Numerical methods - lecture 9
Jiří Zelinka
Autumn 2017
Jiří Zelinka Numerical methods - lecture 9 1/16
Numerical calculation of the derivative
xq, ..., xn - given points,
fo,...,fn- given function values,    = f(*k)
We want to calculate the approximation of f\x) from this data.
Let P be the interpolation polynomial for given data.
f'{x) « P'(x)
Example 1.
n = l,
Data: x0,xi, ŕ0, fľ
Pi*) = *E*;(x - xo) + fo f'(x) « P'(x) =
xi—xq
Jiří Zelinka
Numerical methods - lecture 9
2/16
Example 2.
n = 2, data: x0,xi,x2, ŕ0, /i, £
P(x) = ŕ  (x-xi)(x~x2)    i   f  (x-xq)(x-x2) i   f (x-x0)(x-xi) v   '        0 (x0-xi)(x0-x2)        1 (xi-x0)(xi-x2)        2 (x2-x0)(x2-xi)
pf(x) = ŕ     2x-xi~x2       i   f     2x-xp-x2       i   ^ 2x-x0-xi v   '        0 (x0-xi)(x0-x2)        1 (xi-x0)(xi-x2)        2 (x2-x0)(x2-xi)
Equidistant points: xi — x0 = x2 — xi = h :
P'(v\ _       f 2x-xi-x2       f 2x-x0-x2   i   r 2x-x0-xi
Př(x2) = ±(f0-4f1 + 3f2) P"(x) = ±(f0-2f1 + f2)
Jiří Zelinka
Numerical methods - lecture 9
3/16
Derivation from the Taylor series
/ :   f(x+h) = f{x) + f'{x)h + \f"{x)h2 + \f'"{x)tf + 0(h4) II :   f(x-h) = f{x) - f'\x)h + \f"\x)h2 - ±f'"{x)h3 + 0(/j4) /-// :   f{x+h) - f(x-h) = 2f'{x)h + |P'(x)/73 + 0(/74)
f'{x) = i;[f(x+h)-f{x-h)] + 0(ri>)
l+ll :   f(x+h) + f{x-h) = 2f{x) + f"{x)h2 + 0{h4) f»(x) = ±[f(x+h) - 2f(x) + f(x-h)] + 0{h2)
4/16
Numerical integration - quadrature formulae
x0,..., xn - given points, a < x0 < x\ < • • • < xn < b fo,...,fn- given function values,    = f{xi<)
Let P be the interpolation polynomial for given data.
f(x)dx& / P(x)dx
Example 1.
n = 1, a = x0, b = xi, ŕ(a), P(x) = M(x_a) + f(a)
JP(x)dx=\f^
_ f(a)+f(b)
(b-a)
□ S1
Jiří Zelinka Numerical methods - lecture 9
5/16
Trapezoidal rule
Jiří Zelinka
6/16
Example 2.
n = 2, equidistant points:
a = x0,xi = a + h = ä4^, b = x2 = a + 2h,
^b? fi? 6 ~ function values
Pfx) = ŕ  (x~xi)(x~x2)    i   f  (x-xq)(x-x2)    i   f (x-x0)(x-xi)
V     / 0 ( Yn — Vi Vxn — ^" (*1—Xq)(xi—x2 ) ^ f yv,— YnV Yo —
(x0-xi)(x0-x2) b b
(x2-x0)(x2-x1)
J f{x)dx ^ J P{x)dx  =   ^ [f{a) + 4f{^) + f{b)
=   l[f(a) + 4f(?¥) + f(b)]
Jiří Zelinka
Numerical methods - lecture 9
7/16
Jiří Zelinka
Numerical methods - lecture 9
Composite (chained) trapezoidal rules
Equidistant points:
a = x0 < xi < • • • < b = xm X/+11 = x/ + h, ft = f (x,) We use the trapezoidal rule for every interval [x;,x;+i]:
—r—n H---—h H---—h H-----|----h
= -[fo + 2f1 + 2f2 + --- + 2fn_1 + fn]
Jiří Zelinka Numerical methods - lecture 9
9/16
Composite Simpson's rules
Equidistant points, n - even:
a = x0 < xi < • • • < b = xm X/+11 = x/ + h, ft = f (x,) We use the Simpson's rule for every interval [x2/,x2/+2]
^ [f0 + 4/i + f2] + ^ [6 + 4/3 +      + • • • + +^ [f„-2 + 4/n-l + /n]
= ^[fo + 4/i + 2f2 + 4/3 + 2/4 + • • • + 2fn_2 + 4/:n_i + fn]
11/16
350 r
300 ■
0 1-.............-1
0 2 4 6 8 10
x
Jiří Zelinka
12/16
Monte Carlo integration
Method I
Xi,... ,Xn - random numbers distributed uniformly on [a, b]
b
J f(x)dx « ^ f(X;)
13/16
Monte Carlo integration
Method II
Let f be non-negative on [a, b], f{x) < M for every x G [a, b].
Pd5 Vi]f... ,[X„. Y„] - observations of the random vector [X, Y] distributed uniformly on [a, b] x [0, M]
b
J f{x)dx
1
n
P(Y<f(X))
M{b-a)
a
n
Yl lyi<n><i)
1=1
where / is the indicator function.
14/16
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1 .2
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1 .6
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15/16
Application:
Approximation of tt:
[X, Y] distributed uniformly on [0,1] x [0,1]
P(X2 + Y2 < 1) = f
[Xi, Vi],... \Xn. Yn\\ observations of [X, Y]
n
;'=1
0        0.1       0.2       0.3       0.4       0.5       0.6       0.7       0.8       0.9 1
□ S1
Jiří Zelinka Numerical methods - lecture 9
16/16