Can I calculate the inverse of a matrix using arrayfun?

Dear All,

I have a sparse matrix A. I want to calculate its inverse. I used the following way to calculate invA.

invA = A \ speye(size(A));

My matrix size is 3000 by 3000. I found it took me 1.3 seconds to get invA. It is longer than I expected.

I am wondering if I could use arrayfun to calculate invA. The idea is to calculate each column in invA by invA(:,i) = A \ ei, where ei is a zero column vector except its ith element is 1.0.

I tried the following code to implement the above idea:

invA = arrayfun(@(x) A\x ,speye(size(A,1)));

But I got error message.

Thanks a lot.

Benson

[Matt J]

I don't see how arrayfun could help, unless this is on the GPU.

Answer 1

[Walter Roberson]

invA = cell2mat(arrayfun(@(C) A\sparse(C,1,1,size(A,1),1), 1:size(A,2), 'uniform', 0))

I would be surprised if it is faster.

[Walter Roberson]

A = sprand(20,20, .01)

A = 
   (5,2)       0.2696
   (4,4)       0.9628
   (9,5)       0.0991
  (18,17)      0.8745

tic
invA = cell2mat(arrayfun(@(C) A\sparse(C,1,1,size(A,1),1), 1:size(A,2), 'uniform', 0))

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

invA = 
   (1,1)          NaN
   (3,1)          NaN
   (6,1)          NaN
   (7,1)          NaN
   (8,1)          NaN
   (9,1)          NaN
  (10,1)          NaN
  (11,1)          NaN
  (12,1)          NaN
  (13,1)          NaN
  (14,1)          NaN
  (15,1)          NaN
  (16,1)          NaN
  (18,1)          NaN
  (19,1)          NaN
  (20,1)          Inf
   (1,2)          NaN
   (3,2)          NaN
   (6,2)          NaN
   (7,2)          NaN
   (8,2)          NaN
   (9,2)          NaN
  (10,2)          NaN
  (11,2)          NaN
  (12,2)          NaN
  (13,2)          NaN
  (14,2)          NaN
  (15,2)          NaN
  (16,2)          NaN
  (18,2)          NaN
  (19,2)          Inf
  (20,2)          NaN
   (1,3)          NaN
   (3,3)          NaN
   (6,3)          NaN
   (7,3)          NaN
   (8,3)          NaN
   (9,3)          NaN
  (10,3)          NaN
  (11,3)          NaN
  (12,3)          NaN
  (13,3)          NaN
  (14,3)          NaN
  (15,3)          NaN
  (16,3)          NaN
  (18,3)          Inf
  (19,3)          NaN
  (20,3)          NaN
   (1,4)          NaN
   (3,4)          NaN
   (4,4)       1.0387
   (6,4)          NaN
   (7,4)          NaN
   (8,4)          NaN
   (9,4)          NaN
  (10,4)          NaN
  (11,4)          NaN
  (12,4)          NaN
  (13,4)          NaN
  (14,4)          NaN
  (15,4)          NaN
  (16,4)          NaN
  (18,4)          NaN
  (19,4)          NaN
  (20,4)          NaN
   (1,5)          NaN
   (2,5)       3.7094
   (3,5)          NaN
   (6,5)          NaN
   (7,5)          NaN
   (8,5)          NaN
   (9,5)          NaN
  (10,5)          NaN
  (11,5)          NaN
  (12,5)          NaN
  (13,5)          NaN
  (14,5)          NaN
  (15,5)          NaN
  (16,5)          NaN
  (18,5)          NaN
  (19,5)          NaN
  (20,5)          NaN
   (1,6)          NaN
   (3,6)          NaN
   (6,6)          NaN
   (7,6)          NaN
   (8,6)          NaN
   (9,6)          NaN
  (10,6)          NaN
  (11,6)          NaN
  (12,6)          NaN
  (13,6)          NaN
  (14,6)          NaN
  (15,6)          NaN
  (16,6)          Inf
  (18,6)          NaN
  (19,6)          NaN
  (20,6)          NaN
   (1,7)          NaN
   (3,7)          NaN
   (6,7)          NaN
   (7,7)          NaN
   (8,7)          NaN
   (9,7)          NaN
  (10,7)          NaN
  (11,7)          NaN
  (12,7)          NaN
  (13,7)          NaN
  (14,7)          NaN
  (15,7)          Inf
  (16,7)          NaN
  (18,7)          NaN
  (19,7)          NaN
  (20,7)          NaN
   (1,8)          NaN
   (3,8)          NaN
   (6,8)          NaN
   (7,8)          NaN
   (8,8)          NaN
   (9,8)          NaN
  (10,8)          NaN
  (11,8)          NaN
  (12,8)          NaN
  (13,8)          NaN
  (14,8)          Inf
  (15,8)          NaN
  (16,8)          NaN
  (18,8)          NaN
  (19,8)          NaN
  (20,8)          NaN
   (1,9)          NaN
   (3,9)          NaN
   (5,9)      10.0885
   (6,9)          NaN
   (7,9)          NaN
   (8,9)          NaN
   (9,9)          NaN
  (10,9)          NaN
  (11,9)          NaN
  (12,9)          NaN
  (13,9)          NaN
  (14,9)          NaN
  (15,9)          NaN
  (16,9)          NaN
  (18,9)          NaN
  (19,9)          NaN
  (20,9)          NaN
   (1,10)         NaN
   (3,10)         NaN
   (6,10)         NaN
   (7,10)         NaN
   (8,10)         NaN
   (9,10)         NaN
  (10,10)         NaN
  (11,10)         NaN
  (12,10)         NaN
  (13,10)         Inf
  (14,10)         NaN
  (15,10)         NaN
  (16,10)         NaN
  (18,10)         NaN
  (19,10)         NaN
  (20,10)         NaN
   (1,11)         NaN
   (3,11)         NaN
   (6,11)         NaN
   (7,11)         NaN
   (8,11)         NaN
   (9,11)         NaN
  (10,11)         NaN
  (11,11)         NaN
  (12,11)         Inf
  (13,11)         NaN
  (14,11)         NaN
  (15,11)         NaN
  (16,11)         NaN
  (18,11)         NaN
  (19,11)         NaN
  (20,11)         NaN
   (1,12)         NaN
   (3,12)         NaN
   (6,12)         NaN
   (7,12)         NaN
   (8,12)         NaN
   (9,12)         NaN
  (10,12)         NaN
  (11,12)         Inf
  (12,12)         NaN
  (13,12)         NaN
  (14,12)         NaN
  (15,12)         NaN
  (16,12)         NaN
  (18,12)         NaN
  (19,12)         NaN
  (20,12)         NaN
   (1,13)         NaN
   (3,13)         NaN
   (6,13)         NaN
   (7,13)         NaN
   (8,13)         NaN
   (9,13)         NaN
  (10,13)         Inf
  (11,13)         NaN
  (12,13)         NaN
  (13,13)         NaN
  (14,13)         NaN
  (15,13)         NaN
  (16,13)         NaN
  (18,13)         NaN
  (19,13)         NaN
  (20,13)         NaN
   (1,14)         NaN
   (3,14)         NaN
   (6,14)         NaN
   (7,14)         NaN
   (8,14)         NaN
   (9,14)         Inf
  (10,14)         NaN
  (11,14)         NaN
  (12,14)         NaN
  (13,14)         NaN
  (14,14)         NaN
  (15,14)         NaN
  (16,14)         NaN
  (18,14)         NaN
  (19,14)         NaN
  (20,14)         NaN
   (1,15)         NaN
   (3,15)         NaN
   (6,15)         NaN
   (7,15)         NaN
   (8,15)         Inf
   (9,15)         NaN
  (10,15)         NaN
  (11,15)         NaN
  (12,15)         NaN
  (13,15)         NaN
  (14,15)         NaN
  (15,15)         NaN
  (16,15)         NaN
  (18,15)         NaN
  (19,15)         NaN
  (20,15)         NaN
   (1,16)         NaN
   (3,16)         NaN
   (6,16)         NaN
   (7,16)         Inf
   (8,16)         NaN
   (9,16)         NaN
  (10,16)         NaN
  (11,16)         NaN
  (12,16)         NaN
  (13,16)         NaN
  (14,16)         NaN
  (15,16)         NaN
  (16,16)         NaN
  (18,16)         NaN
  (19,16)         NaN
  (20,16)         NaN
   (1,17)         NaN
   (3,17)         NaN
   (6,17)         Inf
   (7,17)         NaN
   (8,17)         NaN
   (9,17)         NaN
  (10,17)         NaN
  (11,17)         NaN
  (12,17)         NaN
  (13,17)         NaN
  (14,17)         NaN
  (15,17)         NaN
  (16,17)         NaN
  (18,17)         NaN
  (19,17)         NaN
  (20,17)         NaN
   (1,18)         NaN
   (3,18)         NaN
   (6,18)         NaN
   (7,18)         NaN
   (8,18)         NaN
   (9,18)         NaN
  (10,18)         NaN
  (11,18)         NaN
  (12,18)         NaN
  (13,18)         NaN
  (14,18)         NaN
  (15,18)         NaN
  (16,18)         NaN
  (17,18)      1.1435
  (18,18)         NaN
  (19,18)         NaN
  (20,18)         NaN
   (1,19)         NaN
   (3,19)         Inf
   (6,19)         NaN
   (7,19)         NaN
   (8,19)         NaN
   (9,19)         NaN
  (10,19)         NaN
  (11,19)         NaN
  (12,19)         NaN
  (13,19)         NaN
  (14,19)         NaN
  (15,19)         NaN
  (16,19)         NaN
  (18,19)         NaN
  (19,19)         NaN
  (20,19)         NaN
   (1,20)         Inf
   (3,20)         NaN
   (6,20)         NaN
   (7,20)         NaN
   (8,20)         NaN
   (9,20)         NaN
  (10,20)         NaN
  (11,20)         NaN
  (12,20)         NaN
  (13,20)         NaN
  (14,20)         NaN
  (15,20)         NaN
  (16,20)         NaN
  (18,20)         NaN
  (19,20)         NaN
  (20,20)         NaN

toc

Elapsed time is 0.115353 seconds.

tic
invA = A\speye(size(A))

Warning: Matrix is singular to working precision.

invA = 
   (1,1)          NaN
   (3,1)          NaN
   (6,1)          NaN
   (7,1)          NaN
   (8,1)          NaN
   (9,1)          NaN
  (10,1)          NaN
  (11,1)          NaN
  (12,1)          NaN
  (13,1)          NaN
  (14,1)          NaN
  (15,1)          NaN
  (16,1)          NaN
  (18,1)          NaN
  (19,1)          NaN
  (20,1)          Inf
   (1,2)          NaN
   (3,2)          NaN
   (6,2)          NaN
   (7,2)          NaN
   (8,2)          NaN
   (9,2)          NaN
  (10,2)          NaN
  (11,2)          NaN
  (12,2)          NaN
  (13,2)          NaN
  (14,2)          NaN
  (15,2)          NaN
  (16,2)          NaN
  (18,2)          NaN
  (19,2)          Inf
  (20,2)          NaN
   (1,3)          NaN
   (3,3)          NaN
   (6,3)          NaN
   (7,3)          NaN
   (8,3)          NaN
   (9,3)          NaN
  (10,3)          NaN
  (11,3)          NaN
  (12,3)          NaN
  (13,3)          NaN
  (14,3)          NaN
  (15,3)          NaN
  (16,3)          NaN
  (18,3)          Inf
  (19,3)          NaN
  (20,3)          NaN
   (1,4)          NaN
   (3,4)          NaN
   (4,4)       1.0387
   (6,4)          NaN
   (7,4)          NaN
   (8,4)          NaN
   (9,4)          NaN
  (10,4)          NaN
  (11,4)          NaN
  (12,4)          NaN
  (13,4)          NaN
  (14,4)          NaN
  (15,4)          NaN
  (16,4)          NaN
  (18,4)          NaN
  (19,4)          NaN
  (20,4)          NaN
   (1,5)          NaN
   (2,5)       3.7094
   (3,5)          NaN
   (6,5)          NaN
   (7,5)          NaN
   (8,5)          NaN
   (9,5)          NaN
  (10,5)          NaN
  (11,5)          NaN
  (12,5)          NaN
  (13,5)          NaN
  (14,5)          NaN
  (15,5)          NaN
  (16,5)          NaN
  (18,5)          NaN
  (19,5)          NaN
  (20,5)          NaN
   (1,6)          NaN
   (3,6)          NaN
   (6,6)          NaN
   (7,6)          NaN
   (8,6)          NaN
   (9,6)          NaN
  (10,6)          NaN
  (11,6)          NaN
  (12,6)          NaN
  (13,6)          NaN
  (14,6)          NaN
  (15,6)          NaN
  (16,6)          Inf
  (18,6)          NaN
  (19,6)          NaN
  (20,6)          NaN
   (1,7)          NaN
   (3,7)          NaN
   (6,7)          NaN
   (7,7)          NaN
   (8,7)          NaN
   (9,7)          NaN
  (10,7)          NaN
  (11,7)          NaN
  (12,7)          NaN
  (13,7)          NaN
  (14,7)          NaN
  (15,7)          Inf
  (16,7)          NaN
  (18,7)          NaN
  (19,7)          NaN
  (20,7)          NaN
   (1,8)          NaN
   (3,8)          NaN
   (6,8)          NaN
   (7,8)          NaN
   (8,8)          NaN
   (9,8)          NaN
  (10,8)          NaN
  (11,8)          NaN
  (12,8)          NaN
  (13,8)          NaN
  (14,8)          Inf
  (15,8)          NaN
  (16,8)          NaN
  (18,8)          NaN
  (19,8)          NaN
  (20,8)          NaN
   (1,9)          NaN
   (3,9)          NaN
   (5,9)      10.0885
   (6,9)          NaN
   (7,9)          NaN
   (8,9)          NaN
   (9,9)          NaN
  (10,9)          NaN
  (11,9)          NaN
  (12,9)          NaN
  (13,9)          NaN
  (14,9)          NaN
  (15,9)          NaN
  (16,9)          NaN
  (18,9)          NaN
  (19,9)          NaN
  (20,9)          NaN
   (1,10)         NaN
   (3,10)         NaN
   (6,10)         NaN
   (7,10)         NaN
   (8,10)         NaN
   (9,10)         NaN
  (10,10)         NaN
  (11,10)         NaN
  (12,10)         NaN
  (13,10)         Inf
  (14,10)         NaN
  (15,10)         NaN
  (16,10)         NaN
  (18,10)         NaN
  (19,10)         NaN
  (20,10)         NaN
   (1,11)         NaN
   (3,11)         NaN
   (6,11)         NaN
   (7,11)         NaN
   (8,11)         NaN
   (9,11)         NaN
  (10,11)         NaN
  (11,11)         NaN
  (12,11)         Inf
  (13,11)         NaN
  (14,11)         NaN
  (15,11)         NaN
  (16,11)         NaN
  (18,11)         NaN
  (19,11)         NaN
  (20,11)         NaN
   (1,12)         NaN
   (3,12)         NaN
   (6,12)         NaN
   (7,12)         NaN
   (8,12)         NaN
   (9,12)         NaN
  (10,12)         NaN
  (11,12)         Inf
  (12,12)         NaN
  (13,12)         NaN
  (14,12)         NaN
  (15,12)         NaN
  (16,12)         NaN
  (18,12)         NaN
  (19,12)         NaN
  (20,12)         NaN
   (1,13)         NaN
   (3,13)         NaN
   (6,13)         NaN
   (7,13)         NaN
   (8,13)         NaN
   (9,13)         NaN
  (10,13)         Inf
  (11,13)         NaN
  (12,13)         NaN
  (13,13)         NaN
  (14,13)         NaN
  (15,13)         NaN
  (16,13)         NaN
  (18,13)         NaN
  (19,13)         NaN
  (20,13)         NaN
   (1,14)         NaN
   (3,14)         NaN
   (6,14)         NaN
   (7,14)         NaN
   (8,14)         NaN
   (9,14)         Inf
  (10,14)         NaN
  (11,14)         NaN
  (12,14)         NaN
  (13,14)         NaN
  (14,14)         NaN
  (15,14)         NaN
  (16,14)         NaN
  (18,14)         NaN
  (19,14)         NaN
  (20,14)         NaN
   (1,15)         NaN
   (3,15)         NaN
   (6,15)         NaN
   (7,15)         NaN
   (8,15)         Inf
   (9,15)         NaN
  (10,15)         NaN
  (11,15)         NaN
  (12,15)         NaN
  (13,15)         NaN
  (14,15)         NaN
  (15,15)         NaN
  (16,15)         NaN
  (18,15)         NaN
  (19,15)         NaN
  (20,15)         NaN
   (1,16)         NaN
   (3,16)         NaN
   (6,16)         NaN
   (7,16)         Inf
   (8,16)         NaN
   (9,16)         NaN
  (10,16)         NaN
  (11,16)         NaN
  (12,16)         NaN
  (13,16)         NaN
  (14,16)         NaN
  (15,16)         NaN
  (16,16)         NaN
  (18,16)         NaN
  (19,16)         NaN
  (20,16)         NaN
   (1,17)         NaN
   (3,17)         NaN
   (6,17)         Inf
   (7,17)         NaN
   (8,17)         NaN
   (9,17)         NaN
  (10,17)         NaN
  (11,17)         NaN
  (12,17)         NaN
  (13,17)         NaN
  (14,17)         NaN
  (15,17)         NaN
  (16,17)         NaN
  (18,17)         NaN
  (19,17)         NaN
  (20,17)         NaN
   (1,18)         NaN
   (3,18)         NaN
   (6,18)         NaN
   (7,18)         NaN
   (8,18)         NaN
   (9,18)         NaN
  (10,18)         NaN
  (11,18)         NaN
  (12,18)         NaN
  (13,18)         NaN
  (14,18)         NaN
  (15,18)         NaN
  (16,18)         NaN
  (17,18)      1.1435
  (18,18)         NaN
  (19,18)         NaN
  (20,18)         NaN
   (1,19)         NaN
   (3,19)         Inf
   (6,19)         NaN
   (7,19)         NaN
   (8,19)         NaN
   (9,19)         NaN
  (10,19)         NaN
  (11,19)         NaN
  (12,19)         NaN
  (13,19)         NaN
  (14,19)         NaN
  (15,19)         NaN
  (16,19)         NaN
  (18,19)         NaN
  (19,19)         NaN
  (20,19)         NaN
   (1,20)         Inf
   (3,20)         NaN
   (6,20)         NaN
   (7,20)         NaN
   (8,20)         NaN
   (9,20)         NaN
  (10,20)         NaN
  (11,20)         NaN
  (12,20)         NaN
  (13,20)         NaN
  (14,20)         NaN
  (15,20)         NaN
  (16,20)         NaN
  (18,20)         NaN
  (19,20)         NaN
  (20,20)         NaN

toc

Elapsed time is 0.018112 seconds.

[Walter Roberson]

Much slower!

[Benson Gou]

Hi, Walter,

Thanks for your help.

I compared your code and my original code. Your code took 3.7 sceonds while mine took 1.27 seconds.

I am wondering if there exists a faster code than my original code.

Thanks a lot, anyway.

Benso

[Walter Roberson]

Well, you could try matrix decomposition, https://www.mathworks.com/help/matlab/ref/decomposition.html

[Benson Gou]

Hi, Walter,

Yes, decomposition is a good way to solve linear systems. But I need the inverse of the matrix for other calculations, not just solve the linear system.

Thanks a lot.

Benson

[Walter Roberson]

You decompose the matrix. Then you use the decomposition on the left side of \ with speye() on the right hand side.

Most of the time you should not be using the inverse in calculations, and should instead be using the \ operator . inv(A)*b + inv(A)*c --> dA = decompose(A), dA\b +dA\c

[Benson Gou]

Hi, Walter,

I do not have the Image Processing Toolbox. So I cannot run decompose(A).

But the goal that I want to obtain the inverse of A is becaus I need to find out the nonzero elements in invA. So I do not know if decompose(A) is able to be useful for my case.

Thanks a lot.

Benson

[Walter Roberson]

decomposition()

https://www.mathworks.com/help/matlab/ref/decomposition.html

Answer 2

[Sulaymon Eshkabilov]

In fact, you can employ arrayfun for the matric inverse calc, e.g.:

tic;
iA =arrayfun(@inv,A);
toc;

Note that arrayfun is not the best option and does not take the sparse matrix.

Just direct inv() is the fastest so far.

In fact, for solving linear systems, to compute the inverse is not advised.

[Walter Roberson]

iA =arrayfun(@inv,A);

That code would be the equivalent of

iA = 1./A

[Walter Roberson]

A = sprand(20,20, .01)

A = 
  (16,5)       0.5706
  (10,12)      0.5464
  (12,16)      0.7549
   (4,17)      0.6957

iA = arrayfun(@inv, full(A))

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

Warning: Matrix is singular to working precision.

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Warning: Matrix is singular to working precision.

iA = 20×20

Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf 1.4373 Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf 1.8301 Inf Inf Inf Inf Inf Inf Inf Inf

iA2 = 1./A

iA2 = 20×20

Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf 1.4373 Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf Inf 1.8301 Inf Inf Inf Inf Inf Inf Inf Inf

[Walter Roberson]

By the way, you cannot apply arrayfun() directly to a sparse matrix.

[Benson Gou]

Hi, Walter,

Yes, the decomposition is faster than \ operator. Thanks a lot.

Benson