1.
2.
3.
4.
( ) : [a;b] n+1 x 
,x 
,,x 
f(x):
y =f(x ), y =f(x ), , y =f(x ).
f(x) [a;b], .
, . f(x) , F(x), 
y =f(x ), f(x): F(x )=y (i=0,1,,n).
[a;b] f(x)=F(x).
x ,x ,,x ( ) , F(x) .
P (x), n P (x )=y =f(x ), P (x )=y =f(x ),...,P (x )=y =f(x ). (1)
, P (x) n, (1).
() y=f(x) [a;b] P (x), (1).
P (x), (1), f(x), .
y=f(x) f(x)=P (x), [a;b] () .
, (2).
y=f(x) y=P (x) ( n), (x ;y ), (1;1),...,(n;yn).
(2) , [x ;x ], , 
[x ;x ], , (2) . .
.
:
| 0 | 1 | ... | ... | n | |
| Y0 | Y1 | ... | Yi | ... | Yn |
h, 1=0+h, x2=x0+2h,..., xn=xn-1+h=x0+nh. г
y1-y0= 
y0,
y2-y1= y1,
......................... (1)
yi+1-yi= yi,
........................,
yn-yn-1= yn-1.
.
г y1- y0= 2y0, y2- y1= 2y1,..., y+1- y= 2y,..., yn-1- yn-2= 2yn-2 (2)
. , k- (k-1)-
k-1y1- k-1y0= ky0,
k-1y2- k-1y1= ky1,
............................................, (3)
|
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k-1yi+1- k-1yi= kyi, (k=(1,n); 0y=y).
...........................,
k-1yn-(k-1)- k-1yn-k= kyn-k ,
(3) , k- (k-1) . . ij, (1) , yi=yi+1-yi=(+h)-y(xi). (2), (1),
y2i= y+1- y=(+2-+1)-(+1-)=+2-2+1+=(+2h)-2y(xi+h)+y(xi) .
kyi= 
, 
- (0!=1)
, . (1) , 1=0+ y0, 2=1+ y1=(0+ y0)+ y1, (2), 2y0= y1- y0 2=0+2 y0+ 2y0.
, k k y0 k- :
, 
=0.
:
1) 1) =const, =0;
2) 2) (cf(x))=c f(x);
3) 3) (f1(x)+f2(x))= f1(x)+ f2(x);
4) 4) m( nf(x))= m+nf(x).
y=Pn(x)=a0xn+a1xn-1+...+an-1x+an.
= Pn(x)=a0 (xn)+a1 (xn-1)+...+an-1 (x).
, (n)= (x+h)-xn=nhxn-1+ 
,
Pn(x)=a0nhxn-1+(a0 
.
, n 0n n-1 0nhxn-1. , : n(x)- n 0n, k<n kPn(x) n-k n(n-1)...(n-k+1)a0hkxn-k; n- , k>n .
: n h , n.
, , 5 . n , n.
3. . , - . , , . ֳ , .
, h: x0,x1=x0+h, x2=x0+2h,...,xn=x0+nh.
n, :
Pn(x)=q0+q1(x-x0)+...+qk(x-x0)...(x-xk-1)+...+qn(x-x0)...(x-xn-1) (1) q0, q1..., qk,...,qn.
(1) =0, Pn(x1)=qn+q1(x1-x0), Pn(x1)=y1=y0+q1(x1-x0), q1= 
;
=2
Pn(x2)=q0+q1(x2-x0)+q2(x2-x0)(x2-x1);
y2=y0+ 
, 2-0-2 0=q22h2;
q2= 
.
, q3= 
,..., qk= 
,...,qn= 
.
ϳ (1),
Pn(x)=y0+ 
. (2)
. (2), , n, , . (2), . , (2) k!. ֳ k , , , , ( ) . , , n - .
|
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, , ( ). n.
(2) , t= 
. x-x0=th, x-x1=x-x0-h=th-h=(t-1)h, x-x2=(t-2)h,,x-xn-1=(t-(n-1))h, (2),
Pn(x)=Pn(x0+th)=y0+ 
(3)
. (3) n=1, P1(x)=P1(x0+th)=y0+ y0t, n=2 P2(x)=P2(x0+th)=y0+ y0t+ 
n , 
. (2) , .
, , , (3), 
(4), n+1=max 
, a 
x b.
f(n+1)(x) , n+1 Rn(x), 
, 
(5).
, 
.
4. .
Pn(x)=y0+ 
Pn(x)=Pn(x0+th)=y0+ , 0. , 0 ( ). , n, . Pn()
Pn(x)=b0+b1(x-xn)+b2(x-xn)(x-xn-1)++bk(x-xn)(x-xn-(k-1))+bn(x-xn)(x-x1). (6)
ϳ (6) =n, x=xn-1, , x=x0. =n, Pn(xn)=b0, b0=Pn(xn)=yn;
x=xn-1, Pn(xn-1)=yn-1=b0+b1(xn-1-xn)=b0-b1h, yn-1-yn=-b1h, b1= 
;
x=xn-2, Pn(xn-2)=yn-2=b0+b1(xn-2-xn)+b2(xn-2-xn)(xn-2-xn-1), yn-2-yn+ 
, b2= 
. b3= 
,, bn= .
(6) Pn(x)=yn+ 
(7). . t, t= 
, (7) : Pn(x)=Pn(xn+th)=yn+ 
. (8)
(7): 
, n+1=max , a x b.
, 
.
