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I am not able to do complete pivoting in matlab. Say I have some matrix that is not sparse:

A = [0,1,1,1;0,1,0,0;1,1,1,1;0,0,0,-1]

When I try to use the lu command in matlab it does not like it because it is not sparse:

>> [L,U,P,Q] = lu(A) 
Error using lu
Too many output arguments.

Even the docs say that it must be sparse:

[L,U,P,Q] = lu(A) for sparse nonempty A, returns a unit lower triangular matrix L, an upper triangular matrix U, a row permutation matrix P, and a column reordering matrix Q, so that PAQ = L*U. If A is empty or not sparse, lu displays an error message. The statement lu(A,'matrix') returns identical output values.

I have two questions about this:

1) Why must it be sparse? In theory LU decomposition works for non-sparse matrices and so does complete pivoting.

2) What is the appropriate MATLAB method to call to do complete pivoting for non-sparse matrices?

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    Your error message says you have too many output arguments. Try with 2 or 3 outputs. Commented Nov 29, 2017 at 2:58
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    Output Q is specific to sparse matrices. Commented Nov 29, 2017 at 2:58
  • guys. i know. but why? it certainly should theoretically work with a non-sparse matrix ( no reason not to). Commented Nov 29, 2017 at 15:28
  • I would assume it's because columns are not reordered in the case of a full matrix. In the case of a sparse matrix, reordering the columns may lead to a reduction in size of the resulting sparse matrix. But this is pure speculation. Interestingly, the original sparse matrix implementation document referenced here does not mention the Q output, only that A*Q' may be used as an input, so this option must have been added later. Commented Nov 29, 2017 at 16:35
  • If you're really certain that you need complete pivoting, you could try this one: mathworks.com/matlabcentral/fileexchange/… Commented Nov 30, 2017 at 2:42

3 Answers 3

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As mentioned in comments the error is about the number of outputs. Q is for reordering columns which is specific for sparse matrices. Hence, you can get L, U, and P using the following command:

[L,U,P] = lu(A)
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yes, i know. but my second question is why is it only for sparse matrices? it certainly should work with non-sparse matrices
@Hunle Also mentioned in document lu(A,'matrix') works for general matrix and not only for sparse matrices.
you're not answering the question. Why does it not work with non-sparse matrices? I cannot think of a good theoretical reason.
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In theory LU factorization with full pivoting works fine with any matrix (full or sparse). In practice, it depends on the particular library which Matlab uses to compute it. In the book from Quarteroni et al. "Scientific computing with Matlab and Octave" (Springer, 4th edition, pag. 166) it is stated that if the matrix is full and has no particular structure, "a generic triangular factorization is computed by Gaussian elimination with partial pivoting".

My understanding is that in case your matrix is full, Matlab perform LU factorization by using an algorithm which uses only partial pivoting, thus matrix Q is not defined. If the matrix is sparse, a different library is used (UMFpack) which uses full pivoting. That's why matrix Q is returned only for sparse matrix.

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The DATATYPE must be sparse:

>> A = [0,1,1,1;0,1,0,0;1,1,1,1;0,0,0,-1];
>> A = sparse(A);
>> [L,U,P,Q] = lu(A)

L =

(1,1) 1 (2,2) 1 (3,3) 1 (4,4) 1

U =

(1,1) 1 (1,2) 1 (2,2) 1 (1,3) 1 (2,3) 1 (3,3) 1 (1,4) 1 (2,4) 1 (4,4) -1

P =

(2,1) 1 (3,2) 1 (1,3) 1 (4,4) 1

Q =

(1,1) 1 (3,2) 1 (2,3) 1 (4,4) 1

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