Let f RR be continuous. Which of the following statements about the bisection, secant, and Newton methods is/are correct? If f(a) < 0 and f(b) > 0, the error of the bisection method starting from [a, b] decreases exponentially with the number of iterations. For a starting interval [a, b], the bisection method fails if f(a) > f(b). If initialized sufficiently close to a simple root of a smooth (infinitely differentiable) function, the order of convergence of the secant method is approximately 1.618. The secant method cannot be applied with a starting interval [a, b] such that f(a) = f(b). The order of convergence of the Newton method is 1. If f is smooth (infinitely differentiable) on the starting interval [a, b] and contains exactly one root x Є [a, b], then the bisection method will converge to this root. If f contains exactly one root in the starting interval [a, b] with ƒ(a)f (b) < 0, then the secant method will converge to this root.
Let f RR be continuous. Which of the following statements about the bisection, secant, and Newton methods is/are correct? If f(a) < 0 and f(b) > 0, the error of the bisection method starting from [a, b] decreases exponentially with the number of iterations. For a starting interval [a, b], the bisection method fails if f(a) > f(b). If initialized sufficiently close to a simple root of a smooth (infinitely differentiable) function, the order of convergence of the secant method is approximately 1.618. The secant method cannot be applied with a starting interval [a, b] such that f(a) = f(b). The order of convergence of the Newton method is 1. If f is smooth (infinitely differentiable) on the starting interval [a, b] and contains exactly one root x Є [a, b], then the bisection method will converge to this root. If f contains exactly one root in the starting interval [a, b] with ƒ(a)f (b) < 0, then the secant method will converge to this root.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![Let f RR be continuous. Which of the following statements about the bisection, secant, and Newton methods is/are correct?
If f(a) < 0 and f(b) > 0, the error of the bisection method starting from [a, b] decreases exponentially with the number of iterations.
For a starting interval [a, b], the bisection method fails if f(a) > f(b).
If initialized sufficiently close to a simple root of a smooth (infinitely differentiable) function, the order of convergence of the secant method is approximately
1.618.
The secant method cannot be applied with a starting interval [a, b] such that f(a) = f(b).
The order of convergence of the Newton method is 1.
If f is smooth (infinitely differentiable) on the starting interval [a, b] and contains exactly one root x Є [a, b], then the bisection method will converge to this
root.
If f contains exactly one root in the starting interval [a, b] with ƒ(a)f (b) < 0, then the secant method will converge to this root.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F033c0d82-90fb-4c58-9b03-1325bfffdb8d%2Ff895d8eb-0b10-4fd8-a46f-c1bb61284e08%2Folfkuvw_processed.png&w=3840&q=75)
Transcribed Image Text:Let f RR be continuous. Which of the following statements about the bisection, secant, and Newton methods is/are correct?
If f(a) < 0 and f(b) > 0, the error of the bisection method starting from [a, b] decreases exponentially with the number of iterations.
For a starting interval [a, b], the bisection method fails if f(a) > f(b).
If initialized sufficiently close to a simple root of a smooth (infinitely differentiable) function, the order of convergence of the secant method is approximately
1.618.
The secant method cannot be applied with a starting interval [a, b] such that f(a) = f(b).
The order of convergence of the Newton method is 1.
If f is smooth (infinitely differentiable) on the starting interval [a, b] and contains exactly one root x Є [a, b], then the bisection method will converge to this
root.
If f contains exactly one root in the starting interval [a, b] with ƒ(a)f (b) < 0, then the secant method will converge to this root.
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