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## 18.5 Specialized Solvers

__Function File:__**bicgstab***(*`A`,`b`)__Function File:__**bicgstab***(*`A`,`b`,`tol`,`maxit`,`M1`,`M2`,`x0`)This procedure attempts to solve a system of linear equations A*x = b for x. The

`A`must be square, symmetric and positive definite real matrix N*N. The`b`must be a one column vector with a length of N. The`tol`specifies the tolerance of the method, the default value is 1e-6. The`maxit`specifies the maximum number of iterations, the default value is min(20,N). The`M1`specifies a preconditioner, can also be a function handler which returns M\X. The`M2`combined with`M1`defines preconditioner as preconditioner=M1*M2. The`x0`is the initial guess, the default value is zeros(N,1).The value

`x`is a computed result of this procedure. The value`flag`can be 0 when we reach tolerance in`maxit`iterations, 1 when we don't reach tolerance in`maxit`iterations and 3 when the procedure stagnates. The value`relres`is a relative residual - norm(b-A*x)/norm(b). The value`iter`is an iteration number in which x was computed. The value`resvec`is a vector of`relres`for each iteration.

__Function File:__**cgs***(*`A`,`b`)__Function File:__**cgs***(*`A`,`b`,`tol`,`maxit`,`M1`,`M2`,`x0`)This procedure attempts to solve a system of linear equations A*x = b for x. The

`A`must be square, symmetric and positive definite real matrix N*N. The`b`must be a one column vector with a length of N. The`tol`specifies the tolerance of the method, default value is 1e-6. The`maxit`specifies the maximum number of iteration, default value is MIN(20,N). The`M1`specifies a preconditioner, can also be a function handler which returns M\X. The`M2`combined with`M1`defines preconditioner as preconditioner=M1*M2. The`x0`is initial guess, default value is zeros(N,1).

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