Advantage of Numerical Analysis in Scientific Computation and its Applications
Keywords:
numerical, analysis, scientific, computation, advantage
Abstract
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis,[2][3][4] and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
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References
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40. Speed comparison of various number crunching packages Archived 5 October 2006 at the Wayback Machine
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43. Wolfram, Stephen (1999). The MATHEMATICA® book, version 4. Cambridge University Press. ISBN 9781579550042.
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45. Marasco, A.; Romano, A. (2001). Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations. Springer. ISBN 978-0-8176-4205-1.
2. Demmel, J.W. (1997). Applied numerical linear algebra. SIAM. doi:10.1137/1.9781611971446. ISBN 978-1-61197-144-6.
3. Ciarlet, P.G.; Miara, B.; Thomas, J.M. (1989). Introduction to numerical linear algebra and optimization. Cambridge University Press. ISBN 9780521327886. OCLC 877155729.
4. Trefethen, Lloyd; Bau III, David (1997). Numerical Linear Algebra. SIAM. ISBN 978-0-89871-361-9.
5. Brezinski, C.; Wuytack, L. (2012). Numerical analysis: Historical developments in the 20th century. Elsevier. ISBN 978-0-444-59858-5.
6. Stephen Blyth. "An Introduction to Quantitative Finance". 2013. page VII.
7. Watson, G.A. (2010). "The history and development of numerical analysis in Scotland: a personal perspective" (PDF). The Birth of Numerical Analysis. World Scientific. pp. 161–177. ISBN 9789814469456.
8. Bultheel, Adhemar; Cools, Ronald, eds. (2010). The Birth of Numerical Analysis. Vol. 10. World Scientific. ISBN 978-981-283-625-0.
9. Saad, Y. (2003). Iterative methods for sparse linear systems. SIAM. ISBN 978-0-89871-534-7.
10. Hageman, L.A.; Young, D.M. (2012). Applied iterative methods (2nd ed.). Courier Corporation. ISBN 978-0-8284-0312-2.
11. Traub, J.F. (1982). Iterative methods for the solution of equations (2nd ed.). American Mathematical Society. ISBN 978-0-8284-0312-2.
12. Greenbaum, A. (1997). Iterative methods for solving linear systems. SIAM. ISBN 978-0-89871-396-1.
13. Higham 2002
14. Brezinski, C.; Zaglia, M.R. (2013). Extrapolation methods: theory and practice. Elsevier. ISBN 978-0-08-050622-7.
15. Hestenes, Magnus R.; Stiefel, Eduard (December 1952). "Methods of Conjugate Gradients for Solving Linear Systems" (PDF). Journal of Research of the National Bureau of Standards. 49 (6): 409–. doi:10.6028/jres.049.044.
16. Ezquerro Fernández, J.A.; Hernández Verón, M.Á. (2017). Newton's method: An updated approach of Kantorovich's theory. Birkhäuser. ISBN 978-3-319-55976-6.
17. Deuflhard, Peter (2006). Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Computational Mathematics. Vol. 35 (2nd ed.). Springer. ISBN 978-3-540-21099-3.
18. Ogden, C.J.; Huff, T. (1997). "The Singular Value Decomposition and Its Applications in Image Compression" (PDF). Math 45. College of the Redwoods. Archived from the original (PDF) on 25 September 2006.
19. Davis, P.J.; Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation. ISBN 978-0-486-45339-2.
20. Weisstein, Eric W. "Gaussian Quadrature". MathWorld.
21. Geweke, John (1996). "15. Monte carlo simulation and numerical integration". Handbook of Computational Economics. Vol. 1. Elsevier. pp. 731–800. doi:10.1016/S1574-0021(96)01017-9. ISBN 9780444898579.
22. Iserles, A. (2009). A first course in the numerical analysis of differential equations (2nd ed.). Cambridge University Press. ISBN 978-0-521-73490-5.
23. Ames, W.F. (2014). Numerical methods for partial differential equations (3rd ed.). Academic Press. ISBN 978-0-08-057130-0.
24. Johnson, C. (2012). Numerical solution of partial differential equations by the finite element method. Courier Corporation. ISBN 978-0-486-46900-3.
25. Brenner, S.; Scott, R. (2013). The mathematical theory of finite element methods (2nd ed.). Springer. ISBN 978-1-4757-3658-8.
26. Strang, G.; Fix, G.J. (2018) [1973]. An analysis of the finite element method (2nd ed.). Wellesley-Cambridge Press. ISBN 9780980232783. OCLC 1145780513.
27. Strikwerda, J.C. (2004). Finite difference schemes and partial differential equations (2nd ed.). SIAM. ISBN 978-0-89871-793-8.
28. LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press. ISBN 978-1-139-43418-8.
29. Quarteroni, A.; Saleri, F.; Gervasio, P. (2014). Scientific computing with MATLAB and Octave (4th ed.). Springer. ISBN 978-3-642-45367-0.
30. Gander, W.; Hrebicek, J., eds. (2011). Solving problems in scientific computing using Maple and Matlab®. Springer. ISBN 978-3-642-18873-2.
31. Barnes, B.; Fulford, G.R. (2011). Mathematical modelling with case studies: a differential equations approach using Maple and MATLAB (2nd ed.). CRC Press. ISBN 978-1-4200-8350-7. OCLC 1058138488.
32. Gumley, L.E. (2001). Practical IDL programming. Elsevier. ISBN 978-0-08-051444-4.
33. Bunks, C.; Chancelier, J.P.; Delebecque, F.; Goursat, M.; Nikoukhah, R.; Steer, S. (2012). Engineering and scientific computing with Scilab. Springer. ISBN 978-1-4612-7204-5.
34. Thanki, R.M.; Kothari, A.M. (2019). Digital image processing using SCILAB. Springer. ISBN 978-3-319-89533-8.
35. Ihaka, R.; Gentleman, R. (1996). "R: a language for data analysis and graphics" (PDF). Journal of Computational and Graphical Statistics. 5 (3): 299–314. doi:10.1080/10618600.1996.10474713. S2CID 60206680.
36. Bezanson, Jeff; Edelman, Alan; Karpinski, Stefan; Shah, Viral B. (1 January 2017). "Julia: A Fresh Approach to Numerical Computing". SIAM Review. 59 (1): 65–98. doi:10.1137/141000671. hdl:1721.1/110125. ISSN 0036-1445. S2CID 13026838.
37. Jones, E., Oliphant, T., & Peterson, P. (2001). SciPy: Open source scientific tools for Python.
38. Bressert, E. (2012). SciPy and NumPy: an overview for developers. O'Reilly. ISBN 9781306810395.
39. Blanco-Silva, F.J. (2013). Learning SciPy for numerical and scientific computing. Packt. ISBN 9781782161639.
40. Speed comparison of various number crunching packages Archived 5 October 2006 at the Wayback Machine
41. Comparison of mathematical programs for data analysis Archived 18 May 2016 at the Portuguese Web Archive Stefan Steinhaus, ScientificWeb.com
42. Maeder, R.E. (1997). Programming in mathematica (3rd ed.). Addison-Wesley. ISBN 9780201854497. OCLC 1311056676.
43. Wolfram, Stephen (1999). The MATHEMATICA® book, version 4. Cambridge University Press. ISBN 9781579550042.
44. Shaw, W.T.; Tigg, J. (1993). Applied Mathematica: getting started, getting it done (PDF). Addison-Wesley. ISBN 978-0-201-54217-2. OCLC 28149048.
45. Marasco, A.; Romano, A. (2001). Scientific Computing with Mathematica: Mathematical Problems for Ordinary Differential Equations. Springer. ISBN 978-0-8176-4205-1.
Published
2023-06-30
How to Cite
Meena, K. P. (2023). Advantage of Numerical Analysis in Scientific Computation and its Applications. Central Asian Journal of Theoretical and Applied Science, 4(6), 202-209. Retrieved from https://cajotas.centralasianstudies.org/index.php/CAJOTAS/article/view/1235
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