# Cleve’s Corner: Cleve Moler on Mathematics and ComputingScientific computing, math & more

Posts 21 - 30 of 46

# Householder Reflections and the QR Decomposition1

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.... read more >>

# Compare Gram-Schmidt and Householder Orthogonalization Algorithms1

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…. read more >>

# The Graeffe Root-Squaring Method for Computing the Zeros of a Polynomial1

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, $x^3-2x-5$.... read more >>

# 19 Dubious Ways to Compute the Zeros of a Polynomial2

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".... read more >>

# Modernization of Numerical Integration, From Quad to Integral

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

# 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.... read more >>

# Fractal Global Behavior of Newton’s Method

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 $x^3-2x-5$. ... read more >>

# Testing Zero Finders2

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

# Zeroin, Part 3: MATLAB Zero Finder, FZERO

MATLAB adds capability to search for an interval with a sign change.... read more >>

# Zeroin, Part 2: Brent’s Version2

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. ... read more >>

Posts 21 - 30 of 46