(adsbygoogle = window.adsbygoogle || []).push({}); 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 + 10e-10

1.0 10e-10 10e-10 x2 = -10e-10

1.0 10e-10 10e-10] x3] 10e-10]

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 n-long vector of ones. With trivial pivoting, and with

partial pivoting, calculate r = b-Ax. 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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# Gaussian elimination in MATLAB

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