A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions

Sanjit Kumar Mohanty; Rajani Ballav Dash

doi:10.3934/naco.2021031

Article Contents

2022, Volume 12, Issue 4: 705-718. Doi: 10.3934/naco.2021031

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A quadrature rule of Lobatto-Gaussian for numerical integration of analytic functions

Sanjit Kumar Mohanty^1, , and
Rajani Ballav Dash^2,

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
^* Corresponding author: Sanjit Kumar Mohanty

Received: November 30, 2020

Revised: May 31, 2021

Early access: August 2021

Published: December 2022

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Abstract

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.

Keywords:

Gauss-Legendre five point transformed rule,
Lobatto six point transformed rule,
Mixed quadrature rule,
$ SR_{LG}(f) $,
$ ESR_{LG}(f) $.

Mathematics Subject Classification: Primary: 65D30, 65D32; Secondary: 65E05, 03C99.

Citation:

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Figure 1. For the integral $ I_{1}(f) $

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Figure 2. For the integral $ I_{2}(f) $

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Figure 3. For the integral $ I_{3}(f) $

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Figure 4. For the integral $ I_{4}(f) $

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Figure 5. Error comparison for the integral $ I_{1}(f) $

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Figure 6. Error comparison for the integral $ I_{2}(f) $

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Figure 7. Error comparison for the integral $ I_{3}(f) $

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Figure 8. Error comparison for the integral $ I_{4}(f) $

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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

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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

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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	$5.615 \times 10^{-16}$

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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 $\times 10^{-13}$

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