The Solutions To The Asymmetric Systems Of Linear Equations By GMRES And The Conjugate Residual Algorithm

Many scientific and technological problems need to be solved by systems oflinear equations frequently. The Generalized Minimal Residual Method (GMRES) isregarded as one of effective methods to solve large scale asymmetric systems oflinear equations.Using fixed point iterative method to solve system of equations Ax=bwhere r⁽⁰⁾=b-Ax⁽⁰⁾is residual vector, Sr⁽⁰⁾is corrected vector, A is a n order andnon singular matrix, not need to be symmetric. Using Krylov subspace andorthogonalization algorithm (Arnoldi method) we can get vectors v₁,v₂,…,vm forming a orthogonal basis ofκm=Span{v₁,Av₁,…,A^m-1v₁}.Using the FullOrthogonalization method (FOM), we can obtain the approximate solution of Ax=bIOM and DIOM methods can be obtained by using incomplete orthogonalization toKrylov subspace. Put x(m-1) to x(m) and x(m) is written asThe Generalized Minimal Residual Method (GMRES) is to get the unique vectorin x₀+κ_m which minimizes where y is a m dimensional vector, this approximation is to get y_m, satisfyingWhere e₁ is the first arrange of (m+1) th order matrix, andwhere ym is the solution of(m+1)×m least squares problem.There are some the important conclusions below for the GMRES(1)If A is positive definite (A+AT is symmetric and positive definite),GMRES (m) isconvergence for arbitrary m≥1.diagonal matrix of characteristic values of A. Definewhere Pm denotes the set of polynomials of degree less or equal to m. Then theresidual vector norm obtained from the k th step of GMRES satisfies{v_j}^m_j=1 obtained fromright preconditioned GMRES is a orthogonal basis ofright preconditioned Krylov subspace The preconditioned residual vector b- AM^-1u(m)obtained from the algorithm is theresidual vector b-Ax^(m)of variable x. The residual normIs minimized by right preconditioned GMRES , where u belongs to affine subspaceand r⁽⁰⁾=b-AM^-1u(0)=b-Ax⁽⁰⁾.Especially, if A is Hermite matrix ( A H = AT=A), using GMRES can get theConjugate Residual (CR) algorithm. In this case, the residual vectors should beA orthogonal i.e. conjugate. In addition , the vectors Api's(i=0,1,…) areorthogonal. The results below can generalize CR algorithm to the non symmetricsituations (where Hm is not tridiagonal).Let p₀,p₁,…,p_m-1 be a basis of Krylov subspaceκm(A,r⁽⁰⁾) , which areATA-orthogonal,i.e.Then, the approximate solution x^(m) which has the minimal residual norm in theaffine subspace x⁽⁰⁾+κm(A,r⁽⁰⁾) is given byIn addition, x^(m) can be computed from x^(m-1)by...