Numerical methods - lecture 4
Jiří Zelinka
Autumn 2017
Jiří Zelinka Numerical methods - lecture 4 1/14
Repetition
Newton method
f {x) = 0,      xk+1 = xk -
k = 0,1,2,...
9
Jiří Zelinka        Numerical methods - lecture 4
j    "O c^ (|v 2/14
Fourier conditions
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     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.
3/14
Secant methods
Method regula falsi
xk+1 = xk
xk - x£
k — 1,2,...
f(xk) - f(xs) wher s is the largest index for which ŕ(x/c)ŕ(xs) < 0
Jiří Zelinka
< S" ►
Numerical methods - lecture 4
5/14
Order of the convefgence
Let p > 1, xk —> x, ek = xk — x. If
k-too ek+i p
C < oc
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.
p = 1 —)► linear method p = 2 —>> quadratic method
□
umerical methods - lecture 4
6/14
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 ^ = 1.618
Regula falsi 1
Example: geometric sequence
7/14
Acceleration of convergence - Aitken č2-metho
Geometric derivation
Let
e{xk) = *k- */c+i,      s(xk+1) = xk+1 - xk+2-
Points [x/f, s(xk)], [xk+1, s(xk+1)] are connected by the line. Its intersection with the axis x is the approximation of the limit of the sequence xk.
0.25
0.2
0.15
0.1
0.05
-0
-0.05       0       0.05     0.1     0.15     0.2     0.25     0.3 0.35
Jiří Zelinka N umerical methods - lecture 4
8/14
The equation of the line:
y - e(Xk) = £M ~ ^(x - x»)
*k — xk+l
The intersection with the axes x:
~ _    _ s{xk){xk - xk+1) _
xk — Xk ,    . . .   — Xk
e(xk) - £{xk+1)
(xfc+i - xk)
xk+2 - 2xk+1 + xk
Jiří Zelinka
Numerical methods - lecture 4
9/14
Theorem
Let {xk}™=0, lim xk = x, xk ^ x, k — 0,1,2,..., be a sequence and let
Xfr+i-x = (C+7/c)(x/c-x), k = 0,1,2,..., \C\ < 1, lim 7* = 0.
/c—S-oo
Then
Xk = Xk
(xk+1 - xk)
xk+2 - 2xk+1 + xk is defined for k enough large and
lim ^4=0,
/c-^oo Xk — X
i.e., the sequence {x^} converges to x faster than {x^}
10/14
Ordinary differences:
Axk = xk+x - xk
A2xk = Axk+1 - Axk = xk+2 - 2xk+1 + xk A3xk = A2xk+1 - A2xk
Xk = Xk
(Axky A2xk
Jiří Zelinka
□ S1 Numerical methods - lecture 4
11/14
Steffensen method
Let g be iteration function for the equation x = g(x). Let's put
Yk = gM,       zk = g(yk),
_        {yk - Xk)2 *k+i — xk---—■—.
zk - 2yk + xk
This method id called Steffensen method and it can be
described bz the iteration function ip:
Xk+l = vixk),
for
<p(x) = x-
(g(x) ~ x)
xg{g{x)) - g2{x)
g{g{x)) - 2g(x) + x    g{g{x)) - 2g(x) + x'
12/14
Theorem
O If í^(x) = x then g (x) = x.
O If g(x) = x, the derivative g'{x) exits and g\x) ^ 1, then (p(x) = x.
Jiří Zelinka
Numerical methods - lecture 4
5    'O ^ O' 13/14
Systems of non-linear equations
Newton method
F(x) = o, FeC2(0(£))
(dm
dxi dfm{x)
dfi(x) \
dxm
dfm{x)
dx,
m
xk+l = xk - J-\xk)F(xk)
Iteration function
G(x) = x - Jp1(x)F(x)
□ S1
Jiří Zelinka Numerical methods - lecture 4
14/14