
#1
Apr1709, 05:09 PM

P: 109

1. The problem statement, all variables and given/known data
The point of this homework is to experiment with Gaussian elimination, and to develop an appreciation for partial pivoting. Questions 1 & 2 can be done with a calculator (though a computer is preferred). Question 3 requires a computer – you do not necessarily have to write any programs: C++ examples are on the class web page, and Matlab examples are all over the web. Use whatever you like, but give credit to your sources. Q1) [2.0 1:0 1:0 [ x1 [2.0 + 10e10 1.0 10e10 10e10 x2 = 10e10 1.0 10e10 10e10] x3] 10e10] Solve this equation for x using partial pivoting. Strive to achieve the most accurate results possible (e.g., double precision). First obtain the decomposition PA = LR then use this decomposition to determine x with two back substitution steps: Ly = (Pb) Rx = y Q2) The identity matrix I can be thought of as a collection of vectors: I = (e1e2...en) where ei is a vector of length n that is zero, except in the ith component which is 1. If one were to represent the inverse matrix A^1 as a collection of vectors, A^1 = (a1a2...an). Then the ith column of A^1 can be determined by solving Aai = ei. Or, using our LR decomposition, LR(ai )= P(ei). Use this to find the inverse of the matrix in (1) with P,L,R from partial pivoting. Assess the quality of the result by computing (A^1)A. Q3) For n = 60, solve Ax = b where n n matrix A is given by Ai j =r2n + 1sin2i j2n + 1 i; j = 1 to n; and where b is given by the nlong vector of ones. With trivial pivoting, and with partial pivoting, calculate r = bAx. Present sqrt(r*r) (aka the L2 norm krk2), which should be zero with perfect math. 2. Relevant equations I was able to do #1 with a calculator and by hand, and I understand the basic ideas of 2 and 3, but I dont know how to implement them in MATLAB. 3. The attempt at a solution For #1, i get x = 4 2 2 


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