A parallel algorithm for solving block tridiagonal linear systems
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TL;DR: A new form of the arithmetic mean method for solving large block tridiagonal linear systems with coefficient matrices that are symmetric positive definite or positive real or irreducible L-matrices with a strong diagonal dominance is considered.
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Abstract: In this paper, we consider a new form of the arithmetic mean method for solving large block tridiagonal linear systems. The iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible L-matrices with a strong diagonal dominance. When the coefficient matrix is symmetric positive definite, an additive preconditioner for the conjugate gradient method is derived. Both the iterative method and the preconditioner are very suitable for parallel implementation on a multivector computer. Some numerical experiments on systems resulting from the discretization of an elliptic partial differential equation are carried out on the Cray Y-MP.
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Citations
A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method
TL;DR: This study develops a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides and proposes a parallel realization of the domain decomposition method (the Schur complement method).
A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method
Andrew V. Terekhov
- 01 Jun 2013
TL;DR: In this paper, a parallel algorithm for solving block-tridiagonal systems of equations is presented, which is an effective and simple set of procedures for solving engineering tasks on a supercomputer.
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The Newton-arithmetic mean method for the solution of systems of nonlinear equations
TL;DR: The convergence of the method is analysed for the class of systems whose Jacobian matrix satisfies an affine invariant Lipschitz condition and an estimation of the radius of the attraction ball is given.
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The Two-Stage Arithmetic Mean Method
TL;DR: The Two-Stage Arithmetic Mean Method for solving large sparse linear systems has been introduced and analyzed in this paper, where the convergence conditions are derived for symmetric positive definite matrices and M-matrices.
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A parallel solving method for block-tridiagonal equations on CPU---GPU heterogeneous computing systems
Yang Wangdong,Kenli Li,Keqin Li +2 more
TL;DR: A solving method which mixes direct and iterative methods for solving block-tridiagonal systems of linear equations, and an improved algorithm to solve the sub-equations by thread blocks on GPU, so as to significantly reduce the latency of memory access.
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References
•Book
Introduction to Parallel and Vector Solution of Linear Systems
J. M. Ortega
- 30 Apr 1988
TL;DR: The Conjugate Gradient Algorithm and the Iterative Methods for Linear Equations are described, which simplify the derivation of linear algebra to simple linear algebra.
649
Some Aspects of the Cyclic Reduction Algorithm for Block Tridiagonal Linear Systems
TL;DR: The solution of a general block tridiagonal linear system by a cyclic odd-even reduction algorithm is considered, under conditions of diagonal dominance, norms describing the off-diagonal blocks relative to the diagonal blocks decrease quadratically with each reduction.
174
m-Step Preconditioned Conjugate Gradient Methods
TL;DR: This paper discusses preconditioners for the conjugate gradient method which are based on several iterations of stationary iterative methods and efficient computer implementations of these methods are discussed.
An iterative method for large sparse linear systems on a vector computer
TL;DR: This iterative method converges for systems with coefficient matrices that are symmetric positive definite or positive real or irreducible L-matrices with a strong diagonal dominance and is very suitable for parallel implementation on a multiprocessor system, such as the CRAY X-MP.
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