User:Jc161504

Five things I have learned about Wikipedia/Wikiversity:

 * 1) Wikipedia.com was started in 2001.
 * 2) Wikipedia operates under the creative commons license.
 * 3) Wikipedia retains the previous page of any changed page so that the old can be compared to the new.
 * 4) Wikipedia is a non-profit organization.
 * 5) Wikipedia operates in about 250 languages.

Maclaurin series:


 * $$f(0)+f'(0)x + \frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+ \cdots .$$

Homework 2: Problem 1:
Assignment: Find something that is incorrect, incomplete or poorly explained...

(a) A Description of what I think is wrong with the Secant Method:

Under the section titled "Comparisons with other root-finding methods", it should be noted that through iteration implemented by digital computers, algorithms which include any finite and invariable set of evaluations can be thought to occur in O(1) or constant time. Furthermore, due to the Secant Method's iterative process, in which each iteration has 5 operations, round-off error may be greater than that of Newton's iteration which only has two.

(b) My corrected/improved version

If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against α ≈ 1.6). However, Newton's method requires the evaluation of both f and its derivative at every step, while the secant method only requires the evaluation of f. Therefore, the secant method may well be faster in practice when calculated by hand. For instance, if we assume that evaluating f takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor α² ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor 2), so the secant method is faster. If however we consider parallel processing for the evaluation of the derivative, Newton's method proves its worth, being faster in time, though still spending more steps.

Though these additional evaluations may allow the secant method to be more efficient while implemented by hand, with the use of digital computers, an additional yet constant evaluation of an expression causes no such deficiency in time efficiency. Any set of finite and invariable calculations is said to occur in constant time in computer science or $$\mathcal O(1)$$. Furthermore, when one takes into consideration the number of calculations required for each iteration of the secant method, one finds five operations must be completed as opposed to only two with newton's method, which may lead to a slightly more accurate result when considering round-off errors.