Can One Hear the Shape of a Drum? Part 2, Eigenfunctions1

Posted by Cleve Moler,

This is the second part of a series of posts about Marc Kac's 1966 paper in the American Mathematical Monthly [1]. This part is devoted to contour plots of the eigenfunctions. ... read more >>

Can One Hear the Shape of a Drum? Part 1, Eigenvalues2

Posted by Cleve Moler,

The title of this multi-part posting is also the title of a 1966 article by Marc Kac in the American Mathematical Monthly [1]. This first part is about isospectrality. ... read more >>

Posted by Cleve Moler,

How do you compute the hypotenuse of a right triangle without squaring the lengths of the sides and without taking any square roots?... read more >>

A Balancing Act for the Matrix Exponential

Posted by Cleve Moler,

I have been interested in the computation of the matrix exponential, $e^A$, for a long time. A recent query from a user provides a provocative example.... read more >>

Splines and Pchips2

Posted by Cleve Moler,

MATLAB has two different functions for piecewise cubic interpolation, spline and pchip. Why are there two? How do they compare? ... read more >>

Friday the 13th4

Posted by Cleve Moler,

We all know that Friday the 13th is unlucky, but is it unlikely?... read more >>

Symmetric Pair Decomposition1

Posted by Cleve Moler,

An esoteric fact about matrices is that any real matrix can be written as the product of two symmetric matrices. I've known about this fact for years, but never seriously explored the computational aspects. So I'm using this post to clarify my own understanding of what I'll call the symmetric pair decomposition. It turns out that there are open questions. I don't think we know how to reliably compute the factors. But I also have to admit that, even if we could compute them, I don't know of any practical use.... read more >>

Exponential Growth2

Posted by Cleve Moler,

What, exactly, is exponential growth? What is e and what does it have to do with exponential growth? A simple MATLAB interactive graphic introduces these concepts. ... read more >>

Symplectic Spacewar4

Posted by Cleve Moler,

This is the probably the first article ever written with the title "Symplectic Spacewar". If I Google that title today, including the double quotes to keep the two words together, I do not get any hits. But as soon as Google notices this post, I should be able to see at least one hit. ... read more >>

Biorhythms2

Posted by Cleve Moler,

Biorhythms were invented over 100 years ago and entered our popular culture in the 1960s. You can still find many Web sites today that offer to prepare personalized biorhythms, or that sell software to compute them. Biorhythms are based on the notion that three sinusoidal cycles influence our lives. The physical cycle has a period of 23 days, the emotional cycle has a period of 28 days, and the intellectual cycle has a period of 33 days. For any individual, the cycles are initialized at birth. All the people on earth born on one particular day share the biorhythm determined by that date.... read more >>

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