Lecture Notes Course à MA 190 Numerical Mathematics, First ...

1.6.5 Mean value theoremsThe mean value formulas in Subsection 13.1.9 are often used in error estimations.If the interval [a, b] and the functions f and w as well as the points t i andweights w i are given, formulas (13.1), (13.2) and (13.3) define equations whichare satisfied by a number s which may not be uniquely determined. We illustratewith some numerical examples:In (13.1) we set a = −1, b = 1, f(t) = t 3 . Thus we get the equationwhich has the solutions1 − (−1)2= 3s 2s = ±1/ √ 3In (13.2) we put a = −1, b = 1, w(t) = 1, f(t) = t 4 to arrive at the equationwhich has the real solutions0.4 = s 4 · 2s = ± 4√ 0.2In (13.3) we put n = 3, t 1 = −1, t 2 = 0, t 3 = 1, w 1 = 1/4, w 2 = 1/2, w 3 =1/4, f(t) = t 2 Then we find the equation1/4 + 1/4 = s 2which has the solutions = ±1/ √ 21.6.6 Integral estimates for partial sums:Theorem 1.6.2 Let f be a function which is continuous and decreasing overthe positive real halfline and let h be a positive constant. Then for n > m wehave the inequalitiesandh∫ nhmhf(t) dt − hn∑f(rh) =r=m∫ nhmh(f(mh) − f(nh)2+n−1∑r=m+1f(rh))f(mh) − f(nh)≤ h2f(t) dt + h(f(mh) + f(nh))/2 ± h(f(mh) − f(nh))/2,(this is a simple instance of the Euler-Maclaurin summation formula)The proof is based on using the inequalitieshf(h(r + 1)) ≤∫ h(r+1)rhf(t) dt ≤ hf(rh), r = m, m + 1, . . . , n − 1.14