Bisection Method Calculator

The bisection method is a simple way to find the root of a function, which is where the function crosses the x-axis. To use this method, start with two points, a and b, where the function changes sign. This means f(a) and f(b) should have opposite signs, indicating there is a root between them.

Next, calculate the midpoint ccc of a and b and evaluate the function at this point. If f(c) is zero, you've found the root. If not, determine which half of the interval contains the root by checking where the sign changes. Replace either aaa or b with ccc to narrow down the interval, and repeat the process until the interval is small enough to estimate the root accurately.

Yes, the bisection method always converges to a root of the function, provided a few conditions are met:

Sign Change: The method requires that the function changes sign over the interval [a,b]. This ensures that a root exists within this interval, according to the Intermediate Value Theorem.

Continuity: The function must be continuous over the interval [a,b]. Discontinuities can prevent the method from properly narrowing down to the root.

Interval Size: The interval [a,b] must initially contain the root. As long as this condition is met and the function changes sign over the interval, the bisection method will continue to reduce the interval size and approach the root.

The method works by repeatedly halving the interval and choosing the subinterval where the sign change occurs, thus systematically narrowing down the location of the root.

Because the interval size shrinks with each iteration, the method converges to the root, though it might require many iterations to achieve a high level of precision.

To find the number of iterations required in the bisection method to achieve the desired level of accuracy, you can use the following approach:

Formula for Number of Iterations:

The number of iterations n needed to reduce the interval to a specified accuracy ϵ can be approximated using the formula:

$$ n ≥ \frac{log(\frac{b-a}{ϵ})}{log (2)} $$

Where:

• a and b are the initial endpoints of the interval.
• ϵ is the desired accuracy (the width of the interval within which you want the root to be located).

Explanation:

1. Initial Interval: Compute the initial width of the interval, b − a.
2. Desired Accuracy: Determine the desired accuracy ϵ\epsilonϵ. This is the maximum acceptable error in the approximation of the root.
3. Apply the Formula: Use the formula to calculate the minimum number of iterations required to achieve the desired accuracy.

Example:

Suppose you start with an interval [a, b] = [1, 4] and want an accuracy of ϵ = 0.01

1. Initial Width: $$ b − a \;=\; 4 − 1 \;=\; 3 $$
2. Desired Accuracy: ϵ = 0.01.
3. Apply the formula:

$$ n ≥ \frac{log(\frac{3}{0.01})}{log(2)} \;=\; \frac{log(300)}{log(2)} \approx \frac{2.477}{0.301} \approx 8.22 $$

Round up to the next whole number, so you need at least 9 iterations to achieve the desired accuracy.

This formula provides a way to estimate the number of iterations needed to ensure that the width of the interval in which the root is found is less than or equal to the desired accuracy.

In the bisection method, tolerance refers to the acceptable level of error or the desired precision in the approximation of the root. It defines how close the method needs to get to the actual root before stopping. Tolerance helps determine when the interval is sufficiently small to provide an acceptable estimate of the root.

Understanding Tolerance:

• Definition: Tolerance (ϵ) is the maximum allowable difference between the true root and the approximated root. It defines the width of the interval in which the root is located.
• Interval Size: The method continues iterating until the width of the interval [a,b] is less than or equal to the tolerance ϵ. Mathematically, the condition to stop is b-a/2n≤ϵwhere n is the number of iterations.

Example:

If you want to find a root with a tolerance of ϵ=0.01, the bisection method will keep halving the interval until the width of the interval is less than or equal to 0.01. This ensures that the root is estimated within ±0.01 of the actual root.

Importance:

• Accuracy: The choice of tolerance directly affects the accuracy of the root approximation. A smaller tolerance leads to a more precise root estimate but requires more iterations.
• Stopping Criterion: Tolerance serves as a stopping criterion for the algorithm. When the interval size falls below the tolerance, the method concludes that the root has been found with the desired precision.

To find a root using the bisection method, follow these steps:

1. Choose Initial Interval:

Select two points, a and b, such that f(a) and f(b) have opposite signs. This indicates that a root lies between a and b according to the Intermediate Value Theorem.

2. Check for Convergence:

Ensure the function is continuous over the interval [a,b]. If f(a) and f(b) do not have opposite signs, you need to choose a different interval.

3. Calculate the Midpoint:

Compute the midpoint ccc of the interval:

$$ c \;=\; \frac{a+b}{2} $$

4. Evaluate the Function:

Calculate f(c).

• If f(c) = 0, c is the root, and you can stop.
• If f(c) is not zero, proceed to the next step.

5. Update the Interval:

Determine which subinterval contains the root:

• If f(a) and f(c) have opposite signs, the root lies between a and c. Update b to c.
• If f(c) and f(b) have opposite signs, the root lies between c and b. Update a to c.

6. Repeat:

Repeat steps 3 to 5 until the interval [a,b] is reduced to a size smaller than the desired tolerance ϵ, or until the function value f(c) is sufficiently close to zero.