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Using Theorem 5.1 to evaluate the error terms and dividing by h^2 yields the following formula: Three-point centered-difference formula for second derivative f''(x) = (f(x - h) - 2f(x) + f(x + h)) / h^2 - (h^2 / 12)f^(iv)(c) (5.8) for some c between x - h and x + h. 5.1.2 Rounding error So far, all of this chapter's formulas break the rule from Chapter 0 that advises against subtracting nearly equal numbers. This is the greatest difficulty with numerical differentiation, but it is essentially impossible to avoid. To understand the problem better, consider the following example: EXAMPLE 5.3 Approximate the derivative of f(x) = e^x at x = 0. The two-point formula (5.4) gives f'(x) ? (e^(x+h) - e^x) / h, (5.9) and the three-point formula (5.7) yields f'(x) ? (e^(x+h) - e^(x-h)) / 2h. (5.10) The results of these formulas for x = 0 and a wide range of increment size h, along with errors compared with the correct value e^0 = 1, are given in the following table: h | formula (5.9) | error | formula (5.10) | error 10^-1 | 1.05170918075648 | -0.05170918075648 | 1.00166750019844 | -0.00166750019844 10^-2 | 1.00501670841679 | -0.00501670841679 | 1.00001666674999 | -0.00001666674999 10^-3 | 1.00050016670838 | -0.00050016670838 | 1.00000016666668 | -0.00000016666668 10^-4 | 1.00005000166714 | -0.00005000166714 | 1.00000000166689 | -0.00000000166689 10^-5 | 1.00000500000696 | -0.00000500000696 | 1.00000000001210 | -0.00000000001210 10^-6 | 1.00000049996218 | -0.00000049996218 | 0.99999999997324 | 0.00000000002676 10^-7 | 1.00000004943368 | -0.00000004943368 | 0.99999999947364 | 0.00000000052636 10^-8 | 0.99999999392253 | 0.00000000607747 | 0.99999999392253 | 0.00000000607747 10^-9 | 1.00000008274037 | -0.00000008274037 | 1.00000002722922 | -0.00000002722922

          Using Theorem 5.1 to evaluate the error terms and dividing by h^2 yields the following formula:

Three-point centered-difference formula for second derivative

f''(x) = (f(x - h) - 2f(x) + f(x + h)) / h^2 - (h^2 / 12)f^(iv)(c) (5.8)

for some c between x - h and x + h.

5.1.2 Rounding error
So far, all of this chapter's formulas break the rule from Chapter 0 that advises against subtracting nearly equal numbers. This is the greatest difficulty with numerical differentiation, but it is essentially impossible to avoid. To understand the problem better, consider the following example:

EXAMPLE 5.3 Approximate the derivative of f(x) = e^x at x = 0.

The two-point formula (5.4) gives

f'(x) ? (e^(x+h) - e^x) / h, (5.9)

and the three-point formula (5.7) yields

f'(x) ? (e^(x+h) - e^(x-h)) / 2h. (5.10)

The results of these formulas for x = 0 and a wide range of increment size h, along with errors compared with the correct value e^0 = 1, are given in the following table:

h | formula (5.9) | error | formula (5.10) | error
10^-1 | 1.05170918075648 | -0.05170918075648 | 1.00166750019844 | -0.00166750019844
10^-2 | 1.00501670841679 | -0.00501670841679 | 1.00001666674999 | -0.00001666674999
10^-3 | 1.00050016670838 | -0.00050016670838 | 1.00000016666668 | -0.00000016666668
10^-4 | 1.00005000166714 | -0.00005000166714 | 1.00000000166689 | -0.00000000166689
10^-5 | 1.00000500000696 | -0.00000500000696 | 1.00000000001210 | -0.00000000001210
10^-6 | 1.00000049996218 | -0.00000049996218 | 0.99999999997324 | 0.00000000002676
10^-7 | 1.00000004943368 | -0.00000004943368 | 0.99999999947364 | 0.00000000052636
10^-8 | 0.99999999392253 | 0.00000000607747 | 0.99999999392253 | 0.00000000607747
10^-9 | 1.00000008274037 | -0.00000008274037 | 1.00000002722922 | -0.00000002722922

Using Theorem 5.1 to evaluate the error terms and dividing by h^2 yields the following formula:

Three-point centered-difference formula for second derivative

f”(x) = (f(x - h) - 2f(x) + f(x + h)) / h^2 - (h^2 / 12)f^(iv)(c) (5.8)

for some c between x - h and x + h.

5.1.2 Rounding error
So far, all of this chapter's formulas break the rule from Chapter 0 that advises against subtracting nearly equal numbers. This is the greatest difficulty with numerical differentiation, but it is essentially impossible to avoid. To understand the problem better, consider the following example:

EXAMPLE 5.3 Approximate the derivative of f(x) = e^x at x = 0.

The two-point formula (5.4) gives

f'(x) ? (e^(x+h) - e^x) / h, (5.9)

and the three-point formula (5.7) yields

f'(x) ? (e^(x+h) - e^(x-h)) / 2h. (5.10)

The results of these formulas for x = 0 and a wide range of increment size h, along with errors compared with the correct value e^0 = 1, are given in the following table:

h | formula (5.9) | error | formula (5.10) | error
10^-1 | 1.05170918075648 | -0.05170918075648 | 1.00166750019844 | -0.00166750019844
10^-2 | 1.00501670841679 | -0.00501670841679 | 1.00001666674999 | -0.00001666674999
10^-3 | 1.00050016670838 | -0.00050016670838 | 1.00000016666668 | -0.00000016666668
10^-4 | 1.00005000166714 | -0.00005000166714 | 1.00000000166689 | -0.00000000166689
10^-5 | 1.00000500000696 | -0.00000500000696 | 1.00000000001210 | -0.00000000001210
10^-6 | 1.00000049996218 | -0.00000049996218 | 0.99999999997324 | 0.00000000002676
10^-7 | 1.00000004943368 | -0.00000004943368 | 0.99999999947364 | 0.00000000052636
10^-8 | 0.99999999392253 | 0.00000000607747 | 0.99999999392253 | 0.00000000607747
10^-9 | 1.00000008274037 | -0.00000008274037 | 1.00000002722922 | -0.00000002722922

Added by -Ngeles L.

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