### 22.3 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 also includes a number of functions to solve sparse linear equations using iterative techniques.

`x =` pcg `(A, b, tol, maxit, m1, m2, x0, …)` ¶
`x =` pcg `(A, b, tol, maxit, M, [], x0, …)` ¶
`[x, flag, relres, iter, resvec, eigest] =` pcg `(A, b, …)` ¶

  Solve the linear system of equations `A * x = b` by means of the Preconditioned Conjugate Gradient iterative method.

  The input arguments are:

  - A is the matrix of the linear system and it must be square. A can be passed as a matrix, function handle, or inline function `Afcn` such that `Afcn(x) = A * x`. Additional parameters to `Afcn` may be passed after x0.

    A has to be Hermitian and Positive Definite (HPD). If `pcg` detects A not to be positive definite, a warning is printed and the flag output is 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)`. If tol is omitted or empty, then a tolerance of 1e-6 is used.
  - maxit is the maximum allowed number of iterations; if maxit is omitted or empty then a value of 20 is used.
  - m is a HPD preconditioning matrix. For any decomposition `m = p1 * p2` such that `inv (p1) * A * inv (p2)` is HPD, the conjugate gradient method is formally applied to the linear system `inv (p1) * A * inv (p2) * y = inv (p1) * b`, with `x = inv (p2) * y` (split preconditioning). In practice, at each iteration of the conjugate gradient method a linear system with matrix m is solved with `mldivide`. If a particular factorization `m = m1 * m2` is available (for instance, an incomplete Cholesky factorization of a), the two matrices m1 and m2 can be passed and the relative linear systems are solved with the `mldivide` operator. 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. If m1 is omitted or empty `[]`, then no preconditioning is applied. If no factorization of m is available, m2 can be omitted or left [], and the input variable m1 can be used to pass the preconditioner m.
  - x0 is the initial guess. If x0 is omitted or empty then the function sets x0 to a zero vector by default.

  The arguments which follow x0 are treated as parameters, and passed in an appropriate manner to any of the functions (A or m1 or m2) that have been given 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`. If the algorithm did not converge, then x is the iteration which has the minimum residual.
  - flag reports on the convergence:
    - 0: The algorithm converged to within the prescribed tolerance.
    - 1: The algorithm did not converge and it reached the maximum number of iterations.
    - 2: The preconditioner matrix is singular.
    - 3: The algorithm stagnated, i.e., the absolute value of the difference between the current iteration x and the previous is less than `eps * norm (x,2)`.
    - 4: The algorithm detects that the input (preconditioned) matrix is not HPD.
  - relres is the ratio of the final residual to its initial value, measured in the Euclidean norm.
  - iter indicates the iteration of x which it was computed. Since the output x corresponds to the minimal residual solution, the total number of iterations that the method performed is given by `length(resvec) - 1`.
  - 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 `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.

  Let us consider a trivial problem with a tridiagonal matrix

      n = 10;
      A = toeplitz (sparse ([1, 1], [1, 2], [2, 1], 1, n));
      b = A * ones (n, 1);
      M1 = ichol (A); # in this tridiagonal case it corresponds to chol (A)'
      M2 = M1';
      M = M1 * M2;
      Afcn = @(x) A * x;
      Mfcn = @(x) M \ x;
      M1fcn = @(x) M1 \ x;
      M2fcn = @(x) M2 \ x;

  EXAMPLE 1: Simplest use of `pcg`

      x = pcg (A, b)

  EXAMPLE 2: `pcg` with a function which computes `A * x`

      x = pcg (Afcn, b)

  EXAMPLE 3: `pcg` with a preconditioner matrix M

      x = pcg (A, b, 1e-06, 100, M)

  EXAMPLE 4: `pcg` with a function as preconditioner

      x = pcg (Afcn, b, 1e-6, 100, Mfcn)

  EXAMPLE 5: `pcg` with preconditioner matrices M1 and M2

      x = pcg (A, b, 1e-6, 100, M1, M2)

  EXAMPLE 6: `pcg` with functions as preconditioners

      x = pcg (Afcn, b, 1e-6, 100, M1fcn, M2fcn)

  EXAMPLE 7: `pcg` with as input a function requiring an argument

        function y = Ap (A, x, p) # compute A^p * x
           y = x;
           for i = 1:p
             y = A * y;
           endfor
        endfunction
      Apfcn = @(x, p) Ap (A, x, p);
      x = pcg (Apfcn, b, [], [], [], [], [], 2);

  EXAMPLE 8: explicit example to show that `pcg` uses a split preconditioner

      M1 = ichol (A + 0.1 * eye (n)); # factorization of A perturbed
      M2 = M1';
      M = M1 * M2;

      ## reference solution computed by pcg after two iterations
      [x_ref, fl] = pcg (A, b, [], 2, M)

      ## split preconditioning
      [y, fl] = pcg ((M1 \ A) / M2, M1 \ b, [], 2)
      x = M2 \ y # compare x and x_ref

  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) Revised version of this book is available online at https://www-users.cs.umn.edu/~saad/books.html

  See also: sparse, pcr, gmres, bicg, bicgstab, cgs.

`x =` pcr `(A, b, tol, maxit, m, x0, …)` ¶
`[x, flag, relres, iter, resvec] =` pcr `(…)` ¶

  Solve 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 a `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 = sparse (diag (1:n));
      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 = apply_a (x)
        y = [1:10]' .* x;
      endfunction

      x = pcr ("apply_a", 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 = 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);

  EXAMPLE 4: Finally, a preconditioner which depends on a parameter k.

      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)

  Reference:

  W. Hackbusch, Iterative Solution of Large Sparse Systems of Equations, section 9.5.4, Springer, 1994.

  See also: sparse, pcg.

The speed with which an iterative solver converges to a solution can be accelerated with the use of a pre-conditioning matrix M. In this case the linear equation `M^-1 * x = M^-1 * A \ b` is solved instead. Typical pre-conditioning matrices are partial factorizations of the original matrix.

`L =` ichol `(A)` ¶
`L =` ichol `(A, opts)` ¶

  Compute the incomplete Cholesky factorization of the sparse square matrix A.

  By default, `ichol` uses only the lower triangle of A and produces a lower triangular factor L such that `L*L'` approximates A.

  The factor given by this routine may be useful as a preconditioner for a system of linear equations being solved by iterative methods such as PCG (Preconditioned Conjugate Gradient).

  The factorization may be modified by passing options in a structure opts. The option name is a field of the structure and the setting is the value of field. Names and specifiers are case sensitive.

  type

    Type of factorization.

    `"nofill"` (default)

      Incomplete Cholesky factorization with no fill-in (IC(0)).

    `"ict"`

      Incomplete Cholesky factorization with threshold dropping (ICT).

  diagcomp

    A non-negative scalar alpha for incomplete Cholesky factorization of `A + alpha * diag (diag (A))` instead of A. This can be useful when A is not positive definite. The default value is 0.

  droptol

    A non-negative scalar specifying the drop tolerance for factorization if performing ICT. The default value is 0 which produces the complete Cholesky factorization.

    Non-diagonal entries of L are set to 0 unless

    `abs (L(i,j)) >= droptol * norm (A(j:end, j), 1)`.

  michol

    Modified incomplete Cholesky factorization:

    `"off"` (default)

      Row and column sums are not necessarily preserved.

    `"on"`

      The diagonal of L is modified so that row (and column) sums are preserved even when elements have been dropped during the factorization. The relationship preserved is: `A * e = L * L' * e`, where e is a vector of ones.

  shape
    `"lower"` (default)

      Use only the lower triangle of A and return a lower triangular factor L such that `L*L'` approximates A.

    `"upper"`

      Use only the upper triangle of A and return an upper triangular factor U such that `U'*U` approximates A.

  EXAMPLES

  The following problem demonstrates how to factorize a sample symmetric positive definite matrix with the full Cholesky decomposition and with the incomplete one.

      A = [ 0.37, -0.05,  -0.05,  -0.07;
           -0.05,  0.116,  0.0,   -0.05;
           -0.05,  0.0,    0.116, -0.05;
           -0.07, -0.05,  -0.05,   0.202];
      A = sparse (A);
      nnz (tril (A))
      ans =  9
      L = chol (A, "lower");
      nnz (L)
      ans =  10
      norm (A - L * L', "fro") / norm (A, "fro")
      ans =  1.1993e-16
      opts.type = "nofill";
      L = ichol (A, opts);
      nnz (L)
      ans =  9
      norm (A - L * L', "fro") / norm (A, "fro")
      ans =  0.019736

  Another example for decomposition is a finite difference matrix used to solve a boundary value problem on the unit square.

      nx = 400; ny = 200;
      hx = 1 / (nx + 1); hy = 1 / (ny + 1);
      Dxx = spdiags ([ones(nx, 1), -2*ones(nx, 1), ones(nx, 1)],
                     [-1 0 1 ], nx, nx) / (hx ^ 2);
      Dyy = spdiags ([ones(ny, 1), -2*ones(ny, 1), ones(ny, 1)],
                     [-1 0 1 ], ny, ny) / (hy ^ 2);
      A = -kron (Dxx, speye (ny)) - kron (speye (nx), Dyy);
      nnz (tril (A))
      ans =  239400
      opts.type = "nofill";
      L = ichol (A, opts);
      nnz (tril (A))
      ans =  239400
      norm (A - L * L', "fro") / norm (A, "fro")
      ans =  0.062327

  References for implemented algorithms:

  [1] Y. Saad, "Preconditioning Techniques", Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996.

  [2] M. Jones, P. Plassmann, An Improved Incomplete Cholesky Factorization, 1992.

  See also: chol, ilu, pcg.

`LUA =` ilu `(A)` ¶
`LUA =` ilu `(A, opts)` ¶
`[L, U] =` ilu `(…)` ¶
`[L, U, P] =` ilu `(…)` ¶

  Compute the incomplete LU factorization of the sparse square matrix A.

  `ilu` returns a unit lower triangular matrix L, an upper triangular matrix U, and optionally a permutation matrix P, such that `L*U` approximates `P*A`.

  The factors given by this routine may be useful as preconditioners for a system of linear equations being solved by iterative methods such as BICG (BiConjugate Gradients) or GMRES (Generalized Minimum Residual Method).

  The factorization may be modified by passing options in a structure opts. The option name is a field of the structure and the setting is the value of field. Names and specifiers are case sensitive.

  `type`

    Type of factorization.

    `"nofill"` (default)

      ILU factorization with no fill-in (ILU(0)).

      Additional supported options: `milu`.

    `"crout"`

      Crout version of ILU factorization (ILUC).

      Additional supported options: `milu`, `droptol`.

    `"ilutp"`

      ILU factorization with threshold and pivoting.

      Additional supported options: `milu`, `droptol`, `udiag`, `thresh`.

  `droptol`

    A non-negative scalar specifying the drop tolerance for factorization. The default value is 0 which produces the complete LU factorization.

    Non-diagonal entries of U are set to 0 unless

    `abs (U(i,j)) >= droptol * norm (A(:,j))`.

    Non-diagonal entries of L are set to 0 unless

    `abs (L(i,j)) >= droptol * norm (A(:,j))/U(j,j)`.

  `milu`

    Modified incomplete LU factorization:

    `"row"`

      Row-sum modified incomplete LU factorization. The factorization preserves row sums: `A * e = L * U * e`, where e is a vector of ones.

    `"col"`

      Column-sum modified incomplete LU factorization. The factorization preserves column sums: `e' * A = e' * L * U`.

    `"off"` (default)

      Row and column sums are not necessarily preserved.

  `udiag`

    If true, any zeros on the diagonal of the upper triangular factor are replaced by the local drop tolerance `droptol * norm (A(:,j))/U(j,j)`. The default is false.

  `thresh`

    Pivot threshold for factorization. It can range between 0 (diagonal pivoting) and 1 (default), where the maximum magnitude entry in the column is chosen to be the pivot.

  If `ilu` is called with just one output, the returned matrix is `L + U - speye (size (A))`, where L is unit lower triangular and U is upper triangular.

  With two outputs, `ilu` returns a unit lower triangular matrix L and an upper triangular matrix U. For opts.type == `"ilutp"`, one of the factors is permuted based on the value of opts.milu. When opts.milu == `"row"`, U is a column permuted upper triangular factor. Otherwise, L is a row-permuted unit lower triangular factor.

  If there are three named outputs and opts.milu != `"row"`, P is returned such that L and U are incomplete factors of `P*A`. When opts.milu == `"row"`, P is returned such that L and U are incomplete factors of `A*P`.

  EXAMPLES

      A = gallery ("neumann", 1600) + speye (1600);
      opts.type = "nofill";
      nnz (A)
      ans = 7840

      nnz (lu (A))
      ans = 126478

      nnz (ilu (A, opts))
      ans = 7840

  This shows that A has 7,840 nonzeros, the complete LU factorization has 126,478 nonzeros, and the incomplete LU factorization, with 0 level of fill-in, has 7,840 nonzeros, the same amount as A. Taken from: https://www.mathworks.com/help/matlab/ref/ilu.html

      A = gallery ("wathen", 10, 10);
      b = sum (A, 2);
      tol = 1e-8;
      maxit = 50;
      opts.type = "crout";
      opts.droptol = 1e-4;
      [L, U] = ilu (A, opts);
      x = bicg (A, b, tol, maxit, L, U);
      norm (A * x - b, inf)

  This example uses ILU as preconditioner for a random FEM-Matrix, which has a large condition number. Without L and U BICG would not converge.

  See also: lu, ichol, bicg, gmres.