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M begins with a preamble. function y = crypto(x) % CRYPTO Cryptography example. % y = crypto(x) converts an ASCII text string into another % coded string. The function is its own inverse, so % crypto(crypto(x)) gives x back. % See also: ENCRYPT. 30 Chapter 1. Introduction to MATLAB A comment precedes the statement that assigns the prime p. % Use a two-character Hill cipher with arithmetic % modulo 97, a prime. p = 97; Choose two characters above ASCII 128 to expand the size of the character set from 95 to 97.
This is the arithmetic used for almost all scientific and engineering computation, especially for large matrices. But for a 3-by-3 matrix, it is easy to repeat the computations using symbolic arithmetic and the Symbolic Toolbox connection to Maple. The statement A = sym(A) changes the internal representation of A to a symbolic form that is displayed as A [ [ [ = 8, 1, 6] 3, 5, 7] 4, 9, 2] Now commands like sum(A), sum(A’)’, det(A), inv(A), eig(A), svd(A) produce symbolic results. In particular, the eigenvalue problem for this matrix can be solved exactly, and e = [ 15] [ 2*6^(1/2)] [ -2*6^(1/2)] A 4-by-4 magic square is one of several mathematical objects on display in Melancolia, a Renaissance etching by Albrect Durer.
The solution process subtracts a multiple of the first equation from the second. 7/17, which turns out to be the floating-point number obtained by truncating, rather than rounding, the binary expansion of 1/10. The matrix A and the right-hand side b are modified by A(2,:) = A(2,:) - mu*A(1,:) b(2) = b(2) - mu*b(1) With exact computation, both A(2,2) and b(2) would become zero, but with floating-point arithmetic, they both become nonzero multiples of eps. 4408e-16 Matlab notices the tiny value of the new A(2,2) and displays a message warning that the matrix is close to singular.