A highly accurate multi-scale full/half-order polynomial interpolation

Computers, Materials and Continua

Volumn 25, Issue 3, 2011, Pages 239-263

  Liu, Chein Shan ^a  

^a NATIONAL TAIWAN UNIVERSITY   (Taiwan)

Author keywords

Differential quadratures; Half order polynomial interpolation; Multi scale polynomial interpolation; Vandermonde matrices

Indexed keywords

COEFFICIENT MATRIX; COMPUTATIONAL APPLICATIONS; CONDITION NUMBERS; CONSTANT PARAMETERS; DIFFERENTIAL QUADRATURE; FICTITIOUS TIME INTEGRATION METHODS; HALF-ORDER POLYNOMIAL INTERPOLATION; HIGH ORDER POLYNOMIAL; MULTISCALE TECHNIQUE; MULTISCALES; NONLINEAR BOUNDARY VALUE PROBLEMS; NUMERICAL RESULTS; POLYNOMIAL INTERPOLATION; PRECONDITIONERS; VANDERMONDE MATRIX;

BOUNDARY VALUE PROBLEMS; DIFFERENTIATION (CALCULUS); INTERPOLATION; NUMBER THEORY; NUMERICAL METHODS; POLYNOMIALS;

MATRIX ALGEBRA;

EID: 83655182319     PISSN: 15462218     EISSN: 15462226     Source Type: Journal    
DOI: None     Document Type: Article

References (46)

1. Bauer, F. L. (1963): Optimally scaled matrices. Num. Math., vol. 5, pp. 73-87.
2. Beckermann, B. (2000): The condition number of real Vandermonde, Krylov and positive definite Hankel matrices. Num. Math., vol. 85, pp. 553-577.
3. Bellman, R. E.; Casti, J. (1971): Differential quadrature and long-term integration. J. Math. Anal. Appl., vol. 34, pp. 235-238.
4. Bellman, R. E.; Kashef, B. G.; Casti, J. (1972): Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations. J. Comp. Phys., vol. 10, pp. 40-52.
5. Björck, A.; Pereyra, V. (1970): Solution of Vandermonde systems of equations. Math. Comput., vol. 24, pp. 893-903.
6. Calvetti, D.; Reichel, L. (1993): Fast inversion of Vanderomnde-like matrices involving orthogonal polynomials. BIT, vol. 33, pp. 473-484.
7. Chang, C. W.; Liu, C.-S. (2009): A fictitious time integration method for backward advection-dispersion equation. CMES: Computer Modeling in Engineering & Sciences, vol. 51, pp. 261-276.
8. Chen, Y.W.; Liu, C.-S.; Chang, J. R. (2009): Applications of the modified Trefftz method for the Laplace equation. Eng. Anal. Bound. Elem., vol. 33, pp. 137-146.
9. Chen, Y. W.; Liu, C.-S.; Chang, C. M.; Chang, J. R. (2010): Applications of the modified Trefftz method to the simulation of sloshing behaviours. Eng. Anal. Bound. Elem.., vol. 34, pp. 581-598.
10. Chen, Y.W.; Yeih,W.; Liu, C.-S.; Chang, J. R. (2012): Numerical simulation of the two-dimensional sloshing problem using a multi-scaling Trefftz method. Eng. Anal. Bound. Elem., vol. 36, pp. 9-29.
11. Chi, C. C.; Yeih, W.; Liu, C.-S. (2009): A novel method for solving the Cauchy problem of Laplace equation using the fictitious time integration method. CMES: Computer Modeling in Engineering & Sciences, vol. 47, pp. 167-190.
12. Gautschi,W. (1975): Norm estimates for inverses of Vandermonde matrices. Num. Math., vol. 23, pp. 337-347.
13. Gautschi, W.; Inglese, G. (1988): Lower bounds for the condition number of Vandermonde matrices. Num. Math., vol. 52, pp. 241-250.
14. Gohberg, I.; Olshevsky, V. (1997): The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices. J. Complexity, vol. 13, pp. 208-234. (Pubitemid 127343456)
15. Higham, N. J. (1987): Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems. Num. Math., vol. 50, pp. 613-632.
16. Higham, N. J. (1988): Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Num. Anal., vol. 8, pp. 473-486.
17. Jog, C. S. (2004): The accurate inversion of Vandermonde matrices. Comp. Math. Appl., vol. 47, pp. 921-929.
18. Ku, C. Y.; Yeih, W.; Liu, C.-S.; Chi, C. C. (2009): Applications of the fictitious time integration method using a new time-like function. CMES: Computer Modeling in Engineering & Sciences, vol. 43, pp. 173-190.
19. Li, R. C. (2006): Asymptotically optimal lower bounds for the condition number of a real Vandermonde matrix. SIAM J. Matrix Anal. Appl., vol. 28, pp. 829-844.
20. Liu, C.-S. (2006): The Lie-group shooting method for nonlinear two-point boundary value problems exhibiting multiple solutions. CMES: Computer Modeling in Engineering & Sciences, vol. 13, pp. 149-163. (Pubitemid 43863599)
21. Liu, C.-S. (2007a): A modified Trefftz method for two-dimensional Laplace equation considering the domain's characteristic length. CMES: Computer Modeling in Engineering & Sciences, vol. 21, pp. 53-65. (Pubitemid 47606861)
22. Liu, C.-S. (2007b): A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain. CMES: Computer Modeling in Engineering & Sciences, vol. 20, pp. 111-122. (Pubitemid 47229156)
23. Liu, C.-S. (2007c): An effectively modified direct Trefftz method for 2D potential problems considering the domain's characteristic length. Eng. Anal. Bound. Elem., vol. 31, pp. 983-993. (Pubitemid 350026773)
24. Liu, C.-S. (2008a): A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly-connected domains. Numer. Meth. Partial Diff. Eq., vol. 24, pp. 179-192.
25. Liu, C.-S. (2008b): A time-marching algorithm for solving non-linear obstacle problems with the aid of an NCP-function. CMC: Computers, Materials & Continua, vol. 8, pp. 53-65.
26. Liu, C.-S. (2008c): A fictitious time integration method for two-dimensional quasilinear elliptic boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol. 33, pp. 179-198.
27. Liu, C.-S. (2009a): A fictitious time integration method for solving m-point boundary value problems. CMES: Computer Modeling in Engineering & Sciences, vol. 39, pp. 125-154.
28. Liu, C.-S. (2009b): A fictitious time integration method for the Burgers equation. CMC: Computers, Materials & Continua, vol. 9, pp. 229-252.
29. Liu, C.-S. (2009c): A fictitious time integration method for solving delay ordinary differential equations. CMC: Computers, Materials & Continua, vol. 10, pp. 97-116.
30. Liu, C.-S. (2009d): A fictitious time integration method for a quasilinear elliptic boundary value problem, defined in an arbitrary plane domain. CMC: Computers, Materials & Continua, vol. 11, pp. 15-32.
31. Liu, C.-S. (2010): The fictitious time integration method to solve the space-and time-fractional Burgers equations. CMC: Computers, Materials & Continua, vol. 15, pp. 221-240.
32. Liu, C.-S.; Atluri, S. N. (2008a): A novel time integration method for solving a large system of non-linear algebraic equations. CMES: Computer Modeling in Engineering & Sciences, vol. 31, pp. 71-83.
33. Liu, C.-S.; Atluri, S. N. (2008b): A novel fictitious time integration method for solving the discretized inverse Sturm-Liouville problems, for specified eigenvalues. CMES: Computer Modeling in Engineering & Sciences, vol. 36, pp. 261-285.
34. Liu, C.-S.; Atluri, S. N. (2008c): A fictitious time integration method (FTIM) for solving mixed complementarity problems with applications to non-linear optimization. CMES: Computer Modeling in Engineering & Sciences, vol. 34, pp. 155-178.
35. Liu, C.-S.; Atluri, S. N. (2009a): A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems. CMES: Computer Modeling in Engineering & Sciences, vol. 43, pp. 253-276.
36. Liu, C.-S.; Atluri, S. N. (2009b): A Fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiationof noisy data, and its relation to the filter theory. CMES: Computer Modeling in Engineering & Sciences, vol. 41, pp. 243-261.
37. Liu, C.-S.; Hong, H. K.; Atluri, S. N. (2010): Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems. CMES: Computer Modeling in Engineering & Sciences, vol. 60, pp. 279-308.
38. Liu, C.-S.; Yeih, W.; Atluri, S. N. (2009): On solving the ill-conditioned system Ax = b: General-purpose conditioners obtained from the boundary-collocation solution of the Laplace equation, using Trefftz expansions with multiple length scales. CMES: Computer Modeling in Engineering & Sciences, vol. 44, pp. 281-311.
39. Neagoe, V. E. (1996): Inversion of the Van der Monde matrix. IEEE Sign. Proce. Lett., vol. 3, pp. 119-120.
40. Quarteroni, A.; Sacco, R.; Saleri, F. (2000): Numerical Mathematics. Springer-Verlag, Berlin.
41. Shen, Y. H.; Liu, C.-S. (2011): A new insight into the differential quadrature method in solving 2-D elliptic PDEs. CMES: Computer Modeling in Engineering & Sciences, vol. 71, pp. 157-178.
42. Shu, C. (2000): Differential Quadrature and Its Application in Engineering. Springer-Verlag, London
43. Skrzipek, M. R. (2004): Inversion of Vandermonde-like matrices. BIT, vol. 44, pp. 291-306.
44. Tsai, C. C.; Liu, C.-S.; Yeih, W. (2010): Fictitious time integration method of fundamental solutions with Chebyshev polynomials for solving Poisson-type nonlinear PDEs. CMES: Computer Modeling in Engineering & Sciences, vol. 56, pp. 131-151.
45. van der Sluis, A. (1969): Condition numbers and equilibration of matrices. Num. Math., vol. 14, pp. 14-23.
46. Wertz, H. J. (1965): On the numerical inversion of a recurrent problem: The Vandermonde matrix. IEEE Trans. Auto. Control, vol. AC-10, p.492.