Numerical methods - lecture 3
Jiří Zelinka
Autumn 2017
Jiří Zelinka
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Repetition
Equation
f{x) = 0       .—>       x = g(x)
Fixed point method:
Iteration process:
xk+1 = g(xk)
Geometric meaning
The fixed point x is the intersection of the function g and line
y = x.
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The existence and uniqueness of the fixed point
Theorem: If for the function g continuous on / = [a, b] the following condition holds
Vx G / : g (x) e /,
then there exists at least one fixed point x e / of the function g. Moreover, if there exits constant L < 1 that for all x e /
VxG / : |g'(x)| < L,
then there exit one fixed point x and for any xq G / the iteration process given by formula
Xfr+i = g(xk)
converges to this fixed point. Function g is called contraction.
Jiří Zelinka
Numerical methods - lecture 3
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The error of the iteration
1 - L
x0 - x!
Classification of the fixed points
The fixed point x of the function g is called
• attractive if |g'(x)| < 1, then the iterative process converges on some neighborhood of x.
• repelling if |gr'(x)| > 1, then the iterative process doesn't converge.
Creating of the iteration function
g(x) = x -
g(x) = x -
f_(x) K
f_{x) h(x)
Jiří Zelinka
Numerical methods - lecture 3
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Newton(-Raphson) method
Let us return to the equation
f(x) = 0.
Xo - initial iteration, X\ - intersection of the tangent to f in Xq the axis x.
xk+1 = xk
f'(xk)
Iteration function:
Xk+i - intersection of the tangent to f in x^ the axis x —> tangent method
Jiří Zelinka
□ [31 Numerical methods - lecture 3
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Convergence
Theorem 1
Newtom methods converges to the root x if the function f has continuous derivative in some neighborhood of x, f'{x) ^ 0 and the initial iteration xq is close enogh to x.
Theorem 2
If f has continuous the second derivative in some neighborhood of x and f'{x) ^ 0 then g\x) = 0 for the iteration function of Newton method.
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Example 1:
Computation of y(a)
f (x) =x2-a, f'{x) = 2x.
xk+1 = xk
X/.        d        X/. ~\- 3
2xl
2xl
Example :
Computation of - without division:
xk+1 = xk{2 - axk)
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Jiří Zelinka Numerical methods - lecture 3
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Fourier conditions
Theorem 3
O Let f has continuous the second derivative in [a, 6], f (a) • f(b) < 0.
O Let Vx e [a, b] : f'{x) ^ 0 and f" doesn't change its sign in [a, b]
Let's choose x0 G {a, 6} such that f (x0) • f" > 0. Then the sequence generated by Newton method converges monotonously to x.
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Fourier conditions for convex function
f m > o
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Jiří Zelinka
□ [31 Numerical methods - lecture 3
Fourier conditions for concave function
f{x0) < 0
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Jiří Zelinka
□ [31 Numerical methods - lecture 3
Methods derived from Newton method
Secant methods
f(xk) - f (xfc_i) xk - Xk_i
i — 1,2,...
Xk+l = Xk
f(xk) - f(xk-i)
i = 1, 2,...
Jiří Zelinka
□ [31 Numerical methods - lecture 3
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Methods derived from Newton method
False position methods (regula falsi)
Similar to secant method with sign control: f(a)f(b) < 0, f e C[a, b], x0 = a, xx = b, f(x0)f(x1) < 0
Xk+l = *k
xk -xs
f m - f m
f(xk),       k = 0,1,...
wher s is the largest index for which f(xk)f(xs) < 0.
Remark: If f is convex or concave in [a, b] then s = 0 or s = 1 for all iterations.
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Jiří Zelinka Numerical methods - lecture 3
12/15
Regula falsi for convex function
Jiří Zelinka
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Order of the convefgence
Let p > 1, Xk —> x, ek = xk — x. If
lim
	e/c	
e/c+i		p
= C < oo
then p is called the order (rate) of the convergence of the sequence (xk)^=0.
If the sequence (x^)^ is generated by the numerical methods, then p is the order (rate) of the method
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Theorem
Let the derivatives of the iteration function g be continuouns to order q > p. Then the order of the convergence of the sequence (x/c)/£=o generated by the iteration process xk+i — §{xk) is equal to p iff
g(x) = x, g'(x) = 0, g"{x) = 0.....g^(x) = 0,
g{p)(x) / 0,
Orders of methods:
Fixed point 1 Newton 2
Secant ^fi = 1.618
Regula falsi 1
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