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|
function spqr_rank_stats (stats, print_opts)
%SPQR_RANK_STATS prints the statistics from spqr_rank functions
%
% For a detailed description of the meaning of the basic statistic for
% spqr_basic, spqr_null, spqr_pinv or spqr_cod, just type
% 'spqr_rank_stats' with no inputs. Type spqr_rank_stats('ssi') or
% spqr_rank_stats('ssp') for a description of the basic statistics for
% sqpr_ssi or spqr_ssp, respectively. spqr_rank_stats(1) also prints a
% detailed description of all statistics calculated by any spqr_function
% when opts.get_details is 1. To print a summary of the stats struct
% returned by a spqr_rank function, use spqr_rank_stats(stats) or
% spqr_rank_stats(stats,1).
%
% Examples
%
% spqr_rank_stats ; % prints a description of basic statistics
% % from spqr_basic, spqr_null, spqr_pinv, spqr_cod
% spqr_rank_stats('ssi') ; % description of basic statistics from spqr_ssi
% spqr_rank_stats('ssp') ; % description of basic statistics from spqr_ssp
% spqr_rank_stats ( 1 ) ; % prints a verbose description of all statistics
% spqr_rank_stats (stats) ; % prints a short summary of the stats struct
% spqr_rank_stats (stats,1); % prints options used, when opts.get_details is 1
%
% See also spqr_basic, spqr_cod, spqr_pinv, spqr_ssi, spqr_ssp
% spqr_rank, Copyright (c) 2012, Leslie Foster and Timothy A Davis.
% All Rights Reserved.
% SPDX-License-Identifier: BSD-3-clause
if (nargin == 0)
stats = 0 ;
get_details = 0 ;
method = 1 ; % spqr_basic, spqr_null, spqr_cod, spqr_pinv
end
if ( nargin == 1 )
if ischar(stats)
get_details = 0 ;
if ( strcmp(stats,'spqr_ssi') || strcmp(stats,'ssi') )
method = 2 ; % spqr_ssp
elseif ( strcmp(stats,'spqr_ssp') || strcmp(stats,'ssp') )
method = 3 ; % spqr_ssi
else
method = 1; % spqr_basic, spqr_null, spqr_cod, spqr_pinv
end
elseif isreal(stats)
get_details = stats;
method = 4; % any method
end
print_opts = 0 ;
end
if (isstruct (stats))
%---------------------------------------------------------------------------
% print the stats returned by the spqr_* function
%---------------------------------------------------------------------------
%---------------------------------------------------------------------------
% flag
%---------------------------------------------------------------------------
flag = -1 ;
if (isfield (stats, 'flag'))
flag = stats.flag ;
end
if ~isfield( stats, 'est_svals')
% description of flag for calls except for call to spqr_ssp
if (flag == 0)
fprintf ('\n flag : %g : ', flag) ;
fprintf ('ok. stats.rank very likely to be correct.\n') ;
elseif (flag <= 2)
fprintf ('\n flag : %g : ', flag) ;
fprintf ('stats.rank may be correct for tolerance stats.tol,\n') ;
fprintf (' but error bounds are too high to confirm this.\n') ;
if (flag == 1)
fprintf ([' However, stats.rank appears to be correct' ...
' for tolerance stats.tol_alt.\n']) ;
end
elseif (flag == 3)
fprintf ('\n flag : %g : ', flag) ;
fprintf ('poor results. stats.rank is likely too high.\n') ;
elseif (flag == 4)
fprintf ('\n flag : %g : ', flag) ;
fprintf ('failure. Overflow during inverse power method.\n') ;
% elseif (flag == 5)
% % this code is disabled because stats.flag=5 is removed.
% fprintf ('\n flag : %g : ', flag) ;
% fprintf ('failure. Inconsistent rank estimates.\n') ;
else
error ('spqr_rank:invalid', 'invalid stats') ;
end
else
% description of flag for calls to spqr_ssp
fprintf ('\n flag : %g : ', flag) ;
if (flag == 0)
fprintf (['ok. spqr_ssp converged with est. relative error ' ...
'<= opts_ssp.convergence_factor.\n']) ;
else
fprintf (['spqr_ssp did not converge with est. relative error ' ...
'<= opts_ssp.convergence_factor.\n']) ;
end
end
%---------------------------------------------------------------------------
% rank
%---------------------------------------------------------------------------
if (isfield (stats, 'rank'))
fprintf (' rank : %g : estimate of numerical rank.\n', stats.rank) ;
end
%---------------------------------------------------------------------------
% rank_spqr
%---------------------------------------------------------------------------
if (isfield (stats, 'rank_spqr'))
fprintf (' rank_spqr : %g : ', stats.rank_spqr) ;
fprintf ('estimate of numerical rank from spqr.\n') ;
fprintf (' This is normally an upper bound on the true rank.\n') ;
end
%---------------------------------------------------------------------------
% tol
%---------------------------------------------------------------------------
if (isfield (stats, 'tol'))
fprintf (' tol : %g : numerical tolerance used.\n', stats.tol) ;
end
%---------------------------------------------------------------------------
% tol_alt
%---------------------------------------------------------------------------
if (isfield (stats, 'tol_alt'))
fprintf (' tol_alt : %g : alternate numerical tolerance used.\n', ...
stats.tol_alt) ;
end
%---------------------------------------------------------------------------
% normest_A
%---------------------------------------------------------------------------
if (isfield (stats, 'normest_A'))
fprintf (' normest_A : %g : estimate of Euclidean norm of A.\n', ...
stats.normest_A) ;
end
%---------------------------------------------------------------------------
% normest_R
%---------------------------------------------------------------------------
if (isfield (stats, 'normest_R'))
fprintf ([' normest_R : %g : estimate of Euclidean norm of R ' ...
'(calculated for spqr_ssi).\n'], ...
stats.normest_R) ;
end
%---------------------------------------------------------------------------
% est_svals_upper_bounds
%---------------------------------------------------------------------------
if (isfield (stats, 'est_sval_upper_bounds'))
fprintf (' est_sval_upper_bounds : ') ;
print_vector (stats.est_sval_upper_bounds) ;
fprintf (' : estimated upper bounds on singular value(s).\n') ;
end
%---------------------------------------------------------------------------
% est_svals_lower_bounds
%---------------------------------------------------------------------------
if (isfield (stats, 'est_sval_lower_bounds'))
fprintf (' est_sval_lower_bounds : ') ;
print_vector (stats.est_sval_lower_bounds) ;
fprintf (' : estimated lower bounds on singular value(s).\n') ;
end
%---------------------------------------------------------------------------
% est_svals_of_R
%---------------------------------------------------------------------------
if (isfield (stats, 'est_svals_of_R'))
fprintf (' est_svals_of_R : ') ;
print_vector (stats.est_svals_of_R) ;
fprintf ([' : estimated singular value(s) of triangular ' ...
'matrix R.\n']) ;
end
%---------------------------------------------------------------------------
% est_svals
%---------------------------------------------------------------------------
if (isfield (stats, 'est_svals'))
fprintf (' est_svals : ') ;
print_vector (stats.est_svals) ;
fprintf ([' : estimated singular value(s) of A*N or A''*NT, ' ...
'from spqr_ssp.\n']) ;
end
%---------------------------------------------------------------------------
% est_error_bounds
%---------------------------------------------------------------------------
if (isfield (stats, 'est_error_bounds'))
fprintf (' est_error_bounds : ') ;
print_vector (stats.est_error_bounds) ;
fprintf (' : error bounds for each singular value.\n') ;
end
%---------------------------------------------------------------------------
% sval_numbers_for_bounds
%---------------------------------------------------------------------------
if (isfield (stats, 'sval_numbers_for_bounds'))
fprintf (' sval_numbers_for_bounds : ') ;
print_vector (stats.sval_numbers_for_bounds) ;
fprintf (' : index of singular value(s), for bounds.\n') ;
end
%---------------------------------------------------------------------------
% est_norm_A_times_N
%---------------------------------------------------------------------------
if (isfield (stats, 'est_norm_A_times_N'))
fprintf ([' est_norm_A_times_N : %g : estimated' ...
' norm(A*N).\n'], stats.est_norm_A_times_N) ;
end
%---------------------------------------------------------------------------
% est_err_bound_norm_A_times_N
%---------------------------------------------------------------------------
if (isfield (stats, 'est_err_bound_norm_A_times_N'))
fprintf ([' est_err_bound_norm_A_times_N : %g : estimated error ' ...
'bound for norm(A*N).\n'], stats.est_err_bound_norm_A_times_N) ;
end
%---------------------------------------------------------------------------
% est_norm_A_transpose_times_NT
%---------------------------------------------------------------------------
if (isfield (stats, 'est_norm_A_transpose_times_NT'))
fprintf ([' est_norm_A_transpose_times_NT : %g : estimated' ...
' norm(A''*NT).\n'], stats.est_norm_A_transpose_times_NT) ;
end
%---------------------------------------------------------------------------
% est_err_bound_norm_A_transpose_times_NT
%---------------------------------------------------------------------------
if (isfield (stats, 'est_err_bound_norm_A_transpose_times_NT'))
fprintf ([' est_err_bound_norm_A_transpose_times_NT : %g : ' ...
' estimated error bound for norm(A''*NT).\n'], ...
stats.est_err_bound_norm_A_transpose_times_NT) ;
end
%---------------------------------------------------------------------------
% norm_R_times_N
%---------------------------------------------------------------------------
if (isfield (stats, 'norm_R_times_N'))
fprintf ([' norm_R_times_N : %g : norm of (R*N), from ' ...
'spqr_ssi.\n'], stats.norm_R_times_N) ;
end
%---------------------------------------------------------------------------
% norm_R_transpose_times_NT
%---------------------------------------------------------------------------
if (isfield (stats, 'norm_R_transpose_times_NT'))
fprintf ([' norm_R_transpose_times_NT : %g : norm of (R''*NT), '...
'from spqr_ssi.\n'], stats.norm_R_transpose_times_NT) ;
end
%---------------------------------------------------------------------------
% iters
%---------------------------------------------------------------------------
if (isfield (stats, 'iters'))
fprintf (' iters : %g : iterations in spqr_ssi or spqr_ssp.\n', ...
stats.iters) ;
end
%---------------------------------------------------------------------------
% nsvals_large_found
%---------------------------------------------------------------------------
if (isfield (stats, 'nsvals_large_found'))
fprintf ([' nsvals_large_found : %d : number of large singular ' ...
'values found.\n'], stats.nsvals_large_found) ;
end
%---------------------------------------------------------------------------
% final_blocksize
%---------------------------------------------------------------------------
if (isfield (stats, 'final_blocksize'))
fprintf (' final_blocksize : %d : final block size in spqr_ssi.\n', ...
stats.final_blocksize) ;
end
%---------------------------------------------------------------------------
% ssi_max_block_used
%---------------------------------------------------------------------------
if (isfield (stats, 'ssi_max_block_used'))
fprintf ([' ssi_max_block_used : %d : max block size for ' ...
'spqr_ssi.\n'], stats.ssi_max_block_used) ;
end
%---------------------------------------------------------------------------
% ssi_min_block_used
%---------------------------------------------------------------------------
if (isfield (stats, 'ssi_min_block_used'))
fprintf ([' ssi_min_block_used : %d : initial block size for ' ...
'spqr_ssi.\n'], stats.ssi_min_block_used) ;
end
%---------------------------------------------------------------------------
% time
%---------------------------------------------------------------------------
if (isfield (stats, 'time'))
fprintf ([' time : %g : total time taken ' ...
'(includes all timings below).\n'], stats.time) ;
end
%---------------------------------------------------------------------------
% time_initialize
%---------------------------------------------------------------------------
if (isfield (stats, 'time_initialize'))
fprintf (' time_initialize : %g : time to initialize',...
stats.time_initialize) ;
if isfield (stats, 'normest_A') || isfield (stats, 'normest_R')
fprintf (' including estimating the norm of A or R.\n')
else
fprintf('.\n')
end
end
%---------------------------------------------------------------------------
% time_svd
%---------------------------------------------------------------------------
if (isfield (stats, 'time_svd'))
fprintf (' time_svd : %g : total time taken by svd.\n', ...
stats.time_svd) ;
end
%---------------------------------------------------------------------------
% time_basis
%---------------------------------------------------------------------------
if (isfield (stats, 'time_basis'))
fprintf (' time_basis : %g : time to compute basis.\n', ...
stats.time_basis) ;
end
%---------------------------------------------------------------------------
% time_iters
%---------------------------------------------------------------------------
if (isfield (stats, 'time_iters'))
fprintf (' time_iters : %g : time for spqr_ssi iterations.\n', ...
stats.time_iters) ;
end
%---------------------------------------------------------------------------
% time_est_error_bounds
%---------------------------------------------------------------------------
if (isfield (stats, 'time_est_error_bounds'))
fprintf ([' time_est_error_bounds : %g : time taken to estimate '...
'error bounds in spqr_ssi.\n'], stats.time_est_error_bounds) ;
end
%---------------------------------------------------------------------------
% opts_used
%---------------------------------------------------------------------------
if (print_opts && isfield (stats, 'opts_used'))
fprintf ('\nopts_used : ') ;
spqr_rank_opts (stats.opts_used) ;
end
%---------------------------------------------------------------------------
% info_spqr1
%---------------------------------------------------------------------------
if (isfield (stats, 'info_spqr1'))
fprintf ('\ninfo_spqr1 : statistics from first QR factorization.\n\n') ;
disp (stats.info_spqr1) ;
end
%---------------------------------------------------------------------------
% info_spqr2
%---------------------------------------------------------------------------
if (isfield (stats, 'info_spqr2'))
fprintf ('\ninfo_spqr2 : statistics from second QR factorization.\n\n');
disp (stats.info_spqr2) ;
end
%---------------------------------------------------------------------------
% stats_ssi
%---------------------------------------------------------------------------
if (isfield (stats, 'stats_ssi'))
% all opts used by spqr_ssp are the same stats.opts_used
fprintf ('\nstats_ssi : statistics from spqr_ssi.\n') ;
spqr_rank_stats (stats.stats_ssi, 0) ;
end
%---------------------------------------------------------------------------
% stats_ssp_N
%---------------------------------------------------------------------------
if (isfield (stats, 'stats_ssp_N'))
fprintf ('\nstats_ssp_N : statistics from spqr_ssp (A,N).\n') ;
spqr_rank_stats (stats.stats_ssp_N, 0) ;
% all other opts used by spqr_ssp are the same stats.opts_used
if (isfield (stats.stats_ssp_N, 'opts_used') && ...
isfield (stats.stats_ssp_N.opts_used, 'k') && print_opts)
fprintf ([' stats_ssp_N.opts_used.k : %d : number of ' ...
'singular values to compute in spqr_ssp(A,N).\n'], ...
stats.stats_ssp_N.opts_used.k) ;
end
end
%---------------------------------------------------------------------------
% stats_ssp_NT
%---------------------------------------------------------------------------
if (isfield (stats, 'stats_ssp_NT'))
fprintf ('\nstats_ssp_NT : statistics from spqr_ssp (A'',NT).\n') ;
spqr_rank_stats (stats.stats_ssp_NT, 0) ;
% all other opts used by spqr_ssp are the same stats.opts_used
if (isfield (stats.stats_ssp_NT, 'opts_used') && ...
isfield (stats.stats_ssp_NT.opts_used, 'k') && print_opts)
fprintf ([' stats_ssp_NT.opts_used.k : %d : number of ' ...
'singular values to compute in spqr_ssp(A'',NT).\n'], ...
stats.stats_ssp_NT.opts_used.k) ;
end
end
%---------------------------------------------------------------------------
% stats_spqr_basic
%---------------------------------------------------------------------------
if (isfield (stats, 'stats_spqr_basic'))
% all opts used by spqr_basic are the same stats.opts_used
fprintf ('\nstats_spqr_basic : statistics from spqr_basic.\n') ;
spqr_rank_stats (stats.stats_spqr_basic, 0) ;
end
%---------------------------------------------------------------------------
% stats_spqr_null
%---------------------------------------------------------------------------
if (isfield (stats, 'stats_spqr_null'))
fprintf ('\nstats_spqr_null : statistics from spqr_null.\n') ;
% all other opts used by spqr_null are the same stats.opts_used
if (isfield (stats.stats_spqr_null, 'opts_used') && ...
isfield (stats.stats_spqr_null.opts_used, 'ssi_min_block'))
fprintf (['\n stats_spqr_null.opts_used.ssi_min_block : %d'...
' : initial block size in spqr_ssi as used by spqr_null.'], ...
stats.stats_spqr_null.opts_used.ssi_min_block) ;
end
spqr_rank_stats (stats.stats_spqr_null, 0) ;
end
else
%---------------------------------------------------------------------------
% describe each statistic
%---------------------------------------------------------------------------
fprintf ('\nDescription of stats returned by ') ;
if method == 1
fprintf ('spqr_basic, spqr_null, spqr_pinv \nor spqr_cod:\n') ;
elseif method == 2
fprintf ('spqr_ssi:\n') ;
elseif method == 3
fprintf ('spqr_ssp:\n') ;
elseif method == 4
fprintf ('all spqr_functions:\n') ;
end
if ( method == 1 || method == 2 || get_details >= 1 )
fprintf ([ ...
'\nstats.flag (for all routines except spqr_ssp) -- \n' ...
' if stats.flag is 0 if it is likely, although not\n' ...
' guaranteed, that stats.rank is the correct numerical rank for\n' ...
' tolerance stats.tol (i.e. agrees with the numerical rank\n' ...
' determined by the singular values of R).\n' ...
' \n'...
' stats.flag is 1 if the calculated numerical rank stats.rank ' ...
'may\n'...
' be correct for the tolerance stats.tol but the estimated error\n'...
' bounds are too large to confirm this. However stats.rank '...
'appears\n'...
' to be correct for an alternate tolerance stats.tol_alt. More\n'...
' generally stats.rank appears to be correct for any tolerance\n'...
' between stats.est_sval_lower_bounds(nsvals_large) and\n'...
' stats.est_sval_upper_bounds(nsvals_large+1).\n' ...
' \n'...
' stats.flag is 2 if the calculated numerical rank ' ...
'stats.numerical\n'...
' may be correct but estimated error bounds are too large to ' ...
'confirm\n'...
' this. The conditions for stats.flag to be 0 or 1 are not\n'...
' satisfied.\n' ...
' \n'...
' stats.flag is 3 if is likely that the numerical rank returned,\n'...
' stats.rank, is too large.\n'...
' \n'...
' stats.flag is 4 if overflow occurred during the inverse power\n'...
' method. The method fails in this case, and all parameters ' ...
'other\n'...
' stats are returned as empty ([ ]).\n' ...
' \n'...
' stats.flag is 5 if a catastrophic failure occurred.\n']) ;
end
if ( method == 3 || get_details >= 1 )
fprintf ([ ...
'\nstats.flag -- (for spqr_ssp) \n' ...
' stats.flag is 0 if spqr_ssp converged with estimated relative\n' ...
' error in singular value opts.k of A (or of A*N) <=\n' ...
' opts_ssp.convergence_factor. stats.flag is 1 if this is not ' ...
'true.\n'])
end
if ( method == 1 || method == 2 || get_details >= 1 )
fprintf ([ ...
'\nstats.rank -- the estimated numerical rank when stats.flag is\n' ...
' 0, 1 or 2. stats.rank is typically an upper bound on the\n'...
' numerical rank when stats.flag is 3. Note that stats.rank ' ...
'is a\n'...
' correction to the rank returned by spqr (stats.rank_spqr) ' ...
'in the\n'...
' case that the calculations in the routine inidicate that the ' ...
'rank\n'...
' returned by spqr not correct.\n']) ;
end
if ( method == 1 || method == 2 || get_details >= 1 )
fprintf ( ...
'\nstats.tol -- the tolerance used to define the numerical rank.\n') ;
end
if ( method == 1 || get_details >= 1 )
fprintf ([ ...
'\nstat.tol_alt -- an alternate tolerance that corresponds to the\n' ...
' calculated numerical rank when stats.flag is 1.\n']) ;
fprintf ([ ...
'\nstats.est_sval_upper_bounds -- stats.est_sval_upper_bounds(i) ' ...
'is an\n'...
' estimate of an upper bound on singular value number\n' ...
' stats.sval_numbers_for_bounds(i) of A.\n']) ;
fprintf ([ ...
'\nstats.est_sval_lower_bounds -- stats.est_sval_lower_bounds(i) ' ...
'is an\n'...
' estimate of an lower bound on singular value number\n' ...
' stats.sval_numbers_for_bounds(i) of A.\n']) ;
fprintf (['\n' ...
' Note that stats.est_sval_upper_bounds(i) is a rigorous upper '...
'bound\n'...
' on some singular value of (A+E) where where E is ' ...
'O(norm(A)*eps)\n'...
' Also stats.est_sval_lower_bounds(i) is a rigorous lower ' ...
'bound on\n'...
' some singular value of (A+E). In both cases the singular ' ...
'value is\n'...
' normally singular value number sval_numbers_for_bounds(i) ' ...
'of A,\n'...
' but the singular value number is not guaranteed. For i such ' ...
'that\n' ...
' sval_numbers_for_bounds(i) = stats.rank (that is for estimates\n'...
' of singular value stats.rank) if ' ...
'stats.est_sval_upper_bounds(i)\n' ...
' is a large multiple of stats.est_sval_lower_bounds(i) then\n' ...
' solution vectors x produced by spqr_basic may be ' ...
'inferior (i.e.\n'...
' be significanty larger) than solutions produced by ' ...
'spqr_pinv or\n' ...
' spqr_cod.\n']) ;
end
if ( method == 2 || get_details >= 1 )
fprintf ([ ...
'\nstats.est_svals_of_R -- computed by spqr_ssi.\n' ...
' stats.est_svals_of_R contains estimates of the smallest ' ...
'singular\n' ...
' of R.\n']) ;
end
if ( method == 3|| get_details >= 1 )
fprintf ([ ...
'\nstats.est_svals -- computed by spqr_ssp.\n' ...
' stats.est_svals(i) is an estimate of the ith largest ' ...
'singular of\n' ...
' A or of A*N. Also for i = 1:nsval, stats.est_svals(i) is a ' ...
'lower\n' ...
' bound on the ith largest singular value of A (or A*N).\n']) ;
end
if ( method == 2 || method == 3 || get_details >= 1 )
fprintf ([ ...
'\nstats.est_error_bounds -- computed by spqr_ssi and spqr_ssp.\n' ...
' stats.est_error_bounds(i) is an estimated bound on the ' ...
'absolute\n'...
' error in singular value number ' ...
'stats.sval_numbers_for_bounds(i).\n' ...
' of R (for spqr_ssi) or of A or A*N (for spqr_ssp). It is ' ...
'also a\n' ...
' rigorous bound on abs (s(i) - some true singular value of ' ...
'(B+E)),\n'...
' where E is O(norm(B)*eps) and B = R (for spqr_ssi) and B =\n' ...
' A or A*N (for spqr_ssp).\n']) ;
end
fprintf ([ ...
'\nstats.sval_numbers_for_bounds -- component i in the error bounds is ' ...
'an estimated\n'...
' error bound for singular value number sval_numbers_for_bounds(i).\n']) ;
if ( method == 1 || get_details >= 1 )
fprintf ([ ...
'\nstats.est_norm_A_transpose_times_NT -- an estimate of ' ...
'norm(A''*NT).\n']) ;
fprintf ( ...
'\nstats.est_norm_A_times_N -- an estimate of norm(A*N).\n') ;
end
if (get_details >= 1)
fprintf (['\n***** Additional statistics when opts.get_details ' ...
'is 2: *****\n']) ;
fprintf ([ ...
'\nstats.rank_spqr -- the rough estimate of the numerical rank\n'...
' computed by spqr. This is typically correct if the numerical\n'...
' rank is well-defined.\n']) ;
fprintf ('\nstats.stats_ssi -- statistics returned by spqr_ssi.\n') ;
fprintf ([ ...
'\nstats_ssi.ssi_max_block_used -- the maximum block size ' ...
'used by spqr_ssi.\n']) ;
fprintf ([ ...
'\nstats_ssi.ssi_min_block_used -- the initial block size ' ...
'used by spqr_ssi.\n']) ;
end
if (get_details == 1)
fprintf (['\n***** Additions statistics when opts.get_details is ' ...
'1: *****\n']) ;
fprintf ([ ...
'\nstats.normest_A -- an estimate of the Euclidean norm of A. '...
'Calculated using\n' ...
' normest(A,0.01).\n']) ;
fprintf ([ ...
'\nstats.normest_R -- an estimate of the Euclidean norm of R. '...
'Calculated for spqr_ssi\n' ...
' using normest(R,0.01).\n']) ;
fprintf ([ ...
'\nstats.est_err_bound_norm_A_times_N -- an estimate of an\n'...
' error bound on stats.est_norm_A_times_N. It is also a\n'...
' rigorous bound on abs (stats.est_norm_A_times_N - s)\n'...
' where s is some singular value of (A+E)*N and where E is\n' ...
' O(norm(A)*eps). Usually the singular value is the first '...
'singular\n'...
' value but this is not guaranteed.\n']) ;
fprintf ([ ...
'\nstats.est_err_bound_norm_A_transpose_times_NT -- an estimate '...
'of an\n'...
' error bound on stats.est_norm_A_transpose_times_NT. It is '...
'also a\n'...
' rigorous bound on abs (stats.est_norm_A_transpose_times_NT '...
'- s)\n'...
' where s is some singular value of (A+E)''*NT and where E is\n' ...
' O(norm(A)*eps). Usually the singular value is the first '...
'singular\n'...
' value but this is not guaranteed.\n']) ;
fprintf ([ ...
'\nstats_ssi.norm_R_times_N -- Euclidean norm of (R*N), from ' ...
'spqr_ssi.\n']) ;
fprintf ([ ...
'\nstats_ssi.norm_R_transpose_times_NT -- Eucliean norm of ' ...
'(R''*NT), from spqr_ssi.\n']) ;
fprintf ([ ...
'\nstats_ssi.iters or stats_ssp_N.iters or stats_ssp_NT.iters -- ' ...
'number of\n' ...
' iterations for subspace iteration in spqr_ssi or spqr_ssp.\n']) ;
fprintf ([ ...
'\nstats_ssi.nsvals_large_found -- the number of ''large'' (larger ' ...
'than tol) singular\n' ...
' values found, from spqr_ssi.\n']) ;
fprintf ([ ...
'\nstats_ssi.final_blocksize -- final block size for subspace ' ...
'iteration in \n spqr_ssi. \n']) ;
fprintf ([ ...
'\nstats.stats_spqr_basic -- statistics returned when spqr_basic ' ...
'is called by spqr_pinv.\n']) ;
fprintf ([ ...
'\nstats.stats_spqr_null -- statistics returned when spqr_null ' ...
'is called by spqr_pinv.\n']) ;
fprintf ([ ...
'\nstats.info_spqr1 -- statistics from spqr for the first QR ' ...
'factorization.\n' ...
' See ''help spqr'' for details.\n']) ;
fprintf ([ ...
'\nstats.info_spqr2 -- statistics from spqr for the second QR ' ...
'factorization, if\n' ...
' required. See ''help spqr'' for details.\n']) ;
fprintf ([ ...
'\nstats.stats_ssp_N -- statistics from spqr_ssp when calculating ' ...
'the basis\n' ...
' N for the null space of A.\n']) ;
fprintf ([ ...
'\nstats.stats_ssp_NT -- statistics from spqr_ssp when calculating ' ...
'the basis\n' ...
' NT for the null space of A transpose.\n']) ;
fprintf ([ ...
'\nstats.opts_used, stats_ssi.opts_used, or stats_ssp.opts_used -- ' ...
'values of\n' ...
' options used. These can be different from values in opts ' ...
'since, for example,\n' ...
' the size of A can restrict some values in opts.\n']) ;
fprintf ([ ...
'\nstats.time, stats_ssi.time, etc. -- the total time of the ' ...
'routine including\n' ...
' the times described below.\n']) ;
fprintf ([ ...
'\nstats.time_initialize, stats_ssi.time_initialize, etc. -- the '...
'time to\n' ...
' set default values of opts, including calculating ' ...
'normest(A,0.01),\n' ...
' or normest(R,0.01) if needed.\n']) ;
fprintf([ ...
'\nstats.time_basis -- the time to compute the basis for the ' ...
'numerical null space\n' ...
' following any calls to spqr and spqr_ssi. This will be small\n'...
' if the null space basis is returned in implicit form but can, '...
'in some cases,\n' ...
' be significant if the null space basis is returned as an ' ...
'explicit matrix.\n']) ;
fprintf ([ ...
'\nstats_ssi.time_iters, stats_ssp_N.time_iters, etc. -- the time ' ...
'for the\n' ...
' subspace iterations in spqr_ssi or spqr_ssp. Excludes time '...
'for initialization,\n' ...
' error flag calculation, etc..\n']) ;
fprintf ([ ...
'\nstats_ssi.time_est_error_bounds, ' ...
'stats_ssp_N.time_est_error_bounds, etc. -- the time\n' ...
' for estimating the singular value error bounds in ' ...
'spqr_ssi or spqr_ssp.\n']) ;
fprintf ([ ...
'\nstats.time_svd, stats_ssi.time_svd, etc. -- the total time ' ...
'for calls to MATLAB''s SVD\n' ...
' in the current routine and its subroutines.\n']) ;
end
end
%-------------------------------------------------------------------------------
% print_vector
%-------------------------------------------------------------------------------
function print_vector (x)
n = length (x) ;
for k = 1:n
fprintf (' %g', x (k)) ;
end
fprintf ('\n') ;
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