Consider the following approximation:
f′(x) ≈ Af(x − 2h) + B f(x − h) + C f(x) + D f(x + h) + E f(x + 2h).
1) Determine the values of the constants A, B, C, D, and E so that the approximation
is as accurate as possible. What is the order of the error? Please note this is (at least somewhat) a “pencil and paper” exercise.
2) Suppose your answer to part 1) above was that the order of the error was
O(hp). Recall that in class we generated Matlab pictures showing a line of slope
p (approximately) when we plotted log h on the horizontal axis and log error on the vertical axis. Produce such a picture for this approximation and compute the slope.
To determine the values of the constants A, B, C, D, and E in the given approximation, we can use Taylor series expansion. Let's start by expanding f(x ± 2h) and f(x ± h) around x:
f(x ± 2h) = f(x) ± 2hf'(x) + (2h)^2/2! * f''(x) ± (2h)^3/3! * f'''(x) + (2h)^4/4! * f''''(x) ± ...
f(x ± h) = f(x) ± hf'(x) + (h)^2/2! * f''(x) ± (h)^3/3! * f'''(x) + (h)^4/4! * f''''(x) ± ...
Substituting these expansions into the given approximation, we have:
f'(x) ≈ Af(x - 2h) + Bf(x - h) + Cf(x) + Df(x + h) + Ef(x + 2h)
≈ A(f(x) - 2hf'(x) + (2h)^2/2! * f''(x) - (2h)^3/3! * f'''(x) + (2h)^4/4! * f''''(x))
+ B(f(x) - hf'(x) + (h)^2/2! * f''(x) - (h)^3/3! * f'''(x) + (h)^4/4! * f''''(x))
+ Cf(x) + D(f(x) + hf'(x) + (h)^2/2! * f''(x) + (h)^3/3! * f'''(x) + (h)^4/4! * f''''(x))
+ E(f(x) + 2hf'(x) + (2h)^2/2! * f''(x) + (2h)^3/3! * f'''(x) + (2h)^4/4! * f''''(x))
Now, let's group the terms with the same power of h:
f'(x) ≈ (A + B + C + D + E) f(x)
+ (-2A - B + D + 2E) hf'(x)
+ (2A/2! + B/2! + D/2! + 2E/2!) (2h)^2 * f''(x)
+ (-2A/3! - B/3! + D/3! + 2E/3!) (2h)^3 * f'''(x)
+ (2A/4! + B/4! + D/4! + 2E/4!) (2h)^4 * f''''(x)
We can equate the coefficients of each power of h to zero to find the values of the constants:
Coefficients of f'(x):
-2A - B + D + 2E = 0 ——(1)
Coefficients of f''(x):
2A/2! + B/2! + D/2! + 2E/2! = 0 ——(2)
Coefficients of f'''(x):
-2A/3! - B/3! + D/3! + 2E/3! = 0 ——(3)
Coefficients of f''''(x):
2A/4! + B/4! + D/4! + 2E/4! = 0 ——(4)
Solving the system of equations (1)-(4) simultaneously will give us the values of A, B, C, D, and E.
To determine the order of the error, we need to look at the highest power of h that remains in the approximation after the constants have been determined.
Let's assume that the highest power of h in the approximation is h^p. Now, if we multiply h by a factor of k, the approximation becomes:
f'(x) ≈ A(f(x - 2kh) + B(f(x - kh)) + C(f(x)) + D(f(x + kh)) + E(f(x + 2kh))
Comparing this with the original approximation:
f'(x) ≈ Af(x - 2h) + Bf(x - h) + Cf(x) + Df(x + h) + Ef(x + 2h)
We can see that the coefficients A, B, C, D, and E remain the same, but we have multiplied each argument of f by k. As a result, the highest power of h remains h^p, and the order of the error will be O(h^p).
To produce a plot of log h versus log error, we can choose different values of h and compute the error for each value using the original approximation. Then, we take logarithms of both h and the corresponding error and plot them on a graph. The slope of the resulting line approximately represents p.
Note: To compute the error, we need to know the actual value of f'(x) or have a reference solution to compare with.