N-Term Approximation for Matrices

Hello,

Utilizing a function I created to calculate the N-term approximation for a matrix:

function[eA]= N_Approx( N,A,I )
  sum=0;
for i=1:N
    ai=(A^i)/(factorial(i));
    sum=sum+ai;
end
eA=I+sum;

I would like to solve this function over a range of values of N however when I set my range for N (ie N=1:10), the approximation function above only outputs a matrix for the last number of the range. Is there a way to store the matrix created for every iteration?

Any help would be appreciated.

Thank you! CAAJ

[Guillaume]

DO NOT use sum as a variable name. Soon enough, you will try to use the sum function in the same code and end up wondering why it comes up with some weird error. That will be because matlab is trying to index into your sum matrix instead of calling the sum function.

Answer 1

[Jan]

The power operation is expensive and you can avoid the repeated application:

   function Result = N_Approx(N, A, I)
   Result = zeros([size(A), N]);
   Ai     = eye(size(A));
   SumAi  = I;  % If I should be added to each matrix
   facti  = 1;
   for i = 1:N
     facti = facti * i;          % cumulative product instead of factorial
     Ai    = Ai * A;             % cumulative product instead of power
     SumAi = SumAi + Ai / facti;
     Result(:, :, i) = SumAi;
   end

Even if runtime does not matter in your case, it is a good programming practice to reduce the number of power operations, when possible.

If the variable "I" should not be added to each matrix, use: "SumAi = 0" instead of "SumAi = I" and add I to the final Result only.

You can access the intermediate matrices and the last value by:

Result(:, :, k)

with k goes from 1 to N.

As Walter has said already: There are many of questions in the forum caused by using "sum" as name of a variable, when later in the code it is tried to use as function. Therefore it is recommended to avoid this.

[CAAJ]

I'd like to have a range from N=1:10, and see how using 10 terms changes the matrix verses using 9 terms, and so on. I don't really understand how else to output that besides setting the range of N=10 before calling the function. And even when doing that, the function outputs only one matrix rather than the 10 different matrices.

[Jan]

The shown code replies all 10 matrices, but stored in a 3D-array: The first matrix is: Result(:, :, 1) and so on. Now you can inspect the differences between using 9 or 10 terms by comparing Result(:, :, 9) and Result(:, :, 10).

[CAAJ]

Thank you very much Jan!

Answer 2

[John BG]

1.

init sum to all zeros same size as resulting coefficients by replacing

sum=0;

with

sum=zeros([size(A),N]);

2.

in the loop, instead of accumulating in variable sum, stack the iterations indifferent layers of sum. If A is size 3x3, now sum is size 3x3xN

for i=1:N

      ai=(A^i)/(factorial(i));
      sum(:,:,i)=ai;
  end

example

A=magic(3)
N=4
for i=1:N
ai=(A^i)/(factorial(i));
sum(:,:,i)=ai;
end

to read each coefficient use the following

sum(:,:,1)
 =
          8.00          1.00          6.00
          3.00          5.00          7.00
          4.00          9.00          2.00
sum(:,:,2)
ans =
         45.50         33.50         33.50
         33.50         45.50         33.50
         33.50         33.50         45.50
sum(:,:,3)
 =
        199.50        171.50        191.50
        179.50        187.50        195.50
        183.50        203.50        175.50

.

CAAJ

if you find this answer useful would you please be so kind to mark my answer as Accepted Answer?

To any other reader, please if you find this answer of any help solving your question,

please click on the thumbs-up vote link,

thanks in advance

John BG

<mailto:[e-mail] [e-mail]>

[Walter Roberson]

Please do not use sum as the name of a variable, as that interferes with using the key MATLAB function sum(). Chances are high that at some point you will have used sum as a variable and then expect to have sum() be the function and have it mysteriously crash. And if you do manage to avoid that then you are still going to leave other people who read it confused.

[Jan]

@John BG: You forgot the cumulative summation.

[CAAJ]

For your example code above, is it saying that for of the range of N=1:4, that the code outputs the sum of all matrices up to the Nth term inputted, or does it output the matrix at the Nth term inputted.

[Guillaume]

@John,

As an Answerer, try to teach good practices to the people you answer to. One of these is to NEVER use sum as a variable name, as they usually don't have the experience to see where it's going to cause problem. For evidence, see this non-exhaustive list:

• sum-function-not-working
• how-to-use-sum
• problem-with-sum-command
• subscript-indices-must-either-be-real-positive-integers-or-logicals
• index-exceeds-matrix-dimensions
• i-keep-getting-index-exceeds-matrix-dimensions-but-i-don-t-understand-why-i-m-trying-to-sum-the-ele
• why-the-error-is-coming
• why-the-code-gives-an-error
• error-when-sum-a-vector
• subscript-indices-must-either-be-real-positive-integers-or-logicals

[CAAJ]

Thank you Guillaume, I have changed that in my code and seem to still be having the same error.

[John BG]

Guillon,

It's CAAJ that chose that particular name of variable

[Walter Roberson]

John,

Here, we do not believe that our role is only to literally answer Questions; we believe that part of our role is to teach people how to write clear and efficient and bug-free MATLAB programs. Part of that involves pointing out to people where their programs are weak, and suggesting improvements. Your efforts here would be more effective if you were to do that as well.

Answer 3

[CAAJ]

Just to clarify a bit, with inputting a range for my function beforehand, I would like to see how using 10 terms changes the matrix verses using 9 terms, and etc. I don't really understand why, but when I do set my range before hand, the function outputs only one matrix rather than the 10 different matrices created for the range of N values.

[John BG]

this is why I give you a layered 3D matrix as answer, you can then pull each term and do whatever you want, sum it or anything else you find convenient.

[Walter Roberson]

Jan's answer does give a 3D matrix with all of the results.