Numerical methods - lecture 8
Jiří Zelinka
Autumn 2017
Jiří Zelinka
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Spline interpolation
x0,..., xn - given points, x0 < x\ < • • • < xn fo,...,fn- given function values
r, d > 0 natural numbers, r - degree, d - defect
S - spline - piecewise polynomials of degree r S has continuous derivatives up to order r — d
Sr d - space of splines of degree r with defect d
Jiří Zelinka
□ UP Numerical methods - lecture 8
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Example 1.
Si i - piecewise linear continuous functions
0.9 h
0.8 h
0.7 h
0.6 h
0.5 h
0.4 h
0.3 h
0.2 h
0.1
linear spline
function 1/(1 -x )
S G <Si?i - linear spline, it is determined uniquely by the function values /b, • • •, fn-
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Jiří Zelinka Numerical methods - lecture 8
Example 2.
£3,1 - piecewise polynomials of degree 3 with continuous derivatives to order 2.
Number of parameters describing spline S e £3,1:
We have n subintervals //< = [x^, x^+i], k = 0,..., n — 1, in every subinterval the spline is described by 4 parameters:
Forx e //c   S(x) = Sk(x) = a/c+6/c(x-x/c)+c/c(x-x/c)2+c//c(x-x/c)3
The spline S is described by 4n parameters.
These parameters are bound by conditions: S is continuous in xi,..., x„_i:     n — 1 conditions S; is continuous in xi,... , xn_i:    n — 1 conditions S/; is continuous in xi,... , xn_i:   n — 1 conditions S(x/C) = 4, /c = 0,..., n\ n + 1 conditions
4/7 — 2 conditions
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Together
To obtain the unique cubic spline we need two additional boundary conditions:
O Sf(x0) and Sf(xn) are given - complete cubic spline
O S^xq) and S^x^) are given, especially Sf(x0) = Sf(xn) = 0: natural cubic spline
O S//; is continuous in xi and xn_i\ not-a-knot conditions
O S(x0) = S(x„), S'(x0) = Sf(xn), S"(x0) = S'^x^periodic spline
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Interpolation spline with not-a-knot conditions
0.8 -
Jiří Zelinka
Numerical methods - lecture 8
Interpolation complete spline
► < [3" ► - lecture 8
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Approximation of functions
Bernstein polynomials
n e N, b„ik(x) = ("k)xk{l - x)"-k, x 6 [0,1], k = 0,..., n
n
k=0
f is continuous on [0,1], 4 =
Bernstein polynomial of degree n for the function f:
n
k=0
Theorem
Bnj converges uniformly on [0,1] to the function f for
n —>► oc
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Jiří Zelinka Numerical methods - lecture 8
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Least squares method
Theoretical background
A • x = b: system of linear equations
For given x let rx = b — A • x: residue for the vector x
x is called the solution in sense of least squares if
X
<
X
for any x.
1Z(A)\ the range space of the matrix A
1Z^(A)\ the orthogonal complement of 7Z(A)
The vector b can be decomposed in the form b = b\ + b<i,
bi e 11(A), b2 e K^(A)
AT • b2 = o, o is the zero vector
Jiří Zelinka
Numerical methods - lecture 8
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x is the solution of the systém
A ■ x = bx
We have
A-x = b1
AT -A-x = AT ■b1 + o = AT ■ b1 + AT ■ b2 = AT So x is the solution of the systém of normál equations:
AT -A-x = AT ■ b
Jiří Zelinka
□ [31 Numerical methods - lecture 8
Application for the function approximation
xq, ..., xn - given points
/b, • • •, fn ~ given function values
0(x) = c0Oo W + • • • + cn<f>n(x) - given function depending on the parameters Cq, ..., cn.
We want to find the parameters Cq, ..., cn to minimize
n
£ [0w - *f
/c=0
5    ""O O' 11/15
We are looking for the solution in the sense of least squares of the system:
c0%{x0)
Q)<ro(*i) Q)<t>o(x2)
+ + +
ci<t>i(xb) + ci<t>i(xi) +
ci<t>i(x2) +
+ Cm0m(xO) + Cm0m(xx) + Cm0m(x2)
fo
fl
Q)<l>o(Xn)     +     Ci<t>i(xn) +
Let
A =
( <t>o(*0)   QiM •• <t>o(xi)   <l>i(xi)   • •
<t>o(x2)  <Di(x2) ••
+     Cm0m(xn)     = fn
<M*l)
<t>m(x2)
V MXn)    <l>i(xn)    •••     <t>m(x„) J
and   r~ =
í fo\ {fnJ
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Jiří Zelinka Numerical methods - lecture 8
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Then the parameters c = (cq, ..., cm)T are given by the normal equations
AT•A•c=AT•f
i.e.
c =
Jiří Zelinka
□ [31 Numerical methods - lecture 8
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Example:
	1	2	3	4	5	6	7	8	9	10
fi	2.7	5.5	7.5	9.0	11.3	12.6	14.9	17.4	19.3	21.5
Find a linear function approximating data. Solution: <t>0(x) = 1, Oi(x) = x
/i i
i
i
i
i
i
i
i
Vi
A =
1 \		f 21 \
2		5.5
3		7.5
4		9.0
5	f =	11.3
6		12.6
7		14.9
8		17.4
9		19.3
10 /		I 21.5 )
Jiří Zeli	nka         Numerical m	ethods - lecture
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c =
(A
T
A)~1AT.f=(12
1.0267 0261
,   <D(x) = 1.0267+2.0261x .
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□ [31 Numerical methods - lecture 8
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