A DERIVATIVE–FREE BLOCK HYBRID METHOD FOR NUMERICAL QUADRATURE
Keywords:
Block hybrid methods, non-singular, A(α)-stable, Convergence, ill- conditioning
Abstract
We derive a new ninth-order block hybrid method for the numerical solution of systems of differential equations and we compare results of numerical experiments with an already existing method in the literature. Both methods are bye-products of linear multistep methods using the interpolation and collocation approach. We show computationally that in the absence of round-off errors, the solution obtained by solving systems of differential equations by the existing block hybrid method derived by differentiating the continuous scheme at a particular off–grid point is the same as those obtained in the new method which is derivative free. Besides, the new block hybrid method which is derivative free results in a well conditioned system as opposed to the ill–conditioned one in the literature. Therefore, providing an answer to Shampine’s claim that matrices arising from the numerical approximation of stiff initial value problems using Linear Multistep Methods are mostly ill–conditioned. Finally, we showed computationally how an LU- type preconditioned Quasi Minimal Residual with a fixed default tolerance reduced the condition number of the old and new methods, with the latter resulting in the smallest minimum norm of residual.
References
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[9] C. Y. Ishola, O. A. Oyelami, C. C. Elebor. A web-based tool for generating block linear multistep methods. Journal of NAMP 70, (2025) 167-178.
[10] S. A. Odejide, A. O. Adeniran. A Hybrid Linear Collocation Multistep Scheme for Solving First Order Initial Value Problems. Journal of the Nigerian Mathematical Society. Vol. (2012). 31 : 229–241.
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[14] M. Farooq, A. Salhi. Improving the solvability of Ill–conditioned systems of linear equations by reducing the condition number of their matrices. Journal of the Korean Mathematical Society, 2011, 48(48), 939–952.
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[27] D. G. Yakubu, A. Shokri, G. M. Kumleng, D. Marian. Second Derivative Block Hybrid Methods for the the numerical integration of Differential systems. Fractal Fract. 2022, 6, 386. https://doi.org/10.3390/fractalfract6070386.
[28] E. O. Omole, L. A. Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Mathematics Letters. Vol. 6, No. 2, (2020), pp. 13–31. doi: 10.11648/j.ml.20200602.12
[29] W. B. Gragg, H. J. Stetter. Generalized multistep predictorcorrector methods. J. ACM (1964), 11, 188–209.
[30] J. C. Butcher. A multistep generalization of RungeKutta method with four or five stages. J. Ass. Comput. Mach. (1967), 14, 8499.
[31] W. C. Gear. Hybrid multistep method for initial value in ordinary differential equations. J. SIAM Numer. Anal. (1965), 2, 69–86.
[32] M. Urabe. An implicit onestep method of highorder accuracy for the numerical integration of ordinary differential equations. J. Numer. Math. 1970, 15, 151–164.
[33] T. A. Mitsui. Modified version of Urabes implicit singlestep method. J. Comp. Appli. Math. 1987, 20, 325–332.
[34] R. O. Akinola, K. J. Ajibade. A Proof of the Non-Singularity of the D Matrix Used in Deriving the two–Step Butchers Hybrid Scheme for the Solution of Initial Value Problems. Journal of Applied Mathematics and Physics, 9, 3177–3201. (2021).
[35] R. O. Akinola. An accurate implementation of the two-step Butcher’s Hybrid Scheme on Initial Value Problems. BSc Thesis, University of Jos, Jos, Nigeria. (Unpublished) (2001).
[36] P. Onumanyi, D. O. Awoyemi, S. N. Jator, U. W. Sirisena (1994). New Linear Multistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of Nigeria Mathematics Society, 13 : 37-51.
[37] R. O. Akinola, A. Shokri, S. W. Yao, S. Y. Kutchin. Circumventing Ill–Conditioning Arising from Using Linear Multistep Methods In Approximating the Solution of Initial Value Problems. Mathematics 2022, 10, 2910. https://doi.org/10.3390/math10162910
[38] R. O. Akinola, A. S. Akoh. A Seventh–Order Computational Algorithm for the Solution of Stiff Systems of Differential Equations. Journal of the Nigerian Mathematical Society. Vol. 42, Issue 3, pp. 215 - 242. (2023).
[39] P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. John Wiley, New York, p. 407. (1962).
[40] J. D. Lambert. Computational Methods in Ordinary Differential Equations. John Wiley, New York. (1973).
[41] J. C. Butcher. Numerical Methods for Ordinary Differential Equations. John Wiley and Sons Ltd. (2016).
[42] J. C. Butcher. Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, Volume 125, Issues 1-2, (2000), Pages 1-29
[43] S. D. Yakubu, P. Sibanda. One–Step Family of Three Optimized Second–Derivative Hybrid Block Methods for Solving First–Order Stiff Problems. Hindawi Journal of Applied Mathematics (2024), Article ID 5078943, https://doi.org/10.1155/2024/5078943
[44] O. A. Akinfenwa, R. I. Abdulganiy, B. I. Akinnukawe, S. A. Okunuga. Seventh order hybrid block method for solution of rst order sti systems of initial value problems. Journal of the Egyptian Mathematical Society, vol. 28, no. 1, p. 28, 2020.
[45] S. E. Ogunfeyitimi, M. N. O. Ikhile. Generalized second derivative linear multistep methods based on the methods of enright. International Journal of Applied and Computational Mathematics, vol. 6, no. 3, p. 76, 2020.
[46] S. E. Ogunfeyitimi, M. N. O. Ikhile. Multi-block generalized Adams-type integration methods for dierential algebraic equations. International Journal of Applied and Computational Mathematics, vol. 7, no. 5, p. 197, 2021.
[47] P. Kaps, S. W. H. Poon, T. D. Bui. Rosenbrock methods for Stiff ODEs: A comparison of Richard- son extrapolation and embedding technique. Computing 34 (1) (1985) 17–40.
[48] W. C. Gear. DIFSUB for Solution of ordinary differential equations. Comm. ACM 1971, 14,
185190.
[49] W. H. Enright. Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 1974, 11, 321– 341.
[50] S. O. Fatunla. Numerical integrators for stiff and highly oscillatory differential equations. J. Math. Comput. 1980, 34, 373–390.
[2] R. O. Akinola, E. O. Omole, J. Sunday, S. Y. Kutchin. A Ninth-Order First Derivative Method for Numerical Integration. J. Nig. Soc. Phys. Sci. 7 (2025). DOI:10.46481/jnsps.2025.2028.
[3] S. O. Adee, V. O. Atabo. Improved two-Point Block Backward Differentiation Formulae for Solv- ing First Order Stiff Initial Value Problems of Ordinary Differential Equations. Nigerian Annals of Pure and Applied Sciences, Vol 3 No 2. (June 2020), pp. 200–209.
[4] T. Aboiyar, T. Luga, B. V. Iyorter. Derivation of Continuous Linear Multistep Methods Using Hermite Polynomials as Basis Functions. American Journal of Applied Mathematics and Statistics. (2015). 3(6):220–225.
[5] A. U. Fotta, T. J. Alabi, B. Abdulqadir. Block Method with One Hybrid Point for the Solution of First Order Initial Value Problems of Ordinary Differential Equations. International Journal of Pure and Applied Mathematics, Vol. 103 No. 3, 511521.
[6] S. O. Ayinde, E. A. Ibijola. A New Numerical Method for Solving First Order Differential Equations. American Journal of Applied Mathematics and Statistics, Vol. 3, No. 4, 156160. DOI: 10.12691/ajams-3-4-4
[7] P. Oliver. A Family of Linear Multistep Methods for the Solution of Stiff and Non-stiff ODEs. IMA Journal of Numerical Analysis, Volume 2, Issue 3, (July 1982), Pages 289–301.
[8] Q. A. Huang, W. Jiang, J. Z. Yang et al. Linear multi-step methods and their numer- ical stability for solving gradient flow equations. Adv Comput Math 49, 39 (2023). https://doi.org/10.1007/s10444-023-10043-1.
[9] C. Y. Ishola, O. A. Oyelami, C. C. Elebor. A web-based tool for generating block linear multistep methods. Journal of NAMP 70, (2025) 167-178.
[10] S. A. Odejide, A. O. Adeniran. A Hybrid Linear Collocation Multistep Scheme for Solving First Order Initial Value Problems. Journal of the Nigerian Mathematical Society. Vol. (2012). 31 : 229–241.
[11] L. N. Trefethen, D. Bau III. Numerical Linear Algebra. SIAM. (1997).
[12] G. H. Golub, J. H. Wilkinson. Ill-conditioned eigensystems and the com- putation of the Jordan canonical form. SIAM review. (1976). 18 (4), 578–619.
[13] G. Peters, J. H. Wilkinson. Inverse iteration, ill–conditioned equations and Newtons method. SIAM review. (1979). 21(3), 339–360.
[14] M. Farooq, A. Salhi. Improving the solvability of Ill–conditioned systems of linear equations by reducing the condition number of their matrices. Journal of the Korean Mathematical Society, 2011, 48(48), 939–952.
[15] C. C. Douglas, L. Lee, M. Yeung. On Solving Ill Conditioned Linear Systems. Procedia Com- puter Science. (2016). 80, pp. 941–950.
[16] G. H. Golub, C. F. Van Loan. Matrix calculations. p. 548. (1999).
[17] R. W. Freund and N. M. Nachtigal. QMR: a Quasi-Minimal Residual method for non- Hermitian linear systems. Numer. Math., 60 (1991) pp. 315–339.
[18] R. W. Freund. Transpose-Free Quasi-Minimal Residual Methods for Non-Hermitian Linear Systems. SIAM J. Sci. Stat. Comput.
[19] R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhour, R. Pozo, C. Romine, and H. Van der Vorst. Templates for the solution of linear systems: Building blocks for iterative methods. SIAM, 2nd ed., 1994.
[20] Y. Saad. Iterative Methods for Sparse Linear Systems. Minneapolis: University of Minnesota MN. p. 567. (2003).
[21] Y. Saad. ILUT: A dual threshold incomplete ILU factorization. Numer Linear Algebra Appl. 1 739–748, 1994.
[22] M. Balandin, E. P. Shurina. Methods for solving SLAE of large dimension. Novosibirsk: NSTU. p. 70. (2000).
[23] S. E. Saukh. Incomplete column-row factorization of matrices for the iterative solution of large systems of equations. Electronic Modeling 32(6) 3–14. (2010).
[24] T. Y. Chen. Preconditioning Sparse Matrices for Computing Eigenvalues and Solving Linear Systems of Equations. University of California at Berkeley. p. 135, (2011).
[25] S. Y. Gogoleva. Preconditioning Based on LU Factorization in Iterative Method for Solving Sys- tems of Linear Algebraic Equations with Sparse Matrices.
[26] J. Demmel. Computational Linear Algebra. Theory and applications. Moscow: Mir. p. 430, (2001).
[27] D. G. Yakubu, A. Shokri, G. M. Kumleng, D. Marian. Second Derivative Block Hybrid Methods for the the numerical integration of Differential systems. Fractal Fract. 2022, 6, 386. https://doi.org/10.3390/fractalfract6070386.
[28] E. O. Omole, L. A. Ukpebor. A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations. Mathematics Letters. Vol. 6, No. 2, (2020), pp. 13–31. doi: 10.11648/j.ml.20200602.12
[29] W. B. Gragg, H. J. Stetter. Generalized multistep predictorcorrector methods. J. ACM (1964), 11, 188–209.
[30] J. C. Butcher. A multistep generalization of RungeKutta method with four or five stages. J. Ass. Comput. Mach. (1967), 14, 8499.
[31] W. C. Gear. Hybrid multistep method for initial value in ordinary differential equations. J. SIAM Numer. Anal. (1965), 2, 69–86.
[32] M. Urabe. An implicit onestep method of highorder accuracy for the numerical integration of ordinary differential equations. J. Numer. Math. 1970, 15, 151–164.
[33] T. A. Mitsui. Modified version of Urabes implicit singlestep method. J. Comp. Appli. Math. 1987, 20, 325–332.
[34] R. O. Akinola, K. J. Ajibade. A Proof of the Non-Singularity of the D Matrix Used in Deriving the two–Step Butchers Hybrid Scheme for the Solution of Initial Value Problems. Journal of Applied Mathematics and Physics, 9, 3177–3201. (2021).
[35] R. O. Akinola. An accurate implementation of the two-step Butcher’s Hybrid Scheme on Initial Value Problems. BSc Thesis, University of Jos, Jos, Nigeria. (Unpublished) (2001).
[36] P. Onumanyi, D. O. Awoyemi, S. N. Jator, U. W. Sirisena (1994). New Linear Multistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of Nigeria Mathematics Society, 13 : 37-51.
[37] R. O. Akinola, A. Shokri, S. W. Yao, S. Y. Kutchin. Circumventing Ill–Conditioning Arising from Using Linear Multistep Methods In Approximating the Solution of Initial Value Problems. Mathematics 2022, 10, 2910. https://doi.org/10.3390/math10162910
[38] R. O. Akinola, A. S. Akoh. A Seventh–Order Computational Algorithm for the Solution of Stiff Systems of Differential Equations. Journal of the Nigerian Mathematical Society. Vol. 42, Issue 3, pp. 215 - 242. (2023).
[39] P. Henrici. Discrete Variable Methods in Ordinary Differential Equations. John Wiley, New York, p. 407. (1962).
[40] J. D. Lambert. Computational Methods in Ordinary Differential Equations. John Wiley, New York. (1973).
[41] J. C. Butcher. Numerical Methods for Ordinary Differential Equations. John Wiley and Sons Ltd. (2016).
[42] J. C. Butcher. Numerical methods for ordinary differential equations in the 20th century. Journal of Computational and Applied Mathematics, Volume 125, Issues 1-2, (2000), Pages 1-29
[43] S. D. Yakubu, P. Sibanda. One–Step Family of Three Optimized Second–Derivative Hybrid Block Methods for Solving First–Order Stiff Problems. Hindawi Journal of Applied Mathematics (2024), Article ID 5078943, https://doi.org/10.1155/2024/5078943
[44] O. A. Akinfenwa, R. I. Abdulganiy, B. I. Akinnukawe, S. A. Okunuga. Seventh order hybrid block method for solution of rst order sti systems of initial value problems. Journal of the Egyptian Mathematical Society, vol. 28, no. 1, p. 28, 2020.
[45] S. E. Ogunfeyitimi, M. N. O. Ikhile. Generalized second derivative linear multistep methods based on the methods of enright. International Journal of Applied and Computational Mathematics, vol. 6, no. 3, p. 76, 2020.
[46] S. E. Ogunfeyitimi, M. N. O. Ikhile. Multi-block generalized Adams-type integration methods for dierential algebraic equations. International Journal of Applied and Computational Mathematics, vol. 7, no. 5, p. 197, 2021.
[47] P. Kaps, S. W. H. Poon, T. D. Bui. Rosenbrock methods for Stiff ODEs: A comparison of Richard- son extrapolation and embedding technique. Computing 34 (1) (1985) 17–40.
[48] W. C. Gear. DIFSUB for Solution of ordinary differential equations. Comm. ACM 1971, 14,
185190.
[49] W. H. Enright. Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 1974, 11, 321– 341.
[50] S. O. Fatunla. Numerical integrators for stiff and highly oscillatory differential equations. J. Math. Comput. 1980, 34, 373–390.
Published
2025-10-31
How to Cite
AKINOLA, R. O., SUNDAY, J., OMOLE, E. O., & AKOR, E. R. (2025). A DERIVATIVE–FREE BLOCK HYBRID METHOD FOR NUMERICAL QUADRATURE. Unilag Journal of Mathematics and Applications, 5(1), 83 - 109. Retrieved from https://lagjma.unilag.edu.ng/article/view/2777
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Copyright (c) 2025 RICHARD OLATOKUNBO AKINOLA, JOSHUA SUNDAY, EZEKIEL O. OMOLE, ELEOJO RACHEL AKOR
This work is licensed under a Creative Commons Attribution 4.0 International License.
