Fixed Point Iteration Method

Yıl 2013, Cilt: 1 Sayı: 1, 23 - 32, 01.05.2013

Mehmet Karakas

Öz

We discuss the problem of finding approximate solutions of the equation x)0 f()0 (1) In some cases it is possible to find the exact roots of the equation (1) for example when f(x) is a quadratic on cubic polynomial otherwise, in general, is interested in finding approximate solutions using some numerical methods. Here, we will discuss a method called fixed point iteration method and a particular case of this method called Newton’s method

Anahtar Kelimeler

Numerical, Iteration, Alegorithm, Varient

Kaynakça

• [1] Aho A.V., Hopcroft J.E. and Ullman J.D. (1974) The desing and analysis of computer algoritdms addıson Wesley.reading mass. 470 pp. Qa76.6.A.36
• [2] Ames W.F (1977) Numerial methods for partial differential equations (second edition).Academic pres. New York: 365 pp. QA374 A46
• [3] Bailey N.I.J (1967) The mathematical approach to bıology and medicine john wiley&sons london: 296 pp. QH324 B28
• [4] Bailey N.T.J (1957) The mathematical theory of epidemics c.griffin.london: 194 pp. RA652.B3
• [5] Bailey P.B., Shampine L.F and Waltman P.E. (1968) Nonlinear two-point boyndary talue problems academic pres New York:171 pp. QA372 B27
• [6] Bartle R (1976) the elements of real analsysis (second edition) John wiley&sons New York: 480 pp. QA300.B29
• [7] Bekker R.G. (1969) Introduction to terrain vehicle systems. University of Michigan pres An Arbor.Mich: 846 pp. TL243.B39
• [8] Barnadelli H. (1941) “Population Waves” journal of the Burma Research society: 31, 1- 18
• [9] Birkhoff G. and C.De Boor (1964) “Error bounds for spline interpolation” Journal of mathematics and mechanics 13.827-836
• [10] Birkhoff G. and Lynch R.E. (1984) Numerical solution of elliptic problems SIAM publications Philadelphia. Pa: 320 pp. QA374.B57
• [11] Birkhoff G. and Rota G. (1978) Ordinary differential equations.john wiley&sons New York: 342 pp. QA372.B58
• [12] Bracewel R. (1978) The fourier transform and its application (second edition). McGaw Hill.New York: 444 pp. QA403.5.B7
• [13] Brent R. (1973) Algorithms for munimuzation without derivatives. prentice-hall. Englewood cliffs.n.j. 195 pp. QA403.5.B7
• [14] Brigham E.O. (1974) The fast fourier transform prentice-hall.englewood cliffs.NJ; 252 pp. QA403.B74
• [15] Brogan W.L. (1982) Modern control theory prentice-hall.englewood cliffs.N.J; 393 pp. QA402.3.B76
• [16] Brown K.M. (1969) “A quadratically convergent Newton-like method based upon Gaussian elimination” SIAM journal on numerical analysis 6.no 4.560-569.
• [17] Broyden C.G. (1965)”A class of methods for solving nonlinear simultaneous equations.”mathematics of computation.19.577-593
• [18] Belirsch R (1964) “Bemerkungen zur romberg-integration” numerische mathematik 6.6.16
• [19] Fehlberg E. (1964) “New high-order Runge-Kutta formulas with step-size control for systems of first-and second-order differential equations” Zeitschrift für angewandte mathematic and mechanic. 44.17-29.
• [20] Fehlberg E. (1966) “New high-order Runge-Kutta formulas with an arbitrarily small truncation error” Zeitschrift für angewandte mathematic and mechanic. 46.1-16.
• [21] Fehlberg E. (1970) “Klassche Runge-kutta formeln vierter und niedrierer ordnung mit schrittweiten-kontrolle und ihre anwendung auf warmeleitungsprobleme” Computing 6.61-71.
• [22] Fix G. (1975) “A survey of numerical methods for selected problems in continuum mechanics”procedings of a conference on numerical methods of ocean circulation national academy of sciences durham N.H.october 17.20. 1972, 268-283
• [23] Forsythe G.E., Malcolm M.A. and Moler C.A. (1977) Computer methods for mathematical comtations.Prentice-hall.englewood cliffs NJ: 259 pp. QA297.F568.
• [24] Forsythe G.E. and Moler C.B. (1967) Computer solution of linear algebraic systems.prentice-hall.Englewood cliffs.NJ; 148 pp. QA297.F57
• [25] Fulks W. (1978) Advanced calculus (third edition). john wiley&sons. New York; 731 pp. QA303 F568
• [26] Garcia C.B. and Gould F.J. (1980) “Relations between several path-following algorithms and local and global Newton methods” SIAM Review; 22, No.3, 263-274.
• [27] Gear C.W. (1971) Numarical initial-value problems in ordinary differential equations.pretice-hall, Englewood cliffs, N.J: 253 pp. QA372.G4
• [28] Gear C.W. (1981) “Numerical solution of ordinary differential equations: Is there anything left to do?” SIAM review; 23 No.1, 10-24
• [29] George J.A. (1973) “Nested dissection of a regular finite-element mesh” SIAM journal on numerical analysis 10, No.2, 345-362
• [30] George J.A. and Liu J.H. (1981) Computer solutıon of large sparse positive difinite systems. prentice-hall englewood cliffs NJ; 324 pp. QA188.G46
• [31] Gladwell I. and Wait R. (1979) A survey of numerical methods for partial differential equations. oxford university pres; 424 pp. QA377.S96
• [32] Golub G.H. and Van Loan C.F. (1963) Matrix computations john Hopkins university press Baltimore; 476 pp. QA188.G65
• [33] Gragg W.B. (1965) “On extrapolation algorithms for ordinary initial-value problems” SIAM Journal on numerical analysis, 2, 284-403.
• [34] Hageman L.A. and Young D.M. (1981) Applied iterative methods. Acedemic pres. New York; 386 pp. QA297.8.H34
• [35] Hamming R.W. (1973) Numerical methods for scientists and engineers (second edition). McGraw-hill, New York; 721 pp. QA297.H28
• [36] Hatcher T.R. (1982) “An error bound for certain successive overrelaxation schems” SIAM journal on numerical analysis.19. No.5.930-941.
• [37] Henrici P. (1962) Dıscrete variable methods in ordinary differential equations john Wiley&sons New York; 407 pp. QA372.H48
• [38] Householder A.S. (1970) The numerical treatment of a single nonlinear equation McGraw-Hill, New York; 216 pp. QA218.H68
• [39] Watkins D.S. (1982) “Understanding the QR algorithm” SIAM review. 24. No.4, 427-44
• [40] Wendroff B. (1966) Theoretical numerical analysis academic pres New York; 2 pp.QA297.W43
• [41] Wilkinson J.H. (1963) Rounding errors in algebraic processes H.M. stationery Office london; 161 pp. QA76.5.W53
• [42] Wilkinson J.H. and Reinsch V. (1971) Hanbook for automatic computation. Volume linear algebra. springer-verlag. Berlin;439 pp. QA251.W67
• [43] Wilkinson J.H. (1965) The algebraic eigenvalue problem. clarendon pres.oxford; 64 pp.QA218.W5
• [44] Winograd S. (1978) “On computing the discrete fourier transform” mathematics computation, 32, 175-199
• [45] Young D.M. and Gregory R.T. (1972) A survey of numerical mathematics vol. addısonwesley; reading.mass, 533 pp. QA297.Y63.
• [46] Young D.M. (1971) Iterative solutıon of large linear systems. academic pres, New York; 5 pp. QA195.Y68
• [47] Ypma T.J. (1983) “Finding a multiple zero by transformation and Newton –like methods SIAM Review, 25, No.3, 365-378
• [48] Zienkiewicz O. (1977) The finite-element method in engineering science. McGraw-hill london; 787 pp.TA640.2.Z5