* if the answer is true, pls just write \"true\"; however, if it is false, pls g
ID: 2968091 • Letter: #
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* if the answer is true, pls just write "true"; however, if it is false, pls give me your example or explanation(reason) why it is false.
* thank you very much your time and work!!
Answer "True" or "False" to the following. Give reasons for your answers. If a backward stable algorithm is applied to a computational problem, the solution will be accurate. A backward stable algorithm produces a good approximation to an exact solution. Well-conditioning is a good property of an algorithm. Cancellation is always bad. If the zeros of a polynomial are all distinct, then they must be well-conditioned. An efficient algorithm is necessarily a stable algorithm. Backward errors relate the errors to the data of the problem. A backward stable algorithm applied to a well-conditioned problem produces an accurate solution. Stability analysis of an algorithm is performed by means of perturbation analysis. (j) A symmetric matrix must be well-conditioned. If the determinant of a matrix A is small, then it must be close to a singular matrix. One must perform a large amount of computations to obtain a large round-off error. If a matrix A is ill-conditioned, then its smallest singular value is very samll.Explanation / Answer
(a) If a backward stable algorithm is applied to a computational problem, the solution will be accurate - FALSE
When a (backward) stable algorithm is applied to a well-conditioned problem, the computed solution should be near the exact solution. However, if a (backward) stable algorithm is applied to an ill condition problem, accuracy is not guaranteed.
(b) A backward stable algorithm produces a good approximation to an exact solution.- FALSE.
The same reason as the previous problem.
(c) Well-conditioning is a good property of an algorithm.- FALSE.
The conditioning of a problem is a property of the problem itself. It deals with how the solution of the problem changes when the input data is perturbed.
(d) Cancellation is always bad. - FALSE.
Cancellation highlights the earlier errors which is a good thing to happen.
(e) If the zeros of a polynomial are all distinct, then they must be well-conditioned. - FALSE.
Consider the Wilkinson polynomial of degree 20, p(x) = (x?1)(x?2)
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