- #1
aznkid310
- 109
- 1
Homework Statement
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.
Homework 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 don't know how to implement them in MATLAB.
The Attempt at a Solution
For #1, i get
x = 4
-2
2
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