MATLAB doesn’t support sparse matrices computing with high precision.

As a result, we have no other way but compare our performance with famous mathematical packages Maple and Mathematica, which do have support for sparse matrices with arbitrary precision elements.

Table below show timing comparisons of solving sparse linear systems of moderate size (min 1000x1000) by Advanpix toolbox and Maple 17 using quadruple precision.

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Newest version of toolbox includes sparse linear system solver – function mldivide or operator "\".

Solver analyzes input matrix and automatically uses the most suitable decomposition:

1. Cholesky LDLT factorization is applied if input matrix is symmetrical/hermitian and has real positive diagonal.
2. LU factorization is applied if LDLT failed or directly if matrix is obviously non-SPD.
   (Algorithm is based on ideas from SuperLU and uses column approximate minimum degree permutation)
3. QR factorization for rectangular matrices to handle least squares problems (beta).
   (Left-looking QR decomposition, also uses colamd pre-ordering)

All methods are capable to compute with arbitrary precision, in real and complex cases. Tables below present timing for solving sparse linear systems with quadruple precision (34 decimal digits).
See the timing tests

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Starting from version 3.5.0 toolbox has support for sparse matrices with multiple precision elements.

Please note, this is the first beta version with limited functionality – at the moment toolbox can only create and display multiprecision sparse matrices:

>> S = zeros(7,3);
>> S(2,1) = eps;
>> S(5,1) = sqrt(2);
>> S(3,2) = pi;
>> S(2,3) = exp(1);
>> S(5,3) = log(2);
>> S(6,3) = sqrt(-1);
>> S = sparse(S)
 
S =
 
   (2,1)          2.22044604925031e-16                         
   (5,1)               1.4142135623731                         
   (3,2)              3.14159265358979                         
   (2,3)              2.71828182845905                         
   (5,3)             0.693147180559945                         
   (6,3)                             0 +                     1i
 
>> A = mp(S)    % Convert S to multiprecision sparse matrix of 34 digits (default)
                % Note accuracy of all constants are extended automatically.
 
A =
 
   (2,1)    1.925929944387235853055977942584927e-34
   (5,1)    1.414213562373095048801688724209698
   (3,2)    3.141592653589793238462643383279503
   (2,3)    2.718281828459045235360287471352662
   (5,3)    0.693147180559945309417232121458177
   (6,3)    0 +     1i

From now on we will extend sparse matrices support to the point of full transparency & flexibility as we have done for dense matrices.

Next steps are: element-wise arithmetic and mathematical functions, basic matrix-matrix and matrix-vector operations. Major goal after that will be to add iterative Krylov subspace methods.

Although such (currently) basic support may seem to be trivial, its implementation is very far from elementary. In fact, sparse matrices support requires by order of magnitude more efforts than dense matrices. Below we outline the major reasons why it is so.
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