Polynomials

While matrices and vectors are the core of linear algebra, many engineering problems—such as curve fitting, signal processing, and finding eigenvalues— revolve around Polynomials.

In SepalSolver, we represent a polynomial as a specialized class that manages a collection of coefficients and provides methods for evaluation, fitting, convolution, deconvolution, differentiation, and integration of polynomials.

A polynomial \(P(x)=a_0 x^n + a_1 x^{n-1} + a_2 x^{n - 2} + \cdots + a_n\) is defined by its coefficients. In SepalSolver, we store these in a double[] array where the index corresponds to the power of x. This makes the “Degree” of the polynomial exactly coefficients.Length - 1.

• Polynomial Evaluation
  • Polynomial Representation and Order
    • 1. The Descending Order Convention
    • 2. Horner’s Method (Descending)
  • Examples
    • Implementation Tip: Power Mapping
• Polynomial Fitting
  • Polynomial Fitting (Polyfit)
    • 1. Mathematical Objective
    • 2. Coefficient Order
  • Examples
    • Usage Warning
  • Multivariate Polynomial Fitting
    • 1.The Mathematical Model
    • 2.Implementation Logic
  • Examples
    • Exercise: Term Expansion
• Polynomial Differentiation and Integration
  • Polynomial Differentiation and Integration
    • 1. Differentiation
    • 2. Integration
• Polynomial Arithmetics
  • Polynomial Arithmetic
    • 1. Addition and Subtraction (PolyAdd / PolySub)
    • 2. Multiplication (Conv)
    • 3. Division and Remainder (Deconv)
  • Examples
    • Numerical Note: Array Sizing
• Polynomial Roots
  • Root Finding for Polynomials
    • 1. The Companion Matrix Method
    • 2. Iterative Refinement (Newton-Raphson)
  • Examples
    • Numerical Note: Complex Roots
• Exercise On Polynomials
  • Exercise: Polynomial Operations and Calculus