Krylov subspace methods

The matrix M is symmetric if. 4 = A. The matrix is said to be positive definite if the Euclidean inner product (x, Alx) 0 whenever x 0 [205]. The Euclidetm inner product between two 

More complex preconditioners are the incomplete LU-preconditioners (ILU) given on the form  

The most efficient Krylov subspace method is the method of Conjugate Gradients (CG) by Hestenes and Stiefel [84]. If, 4 is symmetric positive definite, the solution of the problem Ax = b corresponds to determining a local minimum of the quadratic function  

The basis used in this method is conjugate search directions and orthogonal residuals which is equivalent to finding a minimum point along the search directions. 

In a linear system Ax = b where the matrix A is symmetric and positive definite, the solution is obtained by minimizing the quadratic form (12.544). This implies that the gradient,/ (x) = Ax-b, is zero. In the iteration procedure an approximate solution, Xm+i, can be expressed as a linear combination of the previous solution and a search direction, p, which is scaled by a scaling factor am.  

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Krylov subspace methods (such as Conjugate Gradient CG, the improved BiCGSTAB, and GMRES) in combination with preconditioners for matrix manipulations aimed at enhanced convergence, and... 

For the RDE, the operating range of rotation frequency is between approximately 1 and 50 Hz and a typical radius is 0.25 cm. Dimensionless rate constants were interpolated from working curves generated from a fully implicit simulation using preconditioned Krylov subspace methods (Alden, unpublished work). 

An effective method to acetderato the convttrgerice of the MRM algorithm is based on the Krylov-subspace method (Kleinman and van den Berg, 1993). VVe introduced the Krylov subspace in Chapter 2 as the finite dirncnsiorial subsjtace A, of the Hilbert space M, spanned by the vec.tors r , Lr . lAr,.. 

The Krylov-subspace method is based on approximating the iteration step, Am , in the recursive formula (4.7) by an element of the Krylov subspace... 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

ILU Incomplete LU (ILU) -preconditioners in Krylov subspace methods outline... 

A solution is to use Krylov subspace methods, such as the conjugate gradient (CG) method, the biconjugate gradient (BiCG)... 

Momentum associated with a macroscopic CV (kg m/s) Generalized momenta in Hamiltonian mechanics Search direction in m-th iteration in Krylov subspace methods outline... 

The generalized minimum residual (GMRES) Krylov subspace method... 

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