MATLAB Function Reference

bicg

BiConjugate Gradients method

Syntax

• x = bicg(A,b)
  bicg(A,b,tol)
  bicg(A,b,tol,maxit)
  bicg(A,b,tol,maxit,M)
  bicg(A,b,tol,maxit,M1,M2)
  bicg(A,b,tol,maxit,M1,M2,x0)
  bicg(afun,b,tol,maxit,mfun1,mfun2,x0,p1,p2,...)
  [x,flag] = bicg(A,b,...)
  [x,flag,relres] = bicg(A,b,...)
  [x,flag,relres,iter] = bicg(A,b,...)
  [x,flag,relres,iter,resvec] = bicg(A,b,...)
  

Description

x = bicg(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 and afun(x,'transp') returns A'*x.

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

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

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

bicg(A,b,tol,maxit,M) and bicg(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 bicg applies no preconditioner. M can be a function mfun such that mfun(x) returns M\x and mfun(x,'transp') returns M'\x.

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

bicg(afun,b,tol,maxit,m1fun,m2fun,x0,p1,p2,...) passes parameters p1,p2,... to functions afun(x,p1,p2,...) and afun(x,p1,p2,...,'transp'), and similarly to the preconditioner functions m1fun and m2fun.

[x,flag] = bicg(A,b,...) also returns a convergence flag.

Flag	Convergence
0	bicg converged to the desired tolerance tol within maxit iterations.
1	bicg iterated maxit times but did not converge.
2	Preconditioner M was ill-conditioned.
3	bicg stagnated. (Two consecutive iterates were the same.)
4	One of the scalar quantities calculated during bicg 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] = bicg(A,b,...) also returns the relative residual norm(b-A*x)/norm(b). If flag is 0, relres <= tol.

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

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

Examples

Example 1.

• n = 100; 
  on = ones(n,1); 
  A = spdiags([-2*on 4*on -on],-1:1,n,n);
  b = sum(A,2); 
  tol = 1e-8; 
  maxit = 15;
  M1 = spdiags([on/(-2) on],-1:0,n,n); 
  M2 = spdiags([4*on -on],0:1,n,n);
  
  x = bicg(A,b,tol,maxit,M1,M2,[]);
  

displays this message

• bicg converged at iteration 9 to a solution with relative
  residual 5.3e-009
  

Alternatively, use this matrix-vector product function

• function y = afun(x,n,transp_flag)
  if (nargin > 2) & strcmp(transp_flag,'transp')
      y = 4 * x;
      y(1:n-1) = y(1:n-1) - 2 * x(2:n);
      y(2:n) = y(2:n) - x(1:n-1);
  else
      y = 4 * x;
      y(2:n) = y(2:n) - 2 * x(1:n-1);
      y(1:n-1) = y(1:n-1) - x(2:n);
  end
  

as input to bicg.

•    x1 = bicg(@afun,b,tol,maxit,M1,M2,[],n);
  

Example 2. This examples demonstrates the use of a preconditioner. Start with A = west0479, a real 479-by-479 sparse matrix, and define b so that the true solution is a vector of all ones.

• load west0479;
  A = west0479;
  b = sum(A,2);
  

You can accurately solve A*x = b using backslash since A is not so large.

• x = A \ b;
  norm(b-A*x) / norm(b)
  
  ans =
      8.3154e-017
  

Now try to solve A*x = b with bicg.

• [x,flag,relres,iter,resvec] = bicg(A,b)
  
  flag =
            1
  relres =
            1
  iter =
            0
  

The value of flag indicates that bicg iterated the default 20 times without converging. The value of iter shows that the method behaved so badly that the initial all-zero guess was better than all the subsequent iterates. The value of relres supports this: relres = norm(b-A*x)/norm(b) = norm(b)/norm(b) = 1. You can confirm that the unpreconditioned method oscillates rather wildly by plotting the relative residuals at each iteration.

• semilogy(0:20,resvec/norm(b),'-o')
  xlabel('Iteration Number')
  ylabel('Relative Residual')
  
  
  

Now, try an incomplete LU factorization with a drop tolerance of 1e-5 for the preconditioner.

• [L1,U1] = luinc(A,1e-5);
  Warning: Incomplete upper triangular factor has 1 zero diagonal.
           It cannot be used as a preconditioner for an iterative 
           method.
  
  nnz(A), nnz(L1), nnz(U1) 
  
  ans =
           1887
  ans =
           5562
  ans =
           4320
  

The zero on the main diagonal of the upper triangular U1 indicates that U1 is singular. If you try to use it as a preconditioner,

• [x,flag,relres,iter,resvec] = bicg(A,b,1e-6,20,L1,U1)
  
  flag =
           2
  relres =
           1
  iter =
           0
  resvec =
           7.0557e+005
  

the method fails in the very first iteration when it tries to solve a system of equations involving the singular U1 using backslash. bicg is forced to return the initial estimate since no other iterates were produced.

Try again with a slightly less sparse preconditioner.

• [L2,U2] = luinc(A,1e-6);
  
  nnz(L2), nnz(U2)
  
  ans =
           6231
  ans =
           4559
  

This time U2 is nonsingular and may be an appropriate preconditioner.

• [x,flag,relres,iter,resvec] = bicg(A,b,1e-15,10,L2,U2)
  
  flag =
           0
  relres =
           2.8664e-016
  iter =
           8
  

and bicg converges to within the desired tolerance at iteration number 8. Decreasing the value of the drop tolerance increases the fill-in of the incomplete factors but also increases the accuracy of the approximation to the original matrix. Thus, the preconditioned system becomes closer to inv(U)*inv(L)*L*U*x = inv(U)*inv(L)*b, where L and U are the true LU factors, and closer to being solved within a single iteration.

The next graph shows the progress of bicg using six different incomplete LU factors as preconditioners. Each line in the graph is labeled with the drop tolerance of the preconditioner used in bicg.

See Also

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

@ (function handle), \ (backslash)

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.

betaln

bicgstab