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…. 更多内容 >>

At a minisymposium honoring Charlie Van Loan this week during the SIAM Annual Meeting, I will describe several dubious methods for computing the zeros of polynomials. One of the methods is the Graeffe Root-squaring method, which I will demonstrate using my favorite cubic, $x^3-2x-5$.... 更多内容 >>

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".... 更多内容 >>

The MATLAB functions for the numerical evaluation of integrals has evolved from quad, through quadl and quadgk, to today's integral. ... 更多内容 >>

When the starting point of Newton's method is not close to a zero of the function, the global behavior can appear to be unpredictable. Contour plots of iteration counts to convergence from a region of starting points in the complex plane generate thought-provoking fractal images. Our examples employ the subject of two recent posts, the historic cubic $x^3-2x-5$. ... 更多内容 >>

Use the historic cubic polynomial $x^3 - 2x - 5$ to test a few zero-finding algorithms. ... 更多内容 >>

The cubic polynomial $x^3 - 2x - 5$ has a unique place in the history of numerical methods.... 更多内容 >>

MATLAB adds capability to search for an interval with a sign change.... 更多内容 >>

Richard Brent's improvements to Dekker's zeroin algorithm, published in 1971, made it faster, safer in floating point arithmetic, and guaranteed not to fail. ... 更多内容 >>

Th. J. Dekker's zeroin algorithm from 1969 is one of my favorite algorithms. An elegant technique combining bisection and the secant method for finding a zero of a function of a real variable, it has become fzero in MATLAB today. This is the first of a three part series.... 更多内容 >>