NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 225 



will be affected by an error in the value of the function. The erroneous numbers 

 in the last column are clearly the second, third, fourth, and fifth. The algebraic 

 sum of these four numbers equals the sum of the four correct numbers, or -18. 

 Their average is -4.5. Hence the central numbers are probably -5 and -4. 

 Since the errors in these numbers are -36 and +36, it follows that e is probably 

 + 2. The errors in the second and fifth numbers are -fe and -e respectively. 

 On making these corrections and working back to the first column, it is found 

 that 7073 should be replaced by 7071. 



10.30 Computation of Definite Integrals by Use of Difference Functions. 



Suppose the values of f(t) are kriown for t = t n -2, J_i, t n , and t n+ i. Suppose 

 it is desired to find the integral 



rtn+i 



I. /= / f(t)dt. 



<Jtn 



The coefficients b Q , b iy b z , and 3 of the polynomial can be determined, as above, 

 so that the function 



2. y = b Q + h(t - t n ) + h(t - t n ) z + b,(t - O 3 



shall take the same values as/(/) for / = / n _ 2 , t n -i, t n , and t n+ i. 



With this approximation to the function /(/), the integral becomes (since 



t n+ i -t n = h) 



3 . I n = M& + h(t - t n ) + b*(t - tn) 2 + h(t - / n ) 3 ] dt 



t Jtn 



+ - M 8 ]. 

 4 



The coefficients b , bi, fa, and b s will now be expressed in terms of y n+ i, 

 and A 3 y+i. It follows from (2) that 



yn-i = b - bih + b*h 2 - W, 

 1 y n = b , 



Then it follows from the rules for determining the difference functions that 



f 



f 

 \ 



