math calculator

Bisection & Newton-Raphson Root Finder

Find roots of nonlinear equations using interval bisection and Newton-Raphson iterative methods with this free numerical solver.

Formula Verified by Dr. Elena Rostova, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Inputs

Root Equation Profile

Interval a

Interval b

Max Iterations

Results

Bisection Method Root

1.521240

Newton-Raphson Method Root1.521380

Equation curve & root

Plot of f(x) showing calculated root intersection

Bisection Method Iteration Log

Step	Bracket a	Bracket b	Midpoint c	f(c)	Width
1	1.0000	2.0000	1.500000	-1.25e-1	1.00e+0
2	1.5000	2.0000	1.750000	1.61e+0	5.00e-1
3	1.5000	1.7500	1.625000	6.66e-1	2.50e-1
4	1.5000	1.6250	1.562500	2.52e-1	1.25e-1
5	1.5000	1.5625	1.531250	5.91e-2	6.25e-2
6	1.5000	1.5313	1.515625	-3.41e-2	3.13e-2
7	1.5156	1.5313	1.523438	1.23e-2	1.56e-2
8	1.5156	1.5234	1.519531	-1.10e-2	7.81e-3
9	1.5195	1.5234	1.521484	6.22e-4	3.91e-3
10	1.5195	1.5215	1.520508	-5.18e-3	1.95e-3
11	1.5205	1.5215	1.520996	-2.28e-3	9.77e-4
12	1.5210	1.5215	1.521240	-8.29e-4	4.88e-4

Newton-Raphson Iteration Log

Step	Estimate x_k	f(x_k)	f'(x_k)	Next x_{k+1}	Difference
1	1.500000	-1.25e-1	5.7500	1.521739	2.17e-2
2	1.521739	2.14e-3	5.9471	1.521380	3.59e-4
3	1.521380	5.89e-7	5.9438	1.521380	9.92e-8
4	1.521380	4.53e-14	5.9438	1.521380	7.55e-15

Step-by-Step Numerical Analysis

Select root equation: \(f(x) = x^3 - x - 2 = 0\)

Intermediate value theorem check passed: \(f(1) = -2.0000\) and \(f(2) = 4.0000\) have opposite signs.

Tip

Save this page to compare numerical approximations, solve non-algebraic equations, or verify calculus roots.

Formula

Numerical Root formulas

c = \frac{a+b}{2}, \quad x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}

Bisection method continuously bisects interval brackets. Newton-Raphson uses tangent slope intercepts to converge quadratically.

Step by step

How the calculation works

1Select the function equation profile (Cubic, Exponential, Trigonometric).
2Set the initial interval brackets [a, b] where the function changes signs.
3Define the maximum iteration limit to observe and compare convergence speed.

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Results

Bisection Method Root

1.521240

Newton-Raphson Method Root1.521380

Equation curve & root

Plot of f(x) showing calculated root intersection

Bisection Method Iteration Log

Step	Bracket a	Bracket b	Midpoint c	f(c)	Width
1	1.0000	2.0000	1.500000	-1.25e-1	1.00e+0
2	1.5000	2.0000	1.750000	1.61e+0	5.00e-1
3	1.5000	1.7500	1.625000	6.66e-1	2.50e-1
4	1.5000	1.6250	1.562500	2.52e-1	1.25e-1
5	1.5000	1.5625	1.531250	5.91e-2	6.25e-2
6	1.5000	1.5313	1.515625	-3.41e-2	3.13e-2
7	1.5156	1.5313	1.523438	1.23e-2	1.56e-2
8	1.5156	1.5234	1.519531	-1.10e-2	7.81e-3
9	1.5195	1.5234	1.521484	6.22e-4	3.91e-3
10	1.5195	1.5215	1.520508	-5.18e-3	1.95e-3
11	1.5205	1.5215	1.520996	-2.28e-3	9.77e-4
12	1.5210	1.5215	1.521240	-8.29e-4	4.88e-4

Newton-Raphson Iteration Log

Step	Estimate x_k	f(x_k)	f'(x_k)	Next x_{k+1}	Difference
1	1.500000	-1.25e-1	5.7500	1.521739	2.17e-2
2	1.521739	2.14e-3	5.9471	1.521380	3.59e-4
3	1.521380	5.89e-7	5.9438	1.521380	9.92e-8
4	1.521380	4.53e-14	5.9438	1.521380	7.55e-15

Step-by-Step Numerical Analysis

Select root equation: \(f(x) = x^3 - x - 2 = 0\)

Intermediate value theorem check passed: \(f(1) = -2.0000\) and \(f(2) = 4.0000\) have opposite signs.

How to understand and use this calculator

Introduction to Numerical Root-Finding Algorithms

In mathematics, a root of a function f(x) is a value of x for which the function equals zero (f(x) = 0). While simple linear and quadratic equations can be solved algebraically, many higher-degree polynomials and transcendental equations (equations containing trigonometric, exponential, or logarithmic terms) cannot be solved analytically. In these cases, we must use numerical methods to approximate the root to a desired level of accuracy.

Numerical root-finding algorithms are iterative procedures that start with an initial guess or interval and refine the approximation step-by-step. The two most popular methods are the Bisection Method (a bracketed method that is guaranteed to converge) and the Newton-Raphson Method (an open method that converges rapidly but requires calculating the function's derivative).

This calculator solves root-finding problems for standard test functions. It displays the step-by-step iteration tables for both Bisection and Newton-Raphson methods, helping you compare their convergence rates and behavior.

How to Use This Calculator

Pick a test function (cubic x^3 - x - 2, exponential e^(-x) - x, or trigonometric cos(x) - x), set the bracket interval [a, b], and choose a maximum iteration count (up to 30). The calculator runs both the Bisection method and Newton-Raphson from the same bracket, showing the full iteration log for each so you can compare how quickly each converges to the root.

Mathematical Formulation of Bisection and Newton-Raphson Methods

The Bisection Method is based on the Intermediate Value Theorem. If a continuous function f(x) changes sign over an interval [a, b] (meaning f(a) · f(b) < 0), there must be at least one root in the interval. The method halves the interval at each step:

c = (a + b)/2

If f(c) = 0, c is the root. Otherwise, we replace either a or b with c depending on the sign of f(c) to maintain a sign change across the new, smaller interval.

The Newton-Raphson Method uses the tangent line at the current guess x_k to find the next approximation x_(k+1):

x_(k+1) = x_k - f(x_k)/f'(x_k)

This method requires that the derivative f'(x_k) is non-zero at each iteration step.

Worked Case Study: Approximating the Root of x^3 - x - 2 = 0

Let us find the root of f(x) = x^3 - x - 2 in the interval [1, 2] using the Bisection Method.

First, we verify a sign change exists: f(1) = 1^3 - 1 - 2 = -2, and f(2) = 2^3 - 2 - 2 = 4. Since f(1) · f(2) < 0, a root exists.

1. Iteration 1: Midpoint c_1 = (1 + 2)/2 = 1.5. Evaluate f(1.5) = 1.5^3 - 1.5 - 2 = -0.125. Since f(1.5) is negative, the root lies in [1.5, 2]. New interval: [1.5, 2] (width = 0.5).

2. Iteration 2: Midpoint c_2 = (1.5 + 2)/2 = 1.75. Evaluate f(1.75) = 1.75^3 - 1.75 - 2 = 1.6094. Since f(1.75) is positive, the root lies in [1.5, 1.75]. New interval: [1.5, 1.75] (width = 0.25).

3. Iteration 3: Midpoint c_3 = (1.5 + 1.75)/2 = 1.625. Evaluate f(1.625) = 1.625^3 - 1.625 - 2 = 0.6660. The root lies in [1.5, 1.625]. New interval: [1.5, 1.625] (width = 0.125).

Repeating this process rapidly narrows the interval. After 10 iterations the bisection estimate is c_(10) ≈ 1.5205, and after the calculator's default 12 iterations it reaches ≈ 1.52124 — both converging toward the true root of approximately 1.52138 (verified independently with Newton-Raphson, which reaches the same value to 6 decimal places in only 4 iterations from the same starting interval).

Comparing Convergence Rates: Linear vs. Quadratic Convergence

The Bisection method has linear convergence, meaning the error is halved at each iteration step. While slow, it is extremely robust and guaranteed to find a root if one exists in the initial interval.

The Newton-Raphson method has quadratic convergence, meaning the number of correct decimal places roughly doubles at each step when close to the root. However, it can fail to converge if the initial guess is far from the root or near a flat region where the derivative is close to zero.

Related Calculators

For polynomials up to degree 2, exact algebraic roots are faster to get from the quadratic solver than a numerical method. For systems of several linear equations rather than a single nonlinear one, use the system of equations calculator.

Related concepts

Intermediate Value Theorem

A calculus theorem guaranteeing the existence of a root for continuous functions with opposite sign endpoints.

Secant Method

A root-finding algorithm that approximates the derivative in Newton-Raphson using secant lines through two points.

Convergence Rate

A measure of how fast the sequence of approximations approaches the exact root value.

Frequently Asked Questions

What is numerical root finding?+

Numerical root finding is an iterative process used to approximate the solutions of equations f(x) = 0 when algebraic solutions are difficult or impossible to find.

How does the Bisection Method work?+

The Bisection Method repeatedly halves an interval [a, b] where the function changes sign, narrowing down the location of the root.

What is the Intermediate Value Theorem?+

A theorem stating that if a continuous function has opposite signs at the endpoints of an interval, it must cross zero at least once within that interval.

How does the Newton-Raphson Method work?+

The Newton-Raphson Method starts with an initial guess and uses the function's value and derivative to find the x-intercept of the tangent line, which serves as the next guess: x_(k+1) = x_k - f(x_k)/f'(x_k).

Why is the Bisection Method guaranteed to converge?+

Because it is a bracketed method that maintains a sign change across a shrinking interval, ensuring it converges to a root of a continuous function.

When can the Newton-Raphson Method fail?+

It can fail if the initial guess is close to a local extremum (where the derivative f'(x) is zero, leading to division by zero) or if the iterations cycle or diverge away from the root.

What is linear convergence?+

Linear convergence means the error decreases by a constant factor at each step (for Bisection, the error is halved at each step).

What is quadratic convergence?+

Quadratic convergence means the error at step k+1 is proportional to the square of the error at step k, leading to rapid convergence near the root.

How do you choose between Bisection and Newton-Raphson?+

Use Bisection when robustness is required and a sign-changing interval is known. Use Newton-Raphson when speed is important and a good initial guess is available.

What is the tolerance parameter in root finding?+

The tolerance is a small number (like 10⁻⁶) representing the target accuracy. Iterations stop when the interval width or function value is smaller than this tolerance.

Can these methods find multiple roots?+

Each run of these algorithms approximates a single root. To find multiple roots, you must run the algorithm with different starting intervals or initial guesses.

What is a double root or root of multiplicity?+

A double root occurs where the function touches the x-axis but does not cross it (such as f(x) = (x-r)²). Bisection cannot find these roots because there is no sign change.

How to understand and use this calculator

Introduction to Numerical Root-Finding Algorithms

How to Use This Calculator

Mathematical Formulation of Bisection and Newton-Raphson Methods

c = (a + b)/2

If f(c) = 0, c is the root. Otherwise, we replace either a or b with c depending on the sign of f(c) to maintain a sign change across the new, smaller interval.

The Newton-Raphson Method uses the tangent line at the current guess x_k to find the next approximation x_(k+1):

x_(k+1) = x_k - f(x_k)/f'(x_k)

This method requires that the derivative f'(x_k) is non-zero at each iteration step.

Worked Case Study: Approximating the Root of x^3 - x - 2 = 0

Let us find the root of f(x) = x^3 - x - 2 in the interval [1, 2] using the Bisection Method.

First, we verify a sign change exists: f(1) = 1^3 - 1 - 2 = -2, and f(2) = 2^3 - 2 - 2 = 4. Since f(1) · f(2) < 0, a root exists.

1. Iteration 1: Midpoint c_1 = (1 + 2)/2 = 1.5. Evaluate f(1.5) = 1.5^3 - 1.5 - 2 = -0.125. Since f(1.5) is negative, the root lies in [1.5, 2]. New interval: [1.5, 2] (width = 0.5).

3. Iteration 3: Midpoint c_3 = (1.5 + 1.75)/2 = 1.625. Evaluate f(1.625) = 1.625^3 - 1.625 - 2 = 0.6660. The root lies in [1.5, 1.625]. New interval: [1.5, 1.625] (width = 0.125).

Comparing Convergence Rates: Linear vs. Quadratic Convergence

Related Calculators

Related concepts

Intermediate Value Theorem

A calculus theorem guaranteeing the existence of a root for continuous functions with opposite sign endpoints.

Secant Method

A root-finding algorithm that approximates the derivative in Newton-Raphson using secant lines through two points.

Convergence Rate

A measure of how fast the sequence of approximations approaches the exact root value.

Frequently Asked Questions

What is numerical root finding?+

Numerical root finding is an iterative process used to approximate the solutions of equations f(x) = 0 when algebraic solutions are difficult or impossible to find.

How does the Bisection Method work?+

The Bisection Method repeatedly halves an interval [a, b] where the function changes sign, narrowing down the location of the root.

What is the Intermediate Value Theorem?+

A theorem stating that if a continuous function has opposite signs at the endpoints of an interval, it must cross zero at least once within that interval.

How does the Newton-Raphson Method work?+

Why is the Bisection Method guaranteed to converge?+

Because it is a bracketed method that maintains a sign change across a shrinking interval, ensuring it converges to a root of a continuous function.

When can the Newton-Raphson Method fail?+

It can fail if the initial guess is close to a local extremum (where the derivative f'(x) is zero, leading to division by zero) or if the iterations cycle or diverge away from the root.

What is linear convergence?+

Linear convergence means the error decreases by a constant factor at each step (for Bisection, the error is halved at each step).

What is quadratic convergence?+

Quadratic convergence means the error at step k+1 is proportional to the square of the error at step k, leading to rapid convergence near the root.

How do you choose between Bisection and Newton-Raphson?+

Use Bisection when robustness is required and a sign-changing interval is known. Use Newton-Raphson when speed is important and a good initial guess is available.

What is the tolerance parameter in root finding?+

The tolerance is a small number (like 10⁻⁶) representing the target accuracy. Iterations stop when the interval width or function value is smaller than this tolerance.

Can these methods find multiple roots?+

Each run of these algorithms approximates a single root. To find multiple roots, you must run the algorithm with different starting intervals or initial guesses.

What is a double root or root of multiplicity?+

A double root occurs where the function touches the x-axis but does not cross it (such as f(x) = (x-r)²). Bisection cannot find these roots because there is no sign change.

Bisection & Newton-Raphson Root Finder

Inputs

Results

Equation curve & root

Bisection Method Iteration Log

Newton-Raphson Iteration Log

Step-by-Step Numerical Analysis

Numerical Root formulas

How the calculation works

Learn more about this calculation

Scientific Notation Guide: Powers of Ten Made Practical

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Scientific Calculator

Quadratic Solver

Bisection & Newton-Raphson Root Finder

Inputs

Results

Equation curve & root

Bisection Method Iteration Log

Newton-Raphson Iteration Log

Step-by-Step Numerical Analysis

Key output metrics

Newton-Raphson Convergence Speed

How to understand and use this calculator

Introduction to Numerical Root-Finding Algorithms

How to Use This Calculator

Mathematical Formulation of Bisection and Newton-Raphson Methods

Worked Case Study: Approximating the Root of x^3 - x - 2 = 0

Comparing Convergence Rates: Linear vs. Quadratic Convergence

Related Calculators

Related concepts

Frequently Asked Questions

Numerical Root formulas

How the calculation works

Learn more about this calculation

Scientific Notation Guide: Powers of Ten Made Practical

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Scientific Calculator

Quadratic Solver

Key output metrics

Newton-Raphson Convergence Speed

How to understand and use this calculator

Introduction to Numerical Root-Finding Algorithms

How to Use This Calculator

Mathematical Formulation of Bisection and Newton-Raphson Methods

Worked Case Study: Approximating the Root of x^3 - x - 2 = 0

Comparing Convergence Rates: Linear vs. Quadratic Convergence

Related Calculators

Related concepts

Frequently Asked Questions