One interesting property of a Fibonacci sequence is that the ratio of the values of adjacent members of the sequence approach a number called “the golden ratio” or PHI. Create a program that accepts the first two numbers of a Fibonacci sequence as a user input and then calculates additional values in the sequence until the ratio of adjacent values converges to within 0.001. You can do this in a WHILE loop by comparing the ratio of element k to element k-1 and the ratio of element k-1 to element k-2. If you call your sequence x, then the code for the WHILE statement is
while abs(x(k)/x(k-1) – x(k-1)/x(k-2))>0.001
I think I'm doing something wrong... Here's my attempt at a solution
a=input('Please enter the first two numbers of the Fibonacci sequence in "[# #]" form. Do not enter [0 1] or [1 1]. Your free to enter anything else though.');
x=[a ones(1,100000)];
z=3;
while abs(x(z)/x(z-1)-x(z-1)/x(z-2))>0.001
x(z)=x(z-2)+x(z-1);
z=z+1;
end
g=x(find(x>1));
It seems like you're on the right track, but there are a few issues with your code. Here's an explanation of the problems and how to fix them.
1. Input format: Your current input prompt suggests entering the first two Fibonacci numbers in a "[# #]" form. However, the expected input format for MATLAB is a numeric array without square brackets. So, you can modify your code to accept input as individual numbers.
```matlab
a = input('Please enter the first number of the Fibonacci sequence: ');
b = input('Please enter the second number of the Fibonacci sequence: ');
```
2. Initialization of Fibonacci sequence: In your code, you initialize the Fibonacci sequence as `x = [a ones(1,100000)]`, but this will create an array of length 100,002. Since you're only interested in convergence, you can start with just the first three values [a, b, a+b].
```matlab
x = [a, b, a+b];
```
3. Loop condition: Your current while loop condition `abs(x(z)/x(z-1)-x(z-1)/x(z-2))>0.001` checks for the convergence condition correctly, but it can return an index out of bounds error because you haven't checked the length of `x` while accessing elements. It's better to use `numel(x)` to check the length.
```matlab
while abs(x(z)/x(z-1)-x(z-1)/x(z-2)) > 0.001 && numel(x) < 100000
```
4. Fibonacci calculation and iteration: Inside the while loop, you correctly calculate the next Fibonacci number as `x(z) = x(z-2) + x(z-1)`, but you forgot to increment `z` after each iteration. Adding `z = z+1` will fix this issue.
```matlab
while abs(x(z)/x(z-1)-x(z-1)/x(z-2)) > 0.001 && numel(x) < 100000
x(z) = x(z-2) + x(z-1);
z = z + 1;
end
```
5. Extracting Fibonacci numbers after convergence: After the loop, you define `g = x(find(x>1))`, but this will return all numbers greater than 1, including the ones before convergence. To only get the Fibonacci numbers after convergence, you can use `x(3:end)`.
```matlab
g = x(3:end);
```
The modified code should look like this:
```matlab
a = input('Please enter the first number of the Fibonacci sequence: ');
b = input('Please enter the second number of the Fibonacci sequence: ');
x = [a, b, a+b];
z = 3;
while abs(x(z)/x(z-1)-x(z-1)/x(z-2)) > 0.001 && numel(x) < 100000
x(z) = x(z-2) + x(z-1);
z = z + 1;
end
g = x(3:end);
```
This code will calculate additional values in the Fibonacci sequence until the ratio of adjacent values converges within 0.001. The resulting Fibonacci sequence after convergence will be stored in the variable `g`.
It seems like there are a couple of errors in your code. Here are the corrections:
1. When accepting user input, the input() function returns a string. You need to convert it to a list of integers using the eval() function. Here's the updated line:
```
a = eval(input('Please enter the first two numbers of the Fibonacci sequence in "[# #]" form. Do not enter [0 1] or [1 1]. Your are free to enter anything else though.'))
```
2. The line `x=[a ones(1,100000)];` is incorrect. Instead, you want to initialize the `x` variable with the first two numbers of the Fibonacci sequence provided by the user. Here's the updated line:
```
x = [a[0], a[1]]
```
3. The line `g=x(find(x>1));` is unnecessary. If you want to print the Fibonacci sequence elements whose adjacent ratios converge to within 0.001, you can add the following line after the WHILE loop:
```
print(x)
```
4. The line `z = 3;` is not necessary. You can use the length of the `x` list to keep track of the current index. Here's the updated WHILE loop:
```
while abs(x[-1] / x[-2] - x[-2] / x[-3]) > 0.001:
x.append(x[-2] + x[-1])
```
With these modifications, your code should work correctly.