Solve *sparse* upper triangular system

Question

If I want to solve a full upper triangular system, I can call linsolve(A,b,'UT') . However this currently is not supported for sparse matrices. How can I overcome this?

Answer 1

UT and LT systems are amongst the easiest systems to solve. Have a read on the wiki about it. Knowing this, it is easy to write your own UT or LT solver:

%# some example data
A = sparse( triu(rand(100)) );
b = rand(100,1);

%# solve UT system by back substitution    
x = zeros(size(b));
for n = size(A,1):-1:1    
    x(n) = (b(n) - A(n,n+1:end)*x(n+1:end) ) / A(n,n);    
end

The procedure is quite similar for LT systems.

Having said that, it is generally much easier and faster to use Matlab's backslash operator:

x = A\b

which also works for spares matrices, as nate already indicated.

Note that this operator also solves UT systems which have non-square A or if A has some elements equal to zero (or < eps ) on the main diagonal. It solves these cases in a least-squares sense, which may or may not be desirable for you. You could check for these cases before carrying out the solve:

if size(A,1)==size(A,2) && all(abs(diag(A)) > eps)
    x = A\b;
else
    %# error, warning, whatever you want
end

Read more about the (back)slash operator by typing

>> help \

or

>> help slash

on the Matlab command prompt.

Answer 2

Edit Since what you need is a triangular solve procedure, also called backward/forward substitution, you can use ordinary MATLAB backslash \\ operator for that:

x = U\b

As mentioned in the original answer, MATLAB will recognise the fact that your matrix is triangular. To be sure of that, you can compare the performance to cs_usolve procedure found in SuiteSparse . It is a mex function implemented in C that computes sparse triangular solve for upper-triangular sparse matrix (there are similar functions there too: cs_lsolve , cs_utsolve and cs_ltsolve ).

You can have a look at a performance comparison of native MATLAB and cs_l(t)solve in the context of sparse Cholesky factorization. Essentially, MATLAB performance is good. The only pitfall is if you want to solve a transposed system

x = U'\b

MATLAB does not recognize that and explicitly creates a transpose of U . In that case you should call cs_utsolve explicitly.

Original answer If your system is symmetric and you only store the upper triangular matrix part (that is how I understood full in your question), and if Cholesky decomposition is suitable for you, chol handles symmetric matrices, if your matrix is positive definite. For indefinite matrices you can use ldl . Both handle sparse storage and work on the symmetric matrix parts.

Newer matlab versions use cholmod and suitesparse for that. That is by far the best performing Cholesky factorization I know of. In matlab it is also parallelised usin parallel BALS.

The factor you obtain from the above functions is upper triangular matrix L such that

A=LL'

All you need to do now is perform forward and backward substitution, which is simple and cheap. In matlab this is automatically done in tha backslash operator

x=L'\(L\b)

the matrix can be sparse, and matlab will recognise that it is upper/lower triangular. You would also use this call together with forward substitution for factors obtained using the cholesky factorization.

Answer 3

你可以在稀疏矩阵上使用MLDIVIDE（\\）或MRDIVIDE（/）运算符......