MATLAB Function Reference    

Conjugate Gradients Squared method



x = cgs(A,b) attempts to solve the system of linear equations A*x = b for x. The n-by-n coefficient matrix A must be square and should be large and sparse. The column vector b must have length n. A can be a function afun such that afun(x) returns A*x.

If cgs converges, a message to that effect is displayed. If cgs fails to converge after the maximum number of iterations or halts for any reason, a warning message is printed displaying the relative residual norm(b-A*x)/norm(b) and the iteration number at which the method stopped or failed.

cgs(A,b,tol) specifies the tolerance of the method, tol. If tol is [], then cgs uses the default, 1e-6.

cgs(A,b,tol,maxit) specifies the maximum number of iterations, maxit. If maxit is [] then cgs uses the default, min(n,20).

cgs(A,b,tol,maxit,M) and cgs(A,b,tol,maxit,M1,M2) use the preconditioner M or M = M1*M2 and effectively solve the system inv(M)*A*x = inv(M)*b for x. If M is [] then cgs applies no preconditioner. M can be a function that returns M\x.

cgs(A,b,tol,maxit,M1,M2,x0) specifies the initial guess x0. If x0 is [], then cgs uses the default, an all-zero vector.

cgs(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) passes parameters p1,p2,... to functions afun(x,p1,p2,...), m1fun(x,p1,p2,...), and m2fun(x,p1,p2,...)

[x,flag] = cgs(A,b,...) returns a solution x and a flag that describes the convergence of cgs.

cgs converged to the desired tolerance tol within maxit iterations.
cgs iterated maxit times but did not converge.
Preconditioner M was ill-conditioned.
cgs stagnated. (Two consecutive iterates were the same.)
One of the scalar quantities calculated during cgs became too small or too large to continue computing.

Whenever flag is not 0, the solution x returned is that with minimal norm residual computed over all the iterations. No messages are displayed if the flag output is specified.

[x,flag,relres] = cgs(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, then relres <= tol.

[x,flag,relres,iter] = cgs(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit.

[x,flag,relres,iter,resvec] = cgs(A,b,...) also returns a vector of the residual norms at each iteration, including norm(b-A*x0).


Example 1.

Alternatively, use this matrix-vector product function

and this preconditioner backsolve function

as inputs to cgs.

Note that both afun and mfun must accept cgs's extra input n=21.

Example 2.

flag is 1 because cgs does not converge to the default tolerance 1e-6 within the default 20 iterations.

flag1 is 2 because the upper triangular U1 has a zero on its diagonal, and cgs fails in the first iteration when it tries to solve a system such as U1*y = r for y with backslash.

flag2 is 0 because cgs converges to the tolerance of 6.344e-16 (the value of relres2) at the fifth iteration (the value of iter2) when preconditioned by the incomplete LU factorization with a drop tolerance of 1e-6. resvec2(1) = norm(b) and resvec2(6) = norm(b-A*x2). You can follow the progress of cgs by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0) with

See Also

bicg, bicgstab, gmres, lsqr, luinc, minres, pcg, qmr, symmlq

@ (function handle), \ (backslash)


[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]  Sonneveld, Peter, "CGS: A fast Lanczos-type solver for nonsymmetric linear systems", SIAM J. Sci. Stat. Comput., January 1989, Vol. 10, No. 1, pp. 36-52.

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