hw4s.pdf - COMPUTER SCIENCE 50100 Computing for Science and...

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COMPUTER SCIENCE 50100 Computing for Science and Engineering FALL 2015 ASSIGNMENT # 4 Solutions (35 points) November 6 This assignment covers Sections 4.2.2–4.5 of the class notes. Your C code should conform to the C11 standard. The use of continue is not permitted. Every loop is permitted to have only one exit. In particular, a break statement is permitted as an exit only for a loop with header “ while (true) ”. Do not use nested if s when short circuit evaluation works. Memory leaks are not permitted. 1. (15 points) A symmetric matrix A is positive definite if and only if it can be factored as A = GG T where the Cholesky factor G is lower triangular with positive diagonal elements. The Cholesky factor G = chol ( A ) can be define recursively by chol([ α ]) = [ α 1 / 2 ] and chol α a T a ˆ A = " α 1 / 2 0 α - 1 / 2 a chol ˆ A - α - 1 aa T # where α is a scalar and a is a column vector. You are to create a file chol.c that defines two different implementations of this algorithm. Both implementations assume that only the lower triangular part of A is given, and both overwrite it with G . Both implementations should halt the calculation if α is ever nonpositive and return the value - 1; otherwise it should return 0. (a) (8 points) The first implementation has prototype int chol(int n, int ndim, double (*a)[ndim]) and is to be a faithful recursive rendition of the alogrithm given above. The array a is assumed to have n or more rows and ndim columns and assumed to contain the lower triangular part of an n by n matrix. (b) (7 points) The second implementation has prototype int chol2(int n, double **a); and is to be a nonrecursive (but otherwise faithful) rendition of the algorithm given above. The array a is assumed to be a one-dimensional array of pointers , with a[i]
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