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doi: 10.3934/naco.2021031
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A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions

1. 

Department of Mathematics, B.S Degree College, Jajpur, Odisha-754296, India

2. 

Department of Mathematics, Ravenshaw Univerty, Cuttack, Odisha-753003, India

* Corresponding author: Sanjit Kumar Mohanty

Received  December 2020 Revised  June 2021 Early access August 2021

A novel quadrature rule is formed combining Lobatto six point transformed rule and Gauss-Legendre five point transformed rule each having precision nine. The mixed rule so formed is of precision eleven. Through asymptotic error estimation the novelty of the quadrature rule is justified. Some test integrals have been evaluated using the mixed rule and its constituents both in non-adaptive and adaptive modes. The results are found to be quite encouraging for the mixed rule which is in conformation with the theoretical prediction.

Citation: Sanjit Kumar Mohanty, Rajani Ballav Dash. A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021031
References:
[1]

Kendall E. Atkinson, An Introduction to Numerical Analysis, 2nd Edition, Wiley Student Edition, 2012. doi: 10.1007/978-3-642-25983-8.  Google Scholar

[2]

D. K. BeheraA. K. Sethi and R. B. Dash, An open type mixed quadrature rule using Fejer and Gaussian quadrature rules, American International Journal of Research in Science, Technology, Engineering & Mathematics, 9 (2015), 265-268.   Google Scholar

[3]

D. CalvettiG. H. GolubW. B. Gragg and L. Reichel, Computation of Gauss-Kronrod quadrature rules, Mathematics of Computation, 69 (2000), 1035-1052.  doi: 10.1090/S0025-5718-00-01174-1.  Google Scholar

[4]

S. Conte and C. de Boor, Elementary Numerical Analysis, Mc-Graw Hill, 1980.  Google Scholar

[5]

R. N. Das and G. Pradhan, A mixed quadrature for approximate evaluation of real and definite integrals, Int. J. Math. Educ. Sci. Technology, 27 (1996), 279-283.   Google Scholar

[6]

R. B. Dash and D. Das, Applicaion of mixed quadrsture rules in adaptive quadrature routines, Gen. Math. Notes, 18 (2013).  Google Scholar

[7] J. Davis Philip and Ph ilip Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, 2007.   Google Scholar
[8]

La urie Dirk, Calculation of Gauss-Kronrod quadrature rules, Mathematics of Computation of the American Mathematical Society, 66 (1997), 1133-1145.  doi: 10.1090/S0025-5718-97-00861-2.  Google Scholar

[9]

H. O. Hera and F. J. Smith, Error estimation in the Clenshaw-Curtis quadrature formula, The Computer Jourrnal, (1968), 213-219.  doi: 10.1093/comjnl/11.2.213.  Google Scholar

[10]

M. K. Jain, S. R. K Iyenger and R. K Jain, Numerical Methods for Scientific and Engineering Computation, 4th Edition, New Age International Publisher, 2003.  Google Scholar

[11]

A. S. Kronrod, Nodes and weights of quadrature formulas, Int. J. Math. Educ. Sci. Technology, (1965), Springer publication.  Google Scholar

[12]

F. G. Lether, On Birkhoff-Young quadrature of analytic functions, J. Comput. Appl. Math., 2 (1976), 81-92.  doi: 10.1016/0771-050X(76)90012-7.  Google Scholar

[13]

J. Lyness and K. Puri, The Euler-Maclaurin expansion for the simplex, Math. Comput., 27 (1996), 273-293.  doi: 10.2307/2005615.  Google Scholar

[14]

S. K. Mohanty, A mixed quadrature rule using Clenshaw-Curtis five point rule modified by richardson extrapolatio, Journal of Ultra Scientist of Physical Sciences, 32 (2020), 6-12.   Google Scholar

[15]

S. K. MohantyD. Das and R. B. Dash, Dual mixed Gaussian quadrature based adaptive scheme for analytic functions, Annals of Pure and Applied Mathematics, 22 (2020), 83-92.   Google Scholar

[16]

S. K. Mohanty and R. B. Dash, A mixed quadrature using Birkhoff-Young rule modified by Richardson extrapolation for numerical integration of Analytic functions, Indian Journal of Mathematics and Mathematical Sciences, 6 (2010), 221-228.   Google Scholar

[17]

S. K. Mohanty and R. B. Dash, A mixed quadrature rule for numerical integration of analytic functions, Bulletin of Pure and Applied Sciences, 27E (2008), 373-376.   Google Scholar

[18]

M. Pal, Numerical Analysis for Scientists and Engineers, Theory and C Programs, Narosa Publishing House, 2008. Google Scholar

[19]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd Edition, Springer International Edition, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[20]

A. K. TripathyR. B. Dash and A. Baral, A mixed quadrature rule blending Lobatto and Gauss-Legendre 3-point rule for approximate evaluation of real definite integrals, Int. J. Computing Scienceand Mathematics, 6 (2015), 366-377.  doi: 10.1504/IJCSM.2015.071809.  Google Scholar

[21]

G. Walter and G. Walter, Adaptive quadrature revisited, BIT Numerical Mathematics, 40 (2000), 84-109.   Google Scholar

show all references

References:
[1]

Kendall E. Atkinson, An Introduction to Numerical Analysis, 2nd Edition, Wiley Student Edition, 2012. doi: 10.1007/978-3-642-25983-8.  Google Scholar

[2]

D. K. BeheraA. K. Sethi and R. B. Dash, An open type mixed quadrature rule using Fejer and Gaussian quadrature rules, American International Journal of Research in Science, Technology, Engineering & Mathematics, 9 (2015), 265-268.   Google Scholar

[3]

D. CalvettiG. H. GolubW. B. Gragg and L. Reichel, Computation of Gauss-Kronrod quadrature rules, Mathematics of Computation, 69 (2000), 1035-1052.  doi: 10.1090/S0025-5718-00-01174-1.  Google Scholar

[4]

S. Conte and C. de Boor, Elementary Numerical Analysis, Mc-Graw Hill, 1980.  Google Scholar

[5]

R. N. Das and G. Pradhan, A mixed quadrature for approximate evaluation of real and definite integrals, Int. J. Math. Educ. Sci. Technology, 27 (1996), 279-283.   Google Scholar

[6]

R. B. Dash and D. Das, Applicaion of mixed quadrsture rules in adaptive quadrature routines, Gen. Math. Notes, 18 (2013).  Google Scholar

[7] J. Davis Philip and Ph ilip Rabinowitz, Methods of Numerical Integration, 2nd Edition, Academic Press, 2007.   Google Scholar
[8]

La urie Dirk, Calculation of Gauss-Kronrod quadrature rules, Mathematics of Computation of the American Mathematical Society, 66 (1997), 1133-1145.  doi: 10.1090/S0025-5718-97-00861-2.  Google Scholar

[9]

H. O. Hera and F. J. Smith, Error estimation in the Clenshaw-Curtis quadrature formula, The Computer Jourrnal, (1968), 213-219.  doi: 10.1093/comjnl/11.2.213.  Google Scholar

[10]

M. K. Jain, S. R. K Iyenger and R. K Jain, Numerical Methods for Scientific and Engineering Computation, 4th Edition, New Age International Publisher, 2003.  Google Scholar

[11]

A. S. Kronrod, Nodes and weights of quadrature formulas, Int. J. Math. Educ. Sci. Technology, (1965), Springer publication.  Google Scholar

[12]

F. G. Lether, On Birkhoff-Young quadrature of analytic functions, J. Comput. Appl. Math., 2 (1976), 81-92.  doi: 10.1016/0771-050X(76)90012-7.  Google Scholar

[13]

J. Lyness and K. Puri, The Euler-Maclaurin expansion for the simplex, Math. Comput., 27 (1996), 273-293.  doi: 10.2307/2005615.  Google Scholar

[14]

S. K. Mohanty, A mixed quadrature rule using Clenshaw-Curtis five point rule modified by richardson extrapolatio, Journal of Ultra Scientist of Physical Sciences, 32 (2020), 6-12.   Google Scholar

[15]

S. K. MohantyD. Das and R. B. Dash, Dual mixed Gaussian quadrature based adaptive scheme for analytic functions, Annals of Pure and Applied Mathematics, 22 (2020), 83-92.   Google Scholar

[16]

S. K. Mohanty and R. B. Dash, A mixed quadrature using Birkhoff-Young rule modified by Richardson extrapolation for numerical integration of Analytic functions, Indian Journal of Mathematics and Mathematical Sciences, 6 (2010), 221-228.   Google Scholar

[17]

S. K. Mohanty and R. B. Dash, A mixed quadrature rule for numerical integration of analytic functions, Bulletin of Pure and Applied Sciences, 27E (2008), 373-376.   Google Scholar

[18]

M. Pal, Numerical Analysis for Scientists and Engineers, Theory and C Programs, Narosa Publishing House, 2008. Google Scholar

[19]

J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd Edition, Springer International Edition, 2002. doi: 10.1007/978-0-387-21738-3.  Google Scholar

[20]

A. K. TripathyR. B. Dash and A. Baral, A mixed quadrature rule blending Lobatto and Gauss-Legendre 3-point rule for approximate evaluation of real definite integrals, Int. J. Computing Scienceand Mathematics, 6 (2015), 366-377.  doi: 10.1504/IJCSM.2015.071809.  Google Scholar

[21]

G. Walter and G. Walter, Adaptive quadrature revisited, BIT Numerical Mathematics, 40 (2000), 84-109.   Google Scholar

Figure 1.  For the integral $ I_{1}(f) $
Figure 2.  For the integral $ I_{2}(f) $
Figure 3.  For the integral $ I_{3}(f) $
Figure 4.  For the integral $ I_{4}(f) $
Figure 5.  Error comparison for the integral $ I_{1}(f) $
Figure 6.  Error comparison for the integral $ I_{2}(f) $
Figure 7.  Error comparison for the integral $ I_{3}(f) $
Figure 8.  Error comparison for the integral $ I_{4}(f) $
Table 1.  Values of different test integrals using Constructed mixed rule and its constituent rules
Integrals Values obtained by different quadrature rules
$I$ $L_{6}(f)$ $GL_{5}(f)$ $SR_{LG}(f)$
$I_{1}=\int_{0}^{i} e^{-z^{2}}dz$ 1.462651839829136i 1.46265166801868i 1.46265174611434181 8181818i
$I_{2}=\int_{-\Pi i}^{\Pi i} sin^{2}z dz$ -131.06185554618i -130.4622662693306i -130.734806849716690 9090i
$I_{3}=\int_{\sqrt{3}i}^{\sqrt{3}i} z^{10}dz$ -78.00591269679589 85i -75.29118816574913 21i -76.5251538616794804 6363i
$I_{4}=\int_{1-\frac{i}{4}}^{1+\frac{i}{4}} \ln z dz$ 0.005113481780491 28i 0.005113481647007 5i 0.00511348170768194 54545454i
$I_{5}=\int_{0}^{2i} \sinh z dz$ -1.41614683574858 -1.4161468372130817 -1.41614683654739910 909090909
$I_{6}=\int_{1-i}^{1+i} \cos zdz$ 1.269927830104607 31i 1.269927829123843 34i 1.26992782956964514 454545454i
Integrals Values obtained by different quadrature rules
$I$ $L_{6}(f)$ $GL_{5}(f)$ $SR_{LG}(f)$
$I_{1}=\int_{0}^{i} e^{-z^{2}}dz$ 1.462651839829136i 1.46265166801868i 1.46265174611434181 8181818i
$I_{2}=\int_{-\Pi i}^{\Pi i} sin^{2}z dz$ -131.06185554618i -130.4622662693306i -130.734806849716690 9090i
$I_{3}=\int_{\sqrt{3}i}^{\sqrt{3}i} z^{10}dz$ -78.00591269679589 85i -75.29118816574913 21i -76.5251538616794804 6363i
$I_{4}=\int_{1-\frac{i}{4}}^{1+\frac{i}{4}} \ln z dz$ 0.005113481780491 28i 0.005113481647007 5i 0.00511348170768194 54545454i
$I_{5}=\int_{0}^{2i} \sinh z dz$ -1.41614683574858 -1.4161468372130817 -1.41614683654739910 909090909
$I_{6}=\int_{1-i}^{1+i} \cos zdz$ 1.269927830104607 31i 1.269927829123843 34i 1.26992782956964514 454545454i
Table 2.  Absolute value of Truncation error due to constructed mixed rule and its constituent rules
Inte grals Exact value $\mid Error\mid $ by different quadrature rules
$I$ $I(f)$ $ \mid EL_{6}(f)\mid$ $\mid EGL_{5}(f)\mid$ $\mid ESR_{LG}(f)\mid$
$I_{1}$ 1.462651745907182i 0.0000000939219 0.0000000778885 0.00000000020715 9818181818181
$I_{2}$ -130.7308543669184i 0.3310011792616 0.2685880975878 0.0039524827982 9090909090909
$I_{3}$ -76.52515386167948 769584i 1.4807588351164 1080416 1.2339656959303 5559584 0.00000000000000 7232203636363
$I_{4}$ 0.005113481707837 01898765i 0.0000000000726 5426101235 0.0000000000608 2951898765 0.00000000000015 507353310454545
$I_{5}$ -1.416146836547142 38 0.0000000007985 6238 0.0000000006659 3932 0.00000000000025 67290909090909
$I_{6}$ 1.269927829569471 7i 0.0000000005351 3561 0.0000000004456 2836 0.00000000000017 3444545454545
Inte grals Exact value $\mid Error\mid $ by different quadrature rules
$I$ $I(f)$ $ \mid EL_{6}(f)\mid$ $\mid EGL_{5}(f)\mid$ $\mid ESR_{LG}(f)\mid$
$I_{1}$ 1.462651745907182i 0.0000000939219 0.0000000778885 0.00000000020715 9818181818181
$I_{2}$ -130.7308543669184i 0.3310011792616 0.2685880975878 0.0039524827982 9090909090909
$I_{3}$ -76.52515386167948 769584i 1.4807588351164 1080416 1.2339656959303 5559584 0.00000000000000 7232203636363
$I_{4}$ 0.005113481707837 01898765i 0.0000000000726 5426101235 0.0000000000608 2951898765 0.00000000000015 507353310454545
$I_{5}$ -1.416146836547142 38 0.0000000007985 6238 0.0000000006659 3932 0.00000000000025 67290909090909
$I_{6}$ 1.269927829569471 7i 0.0000000005351 3561 0.0000000004456 2836 0.00000000000017 3444545454545
Table 3a.  The approximations of the same set of test integrals of Table-1 using the constructed rule $SR_{LG}(f) $ in adaptive scheme
Integrals For the Mixed rule $SR_{GL}(f)$
Approximate value(P) No of steps required $ \mid Error\mid=\mid P-I\mid $
$I_{1}=\int_{0}^{i} e^{-z^{2}}dz$ 1.46265174590728028i 01 $9.828 \times 10^{-14}$
$I_{2}=\int_{-\pi i}^{\pi i}\sin^{2}zdz$ -130.730854366917647i 11 $7.52 \times 10^{-12}$
$I_{3}=\int_{-\sqrt{3} i}^{\sqrt{3} i}z^{10}zdz$ -76.5251538616794411i 01 $0.1 \times 10^{-14}$
$I_{4}=\int_{1-\frac{i}{4}}^{1+\frac{i}{4}} \ln zdz$ 0.0051134817180783701 468i 01 $4.29 \times 10^{-17}$
$I_{5}=\int_{0}^{2i} \sinh zdz$ -1.41614683654714245 01 $6.78 \times 10^{-17}$
$I_{6}=\int_{1-i}^{1+i} \cos zdz$ 1.26992782956947226i 01 $5.615 \times 10^{-16}$
Integrals For the Mixed rule $SR_{GL}(f)$
Approximate value(P) No of steps required $ \mid Error\mid=\mid P-I\mid $
$I_{1}=\int_{0}^{i} e^{-z^{2}}dz$ 1.46265174590728028i 01 $9.828 \times 10^{-14}$
$I_{2}=\int_{-\pi i}^{\pi i}\sin^{2}zdz$ -130.730854366917647i 11 $7.52 \times 10^{-12}$
$I_{3}=\int_{-\sqrt{3} i}^{\sqrt{3} i}z^{10}zdz$ -76.5251538616794411i 01 $0.1 \times 10^{-14}$
$I_{4}=\int_{1-\frac{i}{4}}^{1+\frac{i}{4}} \ln zdz$ 0.0051134817180783701 468i 01 $4.29 \times 10^{-17}$
$I_{5}=\int_{0}^{2i} \sinh zdz$ -1.41614683654714245 01 $6.78 \times 10^{-17}$
$I_{6}=\int_{1-i}^{1+i} \cos zdz$ 1.26992782956947226i 01 $5.615 \times 10^{-16}$
Table 3b.  The approximations of the same set of test integrals of Table-1 using the constituents rules $ L_6(f) $ and $ GL_5(f) $ in the adaptive quadrature routines
Inte grals For the constituent rule $L_6(f)$ For the constituent rule $GL_5(f)$
I Approximate value(P) No of steps required $\mid Error\mid$ = $\mid P-I\mid $ Approximate value(P) No of steps required $\mid Error\mid$ = $\mid P-I\mid $
$I_{1}$ 1.46265174590737 528i 03 1.93 $\times 10^{-13}$ 1.46265174590702 026i 03 1.61 $\times 10^{-13}$
$I_{2}$ -130.730854366918 748i 19 3.48 $\times 10^{-13}$ -130.730854366916 64i 19 1.759 $\times 10^{-12}$
$I_{3}$ -76.5251538616807 84i 15 9.99 $\times 10^{-14}$ -76.5251538616783 175i 15 1.17 $\times 10^{-12}$
$I_{4}$ 0.00511348170786 116954i 01 2.41 $\times 10^{-14}$ 0.00511348170781 688564i 01 2.01 $\times 10^{-14}$
$I_{5}$ -1.41614683654714 172 03 6.57 $\times 10^{-16}$ -1.41614683654714 293 03 5.53 $\times 10^{-16}$
$I_{6}$ 1.26992782956947 28i 03 1.099 $\times 10^{-15}$ 1.26992782956898 902i 01 4.826 $\times 10^{-13}$
Inte grals For the constituent rule $L_6(f)$ For the constituent rule $GL_5(f)$
I Approximate value(P) No of steps required $\mid Error\mid$ = $\mid P-I\mid $ Approximate value(P) No of steps required $\mid Error\mid$ = $\mid P-I\mid $
$I_{1}$ 1.46265174590737 528i 03 1.93 $\times 10^{-13}$ 1.46265174590702 026i 03 1.61 $\times 10^{-13}$
$I_{2}$ -130.730854366918 748i 19 3.48 $\times 10^{-13}$ -130.730854366916 64i 19 1.759 $\times 10^{-12}$
$I_{3}$ -76.5251538616807 84i 15 9.99 $\times 10^{-14}$ -76.5251538616783 175i 15 1.17 $\times 10^{-12}$
$I_{4}$ 0.00511348170786 116954i 01 2.41 $\times 10^{-14}$ 0.00511348170781 688564i 01 2.01 $\times 10^{-14}$
$I_{5}$ -1.41614683654714 172 03 6.57 $\times 10^{-16}$ -1.41614683654714 293 03 5.53 $\times 10^{-16}$
$I_{6}$ 1.26992782956947 28i 03 1.099 $\times 10^{-15}$ 1.26992782956898 902i 01 4.826 $\times 10^{-13}$
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