## The Jacobi Method UpdatedJuly 4, 2014 Mathematics is used extensively in science and engineering. Galileo, who was the first experimental physicist, also had the idea of using mathematics to express the results of his experiments. As he wrote in his 1623 book, The Assayer (Il Saggiatore),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.
## References:- 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.
Linked Keywords: Mathematics; science; engineering; Galileo Galilei; experiment; experimental; physicist; The Assayer; natural philosophy; book; universe; language; alphabet letter; triangle; circle; geometry; geometric; labyrinth; mathematician; scientist; computer; technology; Fourier series; Joseph Fourier; periodic function; sine; cosine; integer; fraction; thermal conduction; signal processing; signal analysis; electronic signal; filter; data compression; algorithm; fast Fourier transform; James Cooley; John Tukey; recursive algorithm; Fourier transform; Carl Friedrich Gauss; special information tone; free and open source; FOSS; Audacity; Gnumeric; engineer; linear equation; Jacobi method; Carl Gustav Jacob Jacobi (1804-1851); German; Jewish; professor; university; iteration; iterate; computer programming; control flow; loop; computer run-time; Gauss–Seidel method; Johns Hopkins University; Rajat Mittal; professor; Department of Mechanical Engineering; graduate school; numerical analysis; numerical method; Xiang Yang; postgraduate education; graduate student; efficiency; efficient; whiteboard; Will Kirk; scientific paper; Journal of Computational Physics; fluid mechanics; calculation; airframe; design; parallel computing; parallel computer; two-dimensional space; two-dimensional; Laplace equation; grid; YouTube Video; undergraduate engineering degree; Peking University; Doctor of Philosophy; doctoral; research; barnacle; ship; hydrodynamics; Office of Naval Research; National Science Foundation; J.W. Cooley and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier," Math. Comput., vol. 19 (1965), pp. 297-301. |
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