TRNG I HC BCH KHOA H NIHM NI SUY SPLINE
LP K S CHT LNG CAO K58
NHM 6NGUYN HUNH CDNG VN NGCNGUYN C TRUNGNGUYN TNG LM
GIO VIN HNG DN: TS. H TH NGC YN
Ta bit, phng php ni suy bng a thc nh xt trong cc tit trc ca
chng, cng thc tnh kh thun li, song nhc im ca n chnh l s mc ni suy
tng ln th bc ca a thc cng tng theo, khng thun li cho tnh ton. Trong
tit ny, xt mt thut ton: t n+1 mc ni suy ta xy dng mt a thc bc thp
hn n trn tng khc, m khi ni chng li vn t trn cao; gi l s ghp trn tng
khc.7.1 Khi nim v s ghp trn (Spline)-Xt mt cch chia on [a,b] nh sau
(hay cn gi l phn hoch)
-nh ngha: Hm ghp trn (spline) S(x) bc m n ( hm ghp trn l trn cp
ty nhng vi m>n th cc o hm cp cao khng cn ngha) trn on l hm s c
tnh cht sau:
(i) l lp hm lin tc v c o hm lin tc n cp (m-1) trn on [a,b]
(ii)Trn mi on nh th S(x) l a thc bc m.
-Gi l tp hp cc a thc bc m, l tp hp cc hm ghp trn trn on delta,
th . D dng ch ra l khng gian tuyn tnh. Gi s , th S(x) c to ra bi n
a thc bc m mi a thc bc m cn m+1 h s cha xc nh. Nh vy c S(x) th cn
xc nh n(m+1) h s.
-Nhng theo cch chua on th c (n-1) im ni (khp): . Ti cc im th hm
S(x) c o hm lin tc n cp (m-1), ngha l c m(n-1) iu kin. V vy cn thiu
n(m+1)-m(n-1)=n+m iu kin. Nhng ti im th c t bng s, nn cn thiu
(n+m)-(n+1)=m-1 iu kin. (m-1) iu kin thiu s c b sung nh cc nt bin v
.-Vy bi ton c gii nh sau:
(i)Gi s, hm s y=f(x) xc nh, lin tc trn on [a,b] hoc cho trong
dng bng s: (1)
(ii)Cc mc ni suy c sp xp theo th t:
(iii)Hy dng hm ghp trn S(x) bc m , trn on [a,b] sao cho (2)- n
gin ta xt hm Spline bc 3 (m=3), thng c s dng trong k thut.7.1 Hm
Spline bc 3 (m=3) (c xy dng bi Ahlberg, Nilson v Walsh)* y ta chn
hm ghp trn bc 3 m khng phi l hm bc 2, hm bc 1 hay hm bc ln hn ba
v:+Hm bc 1 v hm bc 2 thiu s bin thin cn thin c th trn ti cc khp vi
mt s lng mc ni suy khng qu ln.+Hm bc 4 cng l t hm bc 3 m ra, cha k
vic tm ra biu thc cho hm bc 4 ny cng khng phi l iu n gin+Hm bc 3
bin thin nhiu m cng khng phc tp nh hm bc 4, do ta s chn hm bc 3. Hm
bc 3 c y cc tnh cht ca cc hm khc nh c cc tr, li lm, im un
-t: vi Trn on ny th S(x) l a thc bc 3. Nn S(x) phi l a thc bc
nht:
vi
- xc nh v ta ln lt cho v
(i)Khi th
(ii)Khi th
-Vy vi (3)Tch phn ng thc trn hai ln ta c:
(4)
- xc nh v ta cho v
Khi th m
-Vy
-Tng t nh trn, khi
-Thay vo (4) ta c S(x) trn on vi
(5)
Vi
-By gi s dng iu kin ghp trn ti cc im khp () xc nh cc h s ().
T (5) ta c vi :
Vi
-Buc ti im th
(i) s dng S(x) trn
(ii) s dng S(x) trn vi , ta s c
-T vi ta c:
H (n-1) phng trnh (n+1) n
(6)
Vi
-Chia hai v ca (6) cho
(*)
-t v
(7)
-Phng trnh (*) c vit li (vi ):
(8)
-H (8) gm (n-1) phng trnh (n+1) n. Hai phng trnh cn thiu s c b
sung t iu kin bin .
(i)Nu hm s y=f(x) c o hm n cp hai ti x=a & x=b; khi ta buc
hm ni suy Spline bc 3 tha mn iu kin:
+Vy ta c h phng trnh:
(9)
Trong c tnh theo cng thc (7)
(ii)Nu hm s y=f(x) c o hm cp mt ti bin
(10)
+T ng thc S(x), khi ta c:
+M , t suy ra:
+t v phi ca ng thc cui cng l th ta c:
(11)
Trong
+Tng t, ti , t iu kin , suy ra:
(12)
Trong +Vy trong trng hp ny, ta c h:
(13)T (9) v (13) cho c hai trng hp ta c th vit trong dng tng qut
sau:
(14)
Trong -H phng trnh (14) c ma trn h s dng ba ng chp v c ng cho
chnh tri nn p dng c phng php lp hoc cng thc truy ui
-Gii h (14), ta c (). Thay vo (5) ta c hm Spline bc ba trn tng
on .
-Trng hp cc mc ni suy cch u nhau:
th -H (14) c vit li:
(15)TH D:
Cho hm s y=sin(x)trn on . Hy lp hm Spline bc 3 xp x hm sin(x)
trn on cho, vi cc mc ni suy GII: -T hm s ta c bng sau:x0
y=sin(x)010
-Do y=sin(x), s dng iu kin trong trng hp
-Li do (u) theo cng thc (15), ta c h phng trnh:
- y
-Thay vo (5) ta c:
()()Vy
Trn on
CODEfunction S=spline3ddx(X,Y,d2x0,d2xn)% input X la vector
hoanh do% Y la vector tung do% dx0 la dao ham tai x0% dxn la dao
ham tai xn% Output% S la he so cua ham bac 3 noi suy
N=length(X); H=diff(X); G=diff(Y); D(1)=2*d2x0; D(N)=2*d2xn; for
i=2:N-1 D(i)=6/(H(i-1)+H(i))*(G(i)/H(i)-G(i-1)/H(i-1)); end %Phan
tuy chinh Lamda0 va Muyn.(abs(Lamda0)
