GAUSSIAN QUADRATURE

(Redirected from Gauss-Kronrod quadrature)
In numerical analysis, a 'quadrature rule' is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.
(See numerical integration for more on quadrature rules.)
An ''n''-point 'Gaussian quadrature rule', named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2''n'' − 1,
by a suitable choice of the ''n'' points ''x''_''i'' and ''n'' weights ''w''_''i''.
The domain of integration for such a rule is conventionally taken as [−1, 1],
so the rule is stated as
:
It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.

Contents

Rules for the basic problem

Change of interval for Gaussian quadrature

Other forms of Gaussian quadrature

Fundamental theorem

Error estimates

Gauss–Kronrod rules

References

External links

Rules for the basic problem

For the integration problem stated above,
the associated polynomials are Legendre polynomials, ''P''_''n''(''x''). With the ''n''^th polynomial normalized to give ''P''_''n''(1) = 1, the ''i''^th Gauss node, ''x''_''i'', is the ''i''^th root of ''P''_''n''; its weight is given by
:
ight) (P'_n(x_i))^2} ,!
Some low-order rules for solving the integration problem are listed below.
{| class="wikitable" style="margin:auto; background:white;"
! Number of points, ''n'' !! Points, ''x''''i'' !! Weights, ''w''_''i''
|- align="center"
| 1 || 0 || 2
|- align="center"
| 2 || $pmsqrt\{1/3\}$ || 1
|- align="center"
| rowspan="2" | 3 || 0 || ⁸⁄₉
|- align="center"
| $pmsqrt\{3/5\}$ || ⁵⁄₉
|- align="center"
| rowspan="2" | 4 || $pmsqrt\{Big(\; 3\; -\; 2sqrt\{6/5\}\; Big)/7\}$ || $frac\{18+sqrt\{30\}\}\{36\}$
|- align="center"
| $pmsqrt\{Big(\; 3\; +\; 2sqrt\{6/5\}\; Big)/7\}$ || $frac\{18-sqrt\{30\}\}\{36\}$
|- align="center"
| rowspan="3" | 5 || 0 || ¹²⁸⁄₂₂₅
|- align="center"
| $pm\; frac13sqrt\{5-2sqrt\{10/7\}\}$ || $frac\{322+13sqrt\{70\}\}\{900\}$
|- align="center"
| $pm\; frac13sqrt\{5+2sqrt\{10/7\}\}$ || $frac\{322-13sqrt\{70\}\}\{900\}$
|}

Change of interval for Gaussian quadrature

An integral over [''a'', ''b''] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:
:$$
int_a^b f(t),dt = rac{b-a}{2} int_{-1}^1 fleft(rac{b-a}{2}x
+ rac{a+b}{2}
ight),dx

After applying the Gaussian quadrature rule, the following approximation is obtained:
:$$
rac{b-a}{2} sum_{i=1}^n w_i fleft(rac{b-a}{2}x_i + rac{a+b}{2}
ight)

Other forms of Gaussian quadrature

The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand,
and allowing an interval other than [−1, 1].
That is, the problem is to calculate
:$int\_a^b\; omega(x),f(x),dx$
for some choices of ''a'', ''b'', and ω.
For ''a'' = −1, ''b'' = 1, and ω(''x'') = 1,
the problem is the same as that considered above.
Other choices lead to other integration rules.
Some of these are tabulated below.
Equation numbers are given for Abramowitz and Stegun (A & S).
{| class="wikitable" style="margin:auto; background:white;"
! Interval !! ω(''x'') !! Orthogonal polynomials !! A & S
|- align="center"
| [−1, 1] || $1,$ || Legendre polynomials || 25.4.29
|- align="center"
| (−1, 1) || || Jacobi polynomials || 25.4.33 ()
|- align="center"
| (−1, 1) || || Chebyshev polynomials (first kind) || 25.4.38
|- align="center"
| [−1, 1] || $sqrt\{1\; -\; x^2\}$ || Chebyshev polynomials (second kind) || 25.4.40
|- align="center"
| [0, ∞) || $e^\{-x\},$ || Laguerre polynomials || 25.4.45
|- align="center"
| (−∞, ∞) || $e^\{-x^2\}$ || Hermite polynomials || 25.4.46
|}

Fundamental theorem

Let ''q'' be a nontrivial polynomial of degree ''n'' such that
:$$
int_a^b omega(x) , x^k q(x) , dx = 0, quad ext{for all }k=0,1,ldots,n-1.

If we pick the nodes to be the zeros of ''q'', then there exist weights ''w''_''i'' which make the computed integral exact for all polynomials of degree 2''n'' − 1 or less. Furthermore, all these nodes will lie in the open interval (''a'', ''b'') .

Error estimates

The error of a Gaussian quadrature rule can be stated as follows .
For an integrand which has 2''n'' continuous derivatives,
:$int\_a^b\; omega(x),f(x),dx\; -\; sum\_\{i=1\}^n\; w\_i,f(x\_i)$
= rac{f^{(2n)}(xi)}{(2n)!} , (p_n,p_n)
for some ξ in (''a'', ''b''), where ''p''_''n'' is the orthogonal polynomial of order ''n'' and where
:$(f,g)\; =\; int\_a^b\; omega(x)\; f(x)\; g(x)\; ,\; dx\; .\; ,!$
In the important special case of ω(''x'') = 1, we have the error estimate
:
Stoer and Bulirsch remark that this error estimate is inconvenient in practice,
since it may be difficult to estimate the order 2''n'' derivative,
and furthermore the actual error may be much less than a bound established by the derivative.
Another approach is to use two Gaussian quadrature rules of different orders,
and to estimate the error as the difference between the two results.
For this purpose, Gauss-Kronrod quadrature rules can be useful.

Gauss–Kronrod rules

If the interval [''a'', ''b''] is subdivided,
the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers),
and thus the integrand must be evaluated at every point.
''Gauss-Kronrod rules'' are extensions of Gauss quadrature rules generated by adding $n+1$ points to an $n$-point rule in such a way that the resulting rule is of order $3n+1$.
This allows for computing higher-order estimates while re-using the function values of a lower-order estimate.
The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.
The rules are named after Alexander Kronrod who invented them in the 1960s.
The algorithms in QUADPACK (see below) are based on Gauss–Kronrod rules.
A popular example combines a 7-point Gauss rule with a 15-point Kronrod rule . Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations yields both a quadrature estimate and an error estimate.
:{| class="wikitable" style="background-color:white"
|+ (G7,K15) on [−1,1]
|-
! Gauss nodes !! !! Weights
|-
| ±0.94910 79123 42759 || ∗ || 0.12948 49661 68870
|-
| ±0.74153 11855 99394 || ∗ || 0.27970 53914 89277
|-
| ±0.40584 51513 77397 || ∗ || 0.38183 00505 05119
|-
| style="text-align:right" |  0.00000 00000 00000 || ∗ || 0.41795 91836 73469
|-
! Kronrod nodes !! !! Weights
|-
| ±0.99145 53711 20813 || || 0.02293 53220 10529
|-
| ±0.94910 79123 42759 || ∗ || 0.06309 20926 29979
|-
| ±0.86486 44233 59769 || || 0.10479 00103 22250
|-
| ±0.74153 11855 99394 || ∗ || 0.14065 32597 15525
|-
| ±0.58608 72354 67691 || || 0.16900 47266 39267
|-
| ±0.40584 51513 77397 || ∗ || 0.19035 05780 64785
|-
| ±0.20778 49550 07898 || || 0.20443 29400 75298
|-
| style="text-align:right" |  0.00000 00000 00000 || ∗ || 0.20948 21410 84728
|}
Patterson showed how to find further extensions of this type.

References

★

★

★ (Authorized translation from the Russian)

★ (Reference guide for QUADPACK)

★

★ .

★ .

★ (Errata).

External links

★ QUADPACK (part of SLATEC): description [1], source code [2]. QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gauss-Kronrod rules. SLATEC (at Netlib) is a large public domain library for numerical computing.

★ ALGLIB contains a collection of algorithms for numerical integration (in C# / C++ / Delphi / Visual Basic / etc.)

★ GNU Scientific Library - includes C version of QUADPACK algorithms (see also GNU Scientific Library)

★ Gaussian Quadrature Rule of Integration - Notes, PPT, Matlab, Mathematica, Maple, Mathcad at ''Holistic Numerical Methods Institute''

★ Gaussian Quadrature table at sitmo.com

★ Legendre-Gauss Quadrature at MathWorld