Type: Article
Publication Date: 1983-01-01
Citations: 20
DOI: https://doi.org/10.1090/s0025-5718-1983-0689468-1
The paper discusses both theoretical properties and practical implementation of product integration rules of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative normal infinity Superscript normal infinity Baseline k left-parenthesis x right-parenthesis f left-parenthesis x right-parenthesis d x almost-equals sigma-summation Underscript i equals 1 Overscript n Endscripts w Subscript n i Baseline f left-parenthesis x Subscript n i Baseline right-parenthesis comma"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>≈<!-- ≈ --></mml:mo> <mml:munderover> <mml:mo movablelimits="false">∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\int _{ - \infty }^\infty {k(x)f(x)\,dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} }</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <italic>f</italic> is continuous, <italic>k</italic> is absolutely integrable, the nodes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x Subscript n i Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {x_{ni}}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are roots of the Hermite polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript n Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_n}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and the weights <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace w Subscript n i Baseline right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>w</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {w_{ni}}\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are chosen so that the rule is exact if <italic>f</italic> is any polynomial of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than n"> <mml:semantics> <mml:mrow> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">> n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Convergence of the rule to the exact integral as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is proved for a wide class of functions <italic>f</italic> and <italic>k</italic> (including singular or oscillatory functions <italic>k</italic>), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are also discussed.