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Solves the linear system of equations a
*x=b by means of the Preconditioned Conjugate Residuals iterative method. The input arguments are
- 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 a
*x. In principle a should be symmetric and non-singular; ifpcrfinds a to be numerically singular, you will get a warning message and the flag output parameter will be set.- b is the right hand side vector.
- tol is the required relative tolerance for the residual error, b
-a*x. The iteration stops ifnorm (b-a*x) <=tol* norm (b-a*x0). If tol is empty or is omitted, the function sets tol= 1e-6by default.- maxit is the maximum allowable number of iterations; if
[]is supplied formaxit, orpcrhas less arguments, a default value equal to 20 is used.- m is the (left) preconditioning matrix, so that the iteration is (theoretically) equivalent to solving by
pcrP*x=m\b, with P=m\a. Note that a proper choice of the preconditioner may dramatically improve the overall performance of the method. Instead of matrix m, the user may pass a function which returns the results of applying the inverse of m to a vector (usually this is the preferred way of using the preconditioner). If[]is supplied for m, or m is omitted, no preconditioning is applied.- x0 is the initial guess. If x0 is empty or omitted, the function sets x0 to a zero vector by default.
The arguments which follow x0 are treated as parameters, and passed in a proper way to any of the functions (a or m) which are passed to
pcr. See the examples below for further details. The output arguments are
- x is the computed approximation to the solution of a
*x=b.- flag reports on the convergence. flag
= 0means the solution converged and the tolerance criterion given by tol is satisfied. flag= 1means that the maxit limit for the iteration count was reached. flag= 3reports tpcrbreakdown, see [1] for details.- relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.
- iter is the actual number of iterations performed.
- resvec describes the convergence history of the method, so that resvec
(i)contains the Euclidean norms of the residualafter the (i-1)-th iteration, i= 1,2, ...,iter+1.Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of A)
N = 10; A = diag([1:N]); A = sparse(A); b = rand(N,1);Example 1: Simplest use of
pcrx = pcr(A, b)Example 2:
pcrwith a function which computes a*x.function y = applyA(x) y = [1:10]'.*x; endfunction x = pcr('applyA',b)Example 3: Preconditioned iteration, with full diagnostics. The preconditioner (quite strange, because even the original matrix a is trivial) is defined as a function
function y = applyM(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,[],[],'applyM') semilogy([1:iter+1], resvec);Example 4: Finally, a preconditioner which depends on a parameter k.
function y = applyM(x, varargin) K = varargin{1}; y = x; y(1:K) = x(1:K)./[1:K]'; endfunction [x, flag, relres, iter, resvec] = pcr(A,b,[],[],'applyM',[],3)References
[1] W. Hackbusch, "Iterative Solution of Large Sparse Systems of Equations", section 9.5.4; Springer, 1994
See also: sparse, pcg.