For any matrix size n≥2 let A be the matrix of
equation (1), and let the vector
b be given by
For matrix sizes
n = 3, 4, 5, . . . 20
try to find the solution to Ax = b by using
(1) Gaussian elimination.
(2) Gaussian elimination with partial pivoting.
(3) Computing first a Q-R decomposition of A using Gram-Schmidt, and solving
Rx = QTb using only the back-substitution part of the Gaussian
(4) Computing first a Q-R decomposition of A using Householder, and solving
Rx = QTb using only the back-substitution part of the Gaussian elimination algorithm.
(5) Using Jacobi iteration (e.g. start with x0 = 0)
(6) Using Gauss-Seidel iteration (e.g. start with x0 = 0)
Summarize your findings by plotting
against n for all the above
algorithms. Try to explain your result (in a few sentences).
4. Project – Part 3
For any matrix size n≥2 generate the matrix A and the
vector b via the following
algorithm (working C/C++ code, except for setting the matrix/vector elements).
For the matrix sizes
n = 2, 3, . . . , 20, 50, 100
find the solution to the equation Ax = b using the same algorithms as in part
Summarize your results in two plots . One plot is
against n, the other
plot is the run time against n
In order to measure the run time you might want to solve the very same prob-
lem repeatedly (say 105 or so) times. Also, if you are using a compiled language
( like C/C ++ or Java) you probably want to disable optimizations, as this extra
loop might be optimized away by the compiler (unless you know how to trick the
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