reader, Bruno Bazzano, contributed a comment pointing out what he called "a small typo" in my code for the classic Gram-Schmidt algorithm. It is more than a small typo, it is a serious blunder. I must correct the code, then do more careful experiments and reword my conclusions.... 続きを読む >>

Compare Gram-Schmidt and Householder Orthogonalization Algorithms 4

This is a follow-up to my previous post. Classical Gram-Schmidt and Modified Gram-Schmidt are two algorithms for orthogonalizing a set of vectors. Householder elementary reflectors can be used for the same task. The three algorithms have very different roundoff error properties.... 続きを読む >>

Householder Reflections and the QR Decomposition 1

The QR decomposition is often the first step in algorithms for solving many different matrix problems, including linear systems, eigenvalues, and singular values. Householder reflections are the preferred tool for computing the QR decomposition.... 続きを読む >>

Matrix Multiplication Flexes House

A new app employs transformations of a graphic depicting a house to demonstrate matrix multiplication.... 続きを読む >>

Bug Report Revives Interest in SVD Option of “Eigshow”

A few days ago we received email from Mike Hennessey, a mechanical engineering professor at the University of St. Thomas in St. Paul, Minnesota. He has been reading my book “Numerical Computing with MATLAB” very carefully. Chapter 7 is about “Eigenvalues and Singular Values” and section 10.3 is about one of my all-time favorite MATLAB demos, eigshow. Mike discovered an error in my description of the svd option of eigshow that has gone unnoticed in the over ten years that the book has been available from both the MathWorks web site and SIAM…. 続きを読む >>

Compare Gram-Schmidt and Householder Orthogonalization Algorithms 1

Classical Gram-Schmidt and Modified Gram-Schmidt are two algorithms for orthogonalizing a set of vectors. Householder elementary reflectors can be used for the same task. The three algorithms have very different roundoff error properties…. 続きを読む >>

19 Dubious Ways to Compute the Zeros of a Polynomial 2

During the SIAM Annual Meeting this summer in Boston there will be a special minisymposium Wednesday afternoon, July 13, honoring Charlie Van Loan, who is retiring at Cornell. (I use "at" because he's not leaving Ithaca.) I will give a talk titled "19 Dubious Way to Compute the Zeros of a Polynomial", following in the footsteps of the paper about the matrix exponential that Charlie and I wrote in 1978 and updated 25 years later. I really don't have 19 ways to compute polynomial zeros, but then I only have a half hour for my talk. Most of the methods have been described previously in this blog. Today's post is mostly about "roots".... 続きを読む >>

Strang and Moler Video Course on Differential Equations

Gil Strang has produced a MOOC-style video course on Differential Equations and Linear Algebra. I have added some videos about the MATLAB ODE suite. The series is available from the MathWorks Web site, MIT OpenCourseWare and several other popular sources.... 続きを読む >>

Perfect Shuffles of Playing Cards

When a deck of playing cards is shuffled perfectly, the result is not random. A perfect shuffle places the cards in a mathematically precise order. As a result, when the most common version of a perfect shuffle is repeated eight times, the deck returns to its original state.... 続きを読む >>

Golub Matrices, Deceptively Ill Conditioned

I got the idea for these test matrices from Gene Golub years ago, but the mathematical foundation comes from a paper by Divakar Viswanath and Nick Trefethen. ... 続きを読む >>

Posts 61 - 70 of 101