# a1W10 - AMATH 341 / CM 271 / CS 371 Assignment 1 Due :...

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AMATH 341 / CM 271 / CS 371 Assignment 1 Due : Monday January 18, 2010 1. Implement in Matlab two functions function d = det1(A) and function d = det2(A) to compute the determinant of an n × n matrix A using two diﬀerent algorithms. The ﬁrst algorithm, det1(A) , applies lutx to A and then computes det( A ) = det( P ) det( L ) det( U ). It is known that det( U ) is the product of its diagonal entries, and this is not hard to implement in Matlab (see prod and diag ). Next, det( L ) is identically 1 so there is nothing to compute. The determinant of P is either 1 or 1. In more detail, it is 1 if P can be factored as an odd number of swaps while it is +1 if P can be factored as an even number of swaps. An algorithm to compute det( P ) from the vector p returned by lutx is as follows. Initialize a counter c to 0. Initialize a loop index j to 1 and then execute the following repeatedly. If p ( j ) = j , then increment j . If p ( j ) ̸ = j , say p ( j ) = i , then swap entries i and j of p , increment c (which counts the number of swaps), and repeat (without incrementing j ). Terminate the loop when j reaches n + 1. (Use a while loop in Matlab.) The function det2 implements the classical recursive formula for a determinant, det( A ) = n j =1 ( 1) 1+ j a 1 j det( A 1 j ) where A ij is the i,j co-factor of A , i.e. the ( n 1) × ( n 1) matrix obtained by removing row i and column j from A . To implement recursion in matlab, ﬁrst let n be the matrix size (see size ), and take two cases depending on whether n = 1 or n > 1. If n = 1, then det( A ) = A . If

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## This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.

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a1W10 - AMATH 341 / CM 271 / CS 371 Assignment 1 Due :...

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