Making a function sensitive for vector input - matlab

Question

This is for my course in numerical methods. I am trying very hard to understand MATLAB and its syntax but I am 100% self-taught, so please bear with me if my attempts seem ridiculous.
I have written this very easy function to approximate the number e
function e= calcEulerLimit(n)
e = (1 + 1./n).^n;
end
This is the 'basic' definition of the number e using a limit n to infinity approach. For MATLAB I defined the following vector (when I am talking about n in the latter, I am always referring to this vector n)
n=[1:1:10]=[ 1 2 3 4 5 6 7 8 9 10 ]
and the output works just fine as I expected, it is sensitive towards the vector n input when I call my function in MATLAB.
>> calcEulerlimit(n)
ans =
2.0000 2.2500 2.3704 2.4414 2.4883 2.5216 2.5465 2.5658 2.5812 2.5937
Now I want to do the exact same as above with a Taylor Approach, using the infinite summation formula to describe e, here is where I am stuck, the following simple code works:
function e = calcEulerSum(n)
e=1; % base-case, start variable
for i=1:1:n % for loop with step size one
e=e+1/factorial(i)
end
end
but this input, of course, does not work when I want to enter a vector such as n that computes through all variables.
I tried something along the line with another for loop, and a while loop, but the while loop seems to never terminate:
function e = calcEulerSum3(n)
while n
e=1;
e = e + 1./cumprod(n);
end
end
Using cumprod(n) to get the factorial value of each column element of my vector n.

Answer 1

You're trying to vectorize your function. Your solution for scalars works so let's look at what it's doing as i is incremented:
e0 = 1;
i = 1: e(1) = e0 + 1/factorial(1)
i = 2: e(2) = e(1) + 1/factorial(2) = e0 + 1/factorial(1) + 1/factorial(2)
= e0 + sum(1./factorial(1:2))
i = 3: e(3) = e(2) + 1/factorial(3) = ...
= e0 + sum(1./factorial(1:3))
...
i = n: e(n) = e(n-1) + 1/factorial(n) = e0 + 1/factorial(1) + ... + 1/factorial(n)
= e0 + sum(1./factorial(1:n))
So can you come up with a general expression to calculate the vector e given n? The cumsum function will come in handy.

Answer 2

for i=whatsoever,statement(i);end executes the statement on every element of whatsoever. If it is a single number, then on this one, if it is a vector/array, then on every element of it.
1:1:n creates an array of integers from 1 to n on the spot (1:n would do, too). If n already is a vector with the elements you want to iterate over, you can use it directly: for i=n.
However, why did you use the dotted versions of operations in your first and third code block, but not your second? Because you read about MATLAB's vectorization? Then you seem to be on the right track, but remember that the point vectorization is to get rid of an explicit loop entirely.

Answer 3

function e = calcEulerSum(n)
e=nan(1,length(n)); % initialize to nan
for j=1:length(n) % for each element in the input
e(j)=sum(1./factorial(0:n(j))); %each entry is computed in this step, one at a time
end
In this code, each approximation has been vectorized, but I don't see an easy way to vectorize the entire program. There is also no error checking, like making sure the elements of n are non-negative integers, or that n is a vector and not an array.
In the original example, the while never terminates because n never changes.
This should be enough to help get you started.
Cheers!