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Title: Slide 1 Author: mav Last modified by: mav Created Date: 9/18/2002 1:02:11 AM Document presentation format: On-screen Show Company: ITE Other titles – PowerPoint PPT presentation

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Title: 13

1
13 18 ???µß?? 2002
2
?e??e??µe?a

?? p??ß??µa t?? ?a?eµß????
??? e??a? ?a??? ?e??µ???- ?? p??pe? ?????µe?
???????µ??? ?a?eµß???
??p?? ?a?eµß???? Lagrange
?a?eµß??? st?? Maple
S?µpe??f??? ???????µ?? ?a?eµß????
?µ?µat??? ???????µ??? ?a?eµß??? - G?aµµ???
Lagrange
?? F?s???? ??ß???? Splines
??? ?p????µe ?a ????µe µ?a F?s??? Spline?
?d??t?te? F?s???? ??ß???? Splines
? ?a?eµß????sa CatmullRom
? ???????µ??? ?a?eµß??? e??a? G?aµµ??? -
G?aµµ???t?ta t?? ?µ?µat???? ?a?eµß????
S??a?t?se?? ??s?? G?aµµ???? ?a?eµß???? Lagrange

3
?? p??ß??µa t?? ?a?eµß????

???ß??µa G???????µe t?? t?µ? t?? s????t?s??,
x(t), sta s?µe?a t0, t1 , ... , tn. ???a e??a? ?
t?µ? t?? se ??p??? ???? s?µe?? t
?a?ade??µata
????µe µet??se? t? ???µet?? se ???sµ??a s?µe?a
µ?a? pe??????. T????µe ?a µ????µe t? ???µet?? se
??p??a ???a e?d??µesa s?µe?a.
??p??? ?d?a?te?a ?????ß??? p????aµµa ??e?
?p?????se? t?? ta??t?ta µeta????? µ?a? ?µ?da?
d?a??pt?? ?µ?a????? d?af??et???? µe??????.
T????µe ?a µ????µe t?? ta??t?ta µeta?????
?µ?a????? ????? e?d??µes?? µe?e???.
Ta ?p???s??µe ?t? t? t e??a? ??a? p?a?µat????
a???µ??, pa???? p?? st?? p???? µp??e? t? t ?a
e??a? d????sµa p?a?µat???? a???µ??.
S???? ?a as???????µe µe p????? s??a?t?se?? µa??
p? t??
x(t), y(t), z(t), p?? ??????? t?? pa?aµet????
pa??stas? µ?a? ?aµp????.

4
?? ???ß??µa ??? e??a? ?a??? ?e??µ???

??st???? t? p??ß??µa t?? pa?eµß???? de? e??a?
?a??? te??µ???. ?p?????? p??fa??? p?????
s??a?t?se?? p?? d??????ta? ap? ta d????ta s?µe?a.
G?a pa??de??µa
?a p???µata pe??p?????ta? pe??ss?te?? e??
e?et?s??µe pe?????? e?t?? t?? d?ast?µat?? p??
??????? ta d????ta s?µe?a. ??t? t?t??? de?
???eta? p?a pa?eµß??? a??? pa?e?ß???.

5
???p??, t? ?a ?????µe?

?e? µp????µe ?a ??s??µe t? p??ß??µa t??
pa?eµß???? e?t?? ?a? e?µaste d?ate?e?µ???? ?a
d?s??µe ??p??a ep?p??s?et? p????f???a s?et??? µe
t?? s????t?s?.
S??????, ?p???t??µe ?t? ? a???ß?? s????t?s? e??a?
p??a??tata ?µa??, ?at? ??p??a ?????a.
??a stat?st??? p??s????s? ?p???t??µe µ?a
p??a???e???t??? ?ata??µ? s??a?t?se??, ?a? ap?
a?t?? ep??????µe t?? s????t?s? p?? ta?????e? st??
d??e?se? ???st?? t?µ??.
??a p?? s?????? p??s????s? ?p???ste ?t? ?
s????t?s? a???e? se µ?a pe?????sµ??? ???s?
s??a?t?se??, ap? t?? ?p??a µ?a µ??? s????t?s?
ta?????e? µe t?? ded?µ??e? t?µ??.

6
???????µ??? ?a?eµß???

??? ?????µe t?? t?µ? t?? s????t?s?? se n1
d?af??et??? s?µe?a, µp????µe ?a t?? pa?eµß????µe
ß??s???ta? ??a p??????µ? ßa?µ?? t? p??? n t?
?p??? d????eta? ap? ta d????ta s?µe?a.
?? p??????µ? a?t? e??a? µ??ad???, ?a? s??ep?? t?
p??ß??µ? µa? ?a??? te??µ???.
???a? ?????? a?t? ? ??s? ??? a?t??????µe?a
?e????ta
???e s??e??? s????t?s? µp??e? ?a p??se???s?e? µe
?s? a???ße?a (pepe?asµ???) ?????µe µe ??a
p??????µ? ??p???? ßa?µ??.
?? p??????µ? ßa?µ?? t? p??? n p?? pa?eµß??e? t??
n1 t?µ?? µ?a? s????t?s?? µp??e? ?a e??a? p???
d?af??et??? ap? ??t? p?? p??se????e? t??
s????t?s? ?a??.

7
?? Fa???µe?? t?? Runge

runge proc(n) n e??a? ? ßa?µ?? t??
p??????µ??
local f, xp, yp, i, poly, p1, p2, caption
f x -gt 1 / (1x2)
xp seq(-5i10/n, i0..n)
yp map(f,xp)
poly interp(xp,yp,x)
caption cat(degree , n)
p1 plot(f(x), x-5..5, y-2..2,
numpoints401)
p2 plot(poly, x-5..5, y-2..2,
numpoints401, colorred)
plotsdisplay(p1,p2, titlecaption)
end
for n in 2,4,10,20,50,100 do
interface(plotdevicegif, plotoutputrunge.n.
.gif) runge(n)
od

8
??p?? ?a?eµß???? Lagrange

?st? ?t? ?????µe ?a ?????µe p??????µ???
pa?eµß???. ???pe? ?a ß???µe ??a p??????µ? ßa?µ??
n (t? p???) p?? d????eta? ap? ta s?µe?a
(t0, x0), (t1, x1), ... , (tn, xn)
Ta µp????saµe ?a ??s??µe ??a s?st?µa ??aµµ????
e??s?se??
???? ?s?? ?a p??t?µ??saµe ?a ???s?µ?p???s??µe t?
t?p? pa?eµß???? Lagrange

9
???a? ? t?p?? Lagrange ep?t????

?? ?a?ad??µe t?? t?p? pa?eµß???? Lagrange
?a?at????µe ?t? e??a? ??a p??????µ? ßa?µ?? t?
p??? n. T?µ??e?te, ta xi ?a? ti e??a? sta?e???.
?e??? ?µ?? ap? ta d????ta s?µe?a?
G?a n 4, e?et?ste t?? t?µ? ??a t t3. ?? ????
µe i 0,1,2,4 ?a e??a? µ?d??, µ?a ?a? pe???????
t?? ??? (t3 - t3) 0. ? a???µ?t?? t?? ???sµat??
t?? ???? µe i 3 e??a? ?s?? µe t?? pa????µast?.
?p?te ? t?µ? st? t t3 ?a e??a? x3 , t? ?p???
e??a? s?st?!

10
??a ?a??de??µa st?? Maple

gt ti3,2,1,0,1,2
gt xi12,10,5,1,3,4
gt interp(ti,xi,t)
gt with(plots)
gt ptsplot(seq(tii,xii,i1..6),stylepoin
t)
gt interplot(interp(ti,xi,t),t3..3)
gt display(pts,inter)
gt titi,3
gt xixi,6
gt pts2plot(seq(tii,xii,i1..7),stylepoi
nt)
gt inter2plot(interp(ti,xi,t),t3..3)
gt display(pts2,inter2)

11
S?µpe??f??? ???????µ?? ?a?eµß????

? p??????µ??? pa?eµß??? ??e? µe????? ?d??t?te?
p?? e??a? s???? a?ep???µ?te?
? p??s???? ? ? µetaß??? e??? s?µe??? pa?eµß????
µp??e? ?a ep??e?se? ???????? t?? pa?eµß????sa
a??µa ?a? se pe?????? µa???? ap? t? e? ????
s?µe??.
?????? a??a??? sta s?µe?a pa?eµß???? µp????? ?a
????? ?s????? ep?pt?se?? st?? pa?eµß????sa.
? p??sp??e?a pa?e?ß???? µe p??????µa µp??e? ?a
µa? d?se? p??? ?s??µa ap?te??sµata.
?e???? ap? a?t? ta p??ß??µata µp????? ?a
d????????? e?? ???s?µ?p???s??µe tµ?µat???
p??????µ??? pa?eµß??? .

12
?µ?µat??? ???????µ??? ?a?eµß???

??a p??????µ? 9?? ßa?µ?? pa?eµß???? se 10 s?µe?a
? s?µpe??f??? sta ???a µ????? e??a? p??ß??µat???.
??a??a?t???, a? ???s?µ?p???s??µe pe??ss?te?a t??
e??? p??????µa.
???ste ? ?µ?µat??? ??ß??? f?s??? spline p??
pa?eµß??e? ta pa?ap??? ded?µ??a.
?a s?µe?a sta ?p??a pe???µe ap? t? ??a p??????µ?
st? ???? ?????ta? ??µß??. St? pa?ap??? ???e
s?µe?? pa?eµß???? e??a? ??µß??.

13
?µ?µat??? G?aµµ??? ?a?eµß???

?? ap???ste?? s??µa pa?eµß???? e??a? ? tµ?µat???
??aµµ??? pa?eµß??? ap?? e?????µe ta s?µe?a
pa?eµß???? µe e?????aµµa tµ?µata
? µ???d?? a?t? ??e? d?? p??ß??µata
???a??tata e??a? µe??µ???? a???ße?a?.
? pa?eµß????sa ??e? ????e? ? pa??????? t?? de?
e??a? s??e???.
?p? t?? ???? µe??? ??e? ?a? µe???? p??te??µata
????? a??a?? sta s?µe?a pa?eµß???? ep??e??e?
µ???? t?? pe????? p?? ??????? ta a?t?st???a d??
?e?t????? s?µe?a.
? pa?eµß????sa de? ??e? pa?????? s?µpe??f???
e?t?? t?? pe?????? pa?eµß????.

14
?µ?µat??? ?a?eµß??? Lagrange

Ta µp????saµe ?a p??spa??s??µe ?a p????µe µ?a p??
a???ß? tµ?µat??? pa?eµß????sa ???s?µ?p????ta?
pa?eµß??? Lagrange µe pe??ss?te?a ap? d?? s?µe?a.
???ste µ?a tµ?µat??? ??ß??? pa?eµß????sa,
p??sa?µ?sµ??? se µ?a tet??da s?µe??? ???e f???
?d?, ????µe ??µß??? st? 4? ?a? st? 7? s?µe??.
??a????????µe aµ?s?? t? p??ß??µa ?a ??µµ?t?a de?
e?????ta? µeta?? t??? ?µa??.

15
??????ta? ta ??µµ?t?a ?µa??

G?a ?a e??s??µe ta ??µµ?t?a ?µa??, µp????µe ?a
???s?µ?p???s??µe ??ß??? p??????µa, pa?eµß????ta
se d?? µ???? s?µe?a. ???e ??µµ?t? e??a? e?e??e??
?a e???e? µe ta ?e?t????? t?? ?µa??.
??s? e?e??e??a ??e?
???e ??ß??? p??????µ? ??e? 4 pa?aµ?t????. ?? 2
ap? a?t?? ???s?µ?p?????ta? ??a ?a ???p????e? ?
pa?eµß??? sta d?? s?µe?a. ??a pe??sse???? 2
pa??µet??? se ???e ??µµ?t? t?? ?p??e? µp????µe ?a
???s?µ?p???s??µe ??a ?a ?????µe t?? s???????
pa?eµß????sa ?µa??.
Ta d??µe d?? t?t??a s??µata ??a tµ?µat??? ??ß???
pa?eµß???
??? f?s???? ??ß???? splines ?a?
??? pa?eµß????sa CatmullRom.

16
?? F?s???? ??ß???? Splines

?st? ?t? pa?eµß????µe se n1 ded?µ??a s?µe?a
???s?µ?p????ta? n ??ß??? p??????µa. ???ste ??a?
t??p?? ap?te?esµat???? ???s?? t?? d?a??s?µ?? 4n
pa?aµ?t???
?pa?t?se t? ???e ??µµ?t? ?a pa?eµß??e? ta
ded?µ??a s?µe?a p?? t? af?????. ??t? µa? d??e? 2n
pe?????sµ???.
?pa?t?se ta ?e?t????? ??µµ?t?a ?a ????? ?d?e?
t?µ?? p??t?? pa?a????? sta ????? t??? s?µe?a.
????? n-1 pe?????sµ??.
?pa?t?se ta ?e?t????? ??µµ?t?a ?a ????? ?d?e?
t?µ?? ?a? de?te??? pa?a????? sta ????? t???
s?µe?a. ????? n-1 pe?????sµ??.
?a? pe??sse???? d?? a??µa pa??µet???. ?pa?t?se
µ?de????? t?µ?? de?t???? pa?a????? sta ???a.
?? pa?ap??? s??µa µa? d??e? ??ß???? f?s????
splines.

17
??? ?p????µe ?a ????µe µ?a F?s??? Spline?

??? ?a ß???µe t?? f?s??? ??ß??? spline p??
pa?eµß??e? sta s?µe?a (t0,x0 ), (t1,x1 ), (t2,x2
)
H spline ?a ap?te?e?ta? ap? 2 ??µµ?t?a
a3t3 a2t2 a1t a0 (t0,lt t lt t1 )
b3t3 b2t2 b1t b0 (t1,lt t lt t2 )
?p????µe ?a ß???µe ta a ?a? b ?????ta? t?
pa?a??t? s?st?µa t?? 8 ??aµµ9???? e??s?se??
a3t03 a2t02 a1t0 a0 x0
a3t13 a2t12 a1t1 a0 x1
b3t13 b2t12 b1t1 b0 x1
b3t23 b2t22 b1t2 b0 x2
3a3t12 2a2t1 a1 3b3t12 2b2t1 b1
6a3t1 2a2 6b3t1 2b2
6a3t0 2a2 0
6b3t1 2b2 0

18
?d??t?te? F?s???? ??ß???? Splines

? e??es? t?? f?s???? spline ??a ??a s?????
ded?µ???? s?µe??? e??a? ef??t? se ??a ??????
??????? p?a?s??, a??? e??a? µe µ????? ?ß???
t??p?.
??a ta ded?µ??a s?µe?a ep??e????? ??a ta ??µµ?t?a
µ?a f?s???? spline, p??fa??? ta p??s??ste?a
s?µe?a ep??e????? pe??ss?te??.
? f?s??? spline a?t?st???e? st?? s????t?s? p??
??e? t?? pe??ss?te?e? p??a??t?te? ?a pe???e? ap?
ta d????ta s?µe?a, s?µf??a µe µ?a a??et? ??????
p??a???e???t??? ?ata??µ? s??a?t?se??.

19
? ?a?eµß????sa CatmullRom

?s?? ?a p??t?µ??µe µ?a tµ?µat??? ??ß???
pa?eµß????sa p?? e??a? p?? e????a ?a ?p?????s?e?
ap? µ?a f?s??? spline. ?p?s?? ?a ???aµe t????
a??a??? se ??a s?µe?? ?a ep??e??e? µ???? ta
?e?t????? tµ?µat? t??.
? pa?eµß????sa CatmullRom ??e? ?a? t?? d?? a?t??
?d??t?te?. ?p??e? ?a ???s?e? ?? e???
?pa?t?ste sta ???a t?? ???e ??µµat??? ?a
s?µf????? ?? a?????e? t?µ?? sta ded?µ??a s?µe?a.
?pa?t?ste sta ???a t?? ???e ??µµat??? ?a
s?µf????? ?? t?µ?? t?? pa?a????? p?? ?a?????ete
ap? d?? ?e?t????? ded?µ??a s?µe?a.
??a pa??de??µa

20
? ???????µ??? ?a?eµß??? e??a? G?aµµ???

?p???ste ?t? t? p??????µ?
p(t) antn an-1tn-1 a0
pa?eµß??e? sta s?µe?a
(t0,x0 ), (t1,x1), , (tn,xn)
?a? ?t? t? p??????µ?
q(t) bntn bn-1tn-1 b0
pa?eµß??e? sta s?µe?a
(t0,y0 ), (t1,y1), , (tn,yn).
???a? e????? ?a d??µe ?t? t? p??????µ? p(t)
q(t)
pa?eµß??e? sta s?µe?a
(t0, x0y0), (t1, x1y1), , (tn, xnyn).
?a??µ??a, t? cp(t) pa?eµß??e? sta (t0, cx0),
(t1,cx1), , (tn,cxn).
? p??????µ??? pa?eµß??? e??a? µ?a ??aµµ??? p????.
S?µe??s? ? ?????a t?? ???? ??aµµ???'' e??a?
d?af??et??? ed? ap? a?t?? t?? ??aµµ????
pa?eµß????'' sa? pa?eµß??? µe e?????aµµa tµ?µata.

21
G?aµµ???t?ta t?? ?µ?µat???? ?a?eµß????

? tµ?µat??? p??????µ??? pa?eµß??? e??a? ep?s??
??aµµ???, µe t?? p???p??es? ?t? ßas??eta? se
??aµµ???? pa?eµß????se?.
G?a pa??de??µa
? tµ?µat??? pa?eµß????sa Lagrange e??a? ??aµµ???,
epe?d? t? ???e ??µµ?t? t?? e??a? p??????µ???
pa?eµß????sa.
?? f?s???? ??ß???? splines e??a? ??aµµ????,
epe?d? ?? pa??????? st?? ?p??e? ßas????ta? e??a?
??aµµ????, d??.
???ad?, e?? ?? pa??????? d?? splines s?µp?pt???
?a s?µp?pte? ?a? t? ?????sµ? t???.
? pa?eµß????sa CatmullRom e??a? ??aµµ???, µ?a
?a? ßas??eta? se ???se?? p?? ?a???????ta? ap?
?e??? s?µe???, ?? ?p??e? e??a? ??aµµ???? ?? p???
t?? t?µ?? t?? s??a?t?se?? a?t??.

22
S??a?t?se?? ??s?? G?aµµ???? ?a?eµß????

??? ???s?µ?p????µe ??a ??aµµ??? s??µa, µp????µe
?a pa?????µe µ?a s????t?s? pa?eµß???? ap?
s??a?t?se?? ß?s?? ?? ?p??e? pa?eµß????? µ??? se
??a µ?-µ?de???? s?µe??.
G?a ??a s????? s?µe??? t0, t1, ... ,tn, µp????µe
?a ß???µe t?? pa?a??t? s??a?t?se?? ß?s??
p0(t) pa?eµß??e? sta (t0,1), (t1,0), ... ,(tn,0).
p1(t) pa?eµß??e? sta (t0,0), (t1,1), ... ,(tn,0).
pn(t) pa?eµß??e? sta (t0,0), (t1,0), ... ,(tn,1).
???a, ??a ?a pa?eµß????µe sta s?µe?a
(t0, x0), (t1,x1), ... ,(tn,xn).
???e? ?a ?p?????s??µe t?
p(t) x0 p0(t) x1 p1(t) xn pn(t)