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%%%%\addcontentsline{toc}{section}{چکیده انگلیسی} 
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\textbf{Abstract}
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This thesis presents preconditioned Galerkin and minimal residual algorithms for solving Sylvester equations of the form $AX - XB = C$. Given two effective preconditioner matrices $M$ and $N$ for matrices $ A$ and $B$, respectively, we solve the preconditioned Sylvester equations $MAXN - MXBN = MCN$. The algorithms utilize the Arnoldi process to generate orthonormal bases of specific Krylov subspaces, simultaneously reducing the order of the Sylvester equations. Numerical experiments demonstrate that the preconditioned versions of the Galerkin and minimal residual algorithms yield highly accurate solutions and are more robust and efficient compared to their non-preconditioned counterparts. The results highlight the effectiveness of preconditioning in improving convergence and computational efficiency for large-scale Sylvester equations.\vspace{2cm}
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\textbf{Keywords:}
Sylvester matrix equations, Preconditioning, Galerkin method, Minimal residual method, Krylov subspace, Arnoldi process, Numerical algorithms, Iterative methods.

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