Numerical methods - lecture 2
Jiří Zelinka
Autumn 2017
Jiří Zelinka
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Solving nonlinear equations
Equation
f(x) = 0,
x e / = [a, b], f is continuous real function x e / - solution, root of f.
Iterative process: We create sequence (x/c)^0, xk —>► x. (x^)^ - iterative sequence.
Bisection method
f (a) • f (6) < 0, a0 = a, 60 = ft, let x0 = (a0 + 60)/2.
If f (a0) • f (*o) < 0 we choose ai = ao, 6i = x0, else ai = x0, bi = 60, x e [ai,
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Jiří Zelinka
Numerical methods - lecture 2
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Generally: we have ak, bkl f(ak) • f(bk) < 0, x e [ak, bk], let *k = (a* + ^/c)/2.
If f(ak) - f{xi<) < 0 we choose ak+i = a/c, 6/c+i = xk, else
^/c+i = *k, bk+i = ^/c, so x e [a/c+i,
Estimate of the absolute error in k-th step:
xk-x
<
b-a
2k+l
Jiří Zelinka
□ S1 Numerical methods - lecture 2
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Fixed point iteration
• Equation
x = g(x)
• g continuous on / = [a, b]
• Solution x is called the fixed point of the function g
Iteration process
• Let us choose x0 G / and xi = g(x0).
• Generally xk+1 = g(xk).
• Function g is called iteration function.
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Geometric meaning
The fixed point x is the intersection of the function g and line
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Graphical representation of the iteration process:
Jiří Zelinka
□ [31 Numerical methods - lecture 2
The convergence is faster if the derivative of g in t\ intersection is close to 0:
Jiří Zelinka
□ [31 Numerical methods - lecture 2
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.
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Jiří Zelinka Numerical methods - lecture 2
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Estimation of the error
*k-Ú<
L
1 - L
x0 - Xi
Example
x3 + 4x2 - 10 = 0
Classification of the fixed points
The fixed point x of the function g is called
• attractive if < 1, then the iterative process converges on some neighborhood of x.
• repelling if ^(x)! > 1, then the iterative process doesn't converge.
Jiří Zelinka
□ i3i Numerical methods - lecture 2
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le process doesn't converge if Ig"'^)! > 1
Jiří Zelinka
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Creating of the iteration function
f(x) = 0
->• x = g(x)
Generally:
g(x) = x -
g(x) = x -
f_(x) K
f_(x) h(x)
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Jiří Zelinka Numerical methods - lecture 2
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