Question

Assume that two vectors have been defined in MATLAB as follows:

>> a= (8: -0 . 5:4);
>> b = [0.04 1.1143 4.81 7.7 1.05 12.64 5.5836 1.0 6.234 ];
Analyze the following loops:
A)
for k= 1 : 9
v(k) = π/2 ×(a(k)²) × b(k);
end
B)
n = 0;
while n<9
v = π/2 ×(a²) × b;
n = n+1;
end
c)
i=0;
while i<9
i = i + 1;
v(i) = π/2 ×(a(i)²) × b(i);
end

Answer 1

The student's question involves the analysis of for and while loops in MATLAB using two predefined vectors. The for and corrected while loops perform element-wise operations on the vectors. There is an error in the second while loop as it misses indexing the vectors for element-wise calculations.

Step-by-step explanation:

The student is asking about analyzing loops in MATLAB with given vectors a and b. There are three parts to the question:

1. For loop: for k= 1 : 9 v(k) = π/2 ×(a(k)²) × b(k); end
2. While loop with incorrect vector operations: n = 0; while n<9 v = π/2 ×(a²) × b; n = n+1; end
3. Correct while loop: i=0; while i<9 i = i + 1; v(i) = π/2 ×(a(i)²) × b(i); end

The for loop calculates the result for each pair of elements from vectors a and b. The second while loop has an error because it does not index the vectors a and b which should be done for element-wise operations. The third while loop is similar to the for loop and correctly calculates the result for each element pair.

Answer 2

Let's analyze each loop:

A)
```matlab
for k = 1:9
v(k) = (pi/2) * (a(k)^2) * b(k);
end
```
This loop calculates values for the vector \( v \) using the formula \( \frac{\pi}{2} \times (a(k)^2) \times b(k) \) for each element in the vectors \( a \) and \( b \).

B)
```matlab
n = 0;
while n < 9
v = (pi/2) * (a.^2) * b;
n = n + 1;
end
```
This loop calculates the vector \( v \) with the same formula as in Loop A, but it doesn't use the loop variable \( n \). Instead, it repeatedly overwrites the entire vector \( v \) in each iteration, which might not be the intended behavior. It's better to use a vectorized approach without a loop in this case.

C)
```matlab
i = 0;
while i < 9
i = i + 1;
v(i) = (pi/2) * (a(i)^2) * b(i);
end
```
This loop is similar to Loop A, calculating values for the vector \( v \) using the formula \( \frac{\pi}{2} \times (a(i)^2) \times b(i) \) for each element in the vectors \( a \) and \( b \). The loop increments the index \( i \) and calculates \( v(i) \) for each iteration.

Overall, Loop C is more appropriate for this task as it correctly updates individual elements of \( v \) for each iteration. Loop B might not achieve the intended result due to overwriting the entire \( v \) vector in each iteration.