How to Calculate Rate of Convergence

Numerical Convergence Rate Calculator

The rate of convergence is a critical concept in numerical analysis and computational mathematics. It describes how quickly a sequence of approximations approaches its limit (the true solution). Whether you are using the Bisection method, Newton-Raphson method, or the Secant method, understanding the speed of convergence helps in selecting the most efficient algorithm for solving equations.

This calculator helps you determine the Order of Convergence ($p$) and the Asymptotic Error Constant ($\mu$) based on the errors observed in consecutive iterations.

The Mathematical Formula

If a sequence $x_n$ converges to a value $L$, we define the error at step $n$ as $e_n = x_n – L$.

The sequence is said to converge with order $p$ if there exists a constant $\mu > 0$ such that:

lim (n → ∞) |eₙ₊₁| / |eₙ|^p = μ

Where:

• $e_n$: Error at the current iteration.
• $e_{n+1}$: Error at the next iteration.
• $p$: Order of convergence.
• $\mu$: Asymptotic error constant (rate of convergence).

Estimating the Order ($p$)

To calculate the order of convergence experimentally, you need the error values from three consecutive iterations ($e_n, e_{n+1}, e_{n+2}$). The approximate formula is:

p ≈ ln( |eₙ₊₂| / |eₙ₊₁| ) / ln( |eₙ₊₁| / |eₙ| )

Common Convergence Rates

Different numerical methods exhibit different convergence behaviors. Here is a comparison of common methods:

Convergence Type	Order ($p$)	Typical Method	Description
Linear	1	Bisection Method	The error is reduced by a constant factor in each step. Number of correct digits grows linearly.
Superlinear	~1.618	Secant Method	Faster than linear but slower than quadratic. Related to the Golden Ratio.
Quadratic	2	Newton-Raphson	The number of correct digits roughly doubles with every iteration. Very fast convergence.

Step-by-Step Calculation Example

Let's say you are finding the root of a function and you have the following absolute errors for three steps:

• Iteration 1 ($e_1$): 0.1
• Iteration 2 ($e_2$): 0.01
• Iteration 3 ($e_3$): 0.0001

Step 1: Calculate the ratios.

Ratio 1: $0.01 / 0.1 = 0.1$

Ratio 2: $0.0001 / 0.01 = 0.01$

Step 2: Apply the Logarithm formula.

$p \approx \frac{\ln(0.01)}{\ln(0.1)} = \frac{-4.605}{-2.302} = 2$

Since $p=2$, this indicates Quadratic Convergence, likely using Newton's method.

Why is Convergence Rate Important?

In high-performance computing and simulations, the cost of a single iteration can be computationally expensive. Knowing the rate of convergence allows engineers and mathematicians to:

• Predict how many iterations are needed to reach a desired accuracy.
• Compare the efficiency of different algorithms.
• Debug algorithms (if a quadratic method is converging linearly, there may be a bug in the derivative implementation).

Frequently Asked Questions

What if I don't know the true root?

If the true root $L$ is unknown, you cannot calculate the exact error $e_n = x_n – L$. However, for converging sequences, you can approximate the error using the difference between consecutive iterations: $|x_{n+1} – x_n|$. You can use these differences as inputs in the calculator above.

What does linear convergence mean?

Linear convergence ($p=1$) means that the error is multiplied by a constant factor $\mu < 1$ at each step. For example, if $\mu = 0.5$, the error is halved at every iteration. This is reliable but slow compared to quadratic methods.

Can the rate of convergence be negative?

The order of convergence $p$ is always positive. If the sequence is diverging (moving away from the solution), the error ratios will be greater than 1, and the formulas for $p$ may not yield meaningful results for convergence analysis.