Subject: numerical method

1 Question 1 Finite differences with unequal steps . Suppose the forward and backward steps are not equal. Suppose the forward step is hi and the backward step is h2 and h2 h Write the Taylor series for f(h) and f(a h2) up to O(f"(x)) Derive a numerical expression for the first derivative as follows: -f(a) +first two error terms hi + h2 . Show that if h, hi the leading error termin eq. (1.1) ?s 0((hi-ha)f"(z)). Using only f(x), J(x +hi) and f(-h2), derive a finite difference approximation for the first derivative f(), where the leading error term is O(f"(x)). 1. Write your answer up to and including the term is O(""(. 2. Simplify your expression in the special case h h and 2 2h Using only f(x), f(a +h) and f(-h2), derive a finite difference approximation for the second derivative f"(z). The leading error term is 0(f"(2)) if hI h2. 1. Write your answer up to and including the term is O("(x)). 2. Simplify your expression in the special case hi h and h2-2h. . If you have done your work correctly, your answers should reduce to the expressions in the lectures if h1 = h2 = h