Smith 



the variation, d, in S that is important. Shifting the line CD up and down in Fig. 

 15 merely represents a phase shift, but the variation of s gives a measure of the 

 number of radians that are involved in the polynomial approximation of the sine 

 and cosine functions. ^■ 



Each of the terms in the integrands of Eqs. (26a) and (26b) is of the form 

 first treated by Filon. Quadrature formulas with sin \x or cos Xx as weighting 

 functions are readily constructed, since the terms xP sin vx and xP cos \x can 

 be integrated analytically. The details are presented in Ref. 10, which gives 

 both three- and five- point formulas. We shall be content to present here only 

 the three-point formulas. They are , 



- '^iCf i" f-i) cos —^— + 2 (ko + k2) fo sin g^ 



k2( f 2 + f _ i) sin 



■g' 



(27) 



and 



2 (kg + kj) fo cos gg - kjCf 1 + f_i) cos y~~ 



+ k j( f J - f _ 1 ) sin 



1 + g_i 



(28) 



For these formulas: step length = h; complete interval = 2h; ^ = gj - g_i/2; 

 and kg, ki, and k^ are defined in terms of P, which may be large or small. 



Two sets of formulas for the k values are given, for if 9 is small the first 

 set loses accuracy because of roundoff. 



Formulas for k^, kj, and kjfor large values of 0: 



k„ = 



(kg - COS 6) , k , 



-1 



(2k2 + sin 6) 



Formulas for k2, kj, and kg for small values of 6; 



i* 



0' 



5 • 2! 



7 • 4! 



9 • 6! 



11 -8! 13 • 10! 



kj = -^ (5k 2+ sin 0) , k^ = -(e^ki - cos 0) . 



Because the k-f actors must be evaluated in terms of sin 6i and cos 0, practical 

 use of the quadrature formula requires a computer. The cosine formula in Eq. 

 (28) reduces to Simpson's rule as e -► o, as it should. Integration over an extended 

 range is accomplished by repeated application of the formula. 



Figure 16 indicates the formula's accuracy. For this problem, steps can be 

 about 10 times as long as those required by Simpson's rule for the same accuracy. 



354 



