Three-term recurrence relations are the basis for computing Bessel functions.... read more >>

# Three-Term Recurrence Relations and Bessel Functions

Posted by **Cleve Moler**,

Blogs

Posted by **Cleve Moler**,

Three-term recurrence relations are the basis for computing Bessel functions.... read more >>

Posted by **Cleve Moler**,

A binary tree is an elegant way to represent and process Morse code. The new MATLAB graph object provides an elegant way to manipulate binary trees. A new app, `morse_tree`, based on this approach, is now available in version 2.40 of Cleve's Laboratory.... read more >>

**Category:**- Algorithms,
- Fun,
- Graphics,
- History

Posted by **Cleve Moler**,

MIT's Professor Daniel Frey recently introduced me to an ancient mechanical puzzle known as "Chinese Rings", "Patience", or "Baguenaudier." I have modified Daniel's simulator to produce a new app. The state space of the puzzle forms a hypercube.... read more >>

**Category:**- Algorithms,
- Fun,
- Graphics,
- History

Posted by **Cleve Moler**,

"ULP" stands for "unit in the last place." An *ulps plot* samples a fundamental math function such as $\sin{x}$, or a more esoteric function like a Bessel function. The samples are compared with more accurate values obtained from a higher precision computation. A plot of the accuracy, measured in ulps, reveals valuable information about the underlying algorithms.... read more >>

Posted by **Cleve Moler**,

A headline in the *New York Times* at the end of 2016 said "Growth of U.S. Population Is at Slowest Pace Since 1937". This prompted me to revisit an old chestnut about fitting and extrapolating census data. In the process I have added a couple of nonlinear fits, namely the logistic curve and the double exponential Gompertz model.... read more >>

**Category:**- Algorithms,
- Calculus,
- History,
- Numerical Analysis

Posted by **Cleve Moler**,

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.... read more >>

**Category:**- Algorithms,
- Magic Squares,
- Matrices,
- Numerical Analysis

Posted by **Cleve Moler**,

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.... read more >>

**Category:**- Algorithms,
- Magic Squares,
- Matrices,
- Numerical Analysis

Posted by **Cleve Moler**,

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 >>

**Category:**- Algorithms,
- History,
- Matrices,
- People

Posted by **Cleve Moler**,

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 >>

**Category:**- Algorithms,
- History,
- Matrices,
- Numerical Analysis

Posted by **Cleve Moler**,

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 >>

**Category:**- Algorithms,
- Fun,
- History,
- Numerical Analysis,
- People

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