Calculator Input
Example Data Table
| Function | Interval | Tolerance | Expected Order | Error Behavior |
|---|---|---|---|---|
| x² - 2 | [1, 2] | 0.000001 | 1 | Error halves each iteration |
| x³ - x - 2 | [1, 2] | 0.000001 | 1 | Linear convergence |
| cos(x) - x | [0, 1] | 0.000001 | 1 | Guaranteed bracketing |
Formula Used
The bisection midpoint is calculated as:
cₙ = (aₙ + bₙ) / 2
The interval width after each step is:
bₙ - aₙ = (b₀ - a₀) / 2ⁿ
The root error bound is:
|cₙ - α| ≤ (b₀ - a₀) / 2ⁿ⁺¹
The order definition is:
lim |eₙ₊₁| / |eₙ|ᵖ = λ
For bisection, p equals 1 because each new error is proportional to the previous error.
How to Use This Calculator
- Select a predefined function.
- Enter left and right endpoints.
- Make sure the endpoints create a sign change.
- Enter the required tolerance.
- Choose the maximum number of iterations.
- Press the calculate button.
- Read the root approximation and error table.
- Download the result as CSV or PDF.
Understanding the Proof of Bisection Convergence
What This Calculator Shows
The bisection method is a reliable numerical method. It finds a root inside an interval. The method requires a continuous function. The function must change sign across the interval. This calculator displays each iteration clearly. It also shows interval width, midpoint value, function value, and error bound.
Why the Method Converges
The method works by repeatedly cutting the interval into two equal parts. One half keeps the sign change. The other half is discarded. Because the root remains bracketed, the method does not lose the solution. The interval becomes smaller after every iteration. This creates a predictable error pattern.
Order of Convergence
The order of convergence measures how fast the error approaches zero. In many advanced methods, the error may square or shrink faster. Bisection behaves differently. Its error decreases by a constant factor. The factor is one half for the interval width. Therefore the method has linear convergence.
Meaning of Linear Convergence
Linear convergence means the next error is roughly proportional to the current error. The exponent in the convergence definition is one. The constant is finite and positive. For bisection, the error bound follows a simple rule. Every iteration divides the possible error region by two. This is slower than Newton type methods. Still, it is very dependable.
Why the Proof Matters
The proof explains the strength and limitation of bisection. It guarantees progress. It also estimates the number of iterations needed. If a smaller tolerance is required, more interval halvings are needed. This makes the method easy to plan. Engineers, students, and researchers use this proof when checking numerical accuracy.
Practical Interpretation
The table helps you observe the proof numerically. The error ratio tends to stay near one half. That supports the theoretical result. The final root is an approximation, not always the exact root. The error bound tells how close the approximation can be. This makes the bisection method transparent and safe for root finding.
FAQs
1. What is the order of convergence of bisection method?
The bisection method has order one. This means it converges linearly. The error decreases by a fixed factor after each step.
2. Why is bisection called linearly convergent?
It is called linearly convergent because the next error is proportional to the previous error. The interval length halves every iteration.
3. Does bisection always converge?
It converges when the function is continuous and the starting interval has opposite signs at its endpoints.
4. What is the main error bound?
The error bound is based on interval width. After n steps, the possible error is no more than the reduced interval size.
5. Is bisection faster than Newton method?
No. Newton method can be faster. However, bisection is more reliable when a valid bracketing interval is known.
6. What does the error ratio show?
The error ratio compares one error bound with the previous one. For bisection, it usually approaches one half.
7. Can this calculator prove convergence numerically?
Yes. It displays iteration data, shrinking intervals, error bounds, and ratios that support the linear convergence proof.
8. What inputs are required?
You need a function, left endpoint, right endpoint, tolerance, and maximum iteration limit.