牛顿插值多项式

(xi
x0 )
(xi
f (xi ) xi1)( xi
xi1)
(xi
xn )
f [x0 ,L
, xn1]
f n1( )
(n 1)!
推论：若f
(x)
Pn (x),
f
[ x0 ,
,
xk
]
a0n,,kk
n n
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
x
x0
)
L
(x xn )
另一方面
设
{xi
}n i0
Newton插值为N
n
(
x)
则有
{xi
}n i0
U{a}为Nn1(t)
Nn
(t)
f [x0,L
, xn, a](t x0)L
(t xn )
Nn1(a) f (a)
误差
f (a) Nn (a) f [x0,L , xn, a](a x0 )L (a xn )
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
Newton型多项式插值
承袭性： Nn1(x) Nn (x) qn1(x) Pn1
{x0 , x1 , xn1}
{x0 , x1 , xn}
且 Nn (xi ) Nn1(xi ) f (xi ) , qn1(x) an1(x x0 ) (x xn )
而且有：
Nn (x0 ) a0 f (x0)
Nn (x1) a0 a1(x1 x0) f (x1)
Nn (x2 ) a0 a1(x2 x0 ) a2 (x2 x0 )(x2 x1) f (x2 )
Nn (xn )
a0
a1(xn
x0)
an (xn
x0 )
(xn
xn 1 )
f
(xn )
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
这样：
a0 f (x0 )
a1
f (x1) f (x0 ) x1 x0
a2
x2
1
x1
f
(x2 ) x2
f (x0 ) x0
a1
a3
x3
1 x2
f
, xn ]
n i0
( xi
x0 )
( xi
f (xi ) xi 1 ) (xi
xi 1 )
( xi
xn )
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
同样
Nn (x)
的误差为 Rn (x)
f n1( )
(n 1)!
(
%Newton插值法
function y=Newton(x1,y1,x);
m=length(x);
n=length(x1);
for i=2:n
for j=n:-1:i
y1(j)=(y1(j)-y1(j-
1))/(x1(j)-x1(j-i+1));
f [x1,
, xk ] f [x0 , xk x0
, xk1]
称为k阶差商
Sichuan Agricultural University
由归纳：
a0 f (x0 )
数学系
DEPARTMENT OF MATHEMATICS
a1
f (x1) f (x0 ) x1 x0
f [x0 , x1]
同样 Nn (x) Nn1(x) qn (x)
qn (x) an (x x0 ) (x xn1)
i 0,1,L n
为实数
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
Nn (x) a0 a1(x x0 ) an (x x0 ) (x xn1)
f [ x0 , , xn , a]
f n1 ( )
(n 1)!
性质3
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
差商性质总结
f [x0 , , xn ] f [xi0 , , xin ]
f
[ x0 ,
,
xn ]
n i0
源自文库
例子
2点Newton型插值
N1(x)
f (x0 )
f
( x1 ) x1
f (x0 x0
)
(x
x0
)
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
一些性质
Nn (x) Ln (x) xn 的系数一样
性质2
f [x0,
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
2、利用差商表的最外一行，构造插值多项式 Nn (x) f (x0 ) f [x0 , x1]( x x0 ) f [x0 , , xn ]( x x0 ) (x xn1)
此处用到差商的一个性质： （用归纳法易证）
对称性：
f [x0 , , xk ] f [xi0 , , xik ]
定义关键：找不同的元素相减作分母
i0 , , ik是0, , k的任意排列
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
(x3 ) x3
f (x0 ) x0
a1
x1
1 x0
a2
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
定义：差商
f [x0 , x1]
f (x1) f (x0 ) x1 x0
称为1阶差商
f [x0,
, xk ]
a2
x2
1
x1
f
(x2 ) x2
f (x0 ) x0
a1
x2
1 x1
f [x2, x0 ]
f [x1, x0 ]
f [x2, x1, x0 ]
an f [x0 , , xn ]
Sichuan Agricultural University
数学系
DEPARTMENT OF MATHEMATICS
1、先构造差商表Newton插值构造
x0 , f (x0 )
x1, f (x1) f [x0 , x1] x2 , f (x2 ) f [x2 , x1] f [x2 , x1, x0 ]
xn , f (xn ) f [xn1, xn ] f [xn , xn1, xn2 ]
f [xn , , x0 ]