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<p>The vector <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-algorithm of Wynn is a powerful method for accelerating efficiently the convergence of vector sequences. It is also employed for solving nonlinear systems of equations. Also, in the linear case, it corresponds to a specific Krylov-subspace methods. Its kernel is the set of sequences which are transformed into constant sequences whose terms are their limits or antilimits. In 1971, a sufficient condition characterizing sequences in this kernel was given by McLeod. Recently, it has been showed that the Mc Leod’s condition is not necessary. A detailed study for the step <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\varepsilon _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ε</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> of the algorithm has been accomplished, giving rise to an algebraic and geometric formulas for the explicit forms of its kernel. Moreover, a conjecture concerning the explicit algebraic expression of kernel of the vector <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-algorithm has been formulated. In this work, we prove the truthfulness of this conjecture for the dimension <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(d=4.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>4</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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The explicit form of the vector \(\varepsilon \)-algorithm’s kernel : a conjecture and its solution in \(\mathbb {R}^4 \)

  • 期刊论文

  • 发表日期：

    2025年4月23日

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  • Ahmed Salam

摘要

The vector \(\varepsilon \) ε -algorithm of Wynn is a powerful method for accelerating efficiently the convergence of vector sequences. It is also employed for solving nonlinear systems of equations. Also, in the linear case, it corresponds to a specific Krylov-subspace methods. Its kernel is the set of sequences which are transformed into constant sequences whose terms are their limits or antilimits. In 1971, a sufficient condition characterizing sequences in this kernel was given by McLeod. Recently, it has been showed that the Mc Leod’s condition is not necessary. A detailed study for the step \(\varepsilon _2\) ε 2 of the algorithm has been accomplished, giving rise to an algebraic and geometric formulas for the explicit forms of its kernel. Moreover, a conjecture concerning the explicit algebraic expression of kernel of the vector \(\varepsilon \) ε -algorithm has been formulated. In this work, we prove the truthfulness of this conjecture for the dimension \(d=4.\) d = 4 .