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2006
The endeavour to solve systems of linear algebraic systems is already two thousand years old. In the paper we consider the conjugate gradient method that is (theoretically) finite but, in practice, it can be treated as an iterative method. We survey a known modification of the method, the preconditioned conjugate gradient method, that may converge ...
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The endeavour to solve systems of linear algebraic systems is already two thousand years old. In the paper we consider the conjugate gradient method that is (theoretically) finite but, in practice, it can be treated as an iterative method. We survey a known modification of the method, the preconditioned conjugate gradient method, that may converge ...
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Numerische Mathematik, 1963
The CG-algorithm is an iterative method to solve linear systems $$Ax + b = 0$$ (1) where A is a symmetric and positive definite coefficient matrix of order n. The method has been described first by Stiefel and Hesteness [1, 2] and additional information is contained in [3] and [4]. The notations used here coincide partially with those used in
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The CG-algorithm is an iterative method to solve linear systems $$Ax + b = 0$$ (1) where A is a symmetric and positive definite coefficient matrix of order n. The method has been described first by Stiefel and Hesteness [1, 2] and additional information is contained in [3] and [4]. The notations used here coincide partially with those used in
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Superlinear Convergence of Conjugate Gradients
SIAM Journal on Numerical Analysis, 2001The main goal of this paper is to illustrate that some recent results obtained in the logarithmic potential theory can be used for better understanding the phenomenon in the numerical analysis known as superlinear convergence. The authors give a theoretical explanation for superlinear convergence behaviour observed while solving large systems of linear
Beckermann, Bernhard +1 more
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The Kernel Conjugate Gradient Algorithms
IEEE Transactions on Signal Processing, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ming Zhang +3 more
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A Generalized Conjugate Gradient Algorithm
Journal of Optimization Theory and Applications, 1998zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Sanmatías, S., Vercher, E.
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New Hybrid Conjugate Gradient and Broyden–Fletcher–Goldfarb–Shanno Conjugate Gradient Methods
Journal of Optimization Theory and Applications, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Stanimirović, Predrag S. +3 more
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1994
In the following, A ∈ ℝ I x I and b ∈ ℝ I are real. We consider a system $$ Ax\, = \,b $$ (9.1.1) and assume that $$ A\,is\,positive\,definite. $$ (9.1.2) System (1) is associated with the function $$ F\left( x \right): = \,\frac{1}{2}\left\langle {Ax,\,x} \right\rangle \, - \,\left\langle {b,\,x} \right\rangle . $$ (9.1.3)
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In the following, A ∈ ℝ I x I and b ∈ ℝ I are real. We consider a system $$ Ax\, = \,b $$ (9.1.1) and assume that $$ A\,is\,positive\,definite. $$ (9.1.2) System (1) is associated with the function $$ F\left( x \right): = \,\frac{1}{2}\left\langle {Ax,\,x} \right\rangle \, - \,\left\langle {b,\,x} \right\rangle . $$ (9.1.3)
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1980
A first form of the conjugate gradient routine was given in Section 6, Chapter II. It was redeveloped in Section 3, Chapter III, as a CGS-method. In the present chapter we shall study the CG-algorithm in depth, giving several alternative versions of the CG-routine. It is shown that the number of steps required to obtain the minimum point of a quadratic
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A first form of the conjugate gradient routine was given in Section 6, Chapter II. It was redeveloped in Section 3, Chapter III, as a CGS-method. In the present chapter we shall study the CG-algorithm in depth, giving several alternative versions of the CG-routine. It is shown that the number of steps required to obtain the minimum point of a quadratic
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Orderings for Conjugate Gradient Preconditionings
SIAM Journal on Optimization, 1991The paper treats several ordering principles for the solution of Poisson- type elliptic boundary value problems with preconditioned conjugate gradient methods (PCG). As preconditioners SSOR and incomplete Cholesky (IC) are considered. For the solution of the relevant linear systems, which mostly are tridiagonal, on vector or parallel computers ...
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New Imperfect Conjugate Gradient Algorithm
Journal of Optimization Theory and Applications, 1997zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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