Bisección

Summary

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Takeaways

• 😀 The script discusses numerical methods for approximating the roots of non-linear equations of the form f(x) = 0.
• 😀 It introduces two types of root-finding methods: closed methods (bisection and false position) and open methods (fixed point and Newton-Raphson).
• 😀 The bisection method involves iteratively narrowing an interval where the function changes sign, helping to locate the root.
• 😀 To apply the bisection method, two initial values are needed where the function values have opposite signs, ensuring a root exists between them.
• 😀 The process involves calculating the midpoint of the interval and adjusting the interval based on the signs of the function values at the endpoints and the midpoint.
• 😀 After each iteration, the interval is reduced by half, and the function values are re-evaluated at the new midpoint.
• 😀 The method continues until the error tolerance (ε) is met, which is when the difference between successive midpoints is sufficiently small.
• 😀 The script details an example of approximating the square root of 20 using the bisection method with an error tolerance of 10^-4.
• 😀 The minimum number of iterations required to achieve the desired accuracy is calculated using a formula based on the interval size and desired error tolerance.
• 😀 In the example, after 14 iterations, the square root of 20 is approximated as 4.4721, with an error smaller than the set tolerance of 10^-4.

Q & A

What is the main objective of the bisection method described in the script?

The main objective of the bisection method is to systematically narrow down an interval until it is small enough to localize a zero or root of a nonlinear equation.

What is the requirement for selecting the initial values in the bisection method?

The requirement is that the two initial values must be on opposite sides of the root, meaning their functional values must have opposite signs.

What is the general process of the bisection method?

The process involves selecting an interval, calculating its midpoint, evaluating the function at the midpoint, and then adjusting the interval based on whether the midpoint's function value has opposite signs compared to the ends of the interval.

How do we determine if the bisection method has found the root?

We determine the root has been found when the function value at the midpoint equals zero or when the interval size becomes sufficiently small, indicating the approximation is close enough.

What mathematical formula is used to calculate the midpoint in the bisection method?

The formula for calculating the midpoint is C = (A + B) / 2, where A and B are the endpoints of the interval.

What is the significance of the error tolerance (epsilon) in the bisection method?

The error tolerance, epsilon, defines the maximum allowable error for the root approximation. The method stops when the absolute error is less than or equal to epsilon.

How do you calculate the minimum number of iterations required for the bisection method?

The minimum number of iterations is calculated using the formula n ≥ log(B - A) / log(2), where A and B are the initial interval endpoints and n is the number of iterations needed to reduce the interval size sufficiently.

What is the role of the 'fix 4' method mentioned in the script?

'Fix 4' refers to rounding the result to four decimal places to simplify the process and stop when the approximation reaches the desired precision, avoiding excessive iterations.

Why is the interval [4, 5] chosen for approximating the square root of 20 in this example?

The interval [4, 5] is chosen because 4^2 = 16 and 5^2 = 25, and since 20 lies between these values, the square root of 20 is guaranteed to be within this interval.

What is the final result of the bisection method applied to this example, and how do we know it is correct?

The final result is an approximation of the square root of 20, which is 4.4721. This result is verified after 14 iterations, as the error becomes smaller than the tolerance of 10^-4, indicating the approximation is accurate.