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
[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
[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