## Copyright (C) 2004, 2006, 2007, 2009 Piotr Krzyzanowski
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
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## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING.  If not, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {@var{x} =} pcr (@var{a}, @var{b}, @var{tol}, @var{maxit}, @var{m}, @var{x0}, @dots{})
## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} pcr (@dots{})
## 
## Solves the linear system of equations @code{@var{a} * @var{x} =
## @var{b}} by means of the Preconditioned Conjugate Residuals iterative
## method.  The input arguments are
##
## @itemize
## @item
## @var{a} can be either a square (preferably sparse) matrix or a
## function handle, inline function or string containing the name
## of a function which computes @code{@var{a} * @var{x}}.  In principle
## @var{a} should be symmetric and non-singular; if @code{pcr}
## finds @var{a} to be numerically singular, you will get a warning
## message and the @var{flag} output parameter will be set.
## 
## @item
## @var{b} is the right hand side vector.
## 
## @item
## @var{tol} is the required relative tolerance for the residual error,
## @code{@var{b} - @var{a} * @var{x}}.  The iteration stops if @code{norm
## (@var{b} - @var{a} * @var{x}) <= @var{tol} * norm (@var{b} - @var{a} *
## @var{x0})}.  If @var{tol} is empty or is omitted, the function sets
## @code{@var{tol} = 1e-6} by default.
## 
## @item
## @var{maxit} is the maximum allowable number of iterations; if
## @code{[]} is supplied for @code{maxit}, or @code{pcr} has less
## arguments, a default value equal to 20 is used.
##
## @item
## @var{m} is the (left) preconditioning matrix, so that the iteration is
## (theoretically) equivalent to solving by @code{pcr} @code{@var{P} *
## @var{x} = @var{m} \ @var{b}}, with @code{@var{P} = @var{m} \ @var{a}}.
## Note that a proper choice of the preconditioner may dramatically
## improve the overall performance of the method.  Instead of matrix
## @var{m}, the user may pass a function which returns the results of 
## applying the inverse of @var{m} to a vector (usually this is the
## preferred way of using the preconditioner).  If @code{[]} is supplied
## for @var{m}, or @var{m} is omitted, no preconditioning is applied.
## 
## @item
## @var{x0} is the initial guess.  If @var{x0} is empty or omitted, the 
## function sets @var{x0} to a zero vector by default.
## @end itemize
## 
## The arguments which follow @var{x0} are treated as parameters, and
## passed in a proper way to any of the functions (@var{a} or @var{m})
## which are passed to @code{pcr}.  See the examples below for further
## details.  The output arguments are
##
## @itemize
## @item
## @var{x} is the computed approximation to the solution of
## @code{@var{a} * @var{x} = @var{b}}.
## 
## @item
## @var{flag} reports on the convergence.  @code{@var{flag} = 0} means
## the solution converged and the tolerance criterion given by @var{tol}
## is satisfied.  @code{@var{flag} = 1} means that the @var{maxit} limit
## for the iteration count was reached.  @code{@var{flag} = 3} reports t
## @code{pcr} breakdown, see [1] for details.
## 
## @item
## @var{relres} is the ratio of the final residual to its initial value,
## measured in the Euclidean norm.
## 
## @item
## @var{iter} is the actual number of iterations performed.
##
## @item 
## @var{resvec} describes the convergence history of the method,
## so that @code{@var{resvec} (i)} contains the Euclidean norms of the 
## residual after the (@var{i}-1)-th iteration, @code{@var{i} =
## 1,2, @dots{}, @var{iter}+1}.
## @end itemize
## 
## Let us consider a trivial problem with a diagonal matrix (we exploit the
## sparsity of A) 
## 
## @example
## @group
##      n = 10; 
##      a = sparse (diag (1:n));
##      b = rand (N, 1);
## @end group
## @end example
## 
## @sc{Example 1:} Simplest use of @code{pcr}
## 
## @example
##   x = pcr(A, b)
## @end example
## 
## @sc{Example 2:} @code{pcr} with a function which computes
## @code{@var{a} * @var{x}}.
##
## @example
## @group
##   function y = apply_a (x) 
##     y = [1:10]'.*x; 
##   endfunction
## 
##   x = pcr ("apply_a", b)
## @end group
## @end example
## 
## @sc{Example 3:}  Preconditioned iteration, with full diagnostics.  The
## preconditioner (quite strange, because even the original matrix
## @var{a} is trivial) is defined as a function
## 
## @example
## @group
##   function y = apply_m (x)           
##     k = floor (length(x)-2); 
##     y = x; 
##     y(1:k) = x(1:k)./[1:k]'; 
##   endfunction
## 
##   [x, flag, relres, iter, resvec] = ...
##                      pcr (a, b, [], [], "apply_m")
##   semilogy([1:iter+1], resvec);
## @end group
## @end example
##
## @sc{Example 4:} Finally, a preconditioner which depends on a
## parameter @var{k}.
## 
## @example
## @group
##   function y = apply_m (x, varargin)
##     k = varargin@{1@}; 
##     y = x; y(1:k) = x(1:k)./[1:k]';   
##   endfunction
## 
##   [x, flag, relres, iter, resvec] = ...
##                      pcr (a, b, [], [], "apply_m"', [], 3)
## @end group
## @end example
## 
## References:
## 
##      [1] W. Hackbusch, @cite{Iterative Solution of Large Sparse Systems of
##      Equations}, section 9.5.4; Springer, 1994
##
## @seealso{sparse, pcg}
## @end deftypefn

## Author: Piotr Krzyzanowski <piotr.krzyzanowski@mimuw.edu.pl>

function [x, flag, relres, iter, resvec] = pcr (a, b, tol, maxit, m, x0, varargin)

  breakdown = false;

  if (nargin < 6 || isempty (x0))
    x = zeros (size (b));
  else
    x = x0;
  endif

  if (nargin < 5)
    m = [];
  endif

  if (nargin < 4 || isempty (maxit))
    maxit = 20;
  endif

  maxit += 2;

  if (nargin < 3 || isempty (tol))
    tol = 1e-6;
  endif

  if (nargin < 2)
    print_usage ();
  endif

  ##  init
  if (isnumeric (a))            # is A a matrix?
    r = b - a*x;
  else                          # then A should be a function!
    r = b - feval (a, x, varargin{:});
  endif

  if (isnumeric (m))            # is M a matrix?
    if (isempty (m))            # if M is empty, use no precond
      p = r;
    else                        # otherwise, apply the precond
      p = m \ r;
    endif
  else                          # then M should be a function!
    p = feval (m, r, varargin{:});
  endif

  iter = 2;

  b_bot_old = 1;
  q_old = p_old = s_old = zeros (size (x));

  if (isnumeric (a))            # is A a matrix?
    q = a * p;
  else                          # then A should be a function!
    q = feval (a, p, varargin{:});
  endif
        
  resvec(1) = abs (norm (r)); 

  ## iteration
  while (resvec(iter-1) > tol*resvec(1) && iter < maxit)

    if (isnumeric (m))          # is M a matrix?
      if (isempty (m))          # if M is empty, use no precond
        s = q;
      else                      # otherwise, apply the precond
        s = m \ q;
      endif
    else                        # then M should be a function!
      s = feval (m, q, varargin{:});
    endif
    b_top = r' * s;
    b_bot = q' * s;
        
    if (b_bot == 0.0)
      breakdown = true;
      break;
    endif
    lambda = b_top / b_bot;
        
    x += lambda*p;
    r -= lambda*q;
        
    if (isnumeric(a))           # is A a matrix?
      t = a*s;
    else                        # then A should be a function!
      t = feval (a, s, varargin{:});
    endif
        
    alpha0 = (t'*s) / b_bot;
    alpha1 = (t'*s_old) / b_bot_old;
        
    p_temp = p;
    q_temp = q;

    p = s - alpha0*p - alpha1*p_old;
    q = t - alpha0*q - alpha1*q_old;
        
    s_old = s;
    p_old = p_temp;
    q_old = q_temp;
    b_bot_old = b_bot;
        
    resvec(iter) = abs (norm (r));
    iter++;
  endwhile

  flag = 0;
  relres = resvec(iter-1) ./ resvec(1);
  iter -= 2;
  if (iter >= maxit-2)
    flag = 1;
    if (nargout < 2)
      warning ("pcr: maximum number of iterations (%d) reached\n", iter);
      warning ("the initial residual norm was reduced %g times.\n", 1.0/relres);
    endif
  elseif (nargout < 2 && ! breakdown)
    fprintf (stderr, "pcr: converged in %d iterations. \n", iter);
    fprintf (stderr, "the initial residual norm was reduced %g times.\n",
             1.0 / relres);
  endif

  if (breakdown)
    flag = 3;
    if (nargout < 2)
      warning ("pcr: breakdown occurred:\n");
      warning ("system matrix singular or preconditioner indefinite?\n");
    endif
  endif

endfunction

%!demo
%!
%!      # Simplest usage of PCR (see also 'help pcr')
%!
%!      N = 20; 
%!      A = diag(linspace(-3.1,3,N)); b = rand(N,1); y = A\b; #y is the true solution
%!      x = pcr(A,b);
%!      printf('The solution relative error is %g\n', norm(x-y)/norm(y));
%!
%!      # You shouldn't be afraid if PCR issues some warning messages in this
%!      # example: watch out in the second example, why it takes N iterations 
%!      # of PCR to converge to (a very accurate, by the way) solution
%!demo
%!
%!      # Full output from PCR
%!      # We use this output to plot the convergence history  
%!
%!      N = 20; 
%!      A = diag(linspace(-3.1,30,N)); b = rand(N,1); X = A\b; #X is the true solution
%!      [x, flag, relres, iter, resvec] = pcr(A,b);
%!      printf('The solution relative error is %g\n', norm(x-X)/norm(X));
%!      title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||/||b||)');
%!      semilogy([0:iter],resvec/resvec(1),'o-g;relative residual;');
%!demo
%!
%!      # Full output from PCR
%!      # We use indefinite matrix based on the Hilbert matrix, with one
%!      # strongly negative eigenvalue
%!      # Hilbert matrix is extremely ill conditioned, so is ours, 
%!      # and that's why PCR WILL have problems
%!
%!      N = 10; 
%!      A = hilb(N); A(1,1)=-A(1,1); b = rand(N,1); X = A\b; #X is the true solution
%!      printf('Condition number of A is   %g\n', cond(A));
%!      [x, flag, relres, iter, resvec] = pcr(A,b,[],200);
%!      if (flag == 3)
%!        printf('PCR breakdown. System matrix is [close to] singular\n');
%!      end
%!      title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)');
%!      semilogy([0:iter],resvec,'o-g;absolute residual;');
%!demo
%!
%!      # Full output from PCR
%!      # We use an indefinite matrix based on the 1-D Laplacian matrix for A, 
%!      # and here we have cond(A) = O(N^2)
%!      # That's the reason we need some preconditioner; here we take
%!      # a very simple and not powerful Jacobi preconditioner, 
%!      # which is the diagonal of A
%!
%!      # Note that we use here indefinite preconditioners!
%!
%!      N = 100; 
%!      A = zeros(N,N);
%!      for i=1:N-1 # form 1-D Laplacian matrix
%!              A(i:i+1,i:i+1) = [2 -1; -1 2];
%!      endfor
%!      A = [A, zeros(size(A)); zeros(size(A)), -A];
%!      b = rand(2*N,1); X = A\b; #X is the true solution
%!      maxit = 80;
%!      printf('System condition number is %g\n',cond(A));
%!      # No preconditioner: the convergence is very slow!
%!
%!      [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit);
%!      title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)');
%!      semilogy([0:iter],resvec,'o-g;NO preconditioning: absolute residual;');
%!
%!      pause(1);
%!      # Test Jacobi preconditioner: it will not help much!!!
%!
%!      M = diag(diag(A)); # Jacobi preconditioner
%!      [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit,M);
%!      hold on;
%!      semilogy([0:iter],resvec,'o-r;JACOBI preconditioner: absolute residual;');
%!
%!      pause(1);
%!      # Test nonoverlapping block Jacobi preconditioner: this one should give
%!      # some convergence speedup!
%!
%!      M = zeros(N,N);k=4;
%!      for i=1:k:N # get k x k diagonal blocks of A
%!              M(i:i+k-1,i:i+k-1) = A(i:i+k-1,i:i+k-1);
%!      endfor
%!      M = [M, zeros(size(M)); zeros(size(M)), -M];
%!      [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit,M);
%!      semilogy([0:iter],resvec,'o-b;BLOCK JACOBI preconditioner: absolute residual;');
%!      hold off;
%!test
%!
%!      #solve small indefinite diagonal system
%!
%!      N = 10; 
%!      A = diag(linspace(-10.1,10,N)); b = ones(N,1); X = A\b; #X is the true solution
%!      [x, flag] = pcr(A,b,[],N+1);
%!      assert(norm(x-X)/norm(X)<1e-10);
%!      assert(flag,0);
%!
%!test
%!
%!      #solve tridiagonal system, do not converge in default 20 iterations
%!      #should perform max allowable default number of iterations
%!
%!      N = 100; 
%!      A = zeros(N,N);
%!      for i=1:N-1 # form 1-D Laplacian matrix
%!              A(i:i+1,i:i+1) = [2 -1; -1 2];
%!      endfor
%!      b = ones(N,1); X = A\b; #X is the true solution
%!      [x, flag, relres, iter, resvec] = pcr(A,b,1e-12);
%!      assert(flag,1);
%!      assert(relres>0.6);
%!      assert(iter,20);
%!
%!test
%!
%!      #solve tridiagonal system with 'prefect' preconditioner
%!      #converges in one iteration
%!
%!      N = 100; 
%!      A = zeros(N,N);
%!      for i=1:N-1 # form 1-D Laplacian matrix
%!              A(i:i+1,i:i+1) = [2 -1; -1 2];
%!      endfor
%!      b = ones(N,1); X = A\b; #X is the true solution
%!      [x, flag, relres, iter] = pcr(A,b,[],[],A,b);
%!      assert(norm(x-X)/norm(X)<1e-6);
%!      assert(relres<1e-6);
%!      assert(flag,0);
%!      assert(iter,1); #should converge in one iteration
%!
