TD Interpolation

(−1,−1),(0,1),(1,0) (2,0) 3
(x0, y0),
(x1, y1),(x2, y2),(x3, y3).
V
V
x0, x1, x2, . . . , xry0, y1, y2, . . . , yr
Pk,0=ykk= 0,1, . . . , r
Pk,j+1(x) = (xk−x)Pj,j (x)−(xj−x)Pk,j (x)
xk−xj
k=j+ 1, . . . , r j = 0, . . . k −1.
P3,3(x0, y0)=(−1,−1),(x1, y1) = (0,1),(x2, y2) = (1,0) (x3, y3) = (2,0).
Pk,j k≥j x0, x1, . . . xj−1,
xk.
Pk,k
3 (−1,−1),
(0,1),(1,0) (2,0)
x0, x1, x2, x3
ν(m2.s−1)
T(◦C)
T15 16 17 18 19 20 21 22 23 24 25 26 27 28
ν1.14 1.11 1.08 1.06 1.03 1.01 0.983 0.960 0.938 0.917 0.896 0.876 0.857 0.839
26.5
ν= 0.9m2.s−1
f(x) =3√x
(xi, f(xi)), i = 0 4 x0= 0 , x1= 1 , x2= 8 , x3= 27 , x4= 64

f P41
P0(x) = f(x0)
Pk(x) = Pk−1(x)+(x−x0)(x−x1). . . (x−xk−1)f[x0, x1, . . . , xk]
Pi(20) i= 1 4 f(20)
E4(x) = f(x)−P4(x)
E4(20) (2).
f(xi, f(xi)) i= 1 4
(1). Q3Qi(20) i= 1,2,3E3(20) = f(20)−Q3(20)
3√x=β β = 2.71441761659
(∇yi=yi−yi−1)
(f(xi), xi)
i= 0 4
R4
Rk(x) = Rk−1(x)+(x−xn)(x−xn−1). . . (x−xn−(k−1))∇kf(xn)
k!hk
R0(x) = f(xn)
Ri(β)i= 1 4 ε2(β) = f−1(β)−R2(β)
R3(β)−R2(β)
a=x1< x2<··· < xn=b n f : [a, b]→R[xi;xi+1]
s
s(xi) = f(xi), i = 1 . . . n
s0(x−
i) = s0(x+ i) : ,
s00(x−
i) = s00(x+ i) : ,
s00(x1) = s00(xn)=0.
hi=xi+1 −xi, i = 1,2,···n−1; Di=s00(xi), i = 1,2,···n.
s00 [xi;xi+1]
s00(x) = Di+1
hi
(x−xi)−Di
hi
(x−xi+1),
xi≤x≤xi+1, i = 1,2,···n−1
Ai, Bi, i = 1,2,···n−1
s0(x) = Di+1
2hi
(x−xi)2−Di
2hi
(x−xi+1)2+Ai,
xi≤x≤xi+1, i = 1,2,···n−1
s(xi) = f(xi), i = 2,···n−1s(xi+1) = f(xi+1), i = 1,2,···n−1 1 ≤i≤n−1
Ai=f(xi+1)−f(xi)
hi
+hi
6(Di−Di+1), Bi=f(xi)−h2
i
6Di.

s0(x+
i) = s0(x− i)
µiDi−1+ 2Di+λiDi+1 =Fi,2≤i≤n−1
µi=hi−1
hi+hi−1
;λi=hi
hi+hi−1
;
Di
2λ2
µ32λ3
µ42λ4
· · ·
· · ·
µn−22λn−2
µn−12
D2
D3
D4
·
·
Dn−2
Dn−1
=
F2
F3
F4
·
·
Fn−2
Fn−1
s
N
f f0(15) = f0(50) = 0
f[0,6]
P
x0 2 4 6
f(x) 0.5−1.7903 3.3900 −1.2795
f
f(x) = 3 sin(2x) + 1
2cos(3x),
x= 3 x= 5
f
s xi=i, i = 0 . . . 5
Di=s00(xi), i = 0 . . . 5D0D5s
µiDi−1+ 2Di+λiDi+1 =Fi,1≤i≤4,
µi=λi= 1/2, i = 1 . . . 4A A
LU
s
f∈Cn+1([a, b]) a≤x0< x1··· < xn≤b Pnf(xi)
f(x)−Pn(x) = (x−x0)···(x−xn)
(n+ 1)! f(n+1)(ξ)a≤ξ≤b
g: [a, b]→R g0g(n+ 2) g0(n+ 1)
W(t) = f(t)−Pn(t)−(t−x0)···(t−xn)K(x)K W (x)=0
|f(x)−Pn(x)| ≤ Mn+1
(n+ 1)! 
n
Y
i=0
(x−xi)

|Qn
i=0(x−xi)|h θ(x) = Qn i=0(x−xi)h= 1
x0= 0
θ(x+ 1) = θ(x)x+1
x−n
|θ(x)|x0≤x≤x1
maxx0≤x≤x1|θ(x)| ≤ n!hn+1
4
|f(x)−Pn(x)| ≤ Mn+1
4(n+ 1)hn+1
f(x) = ´x
0e−t2/2dt h [0,1]
h f 10−6
arccos cos θ= arccos x⇔x∈[0, π]x= cos θ
Qn(x) = cos(narccos x)Qn[−1,1]
w(x)=1/√1−x2
hQn, Qni=π
2n≥1hQ0, Q0i=π
Qnn Qn+1(x)=2xQn(x)−Qn−1(x)
Qnn
Qnn
xk= cos (2k−1)π
2nk= 1 ···n
Qn[−1,1] n+ 1 yk= cos kπ
nk= 0 ···n
.
Qn=1
2n−1QnQnP n
1
1
2n−1= max
−1≤x≤1
Qn(x) ≤max
−1≤x≤1|P(x)|
|f(x)−P(x)| ≤ Mn+1
(n+ 1)! 
n
Y
i=0
(x−xi)
P f (xi) (xi)
n∈Nf: [a, b]→Rx0≤x1≤. . . ≤xn
[a, b]pn∈Pnf x0, . . . , xn
p(j)
n(z) = f(j)(z), j = 0, . . . , m −1,
z m x0, . . . , xn
pn(x) =
n
X
k=0
f[x0, . . . , xk]vk−1(x),
vk(x)=(x−x0). . . (x−xk)
f[xj] = f(xj)j= 0, . . . , n,
f[x0, . . . , xk] = 
f[x1, . . . , xk]−f[x0, . . . , xk−1]
xk−x0
xk> x0,
f(k)(x0)
k!xk=x0.
f[x0, . . . , xk]xi

p∈P4f(x) = ln x
p(1) = f(1) = 0, p(2) = f(2) = 0.693147,
p0(1) = f0(1) = 1, p0(2) = f0(2) = 0.5,
p00(1) = f00(1) = −1.
x∈[a, b]
f(x)−pn(x) = f[x0, . . . , xn, x]vn(x)
gn(x) = f[x0, . . . , xn, x]
f∈ Ck([a, b]) x0≤x1≤. . . ≤xk[a, b]
•x0≤ξ≤xk
f[x0, . . . , xk] = f(k)(ξ)
k!,
•gk−1: [a, b]→R, x 7→ gk−1(x) = f[x0, . . . , xk−1, x]
d
dxgn(x) = f[x0, . . . , xn, x, x].
α∈[a, b]f0(α) = p0
n(α) + E(f)
E(f) = f(n+2)(ξ)
(n+ 2)! vn(α) + f(n+1)(η) (n+ 1)! v0
n(α)
ξ, η ∈[a, b]
E(f)α xivn(α)=0
xiα n v0
n(α) = 0
n= 1
x0=α x1=α+h
x0=α−h x1=α+h
n= 2 x0=α x1=α+h x2=α+ 2h
f0(α)≈−3f(α)+4f(α+h)−f(α+ 2h)
2h
E(f) = h2
3f000(ξ)
ξ∈[α, α + 2h]