By Clifford A Pickover
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Extra resources for Computers, pattern, chaos and beauty : graphics from an unseen world
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.