Лабораторные работы по численным методам. Пак Т.В. - 10 стр.

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a
i
=
6
h
i+1
h
y
i+1
− y
i
h
i+1
−
2y
0
i
+ y
0
i+1
3
i
, b
i
=
12
h
2
i+1
h
y
0
i
+ y
0
i+1
2
−
y
i+1
− y
i
h
i+1
i
.
¯w
h
y
i
= f(x
i
) i = 0, N
S(x)
f(x) ¯w
h
S(x) ∈ P
3i
(x) ∀x ∈ [x
i
, x
i+1
], i = 0, N − 1
S(x) ∈ C
2
[a, b] S(x) S
0
(x) S
00
(x)
S(x
i
) = f(x
i
) i = 0, N
[a, b] 4N
S(x) ∈ C
2
[a,b]
3(N − 1)
N + 1 3N − 3 + N + 1 = 4N − 2
[a, b]
S
0
(a) = f
0
(a), S
0
(b) = f
0
(b).
S
00
(a) = f
00
(a), S
00
(b) = f
00
(b).
S
i
(a) = S
i
(b), i = 0, 1, 2
S
000
(x
1
+ 0) = S
000
(x
1
− 0), S
000
(x
N−1
− 0) = S
000
(x
N−1
+ 0).
S(x
i
) = y
i
, S
0
(x
i
) = y
0
i
;
S(x) = y
i
+ y
0
i
(x − x
i
) + a
i
(x − x
i
)
2
2
+ b
i
(x − x
i
)
3
6
;
S
0
(x) = y
0
i
+ a
i
(x − x
i
) + b
i
(x − x
i
)
2
2
.
           6 h yi+1 − yi  2y 0 + yi+1
                                  0   i                   12 h yi0 + yi+1
                                                                      0
                                                                            yi+1 − yi i
   ai =                  − i           ,          bi =    2               −             .
          hi+1   hi+1           3                        hi+1       2         hi+1


2. Êóáè÷åñêèå íåëîêàëüíûå ñïëàéíû
Ïóñòü íà ñåòêå w̄h çàäàíû yi = f (xi ), i = 0, N .
    Îïðåäåëåíèå. Ôóíêöèÿ S(x) íàçûâàåòñÿ êóáè÷åñêèì ñïëàéíîì, èíòåð-
ïîëèðóþùèì ôóíêöèþ f (x) â óçëàõ ñåòêè w̄h , åñëè âûïîëíÿþòñÿ ñëåäóþ-
ùèå óñëîâèÿ:
    1. S(x) ∈ P3i (x), äëÿ ∀x ∈ [xi , xi+1 ], i = 0, N − 1,
    2. S(x) ∈ C 2 [a, b], (ò.å. íåïðåðûâíû S(x), S 0 (x), S 00 (x) âî âñåõ âíóòðåííèõ
óçëàõ),
    3. S(xi ) = f (xi ), i = 0, N .
    Ëåãêî ïîêàçàòü, ÷òî ýòèìè óñëîâèÿìè ñïëàéí îïðåäåëÿåòñÿ íåîäíîçíà÷-
íî, à èìåííî, êóáè÷åñêèé ñïëàéí, óäîâëåòâîðÿþùèé äàííîìó îïðåäåëåíèþ,
èìååò åùå 2ñâîáîäíûõ ïàðàìåòðà. Íà êàæäîì èç N îòðåçêîâ ñïëàéí îïðå-
äåëÿåòñÿ 4-ìÿ êîýôôèöèåíòàìè, èòîãî, íà [a, b] âñåãî 4N êîýôôèöèåíòîâ.
                            2
Óñëîâèå 2): S(x) ∈ C[a,b]       äàåò 3(N − 1) ðàâåíñòâ. Óñëîâèå èíòåðïîëÿöèè 1)
äàåò N + 1 ñîîòíîøåíèå. Èòîãî: 3N − 3 + N + 1 = 4N − 2 ñîîòíîøåíèÿ.
    Äâà íåäîñòàþùèõ äîïîëíèòåëüíûõ óñëîâèÿ, êàê ïðàâèëî, çàäàþòñÿ â
âèäå êðàåâûõ óñëîâèé, îïðåäåëÿþùèõ çíà÷åíèå ñïëàéíà èëè åãî ïðîèçâîä-
íûõ íà êîíöàõ îòðåçêà [a, b]:
1. S 0 (a) = f 0 (a), S 0 (b) = f 0 (b).
2. S 00 (a) = f 00 (a), S 00 (b) = f 00 (b).
3. S i (a) = S i (b), i = 0, 1, 2  â ýòîì ñëó÷àå ãîâîðÿò î ïåðèîäè÷åñêîì
ñïëàéíå.
4. S 000 (x1 + 0) = S 000 (x1 − 0), S 000 (xN −1 − 0) = S 000 (xN −1 + 0).


3. Ïîñòðîåíèå êóáè÷åñêîãî ñïëàéíà ÷åðåç íàêëîíû
Êóáè÷åñêèé ñïëàéí ìîæíî ðàññìàòðèâàòü êàê Ýðìèòîâ ñïëàéí, óäîâëåòâî-
ðÿþùèé óñëîâèÿì:
                             S(xi ) = yi ,    S 0 (xi ) = yi0 ;

                                                  (x − xi )2      (x − xi )3
               S(x) = yi + yi0 (x − xi ) + ai                + bi            ;
                                                      2               6

                                                           (x − xi )2
                      S 0 (x) = yi0 + ai (x − xi ) + bi               .
                                                               2

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