An adaptive Richardson iteration method for indefinite linear systems

Calvetti, Daniela; Reichel, Lothar

doi:10.1007/bf02141745

Cited by 20 publications

(11 citation statements)

References 26 publications

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“…In the latter case, the set K m is replaced by two intervals on the real axis, one on each side of the origin. A Richardson method for the solution of (1.1) when A is symmetric indefinite and s = 1 is presented in [5] and the approach there generalizes in an obvious manner to the case s > 1. We therefore omit the details.…”

Section: Algorithm 42 (Adaptive Richardson Method)mentioning

confidence: 99%

Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors

Calvetti

Reichel

1994

Numerische Mathematik

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An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme "learns" about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required. Subject Classification (1991): 65F10, 65F15, 15A24 Mathematics

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Section: Algorithm 42 (Adaptive Richardson Method)mentioning

confidence: 99%

Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors

Calvetti

Reichel

1994

Numerische Mathematik

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“…These well-known facts have been verified for the rectangular cavity [9]. The algorithms that successfully solved all problem sizes were the implicitly restarted Lanczos algorithm [10,21,22] and the Jacobi-Davidson algorithm [11,23,24]. Like Lanczos, these algorithms converge superlinearly but, like subspace iteration, their memory requirements are determined by the order n of the problem times a small multiple of the number p of the desired eigenvalues.…”

Section: Solving the Matrix Eigenvalue Problemmentioning

confidence: 98%

“…A 1 and M 1 have been defined in (21). It turned out that this preconditioner leads to an algorithm that is very similar to the one of the straightforward approach.…”

Section: Solving the Constrained System Of Equation: The Augmentementioning

confidence: 99%

Solving Maxwell eigenvalue problems for accelerating cavities

Arbenz

Geus

Adam

2001

Phys. Rev. ST Accel. Beams

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We investigate algorithms for computing steady state electromagnetic waves in cavities. The Maxwell equations for the strength of the electric field are solved by a mixed method with quadratic finite edge (Nédélec) elements for the field values and corresponding node-based finite elements for the Lagrange multiplier. This approach avoids so-called spurious modes which are introduced if the divergence-free condition for the electric field is not treated properly. To compute a few of the smallest positive eigenvalues and corresponding eigenmodes of the resulting large sparse matrix eigenvalue problems, two algorithms have been used: the implicitly restarted Lanczos algorithm and the Jacobi-Davidson algorithm, both with shift-and-invert spectral transformation. Two-level hierarchical basis preconditioners have been employed for the iterative solution of the resulting systems of equations.

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“…For example, when discretizing the two point boundary value problems or the partial differential equations that frequently appear in oilreservoir engineering, in weather forecasting, or in electronic device modelling among others, linear systems like (1) need to be solved (see, e.g., [1,16]). The well-known Richardson's method (also known as Chebyshev method) and its variations are characterized by using the residual vector, r (x) = b − Ax, as search direction to solve linear systems iteratively (see, e.g., [7,11,12,24,26]). In general, these variations of Richardson's method have not been considered to be competitive with Krylov subspace methods, which represent nowadays the best-known options for solving (1), specially when combined with suitable preconditioning strategies.…”

Section: Introductionmentioning

confidence: 99%

“…The most important is the use of the spectral steplength choice, also known as the BarzilaiBorwein choice, ( [2,13,17,22]) that has proved to yield fast local convergence for the solution of nonlinear optimization problems ( [4,5,6,15,23]). However, this special choice of step size cannot guarantee global convergence by itself, as it usually happens with other variations (see, e.g., [7,11,12,24,26]). For that, we combine its use with a tolerant globalization strategy, that represents the second new feature.…”

Section: Introductionmentioning

confidence: 99%