[alogo] Newton's iterative procedure

The figure below illustrates Newton's theorem for F(x) = a*xn - b*ex (see NewtonIterative.html ). The expression for x - F(x)/F'(x) is the one contained in the formula-object: G(x) = (n-1)*a*xn + b*exp(x)*(1-x))/( n*a*x(n-1) - b*exp(x). The sequence, starting with (that particular) x0 and applying the iterative procedure xn+1 = G(xn) converges rapidly to the zero z = 1.5196... of F(x).

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

Newton's procedure is a special iterative procedure (see Iterative.html ). By moving x0, you can see that the initial x0 for the sequence converging to that root can be arbitrarily selected from the interval [1, 6.5]. But for values as big as x0 = 8.0 the convergence fails.

See Also

Iterative.html
NewtonIterative.html

References

Gisela-Jordan Engeln und Fritz Reutter Numerische Mathematik fuer Ingenieure Mannheim B.I.-Wissenschaftsverlag 1978, p. 43

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