61 



APPENDIX k 



LEAST SQUARE FIT OF A SEVENTH- BY A SIXTH- 

 DEGREE POLYNOMIAL FORM 



The seventh-degree polynomial y^ = f(x) may be written in the form 



f(x) = x(1 - x)g(x) [1] 



where g(x) is a polynomial of the fifth degree. It will be shown that a least 

 square fit to g(x) by a fourth-degree polynomial g(x) is given by 



i(x) = g(x) - Cjx= --|x* + 20x^ -|x^ +-j^x -2^) [2] 



where Cg is the coefficient of x^ in the expansion 



g(x) = Cq + C^x + ... + C^x^ [3] 



Also [2] may be written as 



g(x) = (C^ + ^) + (C^ - ijc^)x + (C, +|c^)x- 

 .(C3 -^5c,)x3.(c^.|c>^ 



[^. 



The corresponding sixth-degree polynomial form will then be given by y^ = f(x) 

 where 



r(x) = x(l - x)g(x) [5] 



Proof : Since g(x) is a polynomial of the fifth degree, it can be expressed as 

 a linear combination of the first five Legendre polynomials. Furthermore, 

 since here the range of x is from to 1 and the Legendre polynomials are or- 

 thogonal over the range -1 to +1 , we express g(x) as 



g(x) = y^P^ + y^P,(0 + ... + y^P^(l) [6] 



where y , y^, ... y are coefficients and P , Pj^(§) ... are the Legendre poly- 

 nomials, and 



? = 2x -^^ [7] 



A theorem on orthogonal functions (Reference 6) then states that the 

 best least-square fit of a fourth-degree polynomial to g(x) is 



