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function x = umfpack_btf (A, b, Control)
% UMFPACK_BTF
%
% x = umfpack_btf (A, b, Control)
%
% solve Ax=b by first permuting the matrix A to block triangular form via dmperm
% and then using UMFPACK to factorize each diagonal block. Adjacent 1-by-1
% blocks are merged into a single upper triangular block, and solved via
% MATLAB's \ operator. The Control parameter is optional (Type umfpack_details
% and umfpack_report for details on its use). A must be square.
%
% See also: umfpack, umfpack_factorize, umfpack_details, dmperm
% UMFPACK Version 4.4, Copyright (c) 2005 by Timothy A. Davis.
% All Rights Reserved. Type umfpack_details for License.
if (nargin < 2)
help umfpack_btf
error ('Usage: x = umfpack_btf (A, b, Control)') ;
end
[m n] = size (A) ;
if (m ~= n)
help umfpack_btf
error ('umfpack_btf: A must be square') ;
end
[m1 n1] = size (b) ;
if (m1 ~= n)
help umfpack_btf
error ('umfpack_btf: b has the wrong dimensions') ;
end
if (nargin < 3)
Control = umfpack ;
end
%-------------------------------------------------------------------------------
% find the block triangular form
%-------------------------------------------------------------------------------
[p,q,r] = dmperm (A) ;
nblocks = length (r) - 1 ;
%-------------------------------------------------------------------------------
% solve the system
%-------------------------------------------------------------------------------
if (nblocks == 1 | sprank (A) < n)
%---------------------------------------------------------------------------
% matrix is irreducible or structurally singular
%---------------------------------------------------------------------------
x = umfpack_solve (A, '\', b, Control) ;
else
%---------------------------------------------------------------------------
% A (p,q) is in block triangular form
%---------------------------------------------------------------------------
b = b (p,:) ;
A = A (p,q) ;
x = zeros (size (b)) ;
%---------------------------------------------------------------------------
% merge adjacent singletons into a single upper triangular block
%---------------------------------------------------------------------------
[r, nblocks, is_triangular] = merge_singletons (r) ;
%---------------------------------------------------------------------------
% solve the system: x (q) = A\b
%---------------------------------------------------------------------------
for k = nblocks:-1:1
% get the kth block
k1 = r (k) ;
k2 = r (k+1) - 1 ;
% solve the system
x (k1:k2,:) = solver (A (k1:k2, k1:k2), b (k1:k2,:), ...
is_triangular (k), Control) ;
% off-diagonal block back substitution
b (1:k1-1,:) = b (1:k1-1,:) - A (1:k1-1, k1:k2) * x (k1:k2,:) ;
end
x (q,:) = x ;
end
%-------------------------------------------------------------------------------
% merge_singletons
%-------------------------------------------------------------------------------
function [r, nblocks, is_triangular] = merge_singletons (r)
%
% Given r from [p,q,r] = dmperm (A), where A is square, return a modified r that
% reflects the merger of adjacent singletons into a single upper triangular
% block. is_triangular (k) is 1 if the kth block is upper triangular. nblocks
% is the number of new blocks.
nblocks = length (r) - 1 ;
bsize = r (2:nblocks+1) - r (1:nblocks) ;
t = [0 (bsize == 1)] ;
z = (t (1:nblocks) == 0 & t (2:nblocks+1) == 1) | t (2:nblocks+1) == 0 ;
y = [(find (z)) nblocks+1] ;
r = r (y) ;
nblocks = length (y) - 1 ;
is_triangular = y (2:nblocks+1) - y (1:nblocks) > 1 ;
%-------------------------------------------------------------------------------
% solve Ax=b, but check for small and/or triangular systems
%-------------------------------------------------------------------------------
function x = solver (A, b, is_triangular, Control)
if (is_triangular)
% back substitution only
x = A \ b ;
elseif (size (A,1) < 4)
% a very small matrix, solve it as a dense linear system
x = full (A) \ b ;
else
% solve it as a sparse linear system
x = umfpack_solve (A, '\', b, Control) ;
end
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