Philosophy is written in this grand book, the universe... But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth."[1]Mathematicians have developed many things that are useful to scientists, but these things have become much more useful in our age of computers and technology. One example is the Fourier series. In 1811, Joseph Fourier showed that many periodic functions could be expressed as the sum of sine and cosine functions of integral fractions of the period. Fourier originally developed his series in the study of thermal conduction. Today, the Fourier series has become especially useful as a means of signal analysis, as a way to separate signals into components for effective filtering, and as a method of data compression. The broader implementation of the Fourier series was facilitated by the development of a rapid algorithm for its calculation. The fast Fourier transform (FFT) algorithm was invented in 1965 by James Cooley and John Tukey. the Cooley-Tukey FFT is a recursive algorithm that breaks the Fourier transform into smaller transforms that are calculated separately and combined to obtain the transform.[2] As I wrote in a previous article (Moona Lisa, January 28, 2013), it was found that the essential trick of the FFT algorithm was discovered by Gauss around 1805 and published after his death.

Frequency spectrum of one of the telephone system special information tones.This FFT spectrum was obtained using the free and open source (FOSS) application, Audacity. (Plot of Audacity data using Gnumeric.) |

Graduate student, Xiang Yang, and Professor Rajat Mittal at a whiteboard.An idea by Yang led to an improvement of the 169-year-old Jacobi method. (Johns Hopkins University photo by Will Kirk.)[5] |

Comparison of linear equation iterative methods of solution. These data are for the solution of the two-dimensional Laplace equation on a 128x128 grid. (Still image from a YouTube Video, modified for clarity.)[6] |

- Galileo Galilei, The Assayer (1623), Stillman Drake, Translator, Doubleday & Co., New York, 1957, p. 237.
- J.W. Cooley and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier," Math. Comput., vol. 19 (1965), pp. 297-301. A PDF file is available here.
- Xiang Yang and Rajat Mittal, "Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation," Journal of Computational Physics, In Press, June 27, 2014, DOI: 10.1016/j.jcp.2014.06.010.
- Xiang Yang and Rajat Mittal, "Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation," PDF Preprint of ref. 3.
- 19th Century Math Tactic Gets a Makeover—and Yields Answers Up to 200 Times Faster, Johns Hopkins University Press Release, June 30, 2014.
- 19th Century Math Tactic Gets a Makeover—and Yields Answers Up to 200 Times Faster, YouTube Video, June 25, 2014.