Two New Calculating Methods for the Irregular Hermite Interpolation Polynomial

Abstract

Based on the theory of the regular Hermite interpolation polynomial, two new calculating methods including undetermined coefficient and Lagrange interpolation have been proposed to solve the complex irregular Hermite interpolation polynomial. KeywordsHermite interpolation polynomial; undetermined coefficient; Lagrange interpolation I. THEORY OF THE HERMITE INTERPOLATION Set ( ) y f x = has definition in interval[ ] , a b , and has corresponding values 0 1 , , , n y y y ⋅ ⋅ ⋅ on the points of 0 1 n a x x x b ≤ < < ⋅⋅ ⋅ < ≤ , if there is a simple function ( ) P x satisfied the following formula: ( ) i i P x y = ( 1, 2, , ) i n = ⋅⋅⋅ (1) where ( ) P x is the interpolation function of ( ) f x , 0 1 , , x x , n x ⋅ ⋅ ⋅ are the interpolation nodes, interval [ ] , a b is the interpolation interval. If ( ) P x is the polynomial with the number not more than n , namely 0 1 ( ) n n P x a a x a x = + + ⋅ ⋅ ⋅ + (2) where ( 0,1, , ) i a i n = ⋅⋅⋅ are the real numbers, we can say ( ) P x is a interpolation polynomial. Suppose ' ( ), ( )( 0,1, , ) j j j j y f x m f x j n = = = ⋅ ⋅ ⋅ on the nodes 0 1 n a x x x b ≤ < < ⋅ ⋅ ⋅ < ≤ , if there is an interpolation polynomial ( ) H x satisfied the conditions (3): ' ( ) , ( ) j j j j H x y H x m = = ( 0,1, , ) j n = ⋅⋅⋅ (3) then ( ) H x is the Hermite interpolation polynomial. If i y and ( 0,1, , ) j m i n = ⋅ ⋅ ⋅ are all known on the nods 0 1 n a x x x b ≤ < < ⋅⋅ ⋅ < ≤ , we can say ( ) H x is the regular Hermite interpolation polynomial, correspondingly, the methods of seeking ( ) H x is the regular Heremite interpolation ; If only some of ( 0,1, , ) j m i n = ⋅ ⋅ ⋅ are known on the nodes 0 1 n a x x x b ≤ < < ⋅ ⋅ ⋅ < ≤ , we can say ( ) H x is the irregular Hermite interpolation polynomial, the methods of seeking ( ) H x is the irregular Heremite interpolation . II. TWO PLANS FOR SOLVING THE IRREGULAR HEREMITE INTERPOLATION POLYNOMIAL If i y ( 0,1, , ) i n = ⋅⋅⋅ and ( 0,1, , ; ) j m j k k n = ⋅ ⋅ ⋅ < are all known on the points 0 1 n a x x x b ≤ < < ⋅⋅ ⋅ < ≤ , and ( 1, , ) j m j k n = + ⋅ ⋅ ⋅ are unknown. In this paper, we want to seek an irregular Heremite interpolation polynomial to meet the following equations: ' ( ) ( 0,1, , ) ( ) ( 0,1, , ; ) i i j j H x y i n H x m j k k n = = ⋅ ⋅ ⋅   = = ⋅ ⋅ ⋅ <  (4) A. Undetermined coefficient Bring the known conditions into the irregular Hermite interpolation polynomial, and to determine the required irregular Hermite interpolation polynomial through solving equations to determine its coefficients. By equations (4), we can know that the equations own 2 n k + + known conditions. Due to 2 n k + + known conditions can uniquely determine a polynomial with the number not more than 1 n k + + , namely 1( ) ( ) n k H x H x + + = (5) so we can suppose the form of 1( ) n k H x + + as follows: 1 1 0 1 1 ( ) n k n k n k H x a a x a x + + + + + + = + + ⋅ ⋅ ⋅ + (6) Bring the unknown conditions ( ) ( 0,1, , ) i i H x y i n = = ⋅⋅⋅ and ( ) ( 0,1, , ; ) j j H x m j k k n = = ⋅⋅⋅ < into the formula (6), we can get the following equations: