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% SPQR_RANK
%
% For sparse, rank deficient matrices the SPQR_RANK package provides
% utilities that are useful for finding solutions to
%
% min || b - A x ||
%
% and to determine the numerical structure of A, including the numerical
% rank of A as well as bases for the numerical null spaces of A and of A
% transpose. The utilities reliably determine the numerical rank in the sense
% that in almost all cases the numerical rank is accurately determined when a
% warning flag returned by the utilities indicates that the numerical rank
% should be correct. Reliable determination of numerical rank is often
% critical to calculations with rank deficient matrices.
%
% See "Algorithm xxx: Reliable Calculation of Numerical Rank, Null Space Bases,
% Pseudoinverse Solutions and Basic Solutions using SuiteSparseQR" by Leslie
% Foster and Timothy Davis, submitted ACM Transactions on Mathematical
% Software, 2011, for detailed discussion of the package.
%
% Files
% spqr_basic - approximate basic solution to min(norm(B-A*x))
% spqr_cod - approximate pseudoinverse solution to min(norm(B-A*x)
% spqr_null - finds an orthonormal basis for numerical null space of a matrix
% spqr_pinv - approx pseudoinverse solution to min(norm(B-A*X))
% spqr_ssi - block power method or subspace iteration applied to inv(R)
% spqr_ssp - block power method or subspace iteration applied to A or A*N
% spqr_null_mult - multiplies a matrix by numerical null space from spqr_rank methods
% spqr_explicit_basis - converts a null space basis to an explicit matrix
% spqr_rank_opts - sets and prints the default options for spqr_rank
% spqr_rank_stats - prints the statistics from spqr_rank functions
%
% quickdemo_spqr_rank - quick demo of the spqr_rank package
% demo_spqr_rank - lengthy demo for spqr_rank functions (requires SJget)
% test_spqr_rank - extensive functionality test of spqr_rank functions
% test_spqr_coverage - statement coverage test of spqr_rank functions
%
% To install this package, simply install all of SuiteSparse from
% http://www.suitesparse.com, the spqr_rank package
% in SuiteSparse/MATLAB_TOOLS/spqr_rank will be installed, along with
% SuiteSparseQR (spqr) and all its dependent packages.
% Copyright 2012, Leslie Foster and Timothy A Davis.
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