Gaussian quadrature

Comparison between 2-point Gaussian and trapezoidal quadrature.
Comparison between 2-point Gaussian and trapezoidal quadrature. The blue line is the polynomial , whose integral in [−1, 1] is 23. The trapezoidal rule returns the integral of the orange dashed line, equal to . The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to . Such a result is exact, since the green region has the same area as the sum of the red regions.

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,[1] is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, …, n. Modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826.[2] The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as

which is exact for polynomials of degree 2n − 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 f(x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].

The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as

where g(x) is well-approximated by a low-degree polynomial, then alternative nodes and weights will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e.,

Common weights include (Chebyshev–Gauss) and . 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 xi are the 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.

Gauss–Legendre quadratureEdit

 
Graphs of Legendre polynomials (up to n = 5)

For the simplest integration problem stated above, i.e., f(x) is well-approximated by polynomials on  , the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x). With the n-th polynomial normalized to give Pn(1) = 1, the i-th Gauss node, xi, is the i-th root of Pn and the weights are given by the formula (Abramowitz & Stegun 1972, p. 887)

 

Some low-order quadrature rules are tabulated below (over interval [−1, 1], see the section below for other intervals).

Number of points, n Points, xi Weights, wi
1 0 2
2   ±0.57735… 1
3 0   0.888889…
  ±0.774597…   0.555556…
4   ±0.339981…   0.652145…
  ±0.861136…   0.347855…
5 0   0.568889…
  ±0.538469…   0.478629…
  ±0.90618…   0.236927…

Change of intervalEdit

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:

 

Applying   point Gaussian quadrature   rule then results in the following approximation:

 

Other formsEdit

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

 

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).

Interval ω(x) Orthogonal polynomials A & S For more information, see …
[−1, 1] 1 Legendre polynomials 25.4.29 § Gauss–Legendre quadrature
(−1, 1)   Jacobi polynomials 25.4.33 (β = 0) Gauss–Jacobi quadrature
(−1, 1)   Chebyshev polynomials (first kind) 25.4.38 Chebyshev–Gauss quadrature
[−1, 1]   Chebyshev polynomials (second kind) 25.4.40 Chebyshev–Gauss quadrature
[0, ∞)   Laguerre polynomials 25.4.45 Gauss–Laguerre quadrature
[0, ∞)   Generalized Laguerre polynomials Gauss–Laguerre quadrature
(−∞, ∞)   Hermite polynomials 25.4.46 Gauss–Hermite quadrature

Fundamental theoremEdit

Let pn be a nontrivial polynomial of degree n such that

 

If we pick the n nodes xi to be the zeros of pn, then there exist n weights wi which make the Gauss-quadrature computed integral exact for all polynomials h(x) of degree 2n − 1 or less. Furthermore, all these nodes xi will lie in the open interval (a, b) (Stoer & Bulirsch 2002, pp. 172–175).

The polynomial pn is said to be an orthogonal polynomial of degree n associated to the weight function ω(x). It is unique up to a constant normalization factor. The idea underlying the proof is that, because of its sufficiently low degree, h(x) can be divided by   to produce a quotient q(x) of degree strictly lower than n, and a remainder r(x) of still lower degree, so that both will be orthogonal to  , by the defining property of  . Thus

 

Because of the choice of nodes xi, the corresponding relation

 

holds also. The exactness of the computed integral for   then follows from corresponding exactness for polynomials of degree only n or less (as is  ).

General formula for the weightsEdit

The weights can be expressed as

 

 

 

 

 

(1)

where   is the coefficient of   in  . To prove this, note that using Lagrange interpolation one can express r(x) in terms of   as

 

because r(x) has degree less than n and is thus fixed by the values it attains at n different points. Multiplying both sides by ω(x) and integrating from a to b yields

 

The weights wi are thus given by

 

This integral expression for   can be expressed in terms of the orthogonal polynomials   and   as follows.

We can write

 

where   is the coefficient of   in  . Taking the limit of x to   yields using L'Hôpital's rule

 

We can thus write the integral expression for the weights as

 

 

 

 

 

(2)

In the integrand, writing

 

yields

 

provided  , because

 

is a polynomial of degree k − 1 which is then orthogonal to  . So, if q(x) is a polynomial of at most nth degree we have

 

We can evaluate the integral on the right hand side for   as follows. Because   is a polynomial of degree n − 1, we have

 

where s(x) is a polynomial of degree  . Since s(x) is orthogonal to   we have

 

We can then write

 

The term in the brackets is a polynomial of degree  , which is therefore orthogonal to  . The integral can thus be written as

 

According to equation (2), the weights are obtained by dividing this by   and that yields the expression in equation (1).

  can also be expressed in terms of the orthogonal polynomials   and now  . In the 3-term recurrence relation   the term with   vanishes, so   in Eq. (1) can be replaced by  .

Proof that the weights are positiveEdit

Consider the following polynomial of degree  

 

where, as above, the xj are the roots of the polynomial  . Clearly  . Since the degree of   is less than  , the Gaussian quadrature formula involving the weights and nodes obtained from   applies. Since   for j not equal to i, we have

 

Since both   and   are non-negative functions, it follows that  .

Computation of Gaussian quadrature rulesEdit

There are many algorithms for computing the nodes xi and weights wi of Gaussian quadrature rules. The most popular are the Golub-Welsch algorithm requiring O(n2) operations, Newton's method for solving   using the three-term recurrence for evaluation requiring O(n2) operations, and asymptotic formulas for large n requiring O(n) operations.

Recurrence relationEdit

Orthogonal polynomials   with   for   for a scalar product  , degree   and leading coefficient one (i.e. monic orthogonal polynomials) satisfy the recurrence relation

 

and scalar product defined

 

for   where n is the maximal degree which can be taken to be infinity, and where  . First of all, the polynomials defined by the recurrence relation starting with   have leading coefficient one and correct degree. Given the starting point by  , the orthogonality of   can be shown by induction. For   one has

 

Now if   are orthogonal, then also  , because in

 

all scalar products vanish except for the first one and the one where   meets the same orthogonal polynomial. Therefore,

 

However, if the scalar product satisfies   (which is the case for Gaussian quadrature), the recurrence relation reduces to a three-term recurrence relation: For   is a polynomial of degree less than or equal to r − 1. On the other hand,   is orthogonal to every polynomial of degree less than or equal to r − 1. Therefore, one has   and   for s < r − 1. The recurrence relation then simplifies to

 

or

 

(with the convention  ) where

 

(the last because of  , since   differs from   by a degree less than r).

The Golub-Welsch algorithmEdit

The three-term recurrence relation can be written in matrix form   where  ,   is the  th standard basis vector, i.e.,  , and J is the so-called Jacobi matrix:

 

The zeros   of the polynomials up to degree n, which are used as nodes for the Gaussian quadrature can be found by computing the eigenvalues of this tridiagonal matrix. This procedure is known as Golub–Welsch algorithm.

For computing the weights and nodes, it is preferable to consider the symmetric tridiagonal matrix   with elements

 

J and   are similar matrices and therefore have the same eigenvalues (the nodes). The weights can be computed from the corresponding eigenvectors: If   is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated to the eigenvalue xj, the corresponding weight can be computed from the first component of this eigenvector, namely:

 

where   is the integral of the weight function

 

See, for instance, (Gil, Segura & Temme 2007) for further details.

Error estimatesEdit

The error of a Gaussian quadrature rule can be stated as follows (Stoer & Bulirsch 2002, Thm 3.6.24). For an integrand which has 2n continuous derivatives,

 

for some ξ in (a, b), where pn is the monic (i.e. the leading coefficient is 1) orthogonal polynomial of degree n and where

 

In the important special case of ω(x) = 1, we have the error estimate (Kahaner, Moler & Nash 1989, §5.2)

 

Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the order 2n 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 rulesEdit

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 2n + 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 is often used as an estimate of the approximation error.

Gauss–Lobatto rulesEdit

Also known as Lobatto quadrature (Abramowitz & Stegun 1972, p. 888) , named after Dutch mathematician Rehuel Lobatto. It is similar to Gaussian quadrature with the following differences:

  1. The integration points include the end points of the integration interval.
  2. It is accurate for polynomials up to degree 2n – 3, where n is the number of integration points (Quarteroni, Sacco & Saleri 2000) .

Lobatto quadrature of function f(x) on interval [−1, 1]:

 

Abscissas: xi is the  st zero of  .

Weights:

 

Remainder:

 

Some of the weights are:

Number of points, n Points, xi Weights, wi
     
   
     
   
     
   
   
     
   
   
     
   
   
   

An adaptive variant of this algorithm with 2 interior nodes[3] is found in GNU Octave and MATLAB as quadl and integrate.[4][5]

See alsoEdit

ReferencesEdit

  • Implementing an Accurate Generalized Gaussian Quadrature Solution to Find the Elastic Field in a Homogeneous Anisotropic Media
  • Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 25.4, Integration". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
  • Anderson, Donald G. (1965). "Gaussian quadrature formulae for  ". Math. Comp. 19 (91): 477–481. doi:10.1090/s0025-5718-1965-0178569-1.
  • Golub, Gene H.; Welsch, John H. (1969), "Calculation of Gauss Quadrature Rules", Mathematics of Computation, 23 (106): 221–230, doi:10.1090/S0025-5718-69-99647-1, JSTOR 2004418
  • Gautschi, Walter (1968). "Construction of Gauss–Christoffel Quadrature Formulas". Math. Comp. 22 (102). pp. 251–270. doi:10.1090/S0025-5718-1968-0228171-0. MR 0228171.
  • Gautschi, Walter (1970). "On the construction of Gaussian quadrature rules from modified moments". Math. Comp. 24. pp. 245–260. doi:10.1090/S0025-5718-1970-0285117-6. MR 0285177.
  • Piessens, R. (1971). "Gaussian quadrature formulas for the numerical integration of Bromwich's integral and the inversion of the laplace transform". J. Eng. Math. 5 (1). pp. 1–9. Bibcode:1971JEnMa...5....1P. doi:10.1007/BF01535429.
  • Danloy, Bernard (1973). "Numerical construction of Gaussian quadrature formulas for   and  ". Math. Comp. 27 (124). pp. 861–869. doi:10.1090/S0025-5718-1973-0331730-X. MR 0331730.
  • Kahaner, David; Moler, Cleve; Nash, Stephen (1989), Numerical Methods and Software, Prentice-Hall, ISBN 978-0-13-627258-8
  • Sagar, Robin P. (1991). "A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals". Comput. Phys. Commun. 66 (2–3): 271–275. Bibcode:1991CoPhC..66..271S. doi:10.1016/0010-4655(91)90076-W.
  • Yakimiw, E. (1996). "Accurate computation of weights in classical Gauss-Christoffel quadrature rules". J. Comput. Phys. 129 (2): 406–430. Bibcode:1996JCoPh.129..406Y. doi:10.1006/jcph.1996.0258.
  • Laurie, Dirk P. (1999), "Accurate recovery of recursion coefficients from Gaussian quadrature formulas", J. Comput. Appl. Math., 112 (1–2): 165–180, doi:10.1016/S0377-0427(99)00228-9
  • Laurie, Dirk P. (2001). "Computation of Gauss-type quadrature formulas". J. Comput. Appl. Math. 127 (1–2): 201–217. Bibcode:2001JCoAM.127..201L. doi:10.1016/S0377-0427(00)00506-9.
  • Riener, Cordian; Schweighofer, Markus (2018). "Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions". Journal of Complexity. 45: 22–54. arXiv:1607.08404. doi:10.1016/j.jco.2017.10.002.
  • Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Springer, ISBN 978-0-387-95452-3.
  • Temme, Nico M. (2010), "§3.5(v): Gauss Quadrature", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
  • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 4.6. Gaussian Quadratures and Orthogonal Polynomials", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
  • Gil, Amparo; Segura, Javier; Temme, Nico M. (2007), "§5.3: Gauss quadrature", Numerical Methods for Special Functions, SIAM, ISBN 978-0-89871-634-4
  • Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2000). Numerical Mathematics. New York: Springer-Verlag. pp. 422, 425. ISBN 0-387-98959-5.
Specific
  1. ^ Methodus nova integralium valores per approximationem inveniendi. In: Comm. Soc. Sci. Göttingen Math. Band 3, 1815, S. 29–76, Gallica, datiert 1814, auch in Werke, Band 3, 1876, S. 163–196.
  2. ^ C. G. J. Jacobi: Ueber Gauß’ neue Methode, die Werthe der Integrale näherungsweise zu finden. In: Journal für Reine und Angewandte Mathematik. Band 1, 1826, S. 301–308, (online), und Werke, Band 6.
  3. ^ Gander, Walter; Gautschi, Walter (2000). "Adaptive Quadrature - Revisited". BIT Numerical Mathematics. 40 (1): 84–101. doi:10.1023/A:1022318402393.
  4. ^ "Numerical integration - MATLAB integral".
  5. ^ "Functions of One Variable (GNU Octave)". Retrieved 28 September 2018.

External linksEdit