Description
We develop MATLAB functions to solve upper and lower triangular systems and then
solve a more general system with LU factorization.
Activities
1. Suppose we have a 4 × 4 upper triangular system Ax = b as below:
A =
−5 5 0 −2
0 6 5 7
0 0 9 −7
0 0 0 −4
, b =
−2
−9
3
−7
.
In the command window, define these matrices.
2. To solve for x using backward substitution, type
>> % Initialize the solution.
>> x = zeros(4,1)
>> % Solve A(4,4)*x(4) = b(4).
>> x(4) = b(4)/A(4,4)
>> % Solve A(3,3)*x(3) + A(3,4)*x(4) = b(3).
>> x(3) = (b(3) – A(3,4)*x(4))/A(3,3)
>> % Solve A(2,2)*x(2) + A(2,3)*x(3) + A(2,4)*x(4) = b(2).
>> x(2) = (b(2) – A(2,3)*x(3) – A(2,4)*x(4))/A(2,2)
>> % Solve A(1,1)*x(1) + A(1,2)*x(2)
>> % + A(1,3)*x(3) + A(1,4)*x(4) = b(1).
>> x(1) = (b(1) – A(1,2)*x(2) – A(1,3)*x(3) – A(1,4)*x(4))/A(1,1)
3. To perform the same task using matrix multiplication, type
>> clear x
>> x = zeros(4,1)
>> x(4) = b(4)/A(4,4)
>> x(3) = (b(3) – A(3,4:4)*x(4:4))/A(3,3)
>> x(2) = (b(2) – A(2,3:4)*x(3:4))/A(2,2)
>> x(1) = (b(1) – A(1,2:4)*x(2:4))/A(1,1)
Remark: When performing matrix multiplication, make sure the matrices have
proper dimension.
MATLAB Lab for Applied Linear Algebra Lab 6
In-Class Exercise
4. Finish the following m-file.
% This function performs backward substitution for 4 x 4 upper
% triangular systems. It solves A*x = b.
% Input: upper triangular matrix A, column vector b
% Output: column vector x that solves A*x = b
function x = backward4(A,b)
x = zeros(4,1);
x(…) = …
for i = 3:-1:1
x(…) = …
end
end
5. Run your function on the same upper triangular system as given above.
6. Write a function forward4 to solve 4 × 4 (or n × n if you want) lower triangular
systems (using forward substitution). Test your program with Bx = b, where
B = AT
, and A and b are given above.
7. We now have backward4.m and forward4.m, as well as MYLU.m from last week’s
lab. Combine these to solve the 4 × 4 system Ax = b as below:
A =
−8 1 5 9
−6 9 3 −4
−5 −2 9 −9
8 −4 3 −3
, b =
−2
−7
9
−5
.
Remark on MATLAB Functions
Inside of a function in MATLAB, you can call and use any other function that you
have, so long as you do one of the following: (a) the function that you are calling
is in the same folder as the function that is using it; (b) the function that you are
calling is beneath the function that is using it in the same script. For example, below
made up function.m uses other function.m. other function.m will have to be in
the same folder as made up function.m, or written beneath made up function.m in
the same script.
function [W,f] = made_up_function(z,p)
x = sin(z)*p/2;
y = other_function(x);
f = y^(1/3);
W = f*x;
end