Iterative Methods for Linear Systems

One of the most important and common applications of numerical linear algebra is the solution of linear systems that can be expressed in the form A*x = b. When A is a large sparse matrix, you can solve the linear system using iterative methods, which enable you to trade off between the run time of the calculation and the precision of the solution. This topic describes the iterative methods available in MATLAB^® to solve the equation A*x = b.

Direct vs. Iterative Methods

There are two types of methods for solving linear equations A*x = b:

• Direct methods are variants of Gaussian elimination. These methods use the individual matrix elements directly, through matrix operations such as LU, QR, or Cholesky factorization. You can use direct methods to solve linear equations with a high level of precision, but these methods can be slow when operating on large sparse matrices. The speed of solving a linear system with a direct method strongly depends on the density and fill pattern of the coefficient matrix.

  For example, this code solves a small linear system.

  A = magic(5);
  b = sum(A,2);
  x = A\b;
  norm(A*x-b)

  ans =
  
     1.4211e-14

  MATLAB implements direct methods through the matrix division operators / and \, as well as functions such as decomposition, lsqminnorm, and linsolve.

• Iterative methods produce an approximate solution to the linear system after a finite number of steps. These methods are useful for large systems of equations where it is reasonable to trade off precision for a shorter run time. Iterative methods use the coefficient matrix only indirectly, through a matrix-vector product or an abstract linear operator. Iterative methods can be used with any matrix, but they are typically applied to large sparse matrices for which direct solves are slow. The speed of solving a linear system with an indirect method does not depend as strongly on the fill pattern of the coefficient matrix as a direct method. However, using an iterative method typically requires tuning parameters for each specific problem.

  For example, this code solves a large sparse linear system that has a symmetric positive definite coefficient matrix.

  A = delsq(numgrid('L',400));
  b = ones(size(A,1),1);
  x = pcg(A,b,[],1000);
  norm(b-A*x)

  pcg converged at iteration 796 to a solution with relative residual 9.9e-07.
  
  ans =
  
     3.4285e-04

  MATLAB implements a variety of iterative methods that have different strengths and weaknesses depending on the properties of the coefficient matrix A.

Direct methods are usually faster and more generally applicable than indirect methods if there is enough storage available to carry them out. Generally, you should attempt to use x = A\b first. If the direct solve is too slow, then you can try using iterative methods.

Generic Iterative Algorithm

Most iterative algorithms that solve linear equations follow a similar process:

1. Start with an initial guess for the solution vector x0. (This is usually a vector of zeros unless you specify a better guess.)

2. Compute the residual norm res = norm(b-A*x0).

3. Compare the residual against the specified tolerance. If res <= tol, end the computation and return the computed answer for x0.

4. Apply A*x0 and update the magnitude and direction of the vector x0 based on the value of the residual and other calculated quantities. This is the step where most computation is done.

5. Repeat Steps 2 through 4 until the value of x0 is good enough to satisfy the tolerance.

Iterative methods differ in how they update the magnitude and direction of x0 in Step 4, and some have slightly different convergence criteria in Steps 2 and 3, but this captures the basic process that all iterative solvers follow.

Summary of Iterative Methods

MATLAB has several functions that implement iterative methods for systems of linear equations. These methods are designed to solve Ax = b or minimize the norm ||b – Ax||. Several of these methods have similarities and are based on the same underlying algorithms, but each algorithm has benefits in certain situations [1], [2].

Description	Notes
pcg (preconditioned conjugate gradients)	Coefficient matrix must be symmetric positive definite. Most effective solver for symmetric positive definite systems since storage for only a limited number of vectors is required.
lsqr (least squares)	The only solver available for rectangular systems. Analytically equivalent to the method of conjugate gradients (PCG) applied to the normal equations (A'*A)*x = A'*b.
minres (minimum residual)	Coefficient matrix must be symmetric but does not need to be positive definite. Each iteration minimizes the residual error in the 2-norm, so the algorithm is guaranteed to make progress from step to step. Does not suffer from breakdowns (when an algorithm becomes unable to make progress toward a solution and halts).
symmlq (symmetric LQ)	Coefficient matrix must be symmetric but does not need to be positive definite. Solves a projected system and keeps the residual orthogonal to all previous ones. Does not suffer from breakdowns (when an algorithm becomes unable to make progress toward a solution and halts).
bicg (biconjugate gradient)	Coefficient matrix must be square. bicg is computationally cheap, but convergence is irregular and unreliable. bicg is historically important because many other iterative algorithms were developed as improvements on it.
bicgstab (biconjugate gradient stabilized)	Coefficient matrix must be square. Uses BiCG steps alternating with GMRES(1) steps for additional stability.
bicgstabl (biconjugate gradient stabilized (l))	Coefficient matrix must be square. Uses BiCG steps alternating with GMRES(2) steps for additional stability.
cgs (conjugate gradient squared)	Coefficient matrix must be square. Requires the same number of operations per iteration as bicg, but avoids using the transpose by working with a squared residual.
gmres (generalized minimum residual)	Coefficient matrix must be square. One of the most dependable algorithms, since the residual norm is minimized in each iteration. Work and required storage rise linearly with iteration count. Choosing an appropriate restart value is essential to avoid unnecessary work and storage.
qmr (quasi-minimal residual)	Coefficient matrix must be square. Overhead per iteration is slightly more than bicg, but this provides more stability.
tfqmr (transpose-free quasi-minimal residual)	Coefficient matrix must be square. Best solver to try for symmetric indefinite systems when memory is limited.

Choosing an Iterative Solver

This flow chart of iterative solvers in MATLAB gives a rough idea of the situations where each solver is useful. You can generally use gmres for almost all square, nonsymmetric problems. There are some cases where the biconjugate gradients algorithms (bicg, bicgstab, cgs, and so on) are more efficient than gmres, but their unpredictable convergence behavior often makes gmres a better initial choice.

Preconditioners

The convergence rate of iterative methods is dependent on the spectrum (eigenvalues) of the coefficient matrix. Therefore, you can improve the convergence and stability of most iterative methods by transforming the linear system to have a more favorable spectrum (clustered eigenvalues or a condition number near 1). This transformation is performed by applying a second matrix, called a preconditioner, to the system. This process transforms the linear system

$$Ax=b$$

into an equivalent system

$$\tilde{A}\tilde{x}=\tilde{b}.$$

The ideal preconditioner transforms the coefficient matrix A into an identity matrix, since any iterative method will converge in one iteration with such a preconditioner. In practice, finding a good preconditioner requires tradeoffs. The transformation is performed in one of three ways: left preconditioning, right preconditioning, or split preconditioning.

The first case is called left preconditioning since the preconditioner matrix M appears on the left of A:

$$\left(M^{-1}A\right)x=\left(M^{-1}b\right).$$

These iterative solvers use left preconditioning:

• bicg

• gmres

• qmr

In right preconditioning, M appears on the right of A:

$$\left(A{M}^{-1}\right)\left(Mx\right)=b.$$

These iterative solvers use right preconditioning:

• lsqr

• bicgstab

• bicgstabl

• cgs

• tfqmr

Finally, for symmetric coefficient matrices A, split preconditioning ensures that the transformed system is still symmetric. The preconditioner $$M=H{H}^T$$ gets split and the factors appear on different sides of A:

$$\left(H^{-1}A{H}^{-T}\right)H^{T}x=\left(H^{-1}b\right)$$

The solver algorithm for split preconditioned systems is based on the above equation, but in practice there is no need to compute H. The solver algorithm multiplies and solves with M directly.

These iterative solvers use split preconditioning:

• pcg

• minres

• symmlq

In all cases, the preconditioner M is chosen to accelerate convergence of the iterative method. When the residual error of an iterative solution stagnates or makes little progress between iterations, it often means you need to generate a preconditioner matrix to incorporate into the problem.

The iterative solvers in MATLAB allow you to specify a single preconditioner matrix M, or two preconditioner matrix factors such that M = M₁M₂. This makes it easy to specify a preconditioner in its factorized form, such as M = LU. Note that in the split preconditioned case, where M = HH^T also holds, there is not a relation between the M1 and M2 inputs and the H factors.

In some cases, preconditioners occur naturally in the mathematical model of a given problem. In the absence of natural preconditioners, you can use one of the incomplete factorizations in this table to generate a preconditioner matrix. Incomplete factorizations are essentially incomplete direct solves that are quick to calculate.

Function	Factorization	Description
ilu	$$\text{A}\approx \text{LU}$$	Incomplete LU factorization for square or rectangular matrices.
ichol	$$\text{A}\approx \text{L}{\text{L}}^{*}$$	Incomplete Cholesky factorization for symmetric positive definite matrices.

See Incomplete Factorizations for more information about ilu and ichol.

Preconditioner Example

Consider the five-point finite difference approximation to Laplace's equation on a square, two-dimensional domain. The following commands use the preconditioned conjugate gradient (PCG) method with the preconditioner M = L*L', where L is the zero-fill incomplete Cholesky factor of A. For this system, pcg is unable to find a solution without specifying a preconditioner matrix.

A = delsq(numgrid('S',250));
b = ones(size(A,1),1);
tol = 1e-3;
maxit = 100;
L = ichol(A);
x = pcg(A,b,tol,maxit,L,L');

pcg converged at iteration 92 to a solution with relative residual 0.00076.

pcg requires 92 iterations to achieve the specified tolerance. However, using a different preconditioner can yield better results. For example, using ichol to construct a modified incomplete Cholesky allows pcg to meet the specified tolerance after only 39 iterations.

L = ichol(A,struct('type','nofill','michol','on'));
x = pcg(A,b,tol,maxit,L,L');

pcg converged at iteration 39 to a solution with relative residual 0.00098.

Equilibration and Reordering

For computationally tough problems, you might need a better preconditioner than the one generated by ilu or ichol directly. For example, you might want to generate a better quality preconditioner or minimize the amount of computation being done. In these cases, you can use equilibration to make the coefficient matrix more diagonally dominant (which can lead to a better quality preconditioner) and reordering to minimize the number of nonzeros in matrix factors (which can reduce memory requirements and may improve the efficiency of subsequent calculations).

If you use both equilibration and reordering to generate a preconditioner, the process is:

1. Use equilibrate on the coefficient matrix.

2. Reorder the equilibrated matrix using a sparse matrix reordering function, such as dissect or symrcm.

3. Generate the final preconditioner using ilu or ichol.

Here is an example that uses equilibration and reordering to generate a preconditioner for a sparse coefficient matrix.

1. Create the coefficient matrix A and a vector of ones b for the right-hand side of the linear equation. Calculate an estimation of the condition number for A.

   load west0479;
   A = west0479;
   b = ones(size(A,1),1);
   condest(A)

   ans =
   
      1.4244e+12

   Use equilibrate to improve the condition number of the coefficient matrix.

   [P,R,C] = equilibrate(A);
   Anew = R*P*A*C;
   bnew = R*P*b;
   condest(Anew)

   ans =
   
      5.1042e+04

2. Reorder the equilibrated matrix using dissect.

   q = dissect(Anew);
   Anew = Anew(q,q);
   bnew = bnew(q);

3. Generate a preconditioner using an incomplete LU factorization.

   [L,U] = ilu(Anew);

4. Solve the linear system with gmres using the preconditioner matrices, a tolerance of 1e-10, 50 maximum outer iterations, and 30 inner iterations.

   tol = 1e-10;
   maxit = 50;
   restart = 30;
   [xnew, flag, relres] = gmres(Anew,bnew,restart,tol,maxit,L,U);
   x(q) = xnew;
   x = C*x(:);

   Now, compare the relres relative residual returned by gmres (which includes the preconditioners) to the relative residual without the preconditioners resnew and the relative residual without equilibration res. The results show that even though the linear systems are all equivalent, the different methods apply different weights to each element, and this can significantly affect the value of the residual.

   relres
   resnew = norm(Anew*xnew - bnew) / norm(bnew)
   res = norm(A*x - b) / norm(b)

   relres =
      8.7537e-11
   resnew =
      3.6805e-08
   res =
      5.1415e-04

Using Linear Operators Instead of Matrices

The iterative solvers in MATLAB do not require that you provide a numeric matrix for A. Since the calculations performed by the solvers use the result of the matrix-vector multiplication A*x or A'*x, you can instead provide a function that calculates the result of those linear operations. A function that calculates these quantities is often called a linear operator.

In addition to using a linear operator instead of a coefficient matrix A, you can also use a linear operator instead of a matrix for the preconditioner M. In that case, the function needs to calculate M\x or M'\x, as indicated on the reference page for the solver.

Using linear operators enables you to exploit patterns in A or M to calculate the value of the linear operations more efficiently than if the solver used the matrix explicitly to carry out the full matrix-vector multiplication. It also means you do not need the memory to store the coefficient or preconditioner matrices, since the linear operator typically calculates the result of the matrix-vector multiplication without forming the matrix at all.

For example, consider the coefficient matrix

A = [2 -1  0  0  0  0;
    -1  2 -1  0  0  0;
     0 -1  2 -1  0  0;
     0  0 -1  2 -1  0;
     0  0  0 -1  2 -1;
     0  0  0  0 -1  2];

When A multiplies a vector, most of the elements in the resulting vector are zeros. The nonzero elements in the result correspond with the nonzero tridiagonal elements of A. So, for a given vector x, the linear operator function simply needs to add together three vectors to calculate the value of A*x:

function y = linearOperatorA(x)
y = -1*[0; x(1:end-1)] ...
  + 2*x ...
  + -1*[x(2:end); 0];
end

Most iterative solvers require the linear operator function for A to return the value of A*x. Likewise, for the preconditioner matrix M, the function generally must calculate M\x. For the solvers lsqr, qmr, and bicg, the linear operator functions must also return the value for A'*x or M'\x when requested. See the iterative solver reference pages for examples and descriptions of linear operator functions.

References

[1] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.

[2] Saad, Yousef, Iterative Methods for Sparse Linear Equations. PWS Publishing Company, 1996.

Related Topics

• Systems of Linear Equations