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