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gmres

Solve system of linear equations — generalized minimum residual method

Description

example

x = gmres(A,b) attempts to solve the system of linear equations A*x = b for x using the Generalized Minimum Residual Method. When the attempt is successful, gmres displays a message to confirm convergence. If gmres fails to converge after the maximum number of iterations or halts for any reason, it displays a diagnostic message that includes the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped. For this syntax, gmres does not restart; the maximum number of iterations is min(size(A,1),10).

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x = gmres(A,b,restart) restarts the method every restart inner iterations. The maximum number of outer iterations is outer = min(size(A,1)/restart,10). The maximum number of total iterations is restart*outer, since gmres performs restart inner iterations for each outer iteration. If restart is size(A,1) or [], then gmres does not restart and the maximum number of total iterations is min(size(A,1),10).

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x = gmres(A,b,restart,tol) specifies a tolerance for the method. The default tolerance is 1e-6.

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x = gmres(A,b,restart,tol,maxit) specifies the maximum number of outer iterations such that the total number of iterations does not exceed restart*maxit. If maxit is [] then gmres uses the default, min(size(A,1)/restart,10). If restart is size(A,1) or [], then the maximum number of total iterations is maxit (instead of restart*maxit). gmres displays a diagnostic message if it fails to converge within the maximum number of total iterations.

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x = gmres(A,b,restart,tol,maxit,M) specifies a preconditioner matrix M and computes x by effectively solving the system M1Ax=M1b. Using a preconditioner matrix can improve the numerical properties of the problem and the efficiency of the calculation.

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x = gmres(A,b,restart,tol,maxit,M1,M2) specifies factors of the preconditioner matrix M such that M = M1*M2.

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x = gmres(A,b,restart,tol,maxit,M1,M2,x0) specifies an initial guess for the solution vector x. The default is a vector of zeros.

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[x,flag] = gmres(___) returns a flag that specifies whether the algorithm successfully converged. When flag = 0, convergence was successful. You can use this output syntax with any of the previous input argument combinations. When you specify the flag output, gmres does not display any diagnostic messages.

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[x,flag,relres] = gmres(___) also returns the relative residual norm(M\(b-A*x))/norm(M\b), which includes the preconditioner matrix M. If flag is 0, then relres <= tol.

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[x,flag,relres,iter] = gmres(___) also returns the inner and outer iteration numbers at which x was computed as a vector [outer inner]. The outer iteration number lies in the range 0 <= iter(1) <= maxit and the inner iteration number is in the range 0 <= iter(2) <= restart.

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[x,flag,relres,iter,resvec] = gmres(___) also returns a vector of the residual norms at each inner iteration, including the first residual norm(M\(b-A*x0)). These are the residual norms for the preconditioned system.

Examples

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Solve a square linear system using gmres with default settings, and then adjust the tolerance and number of iterations used in the solution process.

Create a random sparse matrix A with 50% density and nonzeros on the main diagonal. Also create a random vector b for the right-hand side of Ax=b.

rng default
A = sprandn(400,400,0.5) + 12*speye(400);
b = rand(400,1);

Solve Ax=b using gmres. The output display includes the value of the relative residual error b-Axb.

x = gmres(A,b);
gmres stopped at iteration 10 without converging to the desired tolerance 1e-06
because the maximum number of iterations was reached.
The iterate returned (number 10) has relative residual 0.35.

By default gmres uses 10 iterations and a tolerance of 1e-6, and the algorithm is unable to converge in those 10 iterations for this matrix. Since the residual is still large, it is a good indicator that more iterations (or a preconditioner matrix) are needed. You also can use a larger tolerance to make it easier for the algorithm to converge.

Solve the system again using a tolerance of 1e-4 and 100 iterations.

tol = 1e-4;
maxit = 100;
x = gmres(A,b,[],tol,maxit);
gmres stopped at iteration 100 without converging to the desired tolerance 0.0001
because the maximum number of iterations was reached.
The iterate returned (number 100) has relative residual 0.0045.

Even with a looser tolerance and more iterations the residual error does not improve enough for convergence. When an iterative algorithm stalls in this manner it is a good indication that a preconditioner matrix is needed. However, gmres also has an input that controls the number of inner iterations. By specifying a value for the inner iterations, gmres does more work per outer iteration.

Solve the system again using a restart value of 100 and a maxit value of 20. Rather than doing 100 iterations once, gmres performs 100 iterations between restarts and repeats this 20 times.

restart = 100;
maxit = 20;
x = gmres(A,b,restart,tol,maxit);
gmres(100) converged at outer iteration 2 (inner iteration 75) to a solution with relative residual 9.3e-05.

In this case specifying a large restart value for gmres enables it to converge to a solution within the allowed number of iterations. However, large restart values can consume a lot of memory when A is also large.

Examine the effect of using a preconditioner matrix with non-restarted gmres to solve a linear system.

Load west0479, a real 479-by-479 nonsymmetric sparse matrix.

load west0479
A = west0479;

Define b so that the true solution to Ax=b is a vector of all ones.

b = sum(A,2);

Set the tolerance and maximum number of iterations.

tol = 1e-12;
maxit = 20;

Use gmres to find a solution at the requested tolerance and number of iterations. Specify five outputs to return information about the solution process:

  • x is the computed solution to A*x = b.

  • fl0 is a flag indicating whether the algorithm converged.

  • rr0 is the relative residual of the computed answer x.

  • it0 is a two-element vector [outer inner] indicating the inner and outer iteration numbers when x was computed.

  • rv0 is a vector of the residual history for b-Ax.

[x,fl0,rr0,it0,rv0] = gmres(A,b,[],tol,maxit);
fl0
fl0 = 1
rr0
rr0 = 0.7603
it0
it0 = 1×2

     1    20

fl0 is 1 because gmres does not converge to the requested tolerance 1e-12 within the requested 20 iterations. The best approximate solution that gmres returns is the last one (as indicated by it0(2) = 20). MATLAB stores the residual history in rv0.

To aid with the slow convergence, you can specify a preconditioner matrix. Since A is nonsymmetric, use ilu to generate the preconditioner M=LU. Specify a drop tolerance to ignore nondiagonal entries with values smaller than 1e-6. Solve the preconditioned system M-1Ax=M-1b by specifying L and U as inputs to gmres. Note that when you specify a preconditioner, gmres calculates the residual norm of the preconditioned system for the outputs rr1 and rv1.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6));
[x1,fl1,rr1,it1,rv1] = gmres(A,b,[],tol,maxit,L,U);
fl1
fl1 = 0
rr1
rr1 = 6.9008e-14
it1
it1 = 1×2

     1     6

The use of an ilu preconditioner produces a relative residual less than the prescribed tolerance of 1e-12 at the sixth iteration. The first residual rv1(1) is norm(U\(L\b)), where M = L*U. The last residual rv1(end) is norm(U\(L\(b-A*x1))).

You can follow the progress of gmres by plotting the relative residuals at each iteration. Plot the residual history of each solution with a line for the specified tolerance.

semilogy(0:length(rv0)-1,rv0/norm(b),'-o')
hold on
semilogy(0:length(rv1)-1,rv1/norm(U\(L\b)),'-o')
yline(tol,'r--');
legend('No preconditioner','ILU preconditioner','Tolerance','Location','East')
xlabel('Iteration number')
ylabel('Relative residual')

Using a preconditioner with restarted gmres.

Load west0479, a real 479-by-479 nonsymmetric sparse matrix.

load west0479
A = west0479;

Define b so that the true solution to Ax=b is a vector of all ones.

b = sum(A,2);

Construct an incomplete LU preconditioner with a drop tolerance of 1e-6.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-6));

The benefit to using restarted gmres is to limit the amount of memory required to execute the method. Without restart, gmres requires maxit vectors of storage to keep the basis of the Krylov subspace. Also, gmres must orthogonalize against all of the previous vectors at each step. Restarting limits the amount of workspace used and the amount of work done per outer iteration.

Execute gmres(3), gmres(4), and gmres(5) using the incomplete LU factors as preconditioners. Use a tolerance of 1e-12 and a maximum of 20 outer iterations.

tol = 1e-12;
maxit = 20;
[x3,fl3,rr3,it3,rv3] = gmres(A,b,3,tol,maxit,L,U);
[x4,fl4,rr4,it4,rv4] = gmres(A,b,4,tol,maxit,L,U);
[x5,fl5,rr5,it5,rv5] = gmres(A,b,5,tol,maxit,L,U);
fl3
fl3 = 0
fl4
fl4 = 0
fl5
fl5 = 0

fl3, fl4, and fl5 are all 0 because in each case restarted gmres drives the relative residual to less than the prescribed tolerance of 1e-12.

The following plot shows the convergence history of each restarted gmres method. gmres(3) converges at outer iteration 5, inner iteration 3 (it3 = [5, 3]) which would be the same as outer iteration 6, inner iteration 0, hence the marking of 6 on the final tick mark.

semilogy(1:1/3:6,rv3/norm(U\(L\b)),'-o');
h1 = gca;
h1.XTick = (1:6);
title('gmres(N) for N = 3, 4, 5')
xlabel('Outer iteration number');
ylabel('Relative residual');
hold on
semilogy(1:1/4:3,rv4/norm(U\(L\b)),'-o');
semilogy(1:1/5:2.8,rv5/norm(U\(L\b)),'-o');
yline(tol,'r--');
hold off
legend('gmres(3)','gmres(4)','gmres(5)','Tolerance')
grid on

In general the larger the number of inner iterations, the more work gmres does per outer iteration and the faster it can converge.

Examine the effect of supplying gmres with an initial guess of the solution.

Create a tridiagonal sparse matrix. Use the sum of each row as the vector for the right-hand side of Ax=b so that the expected solution for x is a vector of ones.

n = 900;
e = ones(n,1);
A = spdiags([e 2*e e],-1:1,n,n);
b = sum(A,2);

Use gmres to solve Ax=b twice: one time with the default initial guess, and one time with a good initial guess of the solution. Use 200 iterations and the default tolerance for both solutions. Specify the initial guess in the second solution as a vector with all elements equal to 0.99.

maxit = 200;
x1 = gmres(A,b,[],[],maxit);
gmres converged at iteration 27 to a solution with relative residual 9.5e-07.
x0 = 0.99*e;
x2 = gmres(A,b,[],[],maxit,[],[],x0);
gmres converged at iteration 7 to a solution with relative residual 6.7e-07.

In this case supplying an initial guess enables gmres to converge more quickly.

Returning Intermediate Results

You also can use the initial guess to get intermediate results by calling gmres in a for-loop. Each call to the solver performs a few iterations and stores the calculated solution. Then you use that solution as the initial vector for the next batch of iterations.

For example, this code performs 100 iterations four times and stores the solution vector after each pass in the for-loop:

x0 = zeros(size(A,2),1);
tol = 1e-8;
maxit = 100;
for k = 1:4
    [x,flag,relres] = gmres(A,b,[],tol,maxit,[],[],x0);
    X(:,k) = x;
    R(k) = relres;
    x0 = x;
end

X(:,k) is the solution vector computed at iteration k of the for-loop, and R(k) is the relative residual of that solution.

Solve a linear system by providing gmres with a function handle that computes A*x in place of the coefficient matrix A.

One of the Wilkinson test matrices generated by gallery is a 21-by-21 tridiagonal matrix. Preview the matrix.

A = gallery('wilk',21)
A = 21×21

    10     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     1     9     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     1     8     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     1     7     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     1     6     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     1     5     1     0     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     1     4     1     0     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     1     3     1     0     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     1     2     1     0     0     0     0     0     0     0     0     0     0     0
     0     0     0     0     0     0     0     0     1     1     1     0     0     0     0     0     0     0     0     0     0
      ⋮

The Wilkinson matrix has a special structure, so you can represent the operation A*x with a function handle. 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. Moreover, only the main diagonal has nonzeros that are not equal to 1.

The expression Ax becomes:

Ax=[1010001910001810017100161001510014100130001000110][x1x2x3x4x5x21]=[10x1+x2x1+9x2+x3x2+8x3+x4x19+9x20+x21x20+10x21].

The resulting vector can be written as the sum of three vectors:

Ax=[0+10x1+x2x1+9x2+x3x2+8x3+x4x19+9x20+x21x20+10x21+0]=[0x1x20]+[10x19x210x21]+[x2x210].

In MATLAB®, write a function that creates these vectors and adds them together, thus giving the value of A*x:

function y = afun(x)
y = [0; x(1:20)] + ...
    [(10:-1:0)'; (1:10)'].*x + ...
    [x(2:21); 0];
end

(This function is saved as a local function at the end of the example.)

Now, solve the linear system Ax=b by providing gmres with the function handle that calculates A*x. Use a tolerance of 1e-12, 15 outer iterations, and 10 inner iterations before restart.

b = ones(21,1);
tol = 1e-12;  
maxit = 15;
restart = 10;
x1 = gmres(@afun,b,restart,tol,maxit)
gmres(10) converged at outer iteration 5 (inner iteration 10) to a solution with relative residual 5.3e-13.
x1 = 21×1

    0.0910
    0.0899
    0.0999
    0.1109
    0.1241
    0.1443
    0.1544
    0.2383
    0.1309
    0.5000
      ⋮

Check that afun(x1) produces a vector of ones.

afun(x1)
ans = 21×1

    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
    1.0000
      ⋮

Local Functions

function y = afun(x)
y = [0; x(1:20)] + ...
    [(10:-1:0)'; (1:10)'].*x + ...
    [x(2:21); 0];
end

Input Arguments

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Coefficient matrix, specified as a square matrix or function handle. This matrix is the coefficient matrix in the linear system A*x = b. Generally, A is a large sparse matrix or a function handle that returns the product of a large sparse matrix and column vector.

Specifying A as a Function Handle

You can optionally specify the coefficient matrix as a function handle instead of a matrix. The function handle returns matrix-vector products instead of forming the entire coefficient matrix, making the calculation more efficient.

To use a function handle, use the function signature function y = afun(x). Parameterizing Functions explains how to provide additional parameters to the function afun, if necessary. The function call afun(x) must return the value of A*x.

Data Types: double | function_handle
Complex Number Support: Yes

Right-hand side of linear equation, specified as a column vector. The length of b must be equal to size(A,1).

Data Types: double
Complex Number Support: Yes

Number of inner iterations before restart, specified as a scalar integer. Use this input along with the maxit input to control the maximum number of iterations, restart*maxit. If restart is [] or size(A,1), then gmres does not restart and the total number of iterations is maxit.

A large restart value typically leads to better convergence behavior, but also has higher time and memory requirements.

Data Types: double

Method tolerance, specified as a positive scalar. Use this input to trade-off accuracy and runtime in the calculation. gmres must meet the tolerance within the number of allowed iterations to be successful. A smaller value of tol means the answer must be more precise for the calculation to be successful.

Data Types: double

Maximum number of outer iterations, specified as a positive scalar integer. Increase the value of maxit to allow more iterations for gmres to meet the tolerance tol. Generally, the smaller the value of tol, the more iterations are required to successfully complete the calculation.

If the restart input is also specified, then the total number of iterations is restart*maxit. Otherwise, the total number of iterations is maxit.

The default value of maxit depends on whether restart is specified:

  • If restart is unspecified, or specified as [] or size(A,1), then the default value of maxit is min(size(A,1),10).

  • If restart is specified as a value in the range 1 <= restart < size(A,1), then the default value of maxit is min(ceil(size(A,1)/restart),10).

Data Types: double

Preconditioner matrices, specified as separate arguments of matrices or function handles. You can specify a preconditioner matrix M or its matrix factors M = M1*M2 to improve the numerical aspects of the linear system and make it easier for gmres to converge quickly. You can use the incomplete matrix factorization functions ilu and ichol to generate preconditioner matrices. You also can use equilibrate prior to factorization to improve the condition number of the coefficient matrix. For more information on preconditioners, see Iterative Methods for Linear Systems.

gmres treats unspecified preconditioners as identity matrices.

Specifying M as a Function Handle

You can optionally specify any of M, M1, or M2 as function handles instead of matrices. The function handle performs matrix-vector operations instead of forming the entire preconditioner matrix, making the calculation more efficient.

To use a function handle, use the function signature function y = mfun(x). Parameterizing Functions explains how to provide additional parameters to the function mfun, if necessary. The function call mfun(x) must return the value of M\x or M2\(M1\x).

Data Types: double | function_handle
Complex Number Support: Yes

Initial guess, specified as a column vector with length equal to size(A,2). If you can provide gmres with a more reasonable initial guess x0 than the default vector of zeros, then it can save computation time and help the algorithm converge faster.

Data Types: double
Complex Number Support: Yes

Output Arguments

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Linear system solution, returned as a column vector. This output gives the approximate solution to the linear system A*x = b. If the calculation is successful (flag = 0), then relres is less than or equal to tol.

Whenever the calculation is not successful (flag ~= 0), the solution x returned by gmres is the one with minimal residual norm computed over all the iterations.

Convergence flag, returned as one of the scalar values in this table. The convergence flag indicates whether the calculation was successful and differentiates between several different forms of failure.

Flag Value

Convergence

0

Success — gmres converged to the desired tolerance tol within maxit iterations.

1

Failure — gmres iterated maxit iterations but did not converge.

2

Failure — The preconditioner matrix M or M = M1*M2 is ill conditioned.

3

Failure — gmres stagnated after two consecutive iterations were the same.

4

Failure — One of the scalar quantities calculated by the gmres algorithm became too small or too large to continue computing.

Relative residual error, returned as a scalar. The relative residual error relres = norm(M\(b-A*x))/norm(M\b) is an indication of how accurate the answer is. Note that gmres includes the preconditioner matrix M in the relative residual calculation, while most other iterative solvers do not. If the calculation converges to the tolerance tol within maxit iterations, then relres <= tol.

Data Types: double

Outer and inner iteration numbers, returned as a two-element vector [outer inner]. This output indicates the inner and outer iteration numbers at which the computed answer for x was calculated:

  • If restart is unspecified, or specified as [] or size(A,1), then outer = 1 and all iterations are considered to be inner iterations.

  • If restart is specified as a value in the range 1 <= restart < size(A,1), then the outer iteration number is in the range 0 <= outer <= maxit and the inner iteration number is in the range 0 <= inner <= restart.

Data Types: double

Residual error, returned as a vector. The residual error norm(M\(b-A*x)) reveals how close the algorithm is to converging for a given value of x. Note that gmres includes the preconditioner matrix M in the relative residual calculation, while most other iterative solvers do not. The number of elements in resvec is equal to the total number of iterations (if restart is used, this is at most restart*maxit). You can examine the contents of resvec to help decide whether to change the values of restart, tol, or maxit.

Data Types: double

More About

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Generalized Minimum Residual Method

The generalized minimum residual (GMRES) algorithm was developed to extend the minimum residual (MINRES) algorithm to unsymmetric matrices.

Like conjugate gradients (CG) methods, the GMRES algorithm computes orthogonal sequences, but GMRES needs to store all previous vectors in the sequences. This storage of previous vectors can consume a lot of memory if left unchecked. The "restarted" version of the algorithm controls storage of these sequences by periodically clearing the intermediate sequences and using the results as the initial value in another iteration.

Choosing an appropriate "restart" value is essential to good performance, but choosing such a value is mostly a matter of experience. If the number of iterations before restart is too small, the algorithm might be very slow to converge or fail to converge entirely. But if the restart value is too large, then the algorithm has increased storage requirements and might do unnecessary work [1].

Inner and Outer Iterations

Inner iterations are the iterations that gmres completes before restarting. You can specify the number of inner iterations with the restart argument.

Each time gmres restarts, the outer iteration number advances. You can specify the maximum number of outer iterations with the maxit argument. The default number of outer iterations is min(size(A,1)/restart,10).

For example, if you do not specify restart, then the maximum number of iterations is determined by the value of maxit, and gmres does not restart:

If the restart argument is not specified and the value of maxit is 4, then gmres performs a total of 4 iterations.

However, when you specify restart, the gmres function performs several inner iterations (specified by restart) for each outer iteration (specified by maxit). In this case, the maximum number of total iterations is restart*maxit:

If the restart argument is specified as 4, and the maxit argument is specified as 2, then gmres performs 4 inner iterations for each outer iteration for a total of 8 iterations.

Tips

  • Convergence of most iterative methods depends on the condition number of the coefficient matrix, cond(A). You can use equilibrate to improve the condition number of A, and on its own this makes it easier for most iterative solvers to converge. However, using equilibrate also leads to better quality preconditioner matrices when you subsequently factor the equilibrated matrix B = R*P*A*C.

  • You can use matrix reordering functions such as dissect and symrcm to permute the rows and columns of the coefficient matrix and minimize the number of nonzeros when the coefficient matrix is factored to generate a preconditioner. This can reduce the memory and time required to subsequently solve the preconditioned linear system.

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 and Martin H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., July 1986, Vol. 7, No. 3, pp. 856-869.

Extended Capabilities

Introduced before R2006a