TY  - JOUR
AU  - Fermo, Luisa
AU  - Reichel, Lothar
AU  - Rodriguez, Giuseppe
AU  - Spalević, Miodrag
PY  - 2024
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7357
PB  - Elsevier
T2  - Applied Mathematics and Computation
T1  - Averaged Nystr\" om  interpolants for the solution of Fredholm integral equations of the second kind
DO  - 10.1016/j.amc.2023.128482
ER  - 

@article{
author = "Fermo, Luisa and Reichel, Lothar and Rodriguez, Giuseppe and Spalević, Miodrag",
year = "2024",
publisher = "Elsevier",
journal = "Applied Mathematics and Computation",
title = "Averaged Nystr\" om  interpolants for the solution of Fredholm integral equations of the second kind",
doi = "10.1016/j.amc.2023.128482"
}

Fermo, L., Reichel, L., Rodriguez, G.,& Spalević, M.. (2024). Averaged Nystr\" om  interpolants for the solution of Fredholm integral equations of the second kind. in Applied Mathematics and Computation
Elsevier..
https://doi.org/10.1016/j.amc.2023.128482

Fermo L, Reichel L, Rodriguez G, Spalević M. Averaged Nystr\" om  interpolants for the solution of Fredholm integral equations of the second kind. in Applied Mathematics and Computation. 2024;.
doi:10.1016/j.amc.2023.128482 .

Fermo, Luisa, Reichel, Lothar, Rodriguez, Giuseppe, Spalević, Miodrag, "Averaged Nystr\" om  interpolants for the solution of Fredholm integral equations of the second kind" in Applied Mathematics and Computation (2024),
https://doi.org/10.1016/j.amc.2023.128482 . .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2024
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7068
AB  - Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating
matrix functionals of the form u
T
f (A)v with a large matrix A ∈ R
N×N by lowrank approximations that are obtained by applying a few steps of the symmetric or
nonsymmetric Lanczos processes to A; here u, v ∈ R
N
are vectors. The latter process
is used when the measure associated with the Gauss quadrature rule has support in
the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix,
whose entries determine the recursion coefficients of the monic orthogonal polynomials
associated with the measure, while the nonsymmetric Lanczos process determines a
nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of
sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results
of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated
with a nonnegative measure with support on the real axis, can be expressed as a
weighted sum of two quadrature rules. This decomposition allows faster evaluation of
optimal averaged Gauss quadrature rules than the previously available representation.
The present paper provides a new self-contained proof of this decomposition that
is based on linear algebra techniques. Moreover, these techniques are generalized to
determine a decomposition of the optimal averaged quadrature rules that are associated
with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also,
the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings
allow faster evaluation of optimal averaged Gauss quadrature rules than the previously
available representations. Computational aspects are discussed.
PB  - Elsevier
T2  - Journal of Computational and Applied Mathematics
T1  - Decompositions of optimal averaged Gauss quadrature rules
IS  - Art.  115586
VL  - 438
DO  - 10.1016/j.cam.2023.115586
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2024",
abstract = "Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating
matrix functionals of the form u
T
f (A)v with a large matrix A ∈ R
N×N by lowrank approximations that are obtained by applying a few steps of the symmetric or
nonsymmetric Lanczos processes to A; here u, v ∈ R
N
are vectors. The latter process
is used when the measure associated with the Gauss quadrature rule has support in
the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix,
whose entries determine the recursion coefficients of the monic orthogonal polynomials
associated with the measure, while the nonsymmetric Lanczos process determines a
nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of
sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results
of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated
with a nonnegative measure with support on the real axis, can be expressed as a
weighted sum of two quadrature rules. This decomposition allows faster evaluation of
optimal averaged Gauss quadrature rules than the previously available representation.
The present paper provides a new self-contained proof of this decomposition that
is based on linear algebra techniques. Moreover, these techniques are generalized to
determine a decomposition of the optimal averaged quadrature rules that are associated
with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also,
the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings
allow faster evaluation of optimal averaged Gauss quadrature rules than the previously
available representations. Computational aspects are discussed.",
publisher = "Elsevier",
journal = "Journal of Computational and Applied Mathematics",
title = "Decompositions of optimal averaged Gauss quadrature rules",
number = "Art.  115586",
volume = "438",
doi = "10.1016/j.cam.2023.115586"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2024). Decompositions of optimal averaged Gauss quadrature rules. in Journal of Computational and Applied Mathematics
Elsevier., 438(Art.  115586).
https://doi.org/10.1016/j.cam.2023.115586

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Decompositions of optimal averaged Gauss quadrature rules. in Journal of Computational and Applied Mathematics. 2024;438(Art.  115586).
doi:10.1016/j.cam.2023.115586 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Decompositions of optimal averaged Gauss quadrature rules" in Journal of Computational and Applied Mathematics, 438, no. Art.  115586 (2024),
https://doi.org/10.1016/j.cam.2023.115586 . .

TY  - JOUR
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2024
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7067
AB  - We describe numerical methods for the construction of interpolatory quadrature rules of Radau and Lobatto types. In particular, we are interested in deriving efficient algorithms for computing optimal averaged Gauss–Radau and Gauss–Lobatto type javascript:undefined;quadrature rules. These averaged rules allow us to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto quadrature rules. This is important since the latter rules have higher algebraic degree of exactness than the corresponding Gauss rules, and this makes it possible to construct averaged quadrature rules of higher algebraic degree of exactness than the corresponding “standard” averaged Gauss rules available in the literature.
PB  - Elsevier
T2  - Journal of Computational and Applied Mathematics
T1  - Radau-  and Lobatto-type averaged Gauss rules
IS  - Art  115477
VL  - 437
DO  - 10.1016/j.cam.2023.115475
ER  - 

@article{
author = "Reichel, Lothar and Spalević, Miodrag",
year = "2024",
abstract = "We describe numerical methods for the construction of interpolatory quadrature rules of Radau and Lobatto types. In particular, we are interested in deriving efficient algorithms for computing optimal averaged Gauss–Radau and Gauss–Lobatto type javascript:undefined;quadrature rules. These averaged rules allow us to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto quadrature rules. This is important since the latter rules have higher algebraic degree of exactness than the corresponding Gauss rules, and this makes it possible to construct averaged quadrature rules of higher algebraic degree of exactness than the corresponding “standard” averaged Gauss rules available in the literature.",
publisher = "Elsevier",
journal = "Journal of Computational and Applied Mathematics",
title = "Radau-  and Lobatto-type averaged Gauss rules",
number = "Art  115477",
volume = "437",
doi = "10.1016/j.cam.2023.115475"
}

Reichel, L.,& Spalević, M.. (2024). Radau-  and Lobatto-type averaged Gauss rules. in Journal of Computational and Applied Mathematics
Elsevier., 437(Art  115477).
https://doi.org/10.1016/j.cam.2023.115475

Reichel L, Spalević M. Radau-  and Lobatto-type averaged Gauss rules. in Journal of Computational and Applied Mathematics. 2024;437(Art  115477).
doi:10.1016/j.cam.2023.115475 .

Reichel, Lothar, Spalević, Miodrag, "Radau-  and Lobatto-type averaged Gauss rules" in Journal of Computational and Applied Mathematics, 437, no. Art  115477 (2024),
https://doi.org/10.1016/j.cam.2023.115475 . .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7065
AB  - Pad\'{e}-type approximants are rational functions that approximate a given formal power series. Boutry \cite{Bo} constructed Pad\'{e}-type approximants that correspond to the averaged Gauss quadrature rules introduced by Laurie \cite{La}. More recently, Spalevi\'c \cite{Sp1} proposed optimal averaged Gauss quadrature rules, that have higher degree of precision than the corresponding averaged Gauss rule, with the same number of nodes. This paper defines Pad\'e-type approximants associated with optimal averaged Gauss rules. Numerical examples illustrate their performance.
PB  - the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM)
T2  - ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
T1  - Optimal Averaged Pade-Type Approximants
EP  - 156
SP  - 145
VL  - 59
DO  - 10.1553/etna_vol59s145
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
abstract = "Pad\'{e}-type approximants are rational functions that approximate a given formal power series. Boutry \cite{Bo} constructed Pad\'{e}-type approximants that correspond to the averaged Gauss quadrature rules introduced by Laurie \cite{La}. More recently, Spalevi\'c \cite{Sp1} proposed optimal averaged Gauss quadrature rules, that have higher degree of precision than the corresponding averaged Gauss rule, with the same number of nodes. This paper defines Pad\'e-type approximants associated with optimal averaged Gauss rules. Numerical examples illustrate their performance.",
publisher = "the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM)",
journal = "ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS",
title = "Optimal Averaged Pade-Type Approximants",
pages = "156-145",
volume = "59",
doi = "10.1553/etna_vol59s145"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Optimal Averaged Pade-Type Approximants. in ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM)., 59, 145-156.
https://doi.org/10.1553/etna_vol59s145

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Optimal Averaged Pade-Type Approximants. in ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS. 2023;59:145-156.
doi:10.1553/etna_vol59s145 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Optimal Averaged Pade-Type Approximants" in ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS, 59 (2023):145-156,
https://doi.org/10.1553/etna_vol59s145 . .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7066
AB  - This paper is concerned with the approximation of integrals of a real-valued integrand over
the interval [−1, 1] by Gauss quadrature. The averaged and optimal averaged quadrature
rules ([13,21]) provide a convenient method for approximating the error in the Gauss
quadrature. However, they are applicable to all integrands that are continuous on the
interval [−1, 1] only if their nodes are internal, i.e. if they belong to this interval.
We discuss two approaches to determine averaged quadrature rules with nodes in
[−1, 1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and
(ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of
the first, second, and third kinds that are modified by a linear over linear rational factor,
and discuss the internality of averaged, optimal averaged, and truncated optimal averaged
quadrature rules. Moreover, we show that the weighting yields internal averaged rules
if a weighting parameter is properly chosen, and we provide bounds for this parameter
that guarantee internality. Finally, we illustrate that the weighted averaged rules give more
accurate estimates of the quadrature error than the truncated optimal averaged rules.
PB  - Elsevier
T2  - Applied Numerical Mathematics
T1  - Weighted averaged Gaussian quadrature rules for modified Chebyshev measures
DO  - 10.1016/j.apnum.2023.05.014
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
abstract = "This paper is concerned with the approximation of integrals of a real-valued integrand over
the interval [−1, 1] by Gauss quadrature. The averaged and optimal averaged quadrature
rules ([13,21]) provide a convenient method for approximating the error in the Gauss
quadrature. However, they are applicable to all integrands that are continuous on the
interval [−1, 1] only if their nodes are internal, i.e. if they belong to this interval.
We discuss two approaches to determine averaged quadrature rules with nodes in
[−1, 1]: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and
(ii) weighting the optimal averaged quadrature rule. We consider Chebyshev measures of
the first, second, and third kinds that are modified by a linear over linear rational factor,
and discuss the internality of averaged, optimal averaged, and truncated optimal averaged
quadrature rules. Moreover, we show that the weighting yields internal averaged rules
if a weighting parameter is properly chosen, and we provide bounds for this parameter
that guarantee internality. Finally, we illustrate that the weighted averaged rules give more
accurate estimates of the quadrature error than the truncated optimal averaged rules.",
publisher = "Elsevier",
journal = "Applied Numerical Mathematics",
title = "Weighted averaged Gaussian quadrature rules for modified Chebyshev measures",
doi = "10.1016/j.apnum.2023.05.014"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Weighted averaged Gaussian quadrature rules for modified Chebyshev measures. in Applied Numerical Mathematics
Elsevier..
https://doi.org/10.1016/j.apnum.2023.05.014

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Weighted averaged Gaussian quadrature rules for modified Chebyshev measures. in Applied Numerical Mathematics. 2023;.
doi:10.1016/j.apnum.2023.05.014 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Weighted averaged Gaussian quadrature rules for modified Chebyshev measures" in Applied Numerical Mathematics (2023),
https://doi.org/10.1016/j.apnum.2023.05.014 . .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - http://acmij.az/view.php?lang=az&menu=0
UR  - http://acmij.az/view.php?lang=az&menu=journal&id=624
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7380
AB  - The internality of quadrature rules, i.e., the property that all nodes lie in the interior of the convex hull of the support of the measure, is important in applications, because this allows the application of these quadrature rules to the approximation of integrals with integrands that are defined in the convex hull of the support of the measure only. It is known that the averaged Gauss and optimal averaged Gauss quadrature rules with respect to the four Chebyshev measures modified by a linear divisor are internal. This paper investigates the internality of similarly modified Jacobi measures, namely measures defined by weight functions. With a, b > −1 and z ∈ R, |z| > 1. We will show that in some cases, depending on the exponents a and b, the averaged and optimal averaged Gauss rules for these measures are internal if the number of nodes is large enough.
PB  - Ministry of Communications and Information Technology (Azerbaijan), Azerbaijan National Academy of Sciences and Institute of Applied Mathematics of Baku State University
T2  - Applied and Computational Mathematics
T1  - Internality of averaged Gauss quadrature rules for certain modification of Jacobi measures
EP  - 442
IS  - 4
SP  - 426
VL  - 22
DO  - 10.30546/1683-6154.22.4.2023.426
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
abstract = "The internality of quadrature rules, i.e., the property that all nodes lie in the interior of the convex hull of the support of the measure, is important in applications, because this allows the application of these quadrature rules to the approximation of integrals with integrands that are defined in the convex hull of the support of the measure only. It is known that the averaged Gauss and optimal averaged Gauss quadrature rules with respect to the four Chebyshev measures modified by a linear divisor are internal. This paper investigates the internality of similarly modified Jacobi measures, namely measures defined by weight functions. With a, b > −1 and z ∈ R, |z| > 1. We will show that in some cases, depending on the exponents a and b, the averaged and optimal averaged Gauss rules for these measures are internal if the number of nodes is large enough.",
publisher = "Ministry of Communications and Information Technology (Azerbaijan), Azerbaijan National Academy of Sciences and Institute of Applied Mathematics of Baku State University",
journal = "Applied and Computational Mathematics",
title = "Internality of averaged Gauss quadrature rules for certain modification of Jacobi measures",
pages = "442-426",
number = "4",
volume = "22",
doi = "10.30546/1683-6154.22.4.2023.426"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Internality of averaged Gauss quadrature rules for certain modification of Jacobi measures. in Applied and Computational Mathematics
Ministry of Communications and Information Technology (Azerbaijan), Azerbaijan National Academy of Sciences and Institute of Applied Mathematics of Baku State University., 22(4), 426-442.
https://doi.org/10.30546/1683-6154.22.4.2023.426

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Internality of averaged Gauss quadrature rules for certain modification of Jacobi measures. in Applied and Computational Mathematics. 2023;22(4):426-442.
doi:10.30546/1683-6154.22.4.2023.426 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Internality of averaged Gauss quadrature rules for certain modification of Jacobi measures" in Applied and Computational Mathematics, 22, no. 4 (2023):426-442,
https://doi.org/10.30546/1683-6154.22.4.2023.426 . .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - http://www.ic-mrs.org/
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7662
AB  - The averaged and optimal averaged quadrature rules provide a convenient method of approximating the error in the Gauss quadrature. However, they are fully applicable only if their nodes are internal. We discuss two approaches to determine averaged quadrature rules with internal nodes: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. A survey of our results on internality of averaged Gaussian quadrature rules will be presented.
C3  - 6TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND RELATED SCIENCES  BOOK OF ABSTRACTS
T1  - Internality of Averaged Gaussian Quadrature Rules
UR  - https://hdl.handle.net/21.15107/rcub_machinery_7662
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
abstract = "The averaged and optimal averaged quadrature rules provide a convenient method of approximating the error in the Gauss quadrature. However, they are fully applicable only if their nodes are internal. We discuss two approaches to determine averaged quadrature rules with internal nodes: (i) truncating the Jacobi matrix associated with the optimal averaged rule, and (ii) weighting the optimal averaged quadrature rule. A survey of our results on internality of averaged Gaussian quadrature rules will be presented.",
journal = "6TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND RELATED SCIENCES  BOOK OF ABSTRACTS",
title = "Internality of Averaged Gaussian Quadrature Rules",
url = "https://hdl.handle.net/21.15107/rcub_machinery_7662"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Internality of Averaged Gaussian Quadrature Rules. in 6TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND RELATED SCIENCES  BOOK OF ABSTRACTS.
https://hdl.handle.net/21.15107/rcub_machinery_7662

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Internality of Averaged Gaussian Quadrature Rules. in 6TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND RELATED SCIENCES  BOOK OF ABSTRACTS. 2023;.
https://hdl.handle.net/21.15107/rcub_machinery_7662 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Internality of Averaged Gaussian Quadrature Rules" in 6TH INTERNATIONAL CONFERENCE ON MATHEMATICAL AND RELATED SCIENCES  BOOK OF ABSTRACTS (2023),
https://hdl.handle.net/21.15107/rcub_machinery_7662 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - https://imi.pmf.kg.ac.rs/aaa2023/
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/7216
PB  - Prirodno-matematički fakultet Kragujevac
C3  - International Mathematical Conference Analysis, Approximations and Applications (AAA2023), Vrnjačka Banja
T1  - On Internality of Generalized Averaged Gaussian Quadrature Rules and Their Truncations
UR  - https://hdl.handle.net/21.15107/rcub_machinery_7216
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
publisher = "Prirodno-matematički fakultet Kragujevac",
journal = "International Mathematical Conference Analysis, Approximations and Applications (AAA2023), Vrnjačka Banja",
title = "On Internality of Generalized Averaged Gaussian Quadrature Rules and Their Truncations",
url = "https://hdl.handle.net/21.15107/rcub_machinery_7216"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). On Internality of Generalized Averaged Gaussian Quadrature Rules and Their Truncations. in International Mathematical Conference Analysis, Approximations and Applications (AAA2023), Vrnjačka Banja
Prirodno-matematički fakultet Kragujevac..
https://hdl.handle.net/21.15107/rcub_machinery_7216

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. On Internality of Generalized Averaged Gaussian Quadrature Rules and Their Truncations. in International Mathematical Conference Analysis, Approximations and Applications (AAA2023), Vrnjačka Banja. 2023;.
https://hdl.handle.net/21.15107/rcub_machinery_7216 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "On Internality of Generalized Averaged Gaussian Quadrature Rules and Their Truncations" in International Mathematical Conference Analysis, Approximations and Applications (AAA2023), Vrnjačka Banja (2023),
https://hdl.handle.net/21.15107/rcub_machinery_7216 .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2023
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5111
PB  - Springer
T2  - Numerical Algorithms
T1  - Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds
EP  - 544
SP  - 523
VL  - 92
DO  - 10.1007/s11075-022-01385-w
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2023",
publisher = "Springer",
journal = "Numerical Algorithms",
title = "Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds",
pages = "544-523",
volume = "92",
doi = "10.1007/s11075-022-01385-w"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2023). Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds. in Numerical Algorithms
Springer., 92, 523-544.
https://doi.org/10.1007/s11075-022-01385-w

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds. in Numerical Algorithms. 2023;92:523-544.
doi:10.1007/s11075-022-01385-w .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the third and fourth kinds" in Numerical Algorithms, 92 (2023):523-544,
https://doi.org/10.1007/s11075-022-01385-w . .

TY  - JOUR
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3685
AB  - The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, Spalevic derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. Numerical experiments reported in this paper show both kinds of averaged rules to often give much higher accuracy than can be expected from their degrees of exactness. This is important when estimating the error in a Gauss rule by an associated averaged rule. We use techniques similar to those employed by Trefethen in his investigation of Clenshaw-Curtis rules to shed light on the performance of the averaged rules. The averaged rules are not guaranteed to be internal, i.e., they may have nodes outside the convex hull of the support of the measure. This paper discusses three approaches to modify averaged rules to make them internal.
PB  - Elsevier, Amsterdam
T2  - Journal of Computational and Applied Mathematics
T1  - Averaged Gauss quadrature formulas: Properties and applications
VL  - 410
DO  - 10.1016/j.cam.2022.114232
ER  - 

@article{
author = "Reichel, Lothar and Spalević, Miodrag",
year = "2022",
abstract = "The estimation of the quadrature error of a Gauss quadrature rule when applied to the approximation of an integral determined by a real-valued integrand and a real-valued nonnegative measure with support on the real axis is an important problem in scientific computing. Laurie developed anti-Gauss quadrature rules as an aid to estimate this error. Under suitable conditions the Gauss and associated anti-Gauss rules give upper and lower bounds for the value of the desired integral. It is then natural to use the average of Gauss and anti-Gauss rules as an improved approximation of the integral. Laurie also introduced these averaged rules. More recently, Spalevic derived new averaged Gauss quadrature rules that have higher degree of exactness for the same number of nodes as the averaged rules proposed by Laurie. Numerical experiments reported in this paper show both kinds of averaged rules to often give much higher accuracy than can be expected from their degrees of exactness. This is important when estimating the error in a Gauss rule by an associated averaged rule. We use techniques similar to those employed by Trefethen in his investigation of Clenshaw-Curtis rules to shed light on the performance of the averaged rules. The averaged rules are not guaranteed to be internal, i.e., they may have nodes outside the convex hull of the support of the measure. This paper discusses three approaches to modify averaged rules to make them internal.",
publisher = "Elsevier, Amsterdam",
journal = "Journal of Computational and Applied Mathematics",
title = "Averaged Gauss quadrature formulas: Properties and applications",
volume = "410",
doi = "10.1016/j.cam.2022.114232"
}

Reichel, L.,& Spalević, M.. (2022). Averaged Gauss quadrature formulas: Properties and applications. in Journal of Computational and Applied Mathematics
Elsevier, Amsterdam., 410.
https://doi.org/10.1016/j.cam.2022.114232

Reichel L, Spalević M. Averaged Gauss quadrature formulas: Properties and applications. in Journal of Computational and Applied Mathematics. 2022;410.
doi:10.1016/j.cam.2022.114232 .

Reichel, Lothar, Spalević, Miodrag, "Averaged Gauss quadrature formulas: Properties and applications" in Journal of Computational and Applied Mathematics, 410 (2022),
https://doi.org/10.1016/j.cam.2022.114232 . .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5153
C3  - FAATNA 2020>22 conference, Book of abstracts
T1  - Internality of averaged Gaussian quadrature rules for modified Jacobi measures
EP  - 193
SP  - 193
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5153
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2022",
journal = "FAATNA 2020>22 conference, Book of abstracts",
title = "Internality of averaged Gaussian quadrature rules for modified Jacobi measures",
pages = "193-193",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5153"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2022). Internality of averaged Gaussian quadrature rules for modified Jacobi measures. in FAATNA 2020>22 conference, Book of abstracts, 193-193.
https://hdl.handle.net/21.15107/rcub_machinery_5153

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Internality of averaged Gaussian quadrature rules for modified Jacobi measures. in FAATNA 2020>22 conference, Book of abstracts. 2022;:193-193.
https://hdl.handle.net/21.15107/rcub_machinery_5153 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Internality of averaged Gaussian quadrature rules for modified Jacobi measures" in FAATNA 2020>22 conference, Book of abstracts (2022):193-193,
https://hdl.handle.net/21.15107/rcub_machinery_5153 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5159
PB  - Faculty of Mechanical Engineering, University of Belgrade
C3  - NMLSP conference, Book of abstracts
T1  - Optimal averaged Pade approximants
EP  - 65
SP  - 65
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5159
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2022",
publisher = "Faculty of Mechanical Engineering, University of Belgrade",
journal = "NMLSP conference, Book of abstracts",
title = "Optimal averaged Pade approximants",
pages = "65-65",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5159"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2022). Optimal averaged Pade approximants. in NMLSP conference, Book of abstracts
Faculty of Mechanical Engineering, University of Belgrade., 65-65.
https://hdl.handle.net/21.15107/rcub_machinery_5159

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Optimal averaged Pade approximants. in NMLSP conference, Book of abstracts. 2022;:65-65.
https://hdl.handle.net/21.15107/rcub_machinery_5159 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Optimal averaged Pade approximants" in NMLSP conference, Book of abstracts (2022):65-65,
https://hdl.handle.net/21.15107/rcub_machinery_5159 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5155
C3  - FAATNA 2020>22, Book of abstarcts
T1  - Weighted averaged Gaussian quadrature rules for modified Chebyshev measure
EP  - 197
SP  - 197
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5155
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2022",
journal = "FAATNA 2020>22, Book of abstarcts",
title = "Weighted averaged Gaussian quadrature rules for modified Chebyshev measure",
pages = "197-197",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5155"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2022). Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. in FAATNA 2020>22, Book of abstarcts, 197-197.
https://hdl.handle.net/21.15107/rcub_machinery_5155

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. in FAATNA 2020>22, Book of abstarcts. 2022;:197-197.
https://hdl.handle.net/21.15107/rcub_machinery_5155 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Weighted averaged Gaussian quadrature rules for modified Chebyshev measure" in FAATNA 2020>22, Book of abstarcts (2022):197-197,
https://hdl.handle.net/21.15107/rcub_machinery_5155 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5162
PB  - Faculty of Mechanical Engineering, University of Belgrade
C3  - MNA conference, Book of abstracts
T1  - Weighted averaged Gaussian quadrature rules for modified Chebyshev measure
EP  - 19
SP  - 19
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5162
ER  - 

@conference{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2022",
publisher = "Faculty of Mechanical Engineering, University of Belgrade",
journal = "MNA conference, Book of abstracts",
title = "Weighted averaged Gaussian quadrature rules for modified Chebyshev measure",
pages = "19-19",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5162"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2022). Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. in MNA conference, Book of abstracts
Faculty of Mechanical Engineering, University of Belgrade., 19-19.
https://hdl.handle.net/21.15107/rcub_machinery_5162

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. in MNA conference, Book of abstracts. 2022;:19-19.
https://hdl.handle.net/21.15107/rcub_machinery_5162 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Weighted averaged Gaussian quadrature rules for modified Chebyshev measure" in MNA conference, Book of abstracts (2022):19-19,
https://hdl.handle.net/21.15107/rcub_machinery_5162 .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2021
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3477
AB  - It is desirable that a quadrature rule be internal, i.e., that all nodes of the rule live in the convex hull of the support of the measure. Then the rule can be applied to approximate integrals of functions that have a singularity close to the convex hull of the support of the measure. This paper investigates whether generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the first kind are internal. These rules are applied to estimate the error in Gauss quadrature rules associated with modified Chebyshev measures of the first kind. It is of considerable interest to be able to assess the error in quadrature rules in order to be able to choose a rule that gives an approximation of the desired integral of sufficient accuracy without having to evaluate the integrand at unnecessarily many nodes. Some of the generalized averaged Gauss quadrature formulas considered are found not to be internal. We will show that some truncated variants of these rules are internal, and therefore can be applied to estimate the error in Gauss quadrature rules also when the integrand has singularities on the real axis close to the interval of integration.
PB  - Elsevier, Amsterdam
T2  - Journal of Computational and Applied Mathematics
T1  - Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind
VL  - 398
DO  - 10.1016/j.cam.2021.113696
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Reichel, Lothar and Spalević, Miodrag",
year = "2021",
abstract = "It is desirable that a quadrature rule be internal, i.e., that all nodes of the rule live in the convex hull of the support of the measure. Then the rule can be applied to approximate integrals of functions that have a singularity close to the convex hull of the support of the measure. This paper investigates whether generalized averaged Gauss quadrature formulas for modified Chebyshev measures of the first kind are internal. These rules are applied to estimate the error in Gauss quadrature rules associated with modified Chebyshev measures of the first kind. It is of considerable interest to be able to assess the error in quadrature rules in order to be able to choose a rule that gives an approximation of the desired integral of sufficient accuracy without having to evaluate the integrand at unnecessarily many nodes. Some of the generalized averaged Gauss quadrature formulas considered are found not to be internal. We will show that some truncated variants of these rules are internal, and therefore can be applied to estimate the error in Gauss quadrature rules also when the integrand has singularities on the real axis close to the interval of integration.",
publisher = "Elsevier, Amsterdam",
journal = "Journal of Computational and Applied Mathematics",
title = "Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind",
volume = "398",
doi = "10.1016/j.cam.2021.113696"
}

Đukić, D., Mutavdžić Đukić, R., Reichel, L.,& Spalević, M.. (2021). Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind. in Journal of Computational and Applied Mathematics
Elsevier, Amsterdam., 398.
https://doi.org/10.1016/j.cam.2021.113696

Đukić D, Mutavdžić Đukić R, Reichel L, Spalević M. Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind. in Journal of Computational and Applied Mathematics. 2021;398.
doi:10.1016/j.cam.2021.113696 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Reichel, Lothar, Spalević, Miodrag, "Internality of generalized averaged Gauss quadrature rules and truncated variants for modified Chebyshev measures of the first kind" in Journal of Computational and Applied Mathematics, 398 (2021),
https://doi.org/10.1016/j.cam.2021.113696 . .

TY  - JOUR
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2021
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3597
AB  - Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation.
PB  - Elsevier, Amsterdam
T2  - Applied Numerical Mathematics
T1  - A new representation of generalized averaged Gauss quadrature rules
EP  - 619
SP  - 614
VL  - 165
DO  - 10.1016/j.apnum.2020.11.016
ER  - 

@article{
author = "Reichel, Lothar and Spalević, Miodrag",
year = "2021",
abstract = "Gauss quadrature rules associated with a nonnegative measure with support on (part of) the real axis find many applications in Scientific Computing. It is important to be able to estimate the quadrature error when replacing an integral by an l-node Gauss quadrature rule in order to choose a suitable number of nodes. A classical approach to estimate this error is to evaluate the associated (2l + 1)-node Gauss-Kronrod rule. However, Gauss-Kronrod rules with 2l + 1 real nodes might not exist. The (2l + 1)-node generalized averaged Gauss formula associated with the l-node Gauss rule described in Spalevic (2007) [16] is guaranteed to exist and provides an attractive alternative to the (2l + 1)-node Gauss-Kronrod rule. This paper describes a new representation of generalized averaged Gauss formulas that is cheaper to evaluate than the available representation.",
publisher = "Elsevier, Amsterdam",
journal = "Applied Numerical Mathematics",
title = "A new representation of generalized averaged Gauss quadrature rules",
pages = "619-614",
volume = "165",
doi = "10.1016/j.apnum.2020.11.016"
}

Reichel, L.,& Spalević, M.. (2021). A new representation of generalized averaged Gauss quadrature rules. in Applied Numerical Mathematics
Elsevier, Amsterdam., 165, 614-619.
https://doi.org/10.1016/j.apnum.2020.11.016

Reichel L, Spalević M. A new representation of generalized averaged Gauss quadrature rules. in Applied Numerical Mathematics. 2021;165:614-619.
doi:10.1016/j.apnum.2020.11.016 .

Reichel, Lothar, Spalević, Miodrag, "A new representation of generalized averaged Gauss quadrature rules" in Applied Numerical Mathematics, 165 (2021):614-619,
https://doi.org/10.1016/j.apnum.2020.11.016 . .

TY  - JOUR
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
AU  - Tomanović, Jelena
PY  - 2020
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3329
AB  - It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.
PB  - Univerzitet u Nišu - Prirodno-matematički fakultet - Departmant za matematiku i informatiku, Niš
T2  - Filomat
T1  - Rational Averaged Gauss Quadrature Rules
EP  - 389
IS  - 2
SP  - 379
VL  - 34
DO  - 10.2298/FIL2002379R
ER  - 

@article{
author = "Reichel, Lothar and Spalević, Miodrag and Tomanović, Jelena",
year = "2020",
abstract = "It is important to be able to estimate the quadrature error in Gauss rules. Several approaches have been developed, including the evaluation of associated Gauss-Kronrod rules (if they exist), or the associated averaged Gauss and generalized averaged Gauss rules. Integrals with certain integrands can be approximated more accurately by rational Gauss rules than by Gauss rules. This paper introduces associated rational averaged Gauss rules and rational generalized averaged Gauss rules, which can be used to estimate the error in rational Gauss rules. Also rational Gauss-Kronrod rules are discussed. Computed examples illustrate the accuracy of the error estimates determined by these quadrature rules.",
publisher = "Univerzitet u Nišu - Prirodno-matematički fakultet - Departmant za matematiku i informatiku, Niš",
journal = "Filomat",
title = "Rational Averaged Gauss Quadrature Rules",
pages = "389-379",
number = "2",
volume = "34",
doi = "10.2298/FIL2002379R"
}

Reichel, L., Spalević, M.,& Tomanović, J.. (2020). Rational Averaged Gauss Quadrature Rules. in Filomat
Univerzitet u Nišu - Prirodno-matematički fakultet - Departmant za matematiku i informatiku, Niš., 34(2), 379-389.
https://doi.org/10.2298/FIL2002379R

Reichel L, Spalević M, Tomanović J. Rational Averaged Gauss Quadrature Rules. in Filomat. 2020;34(2):379-389.
doi:10.2298/FIL2002379R .

Reichel, Lothar, Spalević, Miodrag, Tomanović, Jelena, "Rational Averaged Gauss Quadrature Rules" in Filomat, 34, no. 2 (2020):379-389,
https://doi.org/10.2298/FIL2002379R . .

TY  - CONF
AU  - Pranić, Miroslav
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2019
UR  - https://iciam2019.com/images/site/news/ICIAM2019_PROGRAM_ABSTRACTS_BOOK.pdf
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5252
AB  - Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described.
T1  - MS FT-2-2 7  Orthogonal polynomials and quadrature: Theory, computation, and applications
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5252
ER  - 

@conference{
author = "Pranić, Miroslav and Reichel, Lothar and Spalević, Miodrag",
year = "2019",
abstract = "Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described.",
title = "MS FT-2-2 7  Orthogonal polynomials and quadrature: Theory, computation, and applications",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5252"
}

Pranić, M., Reichel, L.,& Spalević, M.. (2019). MS FT-2-2 7  Orthogonal polynomials and quadrature: Theory, computation, and applications. .
https://hdl.handle.net/21.15107/rcub_machinery_5252

Pranić M, Reichel L, Spalević M. MS FT-2-2 7  Orthogonal polynomials and quadrature: Theory, computation, and applications. 2019;.
https://hdl.handle.net/21.15107/rcub_machinery_5252 .

Pranić, Miroslav, Reichel, Lothar, Spalević, Miodrag, "MS FT-2-2 7  Orthogonal polynomials and quadrature: Theory, computation, and applications" (2019),
https://hdl.handle.net/21.15107/rcub_machinery_5252 .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2019
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3017
AB  - Generalized averaged Gaussian quadrature rules and truncated variants associated with a nonnegative measure with support on a real open interval {t : a  lt  t  lt  b} may have nodes outside this interval, in other words the rules may fail to be internal. Such rules cannot be applied when the integrand is defined on {t : a  lt  t  lt  b} only. This paper investigates whether generalized averaged Gaussian quadrature rules and truncated variants are internal for measures induced by Chebyshev polynomials. Our results complement those of Notaris [13] for Gauss-Kronrod quadrature formulas for the same kind of measures.
PB  - Elsevier Science Bv, Amsterdam
T2  - Applied Numerical Mathematics
T1  - Internality of generalized averaged Gaussian quadrature rules and truncated variants for measures induced by Chebyshev polynomials
EP  - 205
SP  - 190
VL  - 142
DO  - 10.1016/j.apnum.2019.03.008
ER  - 

@article{
author = "Đukić, Dušan and Reichel, Lothar and Spalević, Miodrag",
year = "2019",
abstract = "Generalized averaged Gaussian quadrature rules and truncated variants associated with a nonnegative measure with support on a real open interval {t : a  lt  t  lt  b} may have nodes outside this interval, in other words the rules may fail to be internal. Such rules cannot be applied when the integrand is defined on {t : a  lt  t  lt  b} only. This paper investigates whether generalized averaged Gaussian quadrature rules and truncated variants are internal for measures induced by Chebyshev polynomials. Our results complement those of Notaris [13] for Gauss-Kronrod quadrature formulas for the same kind of measures.",
publisher = "Elsevier Science Bv, Amsterdam",
journal = "Applied Numerical Mathematics",
title = "Internality of generalized averaged Gaussian quadrature rules and truncated variants for measures induced by Chebyshev polynomials",
pages = "205-190",
volume = "142",
doi = "10.1016/j.apnum.2019.03.008"
}

Đukić, D., Reichel, L.,& Spalević, M.. (2019). Internality of generalized averaged Gaussian quadrature rules and truncated variants for measures induced by Chebyshev polynomials. in Applied Numerical Mathematics
Elsevier Science Bv, Amsterdam., 142, 190-205.
https://doi.org/10.1016/j.apnum.2019.03.008

Đukić D, Reichel L, Spalević M. Internality of generalized averaged Gaussian quadrature rules and truncated variants for measures induced by Chebyshev polynomials. in Applied Numerical Mathematics. 2019;142:190-205.
doi:10.1016/j.apnum.2019.03.008 .

Đukić, Dušan, Reichel, Lothar, Spalević, Miodrag, "Internality of generalized averaged Gaussian quadrature rules and truncated variants for measures induced by Chebyshev polynomials" in Applied Numerical Mathematics, 142 (2019):190-205,
https://doi.org/10.1016/j.apnum.2019.03.008 . .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
AU  - Tomanović, Jelena
PY  - 2019
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3078
AB  - Generalized averaged Gaussian quadrature rules associated with some measure, and truncated variants of these rules, can be used to estimate the error in Gaussian quadrature rules. However, the former quadrature rules may have nodes outside the interval of integration and, therefore, it may not be possible to apply them when the integrand is defined on the interval of integration only. This paper investigates whether generalized averaged Gaussian quadrature rules associated with modified Chebyshev measures of the second kind, and truncated variants of these rules, are internal, i.e. if all nodes of these quadrature rules are in the interval of integration.
PB  - Elsevier Science Bv, Amsterdam
T2  - Journal of Computational and Applied Mathematics
T1  - Internality of generalized averaged Gaussian quadrature rules and truncated variants for modified Chebyshev measures of the second kind
EP  - 85
SP  - 70
VL  - 345
DO  - 10.1016/j.cam.2018.06.017
ER  - 

@article{
author = "Đukić, Dušan and Reichel, Lothar and Spalević, Miodrag and Tomanović, Jelena",
year = "2019",
abstract = "Generalized averaged Gaussian quadrature rules associated with some measure, and truncated variants of these rules, can be used to estimate the error in Gaussian quadrature rules. However, the former quadrature rules may have nodes outside the interval of integration and, therefore, it may not be possible to apply them when the integrand is defined on the interval of integration only. This paper investigates whether generalized averaged Gaussian quadrature rules associated with modified Chebyshev measures of the second kind, and truncated variants of these rules, are internal, i.e. if all nodes of these quadrature rules are in the interval of integration.",
publisher = "Elsevier Science Bv, Amsterdam",
journal = "Journal of Computational and Applied Mathematics",
title = "Internality of generalized averaged Gaussian quadrature rules and truncated variants for modified Chebyshev measures of the second kind",
pages = "85-70",
volume = "345",
doi = "10.1016/j.cam.2018.06.017"
}

Đukić, D., Reichel, L., Spalević, M.,& Tomanović, J.. (2019). Internality of generalized averaged Gaussian quadrature rules and truncated variants for modified Chebyshev measures of the second kind. in Journal of Computational and Applied Mathematics
Elsevier Science Bv, Amsterdam., 345, 70-85.
https://doi.org/10.1016/j.cam.2018.06.017

Đukić D, Reichel L, Spalević M, Tomanović J. Internality of generalized averaged Gaussian quadrature rules and truncated variants for modified Chebyshev measures of the second kind. in Journal of Computational and Applied Mathematics. 2019;345:70-85.
doi:10.1016/j.cam.2018.06.017 .

Đukić, Dušan, Reichel, Lothar, Spalević, Miodrag, Tomanović, Jelena, "Internality of generalized averaged Gaussian quadrature rules and truncated variants for modified Chebyshev measures of the second kind" in Journal of Computational and Applied Mathematics, 345 (2019):70-85,
https://doi.org/10.1016/j.cam.2018.06.017 . .

TY  - CONF
AU  - Đukić, Dušan
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2018
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6106
AB  - When moments or modi ed moments of the weight function are difficult to
compute, generalized averaged Gaussian quadratures can serve as good substitutes.
These formulas were introduced by Spalević [3], where it was demonstrated
that they may yield a smaller error compared to the Gauss quadrature
rules. However, generalized averaged Gaussian quadratures may have external
nodes. This would make them unusable when the domain of the integrand is
limited to the convex hull of the support of the weight function. In this paper
we investigate whether removing some of the last rows and columns of their
Jacobi matrices (cf. [2]) will produce quadrature rules with no external nodes.
The results that will be presented have been recently published in [1].
PB  - Department of Mathematics, Faculty of Science, Akdeniz University,Turkey
C3  - Proceedings Book of MICOPAM2018 conference
T1  - Internality of truncated averaged Gaussian quadratures
EP  - 66
SP  - 62
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6106
ER  - 

@conference{
author = "Đukić, Dušan and Reichel, Lothar and Spalević, Miodrag",
year = "2018",
abstract = "When moments or modi ed moments of the weight function are difficult to
compute, generalized averaged Gaussian quadratures can serve as good substitutes.
These formulas were introduced by Spalević [3], where it was demonstrated
that they may yield a smaller error compared to the Gauss quadrature
rules. However, generalized averaged Gaussian quadratures may have external
nodes. This would make them unusable when the domain of the integrand is
limited to the convex hull of the support of the weight function. In this paper
we investigate whether removing some of the last rows and columns of their
Jacobi matrices (cf. [2]) will produce quadrature rules with no external nodes.
The results that will be presented have been recently published in [1].",
publisher = "Department of Mathematics, Faculty of Science, Akdeniz University,Turkey",
journal = "Proceedings Book of MICOPAM2018 conference",
title = "Internality of truncated averaged Gaussian quadratures",
pages = "66-62",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6106"
}

Đukić, D., Reichel, L.,& Spalević, M.. (2018). Internality of truncated averaged Gaussian quadratures. in Proceedings Book of MICOPAM2018 conference
Department of Mathematics, Faculty of Science, Akdeniz University,Turkey., 62-66.
https://hdl.handle.net/21.15107/rcub_machinery_6106

Đukić D, Reichel L, Spalević M. Internality of truncated averaged Gaussian quadratures. in Proceedings Book of MICOPAM2018 conference. 2018;:62-66.
https://hdl.handle.net/21.15107/rcub_machinery_6106 .

Đukić, Dušan, Reichel, Lothar, Spalević, Miodrag, "Internality of truncated averaged Gaussian quadratures" in Proceedings Book of MICOPAM2018 conference (2018):62-66,
https://hdl.handle.net/21.15107/rcub_machinery_6106 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2017
UR  - https://easychair.org/smart-program/ACTA2017/index.html
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5175
PB  - Serbian Academy of Sciences and Arts
C3  - ACTA 2017, Book of abstarcts
T1  - Internality of truncated generalized averaged Gaussian quadrature
EP  - 23
SP  - 23
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5175
ER  - 

@conference{
author = "Đukić, Dušan and Reichel, Lothar and Spalević, Miodrag",
year = "2017",
publisher = "Serbian Academy of Sciences and Arts",
journal = "ACTA 2017, Book of abstarcts",
title = "Internality of truncated generalized averaged Gaussian quadrature",
pages = "23-23",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5175"
}

Đukić, D., Reichel, L.,& Spalević, M.. (2017). Internality of truncated generalized averaged Gaussian quadrature. in ACTA 2017, Book of abstarcts
Serbian Academy of Sciences and Arts., 23-23.
https://hdl.handle.net/21.15107/rcub_machinery_5175

Đukić D, Reichel L, Spalević M. Internality of truncated generalized averaged Gaussian quadrature. in ACTA 2017, Book of abstarcts. 2017;:23-23.
https://hdl.handle.net/21.15107/rcub_machinery_5175 .

Đukić, Dušan, Reichel, Lothar, Spalević, Miodrag, "Internality of truncated generalized averaged Gaussian quadrature" in ACTA 2017, Book of abstarcts (2017):23-23,
https://hdl.handle.net/21.15107/rcub_machinery_5175 .

TY  - JOUR
AU  - Eshghi, Nasim
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2017
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/2731
AB  - Matrix functions of the form f (A) v, where A is a large symmetric matrix, f is a function, and v not equal 0 is a vector, are commonly approximated by first applying a few, say n, steps of the symmetric Lanczos process to A with the initial vector v in order to determine an orthogonal section of A. The latter is represented by a (small) n x n tridiagonal matrix to which f is applied. This approach uses the n first Lanczos vectors provided by the Lanczos process. However, n steps of the Lanczos process yield n + 1 Lanczos vectors. This paper discusses how the (n + 1) st Lanczos vector can be used to improve the quality of the computed approximation of f (A) v. Also the approximation of expressions of the form v(T) f (A) v is considered.
PB  - Kent State University
T2  - Electronic Transactions on Numerical Analysis
T1  - Enhanced matrix function approximation
EP  - 205
SP  - 197
VL  - 47
UR  - https://hdl.handle.net/21.15107/rcub_machinery_2731
ER  - 

@article{
author = "Eshghi, Nasim and Reichel, Lothar and Spalević, Miodrag",
year = "2017",
abstract = "Matrix functions of the form f (A) v, where A is a large symmetric matrix, f is a function, and v not equal 0 is a vector, are commonly approximated by first applying a few, say n, steps of the symmetric Lanczos process to A with the initial vector v in order to determine an orthogonal section of A. The latter is represented by a (small) n x n tridiagonal matrix to which f is applied. This approach uses the n first Lanczos vectors provided by the Lanczos process. However, n steps of the Lanczos process yield n + 1 Lanczos vectors. This paper discusses how the (n + 1) st Lanczos vector can be used to improve the quality of the computed approximation of f (A) v. Also the approximation of expressions of the form v(T) f (A) v is considered.",
publisher = "Kent State University",
journal = "Electronic Transactions on Numerical Analysis",
title = "Enhanced matrix function approximation",
pages = "205-197",
volume = "47",
url = "https://hdl.handle.net/21.15107/rcub_machinery_2731"
}

Eshghi, N., Reichel, L.,& Spalević, M.. (2017). Enhanced matrix function approximation. in Electronic Transactions on Numerical Analysis
Kent State University., 47, 197-205.
https://hdl.handle.net/21.15107/rcub_machinery_2731

Eshghi N, Reichel L, Spalević M. Enhanced matrix function approximation. in Electronic Transactions on Numerical Analysis. 2017;47:197-205.
https://hdl.handle.net/21.15107/rcub_machinery_2731 .

Eshghi, Nasim, Reichel, Lothar, Spalević, Miodrag, "Enhanced matrix function approximation" in Electronic Transactions on Numerical Analysis, 47 (2017):197-205,
https://hdl.handle.net/21.15107/rcub_machinery_2731 .

TY  - CONF
AU  - Đukić, Dušan
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
PY  - 2016
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/5138
AB  - Generalized averaged Gaussian quadrature formulas may yield a smaller error than
Gauss quadrature rules. When moments or modified moments are difficult to compute, these formulas can serve as good substitutes. However, generalized averaged
Gaussian quadrature formulas may have external nodes, i.e. nodes outside the convex hull of the measure corresponding to the Gauss rules. This would make them
unusable when the domain of the integrand is limited to this convex hull. In this
paper we investigate whether removing some of the last rows and columns of the
matrices determining generalized averaged Gaussian quadrature rules will produce
quadrature rules with no external nodes.
PB  - UNIVERSITY OF EAST SARAJEVO,  MATHEMATICAL SOCIETY OF THE REPUBLIC OF SRPSKA
C3  - 6th MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA, BOOK OF ABSTRACTS
T1  - GENERALIZED AVERAGED GAUSSIAN QUADRATURE FORMULAS WITH MODIFIED MATRICES
EP  - 27
SP  - 27
UR  - https://hdl.handle.net/21.15107/rcub_machinery_5138
ER  - 

@conference{
author = "Đukić, Dušan and Reichel, Lothar and Spalević, Miodrag",
year = "2016",
abstract = "Generalized averaged Gaussian quadrature formulas may yield a smaller error than
Gauss quadrature rules. When moments or modified moments are difficult to compute, these formulas can serve as good substitutes. However, generalized averaged
Gaussian quadrature formulas may have external nodes, i.e. nodes outside the convex hull of the measure corresponding to the Gauss rules. This would make them
unusable when the domain of the integrand is limited to this convex hull. In this
paper we investigate whether removing some of the last rows and columns of the
matrices determining generalized averaged Gaussian quadrature rules will produce
quadrature rules with no external nodes.",
publisher = "UNIVERSITY OF EAST SARAJEVO,  MATHEMATICAL SOCIETY OF THE REPUBLIC OF SRPSKA",
journal = "6th MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA, BOOK OF ABSTRACTS",
title = "GENERALIZED AVERAGED GAUSSIAN QUADRATURE FORMULAS WITH MODIFIED MATRICES",
pages = "27-27",
url = "https://hdl.handle.net/21.15107/rcub_machinery_5138"
}

Đukić, D., Reichel, L.,& Spalević, M.. (2016). GENERALIZED AVERAGED GAUSSIAN QUADRATURE FORMULAS WITH MODIFIED MATRICES. in 6th MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA, BOOK OF ABSTRACTS
UNIVERSITY OF EAST SARAJEVO,  MATHEMATICAL SOCIETY OF THE REPUBLIC OF SRPSKA., 27-27.
https://hdl.handle.net/21.15107/rcub_machinery_5138

Đukić D, Reichel L, Spalević M. GENERALIZED AVERAGED GAUSSIAN QUADRATURE FORMULAS WITH MODIFIED MATRICES. in 6th MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA, BOOK OF ABSTRACTS. 2016;:27-27.
https://hdl.handle.net/21.15107/rcub_machinery_5138 .

Đukić, Dušan, Reichel, Lothar, Spalević, Miodrag, "GENERALIZED AVERAGED GAUSSIAN QUADRATURE FORMULAS WITH MODIFIED MATRICES" in 6th MATHEMATICAL CONFERENCE OF THE REPUBLIC OF SRPSKA, BOOK OF ABSTRACTS (2016):27-27,
https://hdl.handle.net/21.15107/rcub_machinery_5138 .

TY  - JOUR
AU  - Reichel, Lothar
AU  - Spalević, Miodrag
AU  - Tang, Tunan
PY  - 2016
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/2489
AB  - The need to compute expressions of the form , where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss quadrature formulas.
PB  - Springer, Dordrecht
T2  - Bit Numerical Mathematics
T1  - Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
EP  - 1067
IS  - 3
SP  - 1045
VL  - 56
DO  - 10.1007/s10543-015-0592-7
ER  - 

@article{
author = "Reichel, Lothar and Spalević, Miodrag and Tang, Tunan",
year = "2016",
abstract = "The need to compute expressions of the form , where A is a large square matrix, u and v are vectors, and f is a function, arises in many applications, including network analysis, quantum chromodynamics, and the solution of linear discrete ill-posed problems. Commonly used approaches first reduce A to a small matrix by a few steps of the Hermitian or non-Hermitian Lanczos processes and then evaluate the reduced problem. This paper describes a new method to determine error estimates for computed quantities and shows how to achieve higher accuracy than available methods for essentially the same computational effort. Our methods are based on recently proposed generalized averaged Gauss quadrature formulas.",
publisher = "Springer, Dordrecht",
journal = "Bit Numerical Mathematics",
title = "Generalized averaged Gauss quadrature rules for the approximation of matrix functionals",
pages = "1067-1045",
number = "3",
volume = "56",
doi = "10.1007/s10543-015-0592-7"
}

Reichel, L., Spalević, M.,& Tang, T.. (2016). Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. in Bit Numerical Mathematics
Springer, Dordrecht., 56(3), 1045-1067.
https://doi.org/10.1007/s10543-015-0592-7

Reichel L, Spalević M, Tang T. Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. in Bit Numerical Mathematics. 2016;56(3):1045-1067.
doi:10.1007/s10543-015-0592-7 .

Reichel, Lothar, Spalević, Miodrag, Tang, Tunan, "Generalized averaged Gauss quadrature rules for the approximation of matrix functionals" in Bit Numerical Mathematics, 56, no. 3 (2016):1045-1067,
https://doi.org/10.1007/s10543-015-0592-7 . .