Description
Tasks
Create an M-file named m551lab06.m in which you will complete the following tasks.
1. Start your M-file by defining the variable A to hold the matrix
A =
1 5 1
−2 1 0
3 2 −5
.
Variables: A
2. Add the command
[LA,UA]=lu(A);
1
Variables: LA, UA
Execute the M-file and examine the contents of the matrices LA and UA. Observe that UA is upper
triangular.
Q1: Is LA lower triangular? Why?
Hint: Type doc lu and look at the difference between the commands
[L,U] = lu(A) and [L,U,P] = lu(A)
3. Now add the command
[L,U,P]=lu(A);
Variables: L, U, P
Execute the M-file again and look at the matrices L, U and P.
Q2: Is L lower triangular?
4. In the Matlab command window, type the commands
>> P’*P
>> P*P’
Q3: What do the results tell you about the relationship between P and PT
?
5. In the Matlab command window, type
>> A-P’*L*U
and examine the result. It should be consistent with what you saw in the lu documentation. (Round-off
errors may prevent you from seeing exactly what you expect, but it should be close.)
6. From the previous step we saw that A = PTLU. For your own edification, you should check that this
means that A−1 = U−1L
−1P. One way to compute the matrix on the right-hand side is to use the
backslash operator: U\(L\P). Add a variable named inv LU to your M-file, storing the inverse of A
computed in this way.
Variables: inv LU
7. Add a command to compute the inverse by applying the inv function to A. Name the result inv A.
Variables: inv A
8. Define a variable called inv err holding the difference between inv LU and inv A.
Variables: inv err
Q4: Is inv err the zero matrix? Why not?
9. Create a variable b holding the the column vector
b =
14
0
−8
.
Variables: b
2
10. Now we’re going to solve the linear system
Ax = b
using the LU decomposition. This can be done efficiently in Matlab by observing that
Ax = b ⇐⇒ P
TLUx = b ⇐⇒ x = U−1
L
−1
(Pb)
.
We’ll work our way from the innermost parentheses outward and use the backslash operator instead
of explicitly computing inverses. As we proceed, notice that we never use the expensive operations of
inv or matrix-matrix multiplication.
Now, add the line
Pb = P*b;
beneath the definition of b in your M-file.
Variables: Pb
Run the M-file and compare b and Pb in command window.
Q5: How are the two vectors related?
11. Add two more lines to your M-file, one computing L\Pb and storing it in the variable y, and the next
finishing the computation by computing U\y and storing it in x.
Variables: x, y
Run your M-file, and examine the variable x. If you have completed all steps correctly, x should be
the solution to the linear system.
>> x
x =
1.0000
2.0000
3.0000
If the above isn’t what you see, go back and check your work.
12. Now define a new variable z in your M-file and assign it the value A\b and define xz err to be the
difference between x from the previous steps and z.
Variables: z, xz err
In the command window, examine the value of xz err. This time you should get exactly the zero
vector. The reason is that the steps you took in finding x are exactly the same steps that Matlab took
in finding z.
