On generalized averaged Gaussian formulas

Abstract

We present a simple numerical method for constructing the optimal (generalized) averaged Gaussian quadrature formulas which are the optimal stratified extensions of Gauss quadrature formulas. These extensions exist in many cases in which real positive Kronrod formulas do not exist. For the Jacobi weight functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis x right-parenthesis identical-to w Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis equals left-parenthesis 1 minus x right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus x right-parenthesis Superscript beta"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≡ </mml:mo> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi> α </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi> β </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">w(x)\equiv w^{(\alpha ,\beta )}(x)=(1-x)^\alpha (1+x)^\beta</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="alpha comma beta greater-than negative 1"> <mml:semantics> <mml:mrow> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi> β </mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha ,\beta &gt;-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) we give a necessary and sufficient condition on the parameters <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</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="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the optimal averaged Gaussian quadrature formulas are internal.