The steps for Muller’s Method, a root-finding algorithm, can be summarized as follows:
- Initialization:
- Choose three initial guesses for the root: x₀, x₁, and x₂.
- Calculate the corresponding function values: f(x₀), f(x₁), and f(x₂).
- Quadratic Interpolation:
- Determine the coefficients of the quadratic polynomial that passes through the three points (x₀, f(x₀)), (x₁, f(x₁)), and (x₂, f(x₂)).
- Express the quadratic polynomial as: p(x) = a(x – x₂)² + b(x – x₂) + c, where a, b, and c are the coefficients.
- Compute the discriminant: D = b² – 4ac.
- Root Finding:
- Calculate two possible solutions for the quadratic equation: x⁺ = x₂ – (2c / (b + √D)) and x⁻ = x₂ – (2c / (b – √D)).
- Determine which of the two solutions has a smaller absolute function value: |f(x⁺)| and |f(x⁻)|.
- The smaller absolute function value corresponds to the next approximation of the root: x_new = x₂ + (2c / (b ± √D)).
- Updating:
- Update the values of x₀, x₁, and x₂ for the next iteration: x₀ = x₁, x₁ = x₂, and x₂ = x_new.
- Repeat:
- Repeat steps 2-4 until the desired level of accuracy is achieved or the maximum number of iterations is reached.
- Convergence Check:
- Monitor the convergence of the method by calculating the difference between successive approximations of the root.
- If the difference falls below a specified tolerance, the method can be considered to have converged, and the current approximation is accepted as the root.
These steps outline the iterative process of Muller’s Method for approximating the roots of equations. By repeatedly applying quadratic interpolation and updating the values, the method refines the root estimate until the desired accuracy is reached. It is important to note that choosing appropriate initial guesses and handling complex equations are additional considerations when applying Muller’s Method effectively.
Example:
Example: Approximate a root of the equation f(x) = x^3 – 2x^2 – 5 = 0 using Muller’s Method.
- Initialization:
- Choose three initial guesses for the root: x₀ = 0, x₁ = 1, and x₂ = 2.
- Calculate the corresponding function values: f(x₀) = -5, f(x₁) = -6, and f(x₂) = 2.
- Quadratic Interpolation:
- Determine the coefficients of the quadratic polynomial passing through the points (x₀, f(x₀)), (x₁, f(x₁)), and (x₂, f(x₂)).
- Express the quadratic polynomial as: p(x) = a(x – x₂)² + b(x – x₂) + c.
- Root Finding:
- Calculate two possible solutions for the quadratic equation: x⁺ and x⁻.
- Determine which of the two solutions has a smaller absolute function value: |f(x⁺)| and |f(x⁻)|.
- The smaller absolute function value corresponds to the next approximation of the root: x_new = x₂ + (2c / (b ± √D)).
- Updating:
- Update the values of x₀, x₁, and x₂ for the next iteration: x₀ = x₁, x₁ = x₂, and x₂ = x_new.
- Repeat:
- Repeat steps 2-4 until the desired level of accuracy is achieved or the maximum number of iterations is reached.
Let’s perform one iteration of Muller’s Method to approximate the root:
- Initialization:
- x₀ = 0, x₁ = 1, x₂ = 2
- f(x₀) = -5, f(x₁) = -6, f(x₂) = 2
- Quadratic Interpolation:
- The quadratic polynomial is expressed as p(x) = a(x – x₂)² + b(x – x₂) + c.
- Root Finding:
- Calculate the solutions for the quadratic equation: x⁺ and x⁻.
- Determine which solution has a smaller absolute function value: |f(x⁺)| and |f(x⁻)|.
- Choose the solution with the smaller absolute function value as the next approximation of the root: x_new.
- Updating:
- Update the values of x₀, x₁, and x₂: x₀ = x₁, x₁ = x₂, x₂ = x_new.
- Repeat:
- Repeat steps 2-4 until the desired level of accuracy is achieved or the maximum number of iterations is reached.
Continue the iterations until the desired accuracy is attained or the maximum number of iterations is reached. The process is repeated until the root approximation meets the desired level of precision.
