How to simplify infinite double matrix summation in Matlab.

Question

Hilal Ahmad Bhat 2024-3-2

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https://ww2.mathworks.cn/matlabcentral/answers/2089461-how-to-simplify-infinite-double-matrix-summation-in-matlab

编辑： Torsten 2024-3-5

采纳的回答： Walter Roberson

在 MATLAB Online 中打开

I need to calculate the following sum:

I used the following code:

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1];

syms k l t

f = ((-A).^k./factorial(k)).*((factorial(k+l).*(-R).^l)/gamma((2-beta).*l...

+ 2.*k + 2)).*((t.^((2-beta).*l+2.*k+1))/factorial(l));

F = vpasum(vpasum(f,k,0,inf),l,0,inf);

G = subs(F,t,1)

G =

This yields G as a symbolic sum but I need it as numeric sum. I tried it using symsum function but that too yields a symbolic answer. Using, subs function for t=1 won't change it into a numeric value. Any help would be highly appriciated.

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Walter Roberson 2024-3-2

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(-R).^l

I suspect that is

(-R)^l

and

(-A).^k

I suspect that is

(-A)^k

Hilal Ahmad Bhat 2024-3-3

@Walter Roberson Yes, you are right but that won't make any difference in this case.

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Answer 1

Walter Roberson 2024-3-3

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移动：Walter Roberson 2024-3-3

在 MATLAB Online 中打开

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1];

syms k l t

f = ((-A)^k./factorial(k)).*((factorial(k+l).*(-R)^l)/gamma((2-beta).*l...

+ 2.*k + 2)).*((t.^((2-beta).*l+2.*k+1))/factorial(l));

F = symsum(symsum(f,k,0,inf),l,0,inf);

G = subs(F,t,1)

G =

vpa(G)

ans =

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Hilal Ahmad Bhat 2024-3-5

编辑：Hilal Ahmad Bhat 2024-3-5

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I was trying to solve the question using while loops. I wrote the following code:

clear all; beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1];  %#ok<CLALL>
inn_total = zeros(2); inn_total(:) = inf;
out_total = zeros(2);
l=0; k=0; t=1;
while sqrt(trace(inn_total*inn_total')) > eps % Norm of inn_total should get as much small as it pleases for the convergence of outer summation.
    inn_total = zeros(2);
    term=zeros(2); term(:) = inf;
    while sqrt(trace(term*term')) > eps % Norm of term should get as much small as it pleases for the convergence of inner summation.
        term = ((-A)^k/factorial(k))*((factorial(k+l)*(-R)^l)/...
               gamma((2-beta)*l+ 2*k + 2))*((t^((2-beta)*l+2*k+1))/...
               factorial(l));
        inn_total = inn_total + term;
        l = l+1;
    end    
    out_total = out_total + inn_total;
    k = k+1;
end
out_total %#ok<NOPTS>
out_total = 2×2
    3.1688   -3.6799
   -1.2266    1.9422
[k,l] %#ok<NOPTS>
ans = 1×2
     2    71

The idea that I am using in the convergence of the series is mathematical, i.e.,

converges if norm of each term after some finite 'n' must be less than some predefined small value (eps in my case). For inner sum, while loop stops after value of 'Lth term' is less than 'eps' and the outer while loop stops when the norm of inner sum gets less than 'eps'.

But the values, I get from this method, differ by approx 0.6 than the following two methods - one using the symsum and vpa (@Walter Roberson's method), and the other using the loops (@Torsten's method).

% Walter Roberson Method

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1];

syms k l t

f1 = ((-A)^k/factorial(k))*((factorial(k+l)*(-R)^l)/gamma((2-beta)*l...

+ 2*k + 2))*((t^((2-beta)*l+2*k+1))/factorial(l));

F = symsum(symsum(f1,k,0,inf),l,0,inf);

G = subs(F,t,1);

vpa(G,5)

ans =

% Torsten Method

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1]; t = 1;

N = 4;

M = 36;

mat = zeros(2);

for s1 = 0:N

for s2 = 0:M

mat = mat + f(s1,s2,beta,A,R,t);

end

mat

mat = 2×2

2.5141 -2.8682 -0.7202 1.2210

function mat = f(k,l,beta,A,R,t)

mat = (-A)^k/factorial(k)*(-R)^l/factorial(l)*factorial(k+l)...

*t^((2-beta)*l+2*k+1)/gamma((2-beta)*l+2*k+2);

end

Why does the values differ when I use while loop? Am I missing anything or doing something extra in my code. I need this to run in while loop because it is very useful sometimes. Thanks in advance.

Torsten 2024-3-5

编辑：Torsten 2024-3-5

在 MATLAB Online 中打开

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1]; t = 1;
mat = zeros(2);
mat_inner = Inf(2);
tol = 1e-6;
s1 = 0;
while norm(mat_inner,'fro') > tol
    mat_inner = zeros(2);
    for s2 = 0:s1
        mat_inner = mat_inner + f(s1-s2,s2,beta,A,R,t);
    end
    mat = mat + mat_inner;
    s1 = s1 + 1;
end
s1
s1 = 45
format long
mat
mat = 2x2
   2.514129941450880  -2.868263680288298
  -0.720173713227138   1.221004006453150
function mat = f(k,l,beta,A,R,t)
  mat = (-1)^(k+l)*A^k/factorial(k)*R^l/factorial(l)*factorial(k+l)...
        *t^((2-beta)*l+2*k+1)/gamma((2-beta)*l+2*k+2);
end

Hilal Ahmad Bhat 2024-3-5

@Torsten

This is awesome. Thank you very much.

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Answer 2

Torsten 2024-3-2

2
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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2089461-how-to-simplify-infinite-double-matrix-summation-in-matlab#answer_1420056

编辑：Torsten 2024-3-2

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Are you sure that the matrix potentials for A and R are meant elementwise ?

beta = 1.5; A = [1 0;0 2]; R = [0 3; 1 1]; t = 1;
N = 50;
M = N;
mat = zeros(2);
for s1 = 0:N
    for s2 = 0:M
        mat = mat + f(s1,s2,beta,A,R,t);
    end
end
mat
mat = 2×2
    0.8415    0.2856
    0.5560    0.4253
function mat = f(k,l,beta,A,R,t)
  mat = (-A).^k/factorial(k).*(-R).^l/factorial(l)*factorial(k+l)...
        *t^((2-beta)*l+2*k+1)/gamma((2-beta)*l+2*k+2);
end

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Hilal Ahmad Bhat 2024-3-3

编辑：Hilal Ahmad Bhat 2024-3-3

Oh! It is my mistake. The matrix potential is not elementwise. It is the meant to be the matrix multiplication (i.e., rows × colums). Whatever be the case, I am stuck at finding the infinite sum. I think while loop might help but I don't know how to do it with double summation.

Torsten 2024-3-3

编辑：Torsten 2024-3-3

在 MATLAB Online 中打开

I doubt you will find an analytical expression for your double sum - that's what summing up to infinity would mean. Thus you will have to compute the sum numerically.

Infinite sums are numerically evaluated by building finite sums up to an index N until the result changes only neclectably if N is increased.

That's what I did.

If you meant usual matrix multiplication, replace

function mat = f(k,l,beta,A,R,t)
  mat = (-A).^k/factorial(k).*(-R).^l/factorial(l)*factorial(k+l)...
        *t^((2-beta)*l+2*k+1)/gamma((2-beta)*l+2*k+2);
end

by

function mat = f(k,l,beta,A,R,t)
  mat = (-A)^k/factorial(k)*(-R)^l/factorial(l)*factorial(k+l)...
        *t^((2-beta)*l+2*k+1)/gamma((2-beta)*l+2*k+2);
end

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