## Documentation Center |

Biconjugate gradients method

`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)[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,...)`

`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
handle, `afun`, such that `afun(x,'notransp')` returns `A*x` and `afun(x,'transp')` returns `A'*x`.

Parameterizing Functions explains how to provide additional
parameters to the function `afun`, as well as the
preconditioner function `mfun` described below, if
necessary.

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 handle `mfun`,
such that `mfun(x,'notransp')` 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.

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

Flag | Convergence |
---|---|

| |

| |

Preconditioner | |

| |

One of the scalar quantities calculated during |

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)`.

This example shows how to use `bicg` with
a matrix input. `bicg`. The following code:

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

This example replaces the matrix `A` in the
previous example with a handle to a matrix-vector product function `afun`.
The example is contained in a file `run_bicg` that

Calls

`bicg`with the`@afun`function handle as its first argument.Contains

`afun`as a nested function, so that all variables in`run_bicg`are available to`afun`.

Place the following into a file called `run_bicg`:

function x1 = run_bicg n = 100; on = ones(n,1); b = afun(on,'notransp'); tol = 1e-8; maxit = 15; M1 = spdiags([on/(-2) on],-1:0,n,n); M2 = spdiags([4*on -on],0:1,n,n); x1 = bicg(@afun,b,tol,maxit,M1,M2); function y = afun(x,transp_flag) if strcmp(transp_flag,'transp') % y = A'*x 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); elseif strcmp(transp_flag,'notransp') % y = A*x 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 end end

When you enter

x1 = run_bicg;

MATLAB^{®} software displays the message

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

This example demonstrates the use of a preconditioner.

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

```
load west0479;
A = west0479;
```

Define `b` so that the true solution is a vector of all ones.

b = full(sum(A,2));

Set the tolerance and maximum number of iterations.

tol = 1e-12; maxit = 20;

Use `bicg` to find a solution at the requested tolerance and number of iterations.

[x0,fl0,rr0,it0,rv0] = bicg(A,b,tol,maxit);

`fl0` is 1 because `bicg` does not converge to the requested tolerance `1e-12` within the requested 20 iterations. In fact, the behavior of `bicg` is so poor that the initial guess (`x0 = zeros(size(A,2),1)`) is the best solution and is returned as indicated by `it0 = 0`. MATLAB® stores the residual history in `rv0`.

Plot the behavior of `bicg`.

semilogy(0:maxit,rv0/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');

The plot shows that the solution does not converge. You can use a preconditioner to improve the outcome.

Create the preconditioner with `ilu`, since the matrix `A` is nonsymmetric.

[L,U] = ilu(A,struct('type','ilutp','droptol',1e-5));

Error using ilu There is a pivot equal to zero. Consider decreasing the drop tolerance or consider using the 'udiag' option.

MATLAB cannot construct the incomplete LU as it would result in a singular factor, which is useless as a preconditioner.

You can try again with a reduced drop tolerance, as indicated by the error message.

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

`fl1` is 0 because `bicg` drives the relative residual to `4.1410e-014` (the value of `rr1`). The relative residual is less than the prescribed tolerance of `1e-12` at the sixth iteration (the value of `it1`) when preconditioned by the incomplete LU factorization with a drop tolerance of `1e-6`. The output `rv1(1)` is `norm(b)`, and the output `rv1(7)` is `norm(b-A*x2)`.

You can follow the progress of `bicg` by plotting the relative residuals at each iteration starting from the initial estimate (iterate number 0).

semilogy(0:it1,rv1/norm(b),'-o'); xlabel('Iteration number'); ylabel('Relative residual');

[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.

`bicgstab` | `cgs` | `function_handle` | `gmres` | `ilu` | `lsqr` | `minres` | `mldivide` | `pcg` | `qmr` | `symmlq`

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