Question

Show that the following algorithm for sorting six numbers is suboptimal, regardless of the choices for Algorithms R, S, and T.

Answer 1

we can easily solve a lower triangular system Lx b by simple Consider the problem AX = B with many different right-hand sides associated with the same system matrix. In this case we need to compute the factorization A = LU only once, and then forward substitution In MATLAB this is done with the following function function x -forward (L, x) % FORWARD. For lowe r triangular L, [n, n] We then proceed as before x forward (L, b) solves L*x =b = Solve LY B by many forward substitutions (in parallel) size (L) ; O Solve UX-Y by many back substitutions (in parallel) for k = 2:n % for each row, from the top down % all columns simultaneously -L (k, j ) *x( j) ) /L ( k, k) ; Remark In order to appreciate the usefulness of this approach, note that one can show that the operations count for the matrix factorization is o(, while that for forward and back substitution is O(r2) An upper triangular system Ux- b is solved by back substitution The MATLAB code is: j = 1:x-1; x (k) (x (k) end ote that L (k, j)*x (j) calculates a dot product. Also, note that the ight-hand side vector, here x, is overwritten with the solution function x - backsubs (U, x) % BACKSUBs. For upper triangular U, x backsubs (U,b) solves U*x -b [n, n] x (n) for k size (U) ; x (n) /U (n, n) ; n-1:-1:1 % for each row, % all columns simultaneously from the bottom up x (k) -(x (k) - U (k, j) *x (j))/U(k,k) end Again, U (k, j) *x (j) calculates a dot product, and the right-hand side vector is overwritten with the solution