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# bicgstabl

## Syntax

x = bicgstabl(A,b)
x = bicgstabl(afun,b)
x = bicgstabl(A,b,tol)
x = bicgstabl(A,b,tol,maxit)
x = bicgstabl(A,b,tol,maxit,M)
x = bicgstabl(A,b,tol,maxit,M1,M2)
x = bicgstabl(A,b,tol,maxit,M1,M2,x0)
[x,flag] = bicgstabl(A,b,...)
[x,flag,relres] = bicgstabl(A,b,...)
[x,flag,relres,iter] = bicgstabl(A,b,...)
[x,flag,relres,iter,resvec] = bicgstabl(A,b,...)

## Description

x = bicgstabl(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 the right-hand side column vector b must have length n.

x = bicgstabl(afun,b) accepts a function handle afun instead of the matrix A. afun(x) accepts a vector input x and returns the matrix-vector product A*x. In all of the following syntaxes, you can replace A by afun.

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

x = bicgstabl(A,b,tol,maxit) specifies the maximum number of iterations. If maxit is [] then bicgstabl uses the default, min(N,20).

x = bicgstabl(A,b,tol,maxit,M) and x = bicgstabl(A,b,tol,maxit,M1,M2) use preconditioner M or M=M1*M2 and effectively solve the system A*inv(M)*x = b for x. If M is [] then a preconditioner is not applied. M may be a function handle returning M\x.

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

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

Flag

Convergence

0

bicgstabl converged to the desired tolerance tol within maxit iterations.

1

bicgstabl iterated maxit times but did not converge.

2

Preconditioner M was ill-conditioned.

3

bicgstabl stagnated. (Two consecutive iterates were the same.)

4

One of the scalar quantities calculated during bicgstabl became too small or too large to continue computing.

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

[x,flag,relres,iter] = bicgstabl(A,b,...) also returns the iteration number at which x was computed, where 0 <= iter <= maxit. iter can be k/4 where k is some integer, indicating convergence at a given quarter iteration.

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

## Examples

### Using bicgstabl with Inputs or with a Function

You can pass inputs directly to bicgstabl:

```n = 21;
A = gallery('wilk',n);
b = sum(A,2);
tol = 1e-12;
maxit = 15;
M = diag([10:-1:1 1 1:10]);
x = bicgstabl(A,b,tol,maxit,M);```

You can also use a matrix-vector product function:

```function y = afun(x,n)
y = [0; x(1:n-1)] + [((n-1)/2:-1:0)';
(1:(n-1)/2)'].*x+[x(2:n); 0];```

and a preconditioner backsolve function:

```function y = mfun(r,n)
y = r ./ [((n-1)/2:-1:1)';
1;
(1:(n-1)/2)'];```

as inputs to bicgstabl:

`x1 = bicgstabl(@(x)afun(x,n),b,tol,maxit,@(x)mfun(x,n));`

### Using a Preconditioner

This example demonstrates the use of a preconditioner.

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

```load west0479;
A = west0479;```
2. Define b so that the true solution is a vector of all ones.

`b = full(sum(A,2));`
3. Set the tolerance and maximum number of iterations:

`tol = 1e-12; maxit = 20;`
4. Use bicgstabl to find a solution at the requested tolerance and number of iterations:

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

fl0 is 1 because bicgstabl does not converge to the requested tolerance 1e-12 within the requested 20 iterations. In fact, the behavior of bicgstabl 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.

5. Plot the behavior of bicgstabl:

```semilogy(0:0.25: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.

6. Create a preconditioner with ilu, since 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.

7. 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] = bicgstabl(A,b,tol,maxit,L,U);```

fl1 is 0 because bicgstabl drives the relative residual to 1.0257e-015 (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(9) is norm(b-A*x2) since bicgstabl uses quarter iterations.

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

```semilogy(0:0.25:it1,rv1/norm(b),'-o');
set(gca,'XTick',0:0.25:it1);
xlabel('Iteration number');
ylabel('Relative residual');```