GNU Octave - Iterative Techniques

GNU Octave Manual Version 3
by John W. Eaton, David Bateman, Søren Hauberg
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20.3 Iterative Techniques applied to sparse matrices

The left division \ and right division / operators, discussed in the previous section, use direct solvers to resolve a linear equation of the form x = A \ b or x = b / A. Octave equally includes a number of functions to solve sparse linear equations using iterative techniques.

Function File: x = pcg (a, b, tol, maxit, m1, m2, x0, ...)

Function File: [x, flag, relres, iter, resvec, eigest] = pcg (...)

Solves the linear system of equations a * x = b by means of the Preconditioned Conjugate Gradient 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 positive definite; if pcg finds a to not be positive definite, 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 if norm (b - a * x) <= tol * norm (b - a * x0). If tol is empty or is omitted, the function sets tol = 1e-6 by default.
maxit is the maximum allowable number of iterations; if [] is supplied for maxit, or pcg has less arguments, a default value equal to 20 is used.
m = m1 * m2 is the (left) preconditioning matrix, so that the iteration is (theoretically) equivalent to solving by pcg P * 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 matrices m1 and m2, the user may pass two functions which return the results of applying the inverse of m1 and m2 to a vector (usually this is the preferred way of using the preconditioner). If [] is supplied for m1, or m1 is omitted, no preconditioning is applied. If m2 is omitted, m = m1 will be used as preconditioner.
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 pcg. 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 = 0 means the solution converged and the tolerance criterion given by tol is satisfied. flag = 1 means that the maxit limit for the iteration count was reached. flag = 3 reports that the (preconditioned) matrix was found not positive definite.
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. resvec (i,1) is the Euclidean norm of the residual, and resvec (i,2) is the preconditioned residual norm, after the (i-1)-th iteration, i = 1, 2, ..., iter+1. The preconditioned residual norm is defined as norm (r) ^ 2 = r' * (m \ r) where r = b - a * x, see also the description of m. If eigest is not required, only resvec (:,1) is returned.
eigest returns the estimate for the smallest eigest (1) and largest eigest (2) eigenvalues of the preconditioned matrix P = m \ a. In particular, if no preconditioning is used, the estimates for the extreme eigenvalues of a are returned. eigest (1) is an overestimate and eigest (2) is an underestimate, so that eigest (2) / eigest (1) is a lower bound for cond (P, 2), which nevertheless in the limit should theoretically be equal to the actual value of the condition number. The method which computes eigest works only for symmetric positive definite a and m, and the user is responsible for verifying this assumption.

Let us consider a trivial problem with a diagonal matrix (we exploit the sparsity of A)

    N = 10; 
    A = spdiag ([1:N]);
    b = rand (N, 1);
     [L, U, P, Q] = luinc (A,1.e-3);

Example 1: Simplest use of pcg

x = pcg(A,b)

Example 2: pcg with a function which computes a * x

  function y = applyA (x)
    y = [1:N]'.*x; 
  endfunction

  x = pcg ("applyA", b)

Example 3: pcg with a preconditioner: l * u

x=pcg(A,b,1.e-6,500,L*U);

Example 4: pcg with a preconditioner: l * u. Faster than Example 3 since lower and upper triangular matrices are easier to invert

x=pcg(A,b,1.e-6,500,L,U);

Example 5: 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, eigest] = ...
                     pcg (A, b, [], [], "applyM");
  semilogy (1:iter+1, resvec);

Example 6: 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, eigest] = ...
       pcg (A, b, [], [], "applyM", [], [], 3)

References

[1] C.T.Kelley, “Iterative methods for linear and nonlinear equations”, SIAM, 1995 (the base PCG algorithm) [2] Y.Saad, “Iterative methods for sparse linear systems”, PWS 1996 (condition number estimate from PCG). A revised version of this book is available online at http://www-users.cs.umn.edu/~saad/books.html.

See also sparse, pcr

Function File: x = pcr (a, b, tol, maxit, m, x0, ...)

Function File: [x, flag, relres, iter, resvec] = pcr (...)

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; if pcr finds 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 if norm (b - a * x) <= tol * norm (b - a * x0). If tol is empty or is omitted, the function sets tol = 1e-6 by default.
maxit is the maximum allowable number of iterations; if [] is supplied for maxit, or pcr has 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 pcr P * 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 = 0 means the solution converged and the tolerance criterion given by tol is satisfied. flag = 1 means that the maxit limit for the iteration count was reached. flag = 3 reports t pcr breakdown, 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 residual after 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 pcr

x = pcr(A, b)

Example 2: pcr with 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