Generalized Averaged Gauss Quadrature Rules: A Survey
Abstract
1. Introduction
2. Some Methods for Estimating the Quadrature Error of Gauss Rules
3. Computation of Averaged Gauss Quadrature Rules
4. Some Extensions and Applications of Laurie’s Averaged and Optimal Averaged Gauss Quadrature Rules
- Let be a large symmetric matrix, let , and let the function f be defined on the convex hull of the spectrum of A. The need to evaluate matrix functionals of the form
- The problem of evaluating expressions of the form
- Let be a large symmetric matrix, let , and let f be a function that is defined on the convex hull of the spectrum of A. The need to evaluate expressions of the form
- Averaged quadrature rules associated with Gauss–Radau and Gauss–Lobatto quadrature formulas are described in [53]. They can be applied to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto rules.
- Padé-type approximants are rational functions that approximate a formal series of polynomials; see Djukić et al. [56] describe the construction and performance of Padé-type approximants that correspond to optimal averaged Gauss quadrature rules.
- The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix. This method is closely related to the symmetric Lanczos method and, therefore, to orthogonal polynomials. Almutairi et al. [57] discuss how Gauss quadrature rules, Laurie’s averaged Gauss rules, and optimal averaged Gauss rules can be applied to estimate the norm of the error in approximate solutions computed by the conjugate gradient method.
- The iterative solution of linear systems with a large symmetric, indefinite, nonsingular matrix by a Lanczos-type method is discussed by Alibrahim et al. [58], who describe how the norm of the error in computed approximate solutions can be estimated by Gauss rules, Laurie’s averaged Gauss rules, and optimal averaged Gauss rules.
- Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. Djukić et al. [59], Díaz de Alba et al. [60], and Fermo et al. [61] discuss their numerical solution by Nyström methods that are based on Gauss quadrature rules. It is important to be able to estimate the error in the computed solution because this makes it possible to choose an appropriate number of nodes in the Gauss quadrature rule used. These papers explore the application of anti-Gauss, Laurie’s averaged Gauss, and weighted averaged Gauss quadrature rules for this purpose and analyze the numerical stability of these methods.
- Cubature rules that generalize Laurie’s averaged rule are described in [59,62]. The development of averaged rules for problems in several space-dimensions is still an active area of research. A difficulty is that the domain of integration in higher dimensions may be of a variety of shapes, see, e.g., [63]. Another issue is that when the domain of integration is simple, say, the unit square in the first quadrant, and the integral is approximated by integrating one space-dimension at a time by Gauss quadrature, integration along the first dimension, say, the horizontal axis, yields approximations that are used when integrating along the vertical axis. It remains to be investigated how the errors in the approximations obtained when integrating in the horizontal direction affect the error estimates obtained when integrating in the vertical direction.
5. Internality of Averaged and Optimal Averaged Gauss Rules
5.1. Results for Classical Weight Functions
- if or , then both averaged Gauss formulas are internal;
- if or , then both averaged Gauss formulas are external;
- if , then for , only the optimal averaged Gauss formula is internal, and for , only Laurie’s averaged Gauss formula is internal.
Modifications by Linear Divisors and Factors
5.2. Chebyshev Weight Functions
5.2.1. Modifications by a Linear Divisor
5.2.2. Modifications by a Linear-Over-Linear Factor
5.3. Modifications of the Jacobi Weight Functions
- the largest node internal if , or and ,
- the smallest node internal if , or and .
- the largest node internal if , or and ,
- the smallest node internal if , or and .
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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n | ||||
5 | ||||
10 | ||||
25 | ||||
50 | ||||
, , and . The outermost nodes of and . | ||||
n | ||||
5 | ||||
10 | ||||
25 | ||||
50 | ||||
, , and . The outermost nodes of . |
Property | Gauss–Kronrod | Optimal Averaged Gauss | Averaged Gauss | NAG |
---|---|---|---|---|
existence | not always | yes | yes | yes |
positivity | not always | yes | yes | yes |
internality | not always | not always | not always | not always |
interlacing | yes | yes | yes | no |
complexity | high | low | low | low |
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Djukić, D.L.; Mutavdžić Djukić, R.M.; Reichel, L.; Spalević, M.M. Generalized Averaged Gauss Quadrature Rules: A Survey. Mathematics 2025, 13, 3145. https://doi.org/10.3390/math13193145
Djukić DL, Mutavdžić Djukić RM, Reichel L, Spalević MM. Generalized Averaged Gauss Quadrature Rules: A Survey. Mathematics. 2025; 13(19):3145. https://doi.org/10.3390/math13193145
Chicago/Turabian StyleDjukić, Dušan L., Rada M. Mutavdžić Djukić, Lothar Reichel, and Miodrag M. Spalević. 2025. "Generalized Averaged Gauss Quadrature Rules: A Survey" Mathematics 13, no. 19: 3145. https://doi.org/10.3390/math13193145
APA StyleDjukić, D. L., Mutavdžić Djukić, R. M., Reichel, L., & Spalević, M. M. (2025). Generalized Averaged Gauss Quadrature Rules: A Survey. Mathematics, 13(19), 3145. https://doi.org/10.3390/math13193145