Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Gaussian quadrature
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{Short description|Approximation of the definite integral of a function}} {{Redirect|Gaussian integration|the integral of a Gaussian function|Gaussian integral}} {{more footnotes|date=September 2018}} [[File:Comparison Gaussquad trapezoidal.svg|thumb|upright=1.5|alt=Comparison between 2-point Gaussian and trapezoidal quadrature.|Comparison between 2-point Gaussian and trapezoidal quadrature.<br /> The blue curve shows the function whose definite integral on the interval {{math|[−1, 1]}} is to be calculated (the integrand). The [[trapezoidal rule]] approximates the function with a linear function that coincides with the integrand at the endpoints of the interval and is represented by an orange dashed line. The approximation is apparently not good, so the error is large (the [[trapezoidal rule]] gives an approximation of the integral equal to {{math|1=''y''(−1) + ''y''(1) = −10}}, while the correct value is {{math|{{frac|2|3}}}}). To obtain a more accurate result, the interval must be partitioned into many subintervals and then the ''composite'' trapezoidal rule must be used, which requires many more calculations.<br /> The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the third-degree polynomial {{math|1=''y''(''x'') = 7''x''{{sup|3}} − 8''x''{{sup|2}} − 3''x'' + 3}}, the 2-point Gaussian quadrature rule even returns an exact result.]] In [[numerical analysis]], an {{mvar|n}}-point '''Gaussian quadrature rule''', named after [[Carl Friedrich Gauss]],<ref>{{harvnb|Gauss|1815}}</ref> is a [[quadrature rule]] constructed to yield an exact result for [[polynomial]]s of degree {{math|2''n'' − 1}} or less by a suitable choice of the nodes {{mvar|x{{sub|i}}}} and weights {{mvar|w{{sub|i}}}} for {{math|1=''i'' = 1, ..., ''n''}}. The modern formulation using [[orthogonal polynomial]]s was developed by [[Carl Gustav Jacobi]] in 1826.<ref>{{harvnb|Jacobi|1826}}</ref> The most common domain of integration for such a rule is taken as {{math|[−1, 1]}}, so the rule is stated as <math display="block">\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i),</math> which is exact for polynomials of degree {{math|2''n'' − 1}} or less. This exact rule is known as the [[Gauss–Legendre quadrature]] rule. The quadrature rule will only be an accurate approximation to the integral above if {{math|''f'' (''x'')}} is well-approximated by a polynomial of degree {{math|2''n'' − 1}} or less on {{math|[−1, 1]}}. The Gauss–[[Adrien-Marie Legendre|Legendre]] quadrature rule is not typically used for integrable functions with endpoint [[singularity (math)|singularities]]. Instead, if the integrand can be written as <math display="block">f(x) = \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x),\quad \alpha,\beta > -1,</math> where {{math|''g''(''x'')}} is well-approximated by a low-degree polynomial, then alternative nodes {{mvar|x{{sub|i}}'}} and weights {{mvar|w{{sub|i}}'}} will usually give more accurate quadrature rules. These are known as [[Gauss–Jacobi quadrature]] rules, i.e., <math display="block">\int_{-1}^1 f(x)\,dx = \int_{-1}^1 \left(1 - x\right)^\alpha \left(1 + x\right)^\beta g(x)\,dx \approx \sum_{i=1}^n w_i' g\left(x_i'\right).</math> Common weights include <math display="inline">\frac{1}{\sqrt{1 - x^2}}</math> ([[Chebyshev–Gauss quadrature|Chebyshev–Gauss]]) and <math display="inline">\sqrt{1 - x^2}</math>. One may also want to integrate over semi-infinite ([[Gauss–Laguerre quadrature]]) and infinite intervals ([[Gauss–Hermite quadrature]]). It can be shown (see Press et al., or Stoer and Bulirsch) that the quadrature nodes {{mvar|x{{sub|i}}}} are the [[Root of a function|roots]] of a polynomial belonging to a class of [[orthogonal polynomials]] (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)