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Homework answers / question archive / MAT 275 Laboratory 2 Matrix Computations and Programming in MATLAB In this laboratory session we will learn how to Create and manipulate matrices and vectors
MAT 275 Laboratory 2
Matrix Computations and Programming in MATLAB
In this laboratory session we will learn how to
NOTE: For your lab write-up, follow the instructions of LAB1.
Matrices and Linear Algebra
? Matrices can be constructed in MATLAB in different ways. For example the 3 × 3 matrix
can be entered as
>> A=[8,1,6;3,5,7;4,9,2]
A =
8 1 6
3 5 7
or
>> A=[8,1,6;
3,5,7;
4,9,2]
A =
8 1 6
3 5 7
or defined as the concatenation of 3 rows
>> row1=[8,1,6]; row2=[3,5,7]; row3=[4,9,2]; A=[row1;row2;row3]
8 1 6
or 3 columns
>> col1=[8;3;4]; col2=[1;5;9]; col3=[6;7;2]; A=[col1,col2,col3]
A =
8 1 6
Note the use of , and ;. Concatenated rows/columns must have the same length. Larger matrices can be created from smaller ones in the same way:
>> C=[A,A] % Same as C=[A A]
C =
The matrix C has dimension 3 × 6 (“3 by 6”). On the other hand smaller matrices (submatrices) can be extracted from any given matrix:
>> A(2,3) % coefficient of A in 2nd row, 3rd column ans =
7
>> A(1,:) % 1st row of A ans =
8 1 6
>> A(:,3) % 3rd column of A
ans =
6
7
2
>> A([1,3],[2,3]) % keep coefficients in rows 1 & 3 and columns 2 & 3 ans =
1 6
? Some matrices are already predefined in MATLAB:
>> I=eye(3) % the Identity matrix I =
1 0 0
0 1 0
0 0 1
>> magic(3)
ans =
8 1 6
(what is magic about this matrix?)
? Matrices can be manipulated very easily in MATLAB (unlike Maple). Here are sample commands to exercise with:
>> A=magic(3);
>> B=A’ % transpose of A, i.e, rows of B are columns of A
B =
8 3 4
1 5 9
6 7 2
>> A+B % sum of A and B
ans =
16 4 10 4 10 16
10 16 4
>> A*B % standard linear algebra matrix multiplication ans =
101 71 53
71 83 71
53 71 101
>> A.*B % coefficient-wise multiplication ans =
64 3 24 3 25 63 24 63 4
? One MATLAB command is especially relevant when studying the solution of linear systems of differentials equations: x=A\b determines the solution x = A−1b of the linear system Ax = b. Here is an example:
>> A=magic(3);
>> z=[1,2,3]’ % same as z=[1;2;3] z =
1 2 3
>> b=A*z b =
28
34
28
>> x = A\b % solve the system Ax = b. Compare with the exact solution, z, defined above.
x =
1
2
3
>> y =inv(A)*b % solve the system using the inverse: less efficient and accurate ans =
1.0000
2.0000
3.0000
Now let’s check for accuracy by evaluating the difference z − x and z − y. In exact arithmetic they should both be zero since x, y and z all represent the solution to the system.
>> z - x % error for backslash command ans =
0
0
0
>> z - y % error for inverse ans =
1.0e-015 *
-0.4441
0
-0.8882
Note the multiplicative factor 10−15 in the last computation. MATLAB performs all operations using standard IEEE double precision.
Important!: Because of the finite precision of computer arithmetic and roundoff error, vectors or matrices that are zero (theoretically) may appear in MATLAB in exponential form such as 1.0e-15 M where M is a vector or matrix with entries between −1 and 1. This means that each component of the answer is less than 10−15 in absolute value, so the vector or matrix can be treated as zero (numerically) in comparison to vectors or matrices that are on the order of 1 in size.
EXERCISE 1
Enter the following matrices and vectors in MATLAB
MATLAB Programming
It is often advantageous to be able to execute a segment of a code a number of times. A segment of a code that is executed repeatedly is called a loop.
To understand how loops work, it is important to recognize the difference between an algebraic equality and a MATLAB assignment. Consider the following commands:
>> counter = 2 counter = 2
>> counter = counter +1 counter =
3
The last statement does not say that counter is one more than itself. When MATLAB encounters the second statement, it looks up the present value of counter (2), evaluates the expression counter + 1 (3), and stores the result of the computation in the variable on the left, here counter. The effect of the statement is to increment the variable counter by 1, from 3 to 4. Similarly, consider the commands:
>> v=[1,2,3]
v =
1 2 3
>> v=[v,4]
v =
1 2 3 4
When MATLAB encounters the second statement, it looks up the present value of v, adds the number 4
as entry of the vector, and stores the result in the variable on the left, here v. The effect of the statement is to augment the vector v with the entry 4.
There are two types of loops in MATLAB: for loops and while loops
for loops
When we know exactly how many times to execute the loop, the for loop is often a good implementation choice. One form of the command is as follows:
for k=kmin:kmax
<list of commands> end
The loop index or loop variable is k, and k takes on integer values from the loop’s initial value, kmin, through its terminal value, kmax. For each value of k, MATLAB executes the body of the loop, which is the list of commands.
Here are a few examples:
S = 0; % initialize running sum for k = 1:10
S = S+k^2; end S
Because we are not printing intermediate values of S, we display the final value of S after the loop by typing S on a line by itself. Try removing the “;” inside the loop to see how S is incremented every time we go through the loop.
p = 1; % initialize running product for k = 2:10 p = p*k;
end p
? Whenever possible all these construct should be avoided and built in MATLAB functions used instead to improve efficiency. In particular lengthy loops introduce a substantial overhead.
The value of S in the example above can be evaluated with a single MATLAB statement:
>> S = sum((1:10).^2)
Type help sum to see how the built in sum function works.
Similarly the product p can be evaluated using
>> p = prod(1:10)
Type help prod to see how the built in prod function works.
Recall that a geometric sum is a sum of the form a + ar + ar2 + ar3 + ....
Hint: Start by defining the vector e=0:n-1 and then evaluate the vector R = r.^e. It should be easy to figure out how to find the sum from there.
The counter in a for or while loop can be given explicit increment: for i =m:k:n to advance the counter i by k each time. In this problem we will evaluate the product of the first 10 odd numbers 1 · 3 · 5 · ... · 19 in two ways:
The while loop repeats a sequence of commands as long as some condition is met. The basic structure of a while loop is the following:
while <condition>
<list of commands> end
Here are some examples:
S = 0; % initialize running sum k = 1; % initialize current integer incr = 1; % initialize test value while incr>=1e-10 S = S+incr; k = k+1; incr = 1/k^2;
end
What is the value of S returned by this script? Compare to
.
y = []; % initialize the vector y to the empty vector k = 1; % initialize the counter value = 1; % initialize the test value to be added to the vector y while value < 2000 y = [y, value]; % augment the vector y k = k+1; % update the counter
value = factorial(k); % evaluate the next test value end y
Write a script file that creates a row vector v containing all the powers of 2 below 1000. The output vector should have the form: v = [ 2,4,8,16... ]. Use a while loop.
if statement
The basic structure of an if statement is the following:
if condition
<list of commands> elseif condition <list of commands> :
else
<list of commands> end
Here is an example:
• Evaluate
−
for a given (but unknown) scalar x and, if x = 2, display “y is undefined at x = 2”.
function y=f(x) if x==2 disp(’y is undefined at x = 2’)
elseif x <= 1 y=x^3+2; else y=1/(x-2); end end
We can test the file by evaluating it at different values of x. Below we evaluate the function at x = −1, x = 2 and x = 4.
>> f(-1) ans =
1 >> f(2) y is undefined at x = 2
>> f(4) ans =
0.5000
Write a function file that creates the following piecewise function:
Assume x is a scalar. The function file should contain an
if statement to distinguish between the different cases. The function should also display “the function is undefined at x = 10” if the input is x = 10. Test your function by evaluating f(1), f(4), f(7) and f(10).
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