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Only one of the real roots of f (x) = x6 - x - 1 lies in the interval 1 < x< 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ________.
Correct answer is between '10,10'. Can you explain this answer?
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To find the minimum number of iterations required to achieve an accuracy of 0.001 using the bisection method, we need to understand the concept of the bisection method and how it is applied in this specific case.
The Bisection Method:
The bisection method is an iterative numerical method used to find the root of a function within a given interval. It works by repeatedly dividing the interval in half and selecting the subinterval that contains the root. The process continues until the desired level of accuracy is achieved.
Applying the Bisection Method:
In this case, we are given the function f(x) = x^6 - x - 1 and the interval [1, 2]. We know that there is only one real root within this interval.
To apply the bisection method, we start by evaluating the function at the endpoints of the interval and checking if the signs of the function values differ. If they do, it indicates that there is a root within the interval.
Algorithm:
1. Choose an initial interval [a, b] such that f(a) and f(b) have opposite signs.
2. Calculate the midpoint c = (a + b) / 2.
3. Evaluate f(c) and check the sign.
a. If f(c) = 0, then c is the root.
b. If f(c) and f(a) have opposite signs, set b = c.
c. If f(c) and f(b) have opposite signs, set a = c.
4. Repeat steps 2 and 3 until the desired level of accuracy is achieved.
Calculating the Minimum Number of Iterations:
To calculate the minimum number of iterations required to achieve an accuracy of 0.001, we can use the formula:
n = log2((b - a) / tolerance)
where n is the number of iterations, b and a are the endpoints of the interval, and tolerance is the desired accuracy.
In this case, a = 1, b = 2, and tolerance = 0.001.
Using the formula:
n = log2((2 - 1) / 0.001)
n = log2(1000)
n ≈ 9.97
Rounding up to the nearest integer, the minimum number of iterations required is 10.
Therefore, the required minimum number of iterations to achieve an accuracy of 0.001 using the bisection method is 10.
The Bisection Method:
The bisection method is an iterative numerical method used to find the root of a function within a given interval. It works by repeatedly dividing the interval in half and selecting the subinterval that contains the root. The process continues until the desired level of accuracy is achieved.
Applying the Bisection Method:
In this case, we are given the function f(x) = x^6 - x - 1 and the interval [1, 2]. We know that there is only one real root within this interval.
To apply the bisection method, we start by evaluating the function at the endpoints of the interval and checking if the signs of the function values differ. If they do, it indicates that there is a root within the interval.
Algorithm:
1. Choose an initial interval [a, b] such that f(a) and f(b) have opposite signs.
2. Calculate the midpoint c = (a + b) / 2.
3. Evaluate f(c) and check the sign.
a. If f(c) = 0, then c is the root.
b. If f(c) and f(a) have opposite signs, set b = c.
c. If f(c) and f(b) have opposite signs, set a = c.
4. Repeat steps 2 and 3 until the desired level of accuracy is achieved.
Calculating the Minimum Number of Iterations:
To calculate the minimum number of iterations required to achieve an accuracy of 0.001, we can use the formula:
n = log2((b - a) / tolerance)
where n is the number of iterations, b and a are the endpoints of the interval, and tolerance is the desired accuracy.
In this case, a = 1, b = 2, and tolerance = 0.001.
Using the formula:
n = log2((2 - 1) / 0.001)
n = log2(1000)
n ≈ 9.97
Rounding up to the nearest integer, the minimum number of iterations required is 10.
Therefore, the required minimum number of iterations to achieve an accuracy of 0.001 using the bisection method is 10.
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Only one of the real roots of f (x) = x6 - x - 1 lies in the interval 1 <x< 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ________.Correct answer is between '10,10'. Can you explain this answer?
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Only one of the real roots of f (x) = x6 - x - 1 lies in the interval 1 <x< 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ________.Correct answer is between '10,10'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Only one of the real roots of f (x) = x6 - x - 1 lies in the interval 1 <x< 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ________.Correct answer is between '10,10'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Only one of the real roots of f (x) = x6 - x - 1 lies in the interval 1 <x< 2 and bisection method is used to find its value. For achieving an accuracy of 0.001, the required minimum number of iterations is ________.Correct answer is between '10,10'. Can you explain this answer?.
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