Conte, de Boor - Elementary Numerical Analysis. An Algorithmic Approach (523140), страница 71
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An estimate of the accuracycan then be obtained by comparing these approximate solutions at a fixedset of points on the interval [a,b].418BOUNDARY-VALUE! PROBLEMS IN ORDINARY DIFFERENTIAL EQUATIONSWe turn now to a consideration of the choice of the basis functionsThey are usually chosen so as to have one or more of thefollowing properties:(i) The(ii) Theare continuously differentiable on [a,b]are orthogonal over the interval [a,b], i.e.,are “simple” functions such as polynomials or trigonometric(iii) Thefunctions(iv) Thesatisfy those boundary conditions (if any) which are homogeneous.One commonly used basis is the setwhich is orthogonal over the interval [0, 1]. Note that sin jπx = 0 at x = 0and at x = 1 for all j.
Another important basis set isj = 0, . . . , N where Pj(x) are the Legendre polynomials described inChap. 6. These polynomials are orthogonal over the interval [-1,1].Finally thecan be chosen to be piecewise-cubic polynomials (seeChap. 6).As an example we apply the collocation method to the equation (9.1)which we rewrite asU´´(x) - U(x) = 0(9.20 a)U(0) = 0(9.20b)U(1) = 1We select polynomials for our basis functions and we seek an approximatesolution U N(x) in the form(9.21)U N (x) = c 1 x + c 2 x 2 + c 2 x 3we see that UN(0) = 0 regardless of the choice of the cj’s. Since there arethree coefficients we must impose three conditions on U N(x). One condition is that UN(x) must satisfy the boundary condition at x = 1, hence oneequation for the cj’s is(9.22)U N(1) = cl + c2 + c3 = 1We can impose two additional conditions by insisting that U N(x) satisfythe equation (9.20a) exactly at two points interior to the interval [0,1].
Wechoose, for no special reason, x0 = 1/4 and x1 = 3/4. One computes directlythatU´´N (x) - U N (x) = -c 1 x + (2 - x 2 ) c2 + (6x - x3 )c 3and hence that(9.23)9.3COLLOCATION METHODS419The system of equations (9.22) through (9.23) can be solved directly toyield the solutioncl = 0.852237 l l lc2 = -0.0138527 · · ·c3 = 0.161616 · · ·Substituting these into (9.21) yields the approximate solution(9.24)UN(x) = 0.852237x - 0.0138527x2 + 0.161616x 3This approximate solution can now be used to find an approximate valuefor U(x), or even for U´(x), at any point of the interval [0,1].To see how good an approximation U N(x) is to the exact solutionU(x) = sinh x/sinh 1, we list below a few comparative values (see Table9.1).xUN(X)U(x)0.100.250.500.750.900.0852470.2147190.4246750.6995670.8736110.0853370.2149520.4434090.6997240.873481We thus seem to have two to three digits of agreement, with the worstvalues occurring near the midpoint of the interval.
Considering the smallnumber of basis functions used in U N(x), the results appear to be quitegood. To obtain more accurate results we would simply increase thenumber of basis functions.EXERCISES9.4-l Solve the boundary-value problemU´´(x) - U(x) = xU(0) = 0U (1) = 1by the collocation method. For the trial functions use the polynomial basisU N (x) = c 1 x + c 2 x 2 + c 3 x 3 + · · · + c N x NTake N = 3 first and then N = 4 and compare the results at selected points on the interval.Also compare the approximate results with the exact solution9.4-2 Try to solve the boundary-value problemU´´(x) + U(x) = xU(0) = 0U (1) = 1by the collocation method.
Start with the trial functionwhich automatically satisfies the boundary conditions for all cj 's. Try N = 2 and N = 4 andcompare the results.Previous HomeAPPENDIXSUBROUTINE LIBRARIESListed below are brief descriptions of some major software packageswhich contain tested subroutines for solving all of the major problemsconsidered in this book. Further information as to availability can beobtained from the indicated source.1. IMSL (INTERNATIONAL MATHEMATICAL ANDSTATISTICAL LIBRARY)This is probably the most complete package commercially available. Itcontains some 235 subroutines which are applicable to all of the problemareas discussed in this book and to other areas such as statistical computations and constrained optimization as well.
All of them are written in ANSIFORTRAN and have been adapted to run on all modem large-scalecomputers.SOURCE: IMSL, Inc. GNB Building, 7500 Bellaire Blvd., Houston, Texas77036.2. PORTA fairly complete set of thoroughly tested subroutines for all of thecommonly encountered problems in numerical analysis. It was written in421Next422APPENDIXPFORT, a portable subset of ANSI FORTRAN, and was designed to beeasily portable from one machine to another.SOURCE: Bell Telephone Laboratories, Murray Hill, New Jersey.3. EISPACKA package for solving the standard eigenvalue-eigenvector problem. Itis coded in ANSI FORTRAN in a completely machine-independent form.This is a very high quality software package; it is extremely reliable andcontains numerous diagnostic aids for the user (see [32]).SOURCE: National Energy Software Center, Argonne National Laboratories, 9700 S.
Cass Ave., Argonne, Illinois 60439.4. LINPACKA software package for solving linear systems of equations as well asleast-squares problems. It is written in ANSI FORTRAN, is machineindependent, and is available in real, complex, and double-precisionarithmetic. It has been widely tested at many different computer sites.SOURCE: National Energy Software Center, Argonne National Laboratories, 9700 S. Cass Ave., Argonne, Illinois 60439.Previous HomeREFERENCES1.
Hamming, R. W.: Numerical Methods for Scientists and Engineers, McGraw-Hill, NewYork 1962.2. Henrici, P. K.: Elements of Numerical Analysis, John Wiley, New York, 1964.3. Traub, J. F.: Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey,1963.4. Scarborough, J. B.: Numerical Mathematical Analysis, Johns Hopkins, Baltimore, 1958.5. Hildebrand, F. B.: Introduction to Numerical Analysis, McGraw-Hill, New York, 1956.6. Müller, D. E.: “A method of solving algebraic equations using an automatic computer,”Mathematical Tables and Other Aids to Computation (MTAC), vol.
10, 1956, pp. 208-215.7. Hastings, C. Jr.: Approximations for Digital Computers, Princeton University Press, NewJersey, 1955.8. Milne, W. E.: Numerical calculus, Princeton University Press, New Jersey, 1949.9. Lanczos, C.: Applied Analysis, Prentice-Hall, New Jersey, 1956.10. Householder, A. S.: Principles of Numerical Analysis, McGraw-Hill, New York, 1953.11.
Faddccv, D. K., and V. H. Faddccva: Computational Methods of Linear Algebra, Frccman, San Francisco, 1963.12. Carnahan, B., et al.: Applied Numerical Methods, John Wiley, New York, 1964.13. Modem Computing Methods, Philosophical Library, New York, 1961.14. McCracken, D., and W. S. Dorn: Numerical Methods and Fortran Programming, JohnWiley, New York, 1964.15. Henrici, P. K.: Discrete Variable Methods for Ordinary Differential Equations, John Wiley,New York, 1962.16. Hamming, R. W.: “Stable Predictor-Corrector Methods for Ordinary Differential Equations,” Journal of the Association for Computing Machinery (JACM), vol.
6, no. 1, 1959,pp. 37-47.423Next424REFERENCES17. Rice, J. R.: The Approximation of Functions, vols. 1 and 2, Addison-Wesley, Reading,Mass., 1964.18. Forsythe, G., and C. B. Moler; Computer Solution of Linear Algebraic Systems, PrenticcHall, New Jersey, 1967.19. Isaacson, E., and H. Keller: Analysis of Numerical Methods, John Wiley, New York, 1966.20. Stroud, A.
H., and D. Secrest: Gaussian Quadrature Formulas, Prentice-Hall, New Jersey,1966.21. Johnson, L. W., and R. D. Riess: Numerical Analysis, Addison-Wesley, Reading, Mass,1977.22. Forsythe, G. E., M. A. Malcolm, and C. D. Moler: Computer Methods for MathematicalComputations, Prentice-Hall, New Jersey, 1977.23. Stewart, G. W., Introduction to Matrix Computation, Academic Press, New York, 1973.24. Wilkinson, J. H.: The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.25. Ralston, A.: A First Course in Numerical Analysis, McGraw-Hill, New York, 1965.26.
Shampine, L. and R. Allen: Numerical Computing, Saunders, Philadelphia, 1973.27. Gautschi, W.: "On the Construction of Gaussian Quadrature Rules from ModifiedMomenta,” Math. Comp., vol. 24, 1970, pp. 245-260.28. Fehlberg, E.: “Klassische Runge-Kutta-Formeln vierter und niedriger Ordnung mitSchrittweitenkontrolle und ihre Anwendung auf Wärmeleitungsprobleme,” Computing,vol. 6, 1970, pp. 61-71.29. Hull, T.
E., W. H. Enright, and R. K. Jackson: User’s Guide for DVERK—A Subroutinefor Solving Non-Stiff ODE’s, TR 100, Department of Computer Science, University ofToronto, October, 1976.30. Gear, C. W.: Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, New Jersey, 197 1.31. Strang, G., and G. Fix: An Analysis of the Finite Element Method, Prentice-Hall, NewJersey, 1973.32. Smith, B. T., J. M.
Boyle, J. J. Dongerra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B.Moler: “Matrix Eigensystem routines- EISPACK Guide,” Lecture Notes in ComputerScience, vol. 6, Springer-Verlag, Heidelberg, 1976.33. Ortega, J. M., and W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in SeveralVariables, Academic Press, New York, 1970.34. Robinson, S. R.: Quadratic Interpolation Is Risky,” SIAM J. Numer. Analysis, vol. 16,1979, pp.
377-379.35. Rivlin, T. J.: An Introduction to the Approximation of Functions, Blaisdell, Waltham,Mass., 1969.36. Winograd, S.: “On Computing the Discrete Fourier Transform,” Math. Comp., vol. 32,1978, pp. 175-199.37. Cooley, J. W., and J. W. Tukey: “An Algorithm for the Machine Calculation of ComplexFourier Series,” Math. Comp., vol. 19, 1965, pp. 297-301.38. Ehlich, H., and K. Zeller: “Auswertung der Normen von Interpolationsoperatoren,”Math. Annalen, vol. 164, 1966, pp. 105-112.39. de Boor, C., and A. Pinkus: “Proof of the Conjectures of Bernstein and Erdös,” J.Approximation Theory, vol. 24, 1978, pp.
289-303.40. de Boor, C.: A Practical Guide to Splines, Springer-Verlag, New York, 1978.41. Wendroff, B.: Theoretical Numerical Analysis, Academic Press, New York, 1966.42. Wilkinson, J. H.: Rounding Errors in Algebraic Processes, Prentice-Hall, New Jersey, 1963.Previous HomeINDEXAcceleration, 95ff.( See also Extrapolation to the limit)Adams-Bashforth method, 373 -376predictor form, 383program, 377stability of, 392-394Adams-Moulton method, 382-388program, 387stability of, 394for systems, 399Adaptive quadrature, 328ff.Aitken’s algorithm for polynomial interpolation,50Aitken’s D2-process, 98, 196, 333algorithm, 98Aliasing , 273Alternation in sign, 237Analytic substitution, 294ff., 339Angular frequency, 27 1Approximation, 235ff.Chebyshev , 235 - 244least-squares (see Least-squaresapproximation)uniform, 235-244Back-substitution, 148, 156, 163algorithm, 148.
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