Mathematical Computation June 2015, Volume 4, Issue 2, PP.25-32
The Algebra Theory for Polynomial Interpolation Method Ling Wang, Dianxuan Gong # College of Sciences, Hebei United University, Tangshan 063000, China #
Email: dxgong@heuu.edu.cn
Abstract In this paper, several usually used polynomial interpolation methods are explained in view of vector basis and dimension in linear algebra theory. Using transition matrixes, general conversion formula between the basis function sets of these polynomial interpolation methods are given. An example also shows the effectiveness of the results. Keywords: Vector Basis; Lagrange Interpolation; Newton Interpolation; Taylor Formula; Transition Matrix
1 INTRODUCTION Given a data sample {( x0 , y0 ),( x1 , y1 ),( x2 , y2 ), ( xn , yn )}
from function y = ϕ ( x) , interpolation method[1] is to
construct a simple function y = f ( x) such that yi = f ( xi ), i = 0,1, 2,, n . When we choose polynomial to interpolate the data points, it is called polynomial interpolation method. It is the basis of numerical integration and numerical differential equation algorithm in numerical analysis. Weierstrass’ approximation theorem[2] tells us that continuous functions on closed interval can be uniformly approximated by polynomial function. The commonly used polynomial interpolation methods include undetermined coefficients method (also can be seen as a special case of Taylor formula), Lagrange interpolation method and Newton interpolation method. This document will unify these methods by vector basis and dimension theory in linear algebra. In fact, these methods get a same polynomial which is expressed by different basis functions. The conversion formulas between these methods are given by transition matrixes. In addition, concluding to the results in this paper, one can obtain new interpolation methods by constructing new basis functions set to meet some special demand. The rest of this paper is as follows: Section two and section three will briefly state the relevant methods and concepts of polynomial interpolation and vector space. Unified theory of interpolation methods in view of linear algebra is given in section four also with the conversion formulas. An example is in section five show the effectiveness.
2 COMMONLY USED INTERPOLATION METHODS 2.1 Undetermined Coefficients Method Given function value of y = ϕ ( x) at n + 1 different points that yi = ϕ ( xi ) , i = 0,1, 2,, n . In order to get the approximate expression of y = ϕ ( x) , suppose a polynomial function with order n is write as Pn ( x)= an x n + an −1 x n −1 + + a0 .
Where {a0 , a1 ,, an } are the undetermined coefficients. Polynomial Pn ( x) should meet the following linear function system: an xin + an −1 xin −1 + + a0 = yi for i = 0,1, 2,, n .
Since the points xi are distinct, the only solution will provide the interpolation polynomial Pn ( x) . - 25 www.ivypub.org/mc
2.2 Lagrange Interpolation Method For the above data set { xi , yi } ( i = 0,1, 2,, n ) that yi = ϕ ( xi ) , define the Lagrange interpolation basis function of order n as x − xi , i = 0,1, 2,, n . 0, j ≠ i xi − x j n
li ( x) = ∏ =j
Then the Lagrange interpolation polynomial with degree n satisfying condition Ln ( xi ) = yi for i = 0,1, 2,, n is n
Ln ( x) = ∑ yi li ( x). i =0
2.3 Newton Interpolation Method For the same data set { xi , yi } ( i = 0,1, 2,, n ) from function y = ϕ ( x) , define
ϕ[ x0 , x1 ] =
ϕ ( x1 ) − ϕ ( x0 ) x1 − x0
called the first difference quotient of ϕ ( x) at x0 and x1 . Define
ϕ[ x0 , x1 , x2 ] =
ϕ[ x1 , x2 ] − ϕ[ x0 , x1 ] x2 − x0
called the second difference quotient of ϕ ( x) at x0 x1 and x2 . Define
ϕ[ x0 , x1 ,, xn ] =
ϕ[ x1 , x2 ,, xn ] − ϕ[ x0 , x1 ,, xn −1 ] xn − x0
called the n-th difference quotient of ϕ ( x) at x0 , x1 ,, xn . Then the Newton interpolation polynomial with degree n satisfying condition N n ( xi ) = yi for i = 0,1, 2,, n can be writing as N n ( x) = ϕ ( x0 ) + ϕ[ x0 , x1 ]( x − x0 ) + ϕ[ x0 , x1 , x2 ]( x − x0 )( x − x1 ) + +ϕ[ x0 , x1 ,, xn ]( x − x0 )( x − x1 ) ( x − xn −1 ) .
2.4 Taylor Formula[3] Suppose y = ϕ ( x) is continuous differentiable to order n on closed interval [a, b] containing x* , and has derivative of degree n+1 on interval (a, b) . Then the Taylor formula of ϕ ( x) at x* is
ϕ ( x)= ϕ ( x* ) +
ϕ ′( x* ) 1!
( x − x* ) +
ϕ ′′( x* ) 2!
( x − x* ) 2 + +
ϕ ( n ) ( x* ) n!
( x − x* ) n +
ϕ ( n +1) ( x + θ ( x − x* )) (n + 1)!
( x − x* ) n +1
where 0 < θ < 1 .
3 VECTOR BASIS AND DIMENSION THEORY IN LINEAR ALGEBRA Suppose V ( F ) is a linear space on number field F , and suppose { X1 , X 2 ,, X n } is a set of vectors in space V ( F ) . If the vectors X1 , X 2 ,, X n are linearly independent and can linearly represent all the vectors in V ( F ) , then we call
{ X1 , X 2 ,, X n }
a basis set of space V ( F ) . The number of the vectors in this basis set is called the dimension of the
space, noted as dim V ( F ) = n .
Consider the set of all the polynomials with degree not larger than n
Pn [ x] =
{ p( x) | deg ree( p( x)) ≤ n} .
Pn [ x] make a linear space on real number field under the common addition and multiplication operation, and - 26 www.ivypub.org/mc
dim Pn [ x]= n + 1 .
The basis set of a linear space is not only, a most commonly used basis set is
{1, x, x ,, x } . 2
n
(1)
It is not difficult to prove that the following sets is also a basis set of Pn [ x] respectively = li ( x) | li ( x)
n x − xj = , i 0,1, 2,, n ∏ =j 0, j ≠ i xi − x j
(2)
{1,( x − x0 ),( x − x0 )( x − x1 ),,( x − x0 )( x − x1 )( x − xn −1 )}
(3)
{1, ( x − x ), ( x − x ) ,, ( x − x ) }
(4)
*
* 2
* n
Definition 3.1[4] Suppose two set ( X1 , X 2 ,, X n ) and ( X1′, X 2′ ,, X n′ ) are basis sets of linear space V ( F ) , If a matrix A satisfies ( X1′, X 2′ ,, X n′ )=( X1 , X 2 ,, X n ) A
then the matrix A is call the transition matrix from ( X1 , X 2 ,, X n ) to ( X1′, X 2′ ,, X n′ ) .
4 UNIFICATION OF COMMONLY USED INTERPOLATION METHODS Given function values of y = ϕ ( x) at n + 1 distinct points that yi = ϕ ( xi ) ( i = 0,1, 2,, n ). Using undetermined coefficients method, Lagrange interpolation method and Newton interpolation method, we can get the interpolation polynomials Pn ( x) , Ln ( x) and N n ( x) respectively. In fact, these three polynomials interpolate the same n + 1 data points, in other words, they have the same function values at n + 1 distinct points. So these three polynomials are the same one with different expression, and it is easy to see that they are the special expression according to different basis set (1)(2) and (3). Additionally, we have the following results. (In the rest of this text, we call function set (1) general interpolation basis, call (2) Lagrange interpolation basis, call (3) Newton interpolation basis, call (4) Taylor basis.) Result 4.1 In linear space Pn [ x] , the transition matrix from general interpolation basis to Lagrange interpolation basis is n −1 n−2 n n (−1)n −1 ∏ xi (−1) n − 2 ∏ xi ∑ ∑ (−1) ∏ xi 1 1≤i <i <<i ≤ n 1≤i <i <<i ≤ n k 1= k 1 i == 1 n n n n ∏ ( x0 − xi ) ∏ ( x0 − xi ) ∏ ( x0 − xi ) ∏ ( x0 − xi ) i 1 = =i 1 =i 1 i 1= n −1 n−2 n n −1 n−2 (−1) n ∏ x − x ( 1) − x ( 1) ∑ ∑ ∏ i ∏ i i 1 0≤i <i <<i ≤ n;i ≠1 0≤i <i <<i ≤ n;i ≠1 k =1 k =1 =i 0;i ≠1 n n n n ∏ (x − x ) x − x x − x x − x ( ) ( ) ( ) ∏ ∏ ∏ 1 1 1 i i i =i 0;i ≠1 1 i 0;i ≠1 0;i ≠1 0;i ≠1 i= i= i= . n −1 n−2 n ALG = (−1)n −1 ∏ xi (−1)n − 2 ∏ xi (−1)n ∏ xi ∑ ∑ 1 0≤i <i <<i ≤ n;i ≠ 2 0≤i <i <<i ≤ n;i ≠ 2 k 1= k 1 =i 0;i ≠ 2= n n n n ∏ ( x2 − xi ) ∏ ( x2 − xi ) ∏ ( x2 − xi ) ∏ ( x2 − xi ) i= 0;i ≠ 2 0;i ≠ 2 i= 0;i ≠ 2 i= 0;i ≠ 2 i= n −1 n−2 n −1 − − n 1 n 2 n (−1) ∏ xi (−1) ∏ xi ∑ ∑ (−1) ∏ xi 1 0≤i <i <<i ≤ n −1 0≤i <i <<i ≤ n −1 k 1= k 1 i = 0= n −1 n −1 n −1 n −1 ∏ ( xn − xi ) ∏ ( xn − xi ) ∏ ( xn − xi ) ∏ ( xn − xi ) = =i 0 =i 0 i 0=i 0 k
1
k
n −1
2
1
n−2
2
k
1
2
n −1
k
1
2
n −1
k
k
1
2
n−2
k
1
2
n−2
k
k
k
k
1
2
n −1
k
1
2
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n−2
Result 4.2 In linear space Pn [ x] , the transition matrix from general interpolation basis to Newton interpolation basis is
0 0 0
k
1
k
− ∑ xi
1 0 0 1 0 − x0 1 x0 x1 − x0 − x1 n−2 n−2 n −3 ANG = (−1)n − 2 ∏ xi (−1) n −3 ∏ xi (−1)n −1 ∏ xi ∑ ∑ 0≤i <i <<i ≤ n − 2 0≤i <i <<i ≤ n − 2 = i 0= k 1= k 1 n −1 n −1 n−2 n n −1 n−2 (−1) ∏ xi (−1) ∏ xi ∑ ∑ (−1) ∏ xi 0≤i <i <<i ≤ n −1 0≤i <i <<i ≤ n −1 = i 0= k 1= k 1 k
1
n−2
2
1
2
n −3
k
1
2
n −1
1
2
n−2
n −1
i= 0
0 0 0 . 0 1
Result 4.3[5] In linear space Pn [ x] , the transition matrix from Lagrange interpolation basis to Newton interpolation basis is
ANL
1 1 1 1 1 0 ( x1 − x0 ) ( x2 − x0 ) ( xn −1 − x0 ) ( xn − x0 ) 0 0 ( x2 − x0 )( x2 − x1 ) ( xn −1 − x0 )( xn −1 − x1 ) ( xn − x0 )( xn − x1 ) n−2 n−2 . = 0 0 0 ∏ ( xn −1 − xi ) ∏ ( xn − xi ) =i 0=i 0 n −1 0 0 0 ∏ ( xn − xi ) 0 i =0
Result 4.4[6] In linear space Pn [ x] , the transition matrix from Lagrange interpolation basis to general interpolation basis is
AGL
1 x0 = x02 n x0
1 x1 x12 x1n
1 x2 x22 x2n
1 xn xn2 . xnn
Result 4.5 In linear space Pn [ x] , the transition matrix from Newton interpolation basis to Lagrange interpolation basis is
ALN
1 1 (x − x ) 0 1 1 0 (x − x ) 1 0 0 0 = 0 0 0 0
∏ ( x0 − xi ) ∏ ( x0 − xi ) =i 1 =i 1 1 1 1 n −1 n ( x1 − x0 )( x1 − x2 ) ∏ ( x1 − xi ) ∏ ( x1 − xi ) i= i= 0;i ≠1 0;i ≠1 1 1 ( x2 − x0 )( x2 − x1 ) n −1 n ∏ ( x2 − xi ) ∏ ( x2 − xi ) . i= i= 0;i ≠ 2 0;i ≠ 2 1 1 n−2 0 n ∏ ( xn −1 − xi ) ∏ ( xn −1 − xi ) i = 0;i ≠ n −1 i =0 1 0 0 n −1 ∏ ( xn − xi ) i =0 1 ( x0 − x1 )( x0 − x2 )
1
n −1
1
n
Result 4.6 In linear space Pn [ x] , the transition matrix from Newton interpolation basis to general interpolation basis is
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0 1 x0 1 x2 x0 + x1 0 AGN = n −1 n − 2 n − 2−i i x ∑ x0 x1 0 i 0 = n n −1 n − 2−i i x0 ∑ x0 x1 i 0 =
0 0 1
0 0 0
∏ xi
1
∏ xi
∑ xi
n −3
∑
0≤i1 ≤i2 ≤≤in−3 ≤ 2 k =1 n−2
∑
k
k
0 0 0 . 0 1
n −1
0≤i1 ≤i2 ≤≤in−2 ≤ 2 k =1=i 0
Result 4.7 In linear space Pn [ x] , the transition matrix from general interpolation basis to Taylor basis is
ATG
1 0 − x* 1 x*2 −2 x* = n −1 *n −1 (−1)n − 2 Cn1 −1 x*n − 2 (−1) x (−1)n x*n (−1)n −1 Cn1 x*n −1
0 0 1 (−1)n −3 Cn2−1 x*n −3 (−1) n − 2 Cn2 x*n − 2
0 0 0 1 −Cnn −1 x*
0 0 0 . 0 1
Here Cnk is the binomial coefficient. Result 4.8[7] In linear space Pn [ x] , the transition matrix from Newton interpolation basis to Taylor basis is ATN = 1 0 0 0 −( x* − x0 ) 1 0 0 ( x* − x0 )2 −( x* − x0 ) − ( x* − x1 ) 1 0 n−2 n −3 (−1)n −1 ( x* − x ) n −1 (−1)n −3 ∏ ( x* − xi ) 1 ∑ (−1)n − 2 ∏ ( x* − xi ) ∑ 0 0≤i ≤≤i ≤1 0≤i ≤≤i ≤ 2 = k 1= k 1 n −1 n−2 n n −1 * n−2 * (−1)n ( x* − x0 )n ( − 1) ( x − x ) ( − 1) ( x − x ) − ( x* − xi ) ∑ ∑ ∑ ∏ ∏ i i 0≤i ≤≤i ≤1 0≤i ≤≤i ≤ 2 i= 0 k =1 k =1 k
1
k
n−2
1
n −3
1
n−2
k
1
k
n −1
0 0 0 . 0 1
Result 4.9 In linear space Pn [ x] , the transition matrix from Lagrange interpolation basis to Taylor basis is
ATL
1 −( x* − x0 ) = ( x* − x0 ) 2 (−1)n ( x* − x )n 0
1 −( x* − x1 ) ( x* − x1 )2 (−1)n ( x* − x1 )n
1 −( x* − x2 ) ( x* − x2 )2 (−1)n ( x* − x2 )n
1 −( x* − xn ) ( x* − xn )2 . (−1)n ( x* − xn )n
Result 4.10 In linear space Pn [ x] , the transition matrix from Taylor basis to Newton interpolation basis is 1 0 0 0 ( x* − x0 ) 1 0 0 * * * * ( )( ) ( ) ( ) 1 0 x − x x − x x − x + x − x 0 1 0 1 n−2 n −3 ANT = n − 2 * * * ( x − xi ) 1 ∑ ∑ ∏ ( x − xi ) ∏ ( x − xi ) ∏ = 0≤i <<i ≤ n − 2 k 1 = 0≤i <<i ≤ n − 2 k 1 =i 0 n −1 n −1 n−2 n −1 * * ∏ ( x* − xi ) ( ) ( ) x − x x − x ∑ ∑ ∑ ( x* − xi ) ∏ ∏ i i 0≤i <<i ≤ n −1 k 1 = 0≤i <<i ≤ n −1 k 1 0 i= = i 0 = k
1
n−2
k
1
n −3
1
n−2
k
1
n −1
k
0 0 0 . 0 1
The above transition matrixes are non-singular since xi are distinct. According to the properties of transition matrix in linear algebra, we have the following property Property 4.1 For the transition matrixes between the four basis sets of linear space Pn [ x] , the following hold (1)
AGL = AGN ANL
(2)
ANL ALN = I n +1 - 29 www.ivypub.org/mc
(3)
AGN ANL ALG = I n +1
(4)
ATN ANL ALG = ATG
(5)
ATN ANL is the transition matrix from Lagrange basis to Taylor basis.
According to Property 4.1, when one interpolation polynomial is got, one can obtain the other forms by the transition matrixes. One can also get the Taylor formula at a point by transition matrix. More important, one can get a new interpolation method by construct a new basis set. For example, choose function set
{( x − x ) , ( x − x ) , ( x − x ) ,, ( x − x ) } n
n
0
n
1
2
n
n
(5)
For distinct xi , the set (5) is a basis set for Pn [ x] . It is composed by the translations of a function. The interpolation method by using basis set (5) is in fact a kind of radial basis function interpolation method for even n . Property 4.2 For the transition matrix ATN , the following holds lim ATN = I n +1 .
xi → x* i = 0,1,, n
So, in the Newton interpolation method, if all the interpolation nodes tend to a same point, then limit form of Newton interpolation polynomial is the Taylor polynomial. The same conclusion can be get from the properties of difference quotient[8]. Property 4.3 For the transition matrixes AGL and ATL , the following hold
ATL |x =0 = AGL *
ATN |x =0 = AGN *
ATG |x =0 = I n +1 . *
This shows that the undetermined coefficients method is a special case of Taylor formula.
5 AN EXAMPLE Given five function values of y = ϕ ( x) : ϕ (0)=1 , ϕ (1)=5 , ϕ (2)=5 , ϕ (3)=12 , ϕ (4)=32 . Then the polynomial P4 ( x) which interpolate ϕ ( x) at these five points according to the undetermined coefficients method is P4 ( x)=1+
131 235 2 37 3 5 4 x− x + x − x . 12 24 12 24
L4 ( x) according to Lagrange method has form L4 ( x)=l1 (x)+5l2 ( x) + 5l3 ( x)+12l4 ( x) + 32l5 ( x)
where {l1 ( x), l2 ( x), l3 ( x), l4 ( x), l5 ( x)} is the Lagrange interpolation basis set. The coefficient set is {1,5,5,12,32} . N 4 ( x) according to Newton interpolation method has form
N 4 ( x)=1 + 4x − 2 x( x − 1) +
11 5 x( x − 1)( x − 2) − x( x − 1)( x − 2)( x − 3) . 6 24
11 5 Here the Newton interpolation coefficients are 1, 4, −2, , - . The basis set is 6 24
1 x x( x − 1) x( x − 1)( x − 2) x( x − 1)( x − 2)( x − 3) - 30 www.ivypub.org/mc
It is easy to validate that the transition matrix from general basis set to Lagrange interpolation basis set is
ALG
25 1 − 12 4 0 = 0 −3 4 0 3 1 − 0 4
1 24 1 − 6 1 4 1 − 6 1 24
35 5 − 24 12 13 3 − 3 2 19 −2 4 7 7 − 3 6 11 1 − 24 4
The transition matrix from Lagrange interpolation basis set to Newton interpolation basis set is
ANL
1 0 = 0 0 0
1 1 0 0 0
1 1 3 4 6 12 6 24 0 24
1 2 2 0 0
The transition matrix between Newton interpolation basis set to Taylor basis set at x = 1 are
ATN
1 0 −1 1 = 1 −1 −1 1 1 −1
0 0 1 0 1
0 0 0 1 2
0 1 0 0 0 1 1 0 0 0 0 and ANT = 0 1 1 0 0 −1 0 1 0 0 2 −1 −2 1
0 0 0 . 0 1
Calculate directly can verify that Lagrange interpolation coefficients and undetermined coefficients satisfies 235 131 − 1 12 24
37 12
−
5 (1 5 5 12 32 ) ALG . = 24
Newton interpolation coefficients and Lagrange interpolation coefficients satisfies
(1
11 5 = 5 5 12 32 ) 1 4 -2 − ANL . 6 24
The Taylor formula coefficients can be getting from Newton interpolation coefficients as 1 11 5 − ANT = 1 4 −2 5 − 4 6 2 4
−
43 24
9 4
−
5 . 24
So the Taylor formula of the interpolation polynomial at x = 1 is 1 43 9 5 T ( x)=5 − ( x − 1) − ( x − 1)2 + ( x − 1)3 − ( x − 1)4 +o(( x − 1)4 ) . 4 24 4 24
Finally, we have 1 −1 1 = ATL A= TN ANL −1 1
1 0 0 0 0
1 1 1 1 2 3 1 4 9 . 1 8 27 1 16 81
ACKNOWLEDGMENT This work is supported by the National Natural Science Foundation of China (No. 61170317, 11301120) and Natural Science Foundation of Hebei Province of China (No. 2013209295)
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AUTHORS 2
Dianxuan Gong (1981-), male, Doctor of science, Associate professor, Major in computational geometry,
Graduated from Dalian university of technology. Email: dxgong@heuu.edu.cn
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