搜索结果：Newton ((New York, N.Y.)

The secant equation traditionally constitutes the basis of quasi-Newton methods, as the updated Hessian approximations satisfy the equation on each iteration. Modified versions of the secant relation have recently been the focus of several papers with encouraging outcomes. This paper continues with that idea where a secant-like modification that utilises nonlinear quantities in constructing the Hessian (or its inverse) approximation updates is derived. The technique takes advantage of data readily computed from the two most recent steps. Thus, it offers a substitute to the secant equation to produce better Hessian approximations that result in accelerated convergence to the objective function minimiser. The reported results provide adequate evidence to suggest that the proposed method is promising and deserves attention.

The Newton polytope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript f"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">P_f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a polynomial <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is well known to have a strong impact on its behavior. The Bernstein-Kouchnirenko Theorem asserts that even the number of simultaneous zeros in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript m"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {C}^*)^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a system of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> polynomials depends on their Newton polytopes. In this article, we show that Newton polytopes also have a strong impact on the distribution of zeros and pointwise norms of polynomials, the basic theme being that Newton polytopes determine allowed and forbidden regions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper C Superscript asterisk Baseline right-parenthesis Superscript m"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {C}^*)^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for these distributions. Our results are statistical and asymptotic in the degree of the polynomials. We equip the space of polynomials of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to p"> <mml:semantics> <mml:mrow> <mml:mo> ≤ </mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\leq p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex variables with its usual SU <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We then determine the asymptotics of the conditional expectation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper E Subscript vertical-bar upper N upper P Baseline left-parenthesis upper Z Subscript f 1 comma ellipsis comma f Sub Subscript k Subscript Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">E</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {E}_{|N P}(Z_{f_1, \dots , f_k})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of simultaneous zeros of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:

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