An extension of Ortiz' recursive formulation of the tau method to certain linear systems of ordinary differential equations
Authors:
M. R. Crisci and E. Russo
Journal:
Math. Comp. 41 (1983), 2742
MSC:
Primary 65L05; Secondary 65L07
MathSciNet review:
701622
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Ortiz' stepbystep recursive formulation of the Lanczos tau method is extended to the numerical solution of linear systems of differential equations with polynomial coefficients. Numerical comparisons are made with Gear's and Enright's methods.
 [1]
C. A. Addison, Implementing a Stiff Method Based upon the Second Derivative Formulas, Tech. Rep. No. 130/79, University of Toronto, Dept. of Computer Science, 1979.
 [2]
G.
D. Byrne and A.
C. Hindmarsh, A polyalgorithm for the numerical solution of
ordinary differential equations, ACM Trans. Math. Software
1 (1975), no. 1, 71–96. MR 0378432
(51 #14600)
 [3]
M.
R. Crisci and E.
Russo, 𝐴stability of a class of
methods for the numerical integration of certain linear systems of ordinary
differential equations, Math. Comp.
38 (1982), no. 158, 431–435. MR 645660
(83d:65219), http://dx.doi.org/10.1090/S00255718198206456602
 [4]
W.
H. Enright, Second derivative multistep methods for stiff ordinary
differential equations, SIAM J. Numer. Anal. 11
(1974), 321–331. MR 0351083
(50 #3574)
 [5]
W. H. Enright, T. E. Hull & B. Lindberg, "Comparing numerical methods for stiff systems of O.D.E.: s," BIT, v. 15, 1975, pp. 1048.
 [6]
C.
W. Gear, The automatic integration of stiff ordinary differential
equations., Information Processing 68 (Proc. IFIP Congress, Edinburgh,
1968) NorthHolland, Amsterdam, 1969, pp. 187–193. MR 0260180
(41 #4808)
 [7]
T.
E. Hull, W.
H. Enright, B.
M. Fellen, and A.
E. Sedgwick, Comparing numerical methods for ordinary differential
equations, SIAM J. Numer. Anal. 9 (1972),
603–637; errata, ibid. 11 (1974), 681. MR 0351086
(50 #3577)
 [8]
F. T. Krogh, "On testing a subroutine for the numerical integration of ordinary differential equations," Comm. ACM, v. 20, 1973, pp. 545562.
 [9]
C. Lanczos, "Trigonometric interpolation of empirical and analytical functions," J. Math. Phys., v. 17, 1938, pp. 123199.
 [10]
Cornelius
Lanczos, Applied analysis, Prentice Hall, Inc., Englewood
Cliffs, N. J., 1956. MR 0084175
(18,823c)
 [11]
C.
Lánczos, Legendre versus Chebyshev polynomials, Topics
in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll.,
Dublin, 1972) Academic Press, London, 1973, pp. 191–201. MR 0341880
(49 #6626)
 [12]
Leon
Lapidus and John
H. Seinfeld, Numerical solution of ordinary differential
equations, Mathematics in Science and Engineering, Vol. 74, Academic
Press, New YorkLondon, 1971. MR 0281355
(43 #7073)
 [13]
Eduardo
L. Ortiz, The tau method, SIAM J. Numer. Anal.
6 (1969), 480–492. MR 0258287
(41 #2934)
 [14]
E. Ortiz, W. F. Pursuer & F. J. Canizares, Automation of the Tau Method, Report Math. Dept., University of London, 1972.
 [15]
Eduardo
L. Ortiz, Canonical polynomials in the Lanczos tau method,
Studies in numerical analysis (papers in honour of Cornelius Lanczos on the
occasion of his 80th birthday), Academic Press, London, 1974,
pp. 73–93. MR 0474847
(57 #14478)
 [16]
E.
L. Ortiz and H.
Samara, An operational approach to the tau method for the numerical
solution of nonlinear differential equations, Computing
27 (1981), no. 1, 15–25 (English, with German
summary). MR
623173 (83b:65079), http://dx.doi.org/10.1007/BF02243435
 [17]
E.
L. Ortiz, Step by step Tau method. I. Piecewise polynomial
approximations, Computers and mathematics with applications,
Pergamon, Oxford, 1976, pp. 381–392. MR 0464550
(57 #4480)
 [18]
E.
L. Ortiz, On the numerical solution of nonlinear and functional
differential equations with the tau method, Numerical treatment of
differential equations in applications (Proc. Meeting, Math. Res. Center,
Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer,
Berlin, 1978, pp. 127–139. MR 515576
(80c:65180)
 [1]
 C. A. Addison, Implementing a Stiff Method Based upon the Second Derivative Formulas, Tech. Rep. No. 130/79, University of Toronto, Dept. of Computer Science, 1979.
 [2]
 G. D. Byrne & A. C. Hindmarsh, "A polyalgorithm for the numerical solution of ordinary differential equations," ACM Trans. Math. Software, v. 1, 1975, pp. 7196. MR 0378432 (51:14600)
 [3]
 M. R. Crisci & E. Russo, "Asability of a class of methods for the numerical integration of certain linear systems of ordinary differential equations," Math. Comp., v. 38, 1982, pp. 431435. MR 645660 (83d:65219)
 [4]
 W. H. Enright, "Second derivative multistep methods for stiff ordinary differential equations," SIAM J. Numer. Anal., v. 11, 1974, pp. 321331. MR 0351083 (50:3574)
 [5]
 W. H. Enright, T. E. Hull & B. Lindberg, "Comparing numerical methods for stiff systems of O.D.E.: s," BIT, v. 15, 1975, pp. 1048.
 [6]
 C. W. Gear, "The automatic integration of ordinary differential equations," Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1978), NorthHolland, Amsterdam, 1969. MR 0260180 (41:4808)
 [7]
 T. E. Hull, W. H. Enright, B. M. Fellen & A. E. Sedgwick, "Comparing numerical methods for ordinary differential equations," SIAM J. Numer. Anal., v. 9, 1972, pp. 603637. MR 0351086 (50:3577)
 [8]
 F. T. Krogh, "On testing a subroutine for the numerical integration of ordinary differential equations," Comm. ACM, v. 20, 1973, pp. 545562.
 [9]
 C. Lanczos, "Trigonometric interpolation of empirical and analytical functions," J. Math. Phys., v. 17, 1938, pp. 123199.
 [10]
 C. Lanczos, Applied Analysis, PrenticeHall, Englewood Cliffs, N.J., 1956. MR 0084175 (18:823c)
 [11]
 C. Lanczos, "Legendre versus Chebyshev polynomials," in Miller Topics in Numerical Analysis, Academic Press, London, 1973. MR 0341880 (49:6626)
 [12]
 T. Lapidus & J. H. Seinfeld, "Numerical solutions of ordinary differential equations," in Mathematics in Science and Engineering, vol. 74, Academic Press, New York, 1971. MR 0281355 (43:7073)
 [13]
 E. Ortiz, "The tau method," SIAM J. Numer. Anal., v. 6, 1969, pp. 480492. MR 0258287 (41:2934)
 [14]
 E. Ortiz, W. F. Pursuer & F. J. Canizares, Automation of the Tau Method, Report Math. Dept., University of London, 1972.
 [15]
 E. Ortiz, "Canonical polynomials in the Lanczos tau method," in Studies in Numerical Analysis, Academic Press, London, 1974. MR 0474847 (57:14478)
 [16]
 E. Ortiz & H. Samara, "An operational approach to the tau method for the numerical solution of nonlinear differential equations," Computing, v. 27, 1981, pp. 1525. MR 623173 (83b:65079)
 [17]
 E. Ortiz, "Step by step tau method," Comp. Math. Appl., v. 1, 1975, pp. 381392. MR 0464550 (57:4480)
 [18]
 E. Ortiz, "On the numerical solution of nonlinear and functional differential equations with the tau method," in Numerical Treatment of Differential Equations in Applications, Lecture Notes in Math., vol. 679, SpringerVerlag, Berlin, 1978, pp. 127139. MR 515576 (80c:65180)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65L05,
65L07
Retrieve articles in all journals
with MSC:
65L05,
65L07
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307016229
PII:
S 00255718(1983)07016229
Article copyright:
© Copyright 1983
American Mathematical Society
