function x = r8cbb_sl ( n1, n2, ml, mu, a_lu, b )
%*****************************************************************************80
%
%% R8CBB_SL solves a R8CBB system factored by R8CBB_FA.
%
% Discussion:
%
% Note that in C++ and FORTRAN, we can look at A as an abstract
% vector, but then look at parts of A as storing a two dimensional
% array. MATLAB assigns an inherent dimensionality to a data object,
% and gets very unhappy when you try to manipulate the data yourself.
% This means that the MATLAB implementation of this routine requires
% the use of temporary 2D arrays.
%
% The R8CBB storage format is for a compressed border banded matrix.
% Such a matrix has the logical form:
%
% A1 | A2
% ---+---
% A3 | A4
%
% with A1 a (usually large) N1 by N1 banded matrix, while A2, A3 and A4
% are dense rectangular matrices of orders N1 by N2, N2 by N1, and N2 by N2,
% respectively.
%
% The R8CBB format is the same as the DBB format, except that the banded
% matrix A1 is stored in compressed band form rather than standard
% banded form. In other words, we do not include the extra room
% set aside for fill in during pivoting.
%
% A should be defined as a vector. The user must then store
% the entries of the four blocks of the matrix into the vector A.
% Each block is stored by columns.
%
% A1, the banded portion of the matrix, is stored in
% the first (ML+MU+1)*N1 entries of A, using the obvious variant
% of the LINPACK general band format.
%
% The following formulas should be used to determine how to store
% the entry corresponding to row I and column J in the original matrix:
%
% Entries of A1:
%
% 1 <= I <= N1, 1 <= J <= N1, (J-I) <= MU and (I-J) <= ML.
%
% Store the I, J entry into location
% (I-J+MU+1)+(J-1)*(ML+MU+1).
%
% Entries of A2:
%
% 1 <= I <= N1, N1+1 <= J <= N1+N2.
%
% Store the I, J entry into location
% (ML+MU+1)*N1+(J-N1-1)*N1+I.
%
% Entries of A3:
%
% N1+1 <= I <= N1+N2, 1 <= J <= N1.
%
% Store the I, J entry into location
% (ML+MU+1)*N1+N1*N2+(J-1)*N2+(I-N1).
%
% Entries of A4:
%
% N1+1 <= I <= N1+N2, N1+1 <= J <= N1+N2
%
% Store the I, J entry into location
% (ML+MU+1)*N1+N1*N2+(J-1)*N2+(I-N1).
% (same formula used for A3).
%
% The linear system A * x = b is decomposable into the block system:
%
% ( A1 A2 ) * (X1) = (B1)
% ( A3 A4 ) (X2) (B2)
%
% where A1 is a (usually big) banded square matrix, A2 and A3 are
% column and row strips which may be nonzero, and A4 is a dense
% square matrix.
%
% All the arguments except B are input quantities only, which are
% not changed by the routine. They should have exactly the same values
% they had on exit from R8CBB_FA.
%
% If more than one right hand side is to be solved, with the same
% matrix, R8CBB_SL should be called repeatedly. However, R8CBB_FA only
% needs to be called once to create the factorization.
%
% See the documentation of R8CBB_FA for details on the matrix storage.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 24 March 2004
%
% Author:
%
% John Burkardt
%
% Parameters:
%
% Input, integer N1, N2, the order of the banded and dense blocks.
% N1 and N2 must be nonnegative, and at least one must be positive.
%
% Input, integer ML, MU, the lower and upper bandwidths.
% ML and MU must be nonnegative, and no greater than N1-1.
%
% Input, real A_LU( (ML+MU+1)*N1 + 2*N1*N2 + N2*N2).
% the compact border banded matrix, as factored by R8CBB_FA.
%
% Input, real B(N1+N2), the right hand side of the linear system.
%
% Output, real X(N1+N2), the solution.
%
nband = (ml+mu+1)*n1;
%
% Set B1 := inverse(A1) * B1.
%
if ( 0 < n1 )
a1_lu(1:ml+mu+1,1:n1) = r8vec_to_r8cb ( n1, n1, ml, mu, a_lu(1:nband) );
job = 0;
x(1:n1) = r8cb_np_sl ( n1, ml, mu, a1_lu, b(1:n1), job );
end
%
% Modify the right hand side of the second linear subsystem.
% Replace B2 by B2-A3*B1.
%
for j = 1 : n1
for i = 1 : n2
ij = nband + n1*n2 + (j-1)*n2 + i;
b(n1+i) = b(n1+i) - a_lu(ij) * x(j);
end
end
%
% Solve A4*B2 = B2.
%
if ( 0 < n2 )
a4_lu(1:n2,1:n2) = r8vec_to_r8ge ( ...
n2, n2, a_lu(nband+2*n1*n2+1:nband+2*n1*n2+n2*n2) );
job = 0;
x(n1+1:n1+n2) = r8ge_np_sl ( n2, a4_lu, b(n1+1:n1+n2), job );
end
%
% Modify the first subsolution.
% Set B1 = B1+A2*B2.
%
for i = 1 : n1
for j = 1 : n2
ij = nband + (j-1)*n1 + i;
x(i) = x(i) + a_lu(ij) * x(n1+j);
end
end
return
end