1% SPQR_RANK
2%
3% For sparse, rank deficient matrices the SPQR_RANK package provides
4% utilities that are useful for finding solutions to
5%
6%                min || b - A x ||
7%
8% and to determine the numerical structure of A, including the numerical
9% rank of A as well as bases for the numerical null spaces of A and of A
10% transpose.  The utilities reliably determine the numerical rank in the sense
11% that in almost all cases the numerical rank is accurately determined when a
12% warning flag returned by the utilities indicates that the numerical rank
13% should be correct. Reliable determination of numerical rank is often
14% critical to calculations with rank deficient matrices.
15%
16% See "Algorithm xxx: Reliable Calculation of Numerical Rank, Null Space Bases,
17% Pseudoinverse Solutions and  Basic Solutions using SuiteSparseQR" by Leslie
18% Foster and Timothy Davis, submitted ACM Transactions on Mathematical
19% Software, 2011, for detailed discussion of the package.
20%
21% Files
22%   spqr_basic          - approximate basic solution to min(norm(B-A*x))
23%   spqr_cod            - approximate pseudoinverse solution to min(norm(B-A*x)
24%   spqr_null           - finds an orthonormal basis for numerical null space of a matrix
25%   spqr_pinv           - approx pseudoinverse solution to min(norm(B-A*X))
26%   spqr_ssi            - block power method or subspace iteration applied to inv(R)
27%   spqr_ssp            - block power method or subspace iteration applied to A or A*N
28%   spqr_null_mult      - multiplies a matrix by numerical null space from spqr_rank methods
29%   spqr_explicit_basis - converts a null space basis to an explicit matrix
30%   spqr_rank_opts      - sets and prints the default options for spqr_rank
31%   spqr_rank_stats     - prints the statistics from spqr_rank functions
32%
33%   quickdemo_spqr_rank - quick demo of the spqr_rank package
34%   demo_spqr_rank      - lengthy demo for spqr_rank functions (requires SJget)
35%   test_spqr_rank      - extensive functionality test of spqr_rank functions
36%   test_spqr_coverage  - statement coverage test of spqr_rank functions
37%
38% To install this package, simply install all of SuiteSparse from
39% http://www.suitesparse.com, the spqr_rank package
40% in SuiteSparse/MATLAB_TOOLS/spqr_rank will be installed, along with
41% SuiteSparseQR (spqr) and all its dependent packages.
42
43% Copyright 2012, Leslie Foster and Timothy A Davis.
44