Introduction to Computational Linear Algebra

174 Introduction to Computational Linear Algebra

The procedure is depicted in Algorithm 6.7.

Algorithm 6.7 Preconditioned Bi-Conjugate Gradient Algorithm

% x0,b given; eps1 given tolerance

% kmax maximum number of iterations,

r=b-A*x0;beta=0;r1=r;p=0;p1=0;

x=x0;k=0;e=1;nb=norm(b);

while e>eps1 & k<=kmax

k=k+1;z=M\r;p=z+beta*p;

z1=M^T \ r1;p1=z1+beta*p1;

q=A*p;q1=A^T p1;

nr=r1’*z;nap=p1’*q;alpha=nr/nap;

x=x+alpha*p;r=r-alpha*q;

r1=r1-alpha*q1;nr1=r1’*z;

beta=nr1/nr;nr=nr1;e=norm(r)/nb;

end

6.7 Preconditioning Issues

Preconditioning is essential in practice in order to get a competitive Krylov

method with fast convergence [9]. A first class of preconditioning