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  - 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 . .

Spalević, M., Aranđelović, I., Pejčev, A., Đukić, D., Tomanović, J.,& Mutavdžić Đukić, R.. (2023). Višestruki, krivolinijski i površinski integrali. in Univerzitet u Beogradu - Mašinski fakultet
Univerzitet u Beogradu - Mašinski fakultet..
https://hdl.handle.net/21.15107/rcub_machinery_7221

Spalević M, Aranđelović I, Pejčev A, Đukić D, Tomanović J, Mutavdžić Đukić R. Višestruki, krivolinijski i površinski integrali. in Univerzitet u Beogradu - Mašinski fakultet. 2023;.
https://hdl.handle.net/21.15107/rcub_machinery_7221 .

Spalević, Miodrag, Aranđelović, Ivan, Pejčev, Aleksandar, Đukić, Dušan, Tomanović, Jelena, Mutavdžić Đukić, Rada, "Višestruki, krivolinijski i površinski integrali" in Univerzitet u Beogradu - Mašinski fakultet (2023),
https://hdl.handle.net/21.15107/rcub_machinery_7221 .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Fermo, Luisa
AU  - Mutavdžić Đukić, Rada
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3963
AB  - Stratified cubature rules are proposed to approximate double integrals defined on the real positive semiaxis. In particular, anti-Gauss cubature formulae are introduced and averaged cubature schemes are developed. Some of their appropriate modifications are also studied. Several numerical experiments are given to testify the performance of all the formulae.
PB  - Springer
T2  - Numerical Algorithms
T1  - Averaged cubature schemes on the real positive semiaxis
VL  - 92, 545–569
DO  - 10.1007/s11075-022-01408-6
ER  - 

@article{
author = "Đukić, Dušan and Fermo, Luisa and Mutavdžić Đukić, Rada",
year = "2022",
abstract = "Stratified cubature rules are proposed to approximate double integrals defined on the real positive semiaxis. In particular, anti-Gauss cubature formulae are introduced and averaged cubature schemes are developed. Some of their appropriate modifications are also studied. Several numerical experiments are given to testify the performance of all the formulae.",
publisher = "Springer",
journal = "Numerical Algorithms",
title = "Averaged cubature schemes on the real positive semiaxis",
volume = "92, 545–569",
doi = "10.1007/s11075-022-01408-6"
}

Đukić, D., Fermo, L.,& Mutavdžić Đukić, R.. (2022). Averaged cubature schemes on the real positive semiaxis. in Numerical Algorithms
Springer., 92, 545–569.
https://doi.org/10.1007/s11075-022-01408-6

Đukić D, Fermo L, Mutavdžić Đukić R. Averaged cubature schemes on the real positive semiaxis. in Numerical Algorithms. 2022;92, 545–569.
doi:10.1007/s11075-022-01408-6 .

Đukić, Dušan, Fermo, Luisa, Mutavdžić Đukić, Rada, "Averaged cubature schemes on the real positive semiaxis" in Numerical Algorithms, 92, 545–569 (2022),
https://doi.org/10.1007/s11075-022-01408-6 . .

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  - GEN
AU  - Đukić, Dušan
AU  - Pejčev, Aleksandar
AU  - Tomanović, Jelena
AU  - Mutavdžić Đukić, Rada
PY  - 2022
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6238
PB  - Univerzitet u Beogradu - Mašinski fakultet
T2  - Univerzitet u Beogradu - Mašinski fakultet
T1  - Zbirka zadataka za pripremu kvalifikacionog ispita za upis na Mašinski fakultet u Beogradu
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6238
ER  - 

@misc{
author = "Đukić, Dušan and Pejčev, Aleksandar and Tomanović, Jelena and Mutavdžić Đukić, Rada",
year = "2022",
publisher = "Univerzitet u Beogradu - Mašinski fakultet",
journal = "Univerzitet u Beogradu - Mašinski fakultet",
title = "Zbirka zadataka za pripremu kvalifikacionog ispita za upis na Mašinski fakultet u Beogradu",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6238"
}

Đukić, D., Pejčev, A., Tomanović, J.,& Mutavdžić Đukić, R.. (2022). Zbirka zadataka za pripremu kvalifikacionog ispita za upis na Mašinski fakultet u Beogradu. in Univerzitet u Beogradu - Mašinski fakultet
Univerzitet u Beogradu - Mašinski fakultet..
https://hdl.handle.net/21.15107/rcub_machinery_6238

Đukić D, Pejčev A, Tomanović J, Mutavdžić Đukić R. Zbirka zadataka za pripremu kvalifikacionog ispita za upis na Mašinski fakultet u Beogradu. in Univerzitet u Beogradu - Mašinski fakultet. 2022;.
https://hdl.handle.net/21.15107/rcub_machinery_6238 .

Đukić, Dušan, Pejčev, Aleksandar, Tomanović, Jelena, Mutavdžić Đukić, Rada, "Zbirka zadataka za pripremu kvalifikacionog ispita za upis na Mašinski fakultet u Beogradu" in Univerzitet u Beogradu - Mašinski fakultet (2022),
https://hdl.handle.net/21.15107/rcub_machinery_6238 .

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  - Đukić, Dušan
AU  - Mutavdžić Đukić, Rada
AU  - Pejčev, Aleksandar
AU  - Spalević, Miodrag
PY  - 2020
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/3260
AB  - This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses.
PB  - Kent State University, Kent
T2  - Electronic Transactions on Numerical Analysis
T1  - Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results
EP  - 382
SP  - 352
VL  - 53
DO  - 10.1553/etna_vol53s352
ER  - 

@article{
author = "Đukić, Dušan and Mutavdžić Đukić, Rada and Pejčev, Aleksandar and Spalević, Miodrag",
year = "2020",
abstract = "This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses.",
publisher = "Kent State University, Kent",
journal = "Electronic Transactions on Numerical Analysis",
title = "Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results",
pages = "382-352",
volume = "53",
doi = "10.1553/etna_vol53s352"
}

Đukić, D., Mutavdžić Đukić, R., Pejčev, A.,& Spalević, M.. (2020). Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results. in Electronic Transactions on Numerical Analysis
Kent State University, Kent., 53, 352-382.
https://doi.org/10.1553/etna_vol53s352

Đukić D, Mutavdžić Đukić R, Pejčev A, Spalević M. Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results. in Electronic Transactions on Numerical Analysis. 2020;53:352-382.
doi:10.1553/etna_vol53s352 .

Đukić, Dušan, Mutavdžić Đukić, Rada, Pejčev, Aleksandar, Spalević, Miodrag, "Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results" in Electronic Transactions on Numerical Analysis, 53 (2020):352-382,
https://doi.org/10.1553/etna_vol53s352 . .

TY  - BOOK
AU  - Aranđelović, Ivan
AU  - Pejčev, Aleksandar
AU  - Đukić, Dušan
AU  - Jandrlić, Davorka
AU  - Tomanović, Jelena
AU  - Mutavdžić Đukić, Rada
AU  - Vučić, Miloš
PY  - 2020
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6256
PB  - Univerzitet u Beogradu - Mašinski fakultet
T2  - Univerzitet u Beogradu - Mašinski fakultet
T1  - Matematika 1: udžbenik i zbirka zadataka
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6256
ER  - 

@book{
author = "Aranđelović, Ivan and Pejčev, Aleksandar and Đukić, Dušan and Jandrlić, Davorka and Tomanović, Jelena and Mutavdžić Đukić, Rada and Vučić, Miloš",
year = "2020",
publisher = "Univerzitet u Beogradu - Mašinski fakultet",
journal = "Univerzitet u Beogradu - Mašinski fakultet",
title = "Matematika 1: udžbenik i zbirka zadataka",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6256"
}

Aranđelović, I., Pejčev, A., Đukić, D., Jandrlić, D., Tomanović, J., Mutavdžić Đukić, R.,& Vučić, M.. (2020). Matematika 1: udžbenik i zbirka zadataka. in Univerzitet u Beogradu - Mašinski fakultet
Univerzitet u Beogradu - Mašinski fakultet..
https://hdl.handle.net/21.15107/rcub_machinery_6256

Aranđelović I, Pejčev A, Đukić D, Jandrlić D, Tomanović J, Mutavdžić Đukić R, Vučić M. Matematika 1: udžbenik i zbirka zadataka. in Univerzitet u Beogradu - Mašinski fakultet. 2020;.
https://hdl.handle.net/21.15107/rcub_machinery_6256 .

Aranđelović, Ivan, Pejčev, Aleksandar, Đukić, Dušan, Jandrlić, Davorka, Tomanović, Jelena, Mutavdžić Đukić, Rada, Vučić, Miloš, "Matematika 1: udžbenik i zbirka zadataka" in Univerzitet u Beogradu - Mašinski fakultet (2020),
https://hdl.handle.net/21.15107/rcub_machinery_6256 .

TY  - BOOK
AU  - Aranđelović, Ivan
AU  - Jandrlić, Davorka
AU  - Pejčev, Aleksandar
AU  - Đukić, Dušan
AU  - Tomanović, Jelena
AU  - Mutavdžić Đukić, Rada
PY  - 2019
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6260
PB  - Univerzitet u Beogradu - Mašinski fakultet
T2  - Univerzitet u Beogradu - Mašinski fakultet
T1  - Matematika 2
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6260
ER  - 

@book{
author = "Aranđelović, Ivan and Jandrlić, Davorka and Pejčev, Aleksandar and Đukić, Dušan and Tomanović, Jelena and Mutavdžić Đukić, Rada",
year = "2019",
publisher = "Univerzitet u Beogradu - Mašinski fakultet",
journal = "Univerzitet u Beogradu - Mašinski fakultet",
title = "Matematika 2",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6260"
}

Aranđelović, I., Jandrlić, D., Pejčev, A., Đukić, D., Tomanović, J.,& Mutavdžić Đukić, R.. (2019). Matematika 2. in Univerzitet u Beogradu - Mašinski fakultet
Univerzitet u Beogradu - Mašinski fakultet..
https://hdl.handle.net/21.15107/rcub_machinery_6260

Aranđelović I, Jandrlić D, Pejčev A, Đukić D, Tomanović J, Mutavdžić Đukić R. Matematika 2. in Univerzitet u Beogradu - Mašinski fakultet. 2019;.
https://hdl.handle.net/21.15107/rcub_machinery_6260 .

Aranđelović, Ivan, Jandrlić, Davorka, Pejčev, Aleksandar, Đukić, Dušan, Tomanović, Jelena, Mutavdžić Đukić, Rada, "Matematika 2" in Univerzitet u Beogradu - Mašinski fakultet (2019),
https://hdl.handle.net/21.15107/rcub_machinery_6260 .

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  - THES
AU  - Đukić, Dušan
PY  - 2018
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6114
AB  - Онда када функција није позната аналитички, већ су познате само њене вредности добијене експериментално у неким тачкама, или пак њен интеграл није елементарно израчунљив, на располагању су различите методе квадратуре, тј. нумеричке интеграције. Многе од ових метода одређују приближну вредност интеграла коришћењем вредности функције у појединим тачкама - чворовима. Проблеми који се при томе јављају су разноврсни: од проналажења оптималних чворова и тежинских коефицијената (који су често такође нумерички одређени), преко што тачнијег израчунавања вредности функције у тим чворовима (ако је у њима функција уопште дефинисана), до процене грешке квадратурне формуле (која зависи како од квадратурне формуле, тако и од природе функције). Једна од метода процене квадратурне грешке користи Гаус-Кронродова раширења Гаусових квадратурних формула. У овој дисертацији је разматрана једна модификација Гаусових квадратура, у виду тзв. уопштене усредњене гаусовске квадратуре, која може послужити као замена онда када Гаус-Кронродова квадратура не постоји или није практична. За ове квадратуре дати су неки услови под којима су сви њихови чворови унутар интервала интеграције. Такође су посматране Гаус-Кронродове формуле за тзв. Бернштајн-Сегеове тежинске функције и у тим случајевима су дате експлицитне оцене квадратурне грешке.
AB  - When a function is not known analytically but only by a set of sampled values,
or its integral is not an elementary function, various methods for quadrature, i.e.
numerical integration can be used instead. Many of these methods use the values of
the function at a finite set of points (nodes) to compute the approximate value of
the integral. A variety of problems can arise throughout the process. These include
finding optimal nodes and weights (often numerically, as well), evaluating the function
accurately enough at the nodes (provided that these nodes are actually in its domain),
or finding effective bounds for the quadrature error (which clearly depends on natures
of both the quadrature and the function itself).
One useful method of estimating the quadrature error involves Gauss-Kronrod
extensions of Gaussian quadrature formulae. This thesis discusses a modification of
Gaussian quadratures, known as generalized averaged Gaussian quadratures, which
may serve as a substitute when Gauss-Kronrod quadratures are not available. For
these quadratures, some conditions are given under which all their nodes lie inside the
domain of integration. Also, the thesis studies Gauss-Kronrod quadrature formulae
in the case of Bernstein-Szeg˝o weight functions and gives explicit bounds for the
quadrature error.
T2  - Univerzitet u Kragujevcu Prirodno-matematički fakultet, 19.04.2018.
T1  - Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6114
ER  - 

@phdthesis{
author = "Đukić, Dušan",
year = "2018",
abstract = "Онда када функција није позната аналитички, већ су познате само њене вредности добијене експериментално у неким тачкама, или пак њен интеграл није елементарно израчунљив, на располагању су различите методе квадратуре, тј. нумеричке интеграције. Многе од ових метода одређују приближну вредност интеграла коришћењем вредности функције у појединим тачкама - чворовима. Проблеми који се при томе јављају су разноврсни: од проналажења оптималних чворова и тежинских коефицијената (који су често такође нумерички одређени), преко што тачнијег израчунавања вредности функције у тим чворовима (ако је у њима функција уопште дефинисана), до процене грешке квадратурне формуле (која зависи како од квадратурне формуле, тако и од природе функције). Једна од метода процене квадратурне грешке користи Гаус-Кронродова раширења Гаусових квадратурних формула. У овој дисертацији је разматрана једна модификација Гаусових квадратура, у виду тзв. уопштене усредњене гаусовске квадратуре, која може послужити као замена онда када Гаус-Кронродова квадратура не постоји или није практична. За ове квадратуре дати су неки услови под којима су сви њихови чворови унутар интервала интеграције. Такође су посматране Гаус-Кронродове формуле за тзв. Бернштајн-Сегеове тежинске функције и у тим случајевима су дате експлицитне оцене квадратурне грешке., When a function is not known analytically but only by a set of sampled values,
or its integral is not an elementary function, various methods for quadrature, i.e.
numerical integration can be used instead. Many of these methods use the values of
the function at a finite set of points (nodes) to compute the approximate value of
the integral. A variety of problems can arise throughout the process. These include
finding optimal nodes and weights (often numerically, as well), evaluating the function
accurately enough at the nodes (provided that these nodes are actually in its domain),
or finding effective bounds for the quadrature error (which clearly depends on natures
of both the quadrature and the function itself).
One useful method of estimating the quadrature error involves Gauss-Kronrod
extensions of Gaussian quadrature formulae. This thesis discusses a modification of
Gaussian quadratures, known as generalized averaged Gaussian quadratures, which
may serve as a substitute when Gauss-Kronrod quadratures are not available. For
these quadratures, some conditions are given under which all their nodes lie inside the
domain of integration. Also, the thesis studies Gauss-Kronrod quadrature formulae
in the case of Bernstein-Szeg˝o weight functions and gives explicit bounds for the
quadrature error.",
journal = "Univerzitet u Kragujevcu Prirodno-matematički fakultet, 19.04.2018.",
title = "Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6114"
}

Đukić, D.. (2018). Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура. in Univerzitet u Kragujevcu Prirodno-matematički fakultet, 19.04.2018..
https://hdl.handle.net/21.15107/rcub_machinery_6114

Đukić D. Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура. in Univerzitet u Kragujevcu Prirodno-matematički fakultet, 19.04.2018.. 2018;.
https://hdl.handle.net/21.15107/rcub_machinery_6114 .

Đukić, Dušan, "Унутрашњост скраћених усредњених гаусовских квадратура и оцена грешке Гаус-Кронродових квадратура" in Univerzitet u Kragujevcu Prirodno-matematički fakultet, 19.04.2018. (2018),
https://hdl.handle.net/21.15107/rcub_machinery_6114 .

TY  - JOUR
AU  - Đukić, Dušan
AU  - Pejčev, Aleksandar
AU  - Spalević, Miodrag
PY  - 2018
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/2933
AB  - We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points a" 1 and sum of semi-axes rho > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99-127, 2006).
PB  - Springer, Dordrecht
T2  - Numerical Algorithms
T1  - The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type
EP  - 1028
IS  - 4
SP  - 1003
VL  - 77
DO  - 10.1007/s11075-017-0351-8
ER  - 

@article{
author = "Đukić, Dušan and Pejčev, Aleksandar and Spalević, Miodrag",
year = "2018",
abstract = "We consider the Gauss-Kronrod quadrature formulae for the Bernstein-SzegoIi weight functions consisting of any one of the four Chebyshev weights divided by the polynomial . For analytic functions, the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points a" 1 and sum of semi-axes rho > 1, for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed by S. Notaris (Numer. Math. 103, 99-127, 2006).",
publisher = "Springer, Dordrecht",
journal = "Numerical Algorithms",
title = "The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type",
pages = "1028-1003",
number = "4",
volume = "77",
doi = "10.1007/s11075-017-0351-8"
}

Đukić, D., Pejčev, A.,& Spalević, M.. (2018). The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type. in Numerical Algorithms
Springer, Dordrecht., 77(4), 1003-1028.
https://doi.org/10.1007/s11075-017-0351-8

Đukić D, Pejčev A, Spalević M. The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type. in Numerical Algorithms. 2018;77(4):1003-1028.
doi:10.1007/s11075-017-0351-8 .

Đukić, Dušan, Pejčev, Aleksandar, Spalević, Miodrag, "The error bounds of Gauss-Kronrod quadrature formulae for weight functions of Bernstein-SzegoIi type" in Numerical Algorithms, 77, no. 4 (2018):1003-1028,
https://doi.org/10.1007/s11075-017-0351-8 . .

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  - BOOK
AU  - Spalević, Miodrag
AU  - Aranđelović, Ivan
AU  - Pejčev, Aleksandar
AU  - Doder, Dragan
AU  - Đukić, Dušan
AU  - Tomanović, Jelena
PY  - 2017
UR  - https://machinery.mas.bg.ac.rs/handle/123456789/6271
PB  - Univerzitet u Beogradu - Mašinski fakultet
T2  - Univerzitet u Beogradu - Mašinski fakultet
T1  - Diferencijalne jednačine
UR  - https://hdl.handle.net/21.15107/rcub_machinery_6271
ER  - 

@book{
author = "Spalević, Miodrag and Aranđelović, Ivan and Pejčev, Aleksandar and Doder, Dragan and Đukić, Dušan and Tomanović, Jelena",
year = "2017",
publisher = "Univerzitet u Beogradu - Mašinski fakultet",
journal = "Univerzitet u Beogradu - Mašinski fakultet",
title = "Diferencijalne jednačine",
url = "https://hdl.handle.net/21.15107/rcub_machinery_6271"
}

Spalević, M., Aranđelović, I., Pejčev, A., Doder, D., Đukić, D.,& Tomanović, J.. (2017). Diferencijalne jednačine. in Univerzitet u Beogradu - Mašinski fakultet
Univerzitet u Beogradu - Mašinski fakultet..
https://hdl.handle.net/21.15107/rcub_machinery_6271

Spalević M, Aranđelović I, Pejčev A, Doder D, Đukić D, Tomanović J. Diferencijalne jednačine. in Univerzitet u Beogradu - Mašinski fakultet. 2017;.
https://hdl.handle.net/21.15107/rcub_machinery_6271 .

Spalević, Miodrag, Aranđelović, Ivan, Pejčev, Aleksandar, Doder, Dragan, Đukić, Dušan, Tomanović, Jelena, "Diferencijalne jednačine" in Univerzitet u Beogradu - Mašinski fakultet (2017),
https://hdl.handle.net/21.15107/rcub_machinery_6271 .