Computational Project # 2
Due February 9
Develop, debug, and test a program (MATLAB m-file) that solves a system of linear
equations
=
Ax b
using Doolittle LU decomposition. (Important: do not perform Crout LU
decomposition, you will be given no credits for it.)
The code should be able to solve for arbitrary rectangular
n n
Ć
matrix A and right hand-
side b. Organize your code such a way that both matrix A and vector b are the input
parameters and vector x and matrices L and U are your output arguments.
Consider the possibility that matrix A is rank-deficient and the system does not have the
solution. This will happen when in the process of LU decomposition you will encounter
zero diagonal element. This can only happen if the rows or columns are linear dependent.
If this happen, make your program display the error message like āerror: matrix is rank
deficientā and stop.
If possible, solve the following systems of linear equations with your own program:
a)
1
2
3
0 1 1 4
1 3 2 4
1 2 1 8
x
x
x
ļ£® ļ£¹
ā
ļ£® ļ£¹ ļ£® ļ£¹
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
=
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ā
ļ£° ļ£» ļ£° ļ£»
ļ£° ļ£»
b)
1
2
3
1 1 0 1
1 3 2 0
0 2 2 1
x
x
x
ļ£® ļ£¹
ļ£® ļ£¹ ļ£® ļ£¹
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
=
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ļ£° ļ£» ļ£° ļ£»
ļ£° ļ£»
c)
1
2
3
4
2 2 1 3 1
2 1 1 3 3
1 1 1 4 0
3 3 4 1 9
x
x
x
x
ļ£® ļ£¹
ļ£® ļ£¹ ļ£® ļ£¹
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ā ā ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
=
ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ā ā ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ ļ£Æ ļ£ŗ
ļ£Æ ļ£ŗ
ļ£° ļ£» ļ£° ļ£»
ļ£° ļ£»
Check you results by substituting x into Ax and compare it with b. If your code is correct,
Ax should be identical to b. If your code can not solve the problem, see whether you can
invert the matrix using MATLAB internal procedure, inv(A). If you can, then the
problem definitely has the solution and something is wrong with your code.
P.S. Email your code. The program that you send should be a working program All the
codes will be checked whether they run or not. . If they are erroneous, but run, points will
be taken for the errors. If the code is not running (it has some syntax errors), an
additional 25% will be taken out. The goal of this class is for you to be comfortable
solving engineering problems. Please take your time and learn how to trust the computer.