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• W1997228246 abstract "The existence of a fast algorithm with an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O Subscript upper A Baseline left-parenthesis n left-parenthesis log n right-parenthesis squared right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{O_A}(n{(log n)^2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> time complexity for the multiplication of generalized Hilbert matrices with vectors is shown. These matrices are defined by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper B Subscript p Baseline right-parenthesis Subscript i comma j Baseline equals 1 slash left-parenthesis t Subscript i Baseline minus s Subscript j Baseline right-parenthesis Superscript p> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{({B_p})_{i,j}} = 1/{({t_i} - {s_j})^p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=i comma j equals 1 comma ellipsis comma n> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>i,j = 1, ldots ,n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals 1 comma ellipsis comma q> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p = 1, ldots ,q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=q much-less-than n> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≪<!-- ≪ --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>q ll n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript i> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{t_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s Subscript i> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{s_i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are distinct points in the complex plane and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t Subscript i Baseline not-equals s Subscript j> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mo>≠<!-- ≠ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{t_i} ne {s_j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=i comma j equals 1 comma ellipsis comma n> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>i,j = 1, ldots ,n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The major contribution of the paper is the stable implementation of the algorithm for two important sets of points, the Chebyshev points and the <italic>n</italic>th roots of unity. Such points have applications in the numerical approximation of Cauchy singular integral equations. The time complexity of the algorithm, for these special sets of points, reduces to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O Subscript upper A Baseline left-parenthesis n log n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{O_A}(nlog n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1997228246 title "A fast algorithm for the multiplication of generalized Hilbert matrices with vectors" @default.
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