Download Computers, pattern, chaos and beauty : graphics from an by Clifford A Pickover PDF

By Clifford A Pickover

Combining fractal conception with computing device artwork, this booklet introduces an inventive use of pcs. It describes photograph equipment for detecting styles in advanced info and illustrates easy innovations for visualizing chaotic habit. "Beautiful." — Martin Gardner, Scientific American. Over 275 illustrations, 29 in color.

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Extra resources for Computers, pattern, chaos and beauty : graphics from an unseen world

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8. The bee design in a local coordinate system (left) and placement of several bees in global coordinate systems (right). axes, and normally it encloses the extents of the geometry (the bee). 8 (left): the lower-left point has coordinates (0, 0) and the upper-right point has coordinates (1, 1). Let (u1 , u2 ) be a point in the local system. The local coordinates are also known as parameters. The area on the page where you will place a copy of the bee will define a global coordinate system. Let’s define this system by the minmax box with extents (min1 , min2 ) and (max1 , max2 ).

Iterative System Solvers 49 for our first guess. We note that it clearly is not the solution to our system: Au(1) = b. A better guess ought to be obtained by using the current guess (2) and solving the first equation for a new u1 , the second for a new (2) u2 , and so on. 6⎦ . 1 ⎦ . 333⎦ . 0 Try one more iteration for yourself. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 50 5. Solving Linear Systems This iterative method is known as the Gauss-Jacobi iteration. In the general case, we are given a linear system with n equations and n unknowns ui ; i = 1, .

U1 ⎥ ⎥ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎥ . ⎥⎣ . ⎦ = ⎢ ⎢ .. ⎥ . ⎢ ⎥ ⎥ .. ⎥ u ⎢ .. ⎥ . ⎦ n ⎣ . 4) This also has the matrix form Au = b. 5) However, A is not square. An overdetermined system, as the name implies, will in general not have a solution. Yet an approximate solution may be found as follows. 5) by A’s transpose, AT , we obtain AT Au = AT b. 6) The matrix AT A is square and typically also nonsingular. 6) is known as the normal equations, which form a standard linear system that may be solved by Gauss elimination.

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