Numerical Computation Methods Summary

Finite Difference and Linear Equations

Linear Homogeneous/Inhomogeneous Equations

Linear Equations
Equations where the unknown function and its derivatives appear only in the first degree, with no products or higher powers. They have a straight forward structure, are easy to solve, and ofter have analytical solution.

Homogeneous
All terms in the equation contain the unknow function or its derivatives; there are no constant terms. If the equation includes constant terms, then it is a Non-homogeneous equation.

Finite Difference Method and Linear Equations
The finite difference method is an effective numerical technique for solving various types of differential equations, including linear equations. With linear equations, we can directly replace the derivatives in the equation with their finite difference approximations, resulting in a system of linear algebraic equations.

Solving Methods

1.Finite Difference Method (FDM)(Numerical Approxmation)

  • Forward finite difference method

  • Backward finite difference method

  • Centered finite difference method

2. Characteristic Equation Method

Numerical Integration

Composite trapezoidal rule

Composite Simpson rule

Numerical Interpolating polynomial

For small datasets and when conceptual simplicity is important, Lagrange is fine.
For larger datasets or when you need to add/remove data points frequently, Newton’s divided difference is ususlly the best choice.
Vandermonde matrix is generally not recommended for practical use due to its computational cost and potential numerical instability. It’s more of a theoretical tool.

Vandermonde Matrix

Newton’s divided difference method

Lagrange interpolating polynomial

cardinal functions
Given a set of data points (x0,y0),(x1,y1)...,(xn,yn)(x_0, y_0), (x_1, y_1) \text{...}, (x_n, y_n) , the cardinal functions Li(x)L_i(x) have the following properties:

  • Li(xj)=1L_i(x_j) = 1, when i = j
  • Li(xj)L_i(x_j) = 0, when i != j

The cardinal functions serve to “select” or “active” the corresponding yiy_i values. In the construction of the Lagrange interpolating polynomial, each cardinal function Li(x)L_i(x) is multiplied by a yiy_i . Due to the special properties of the cardinal functions, when x=xix = x_i , only the yiy_i corresponding to Li(x)L_i(x) is “selected”, while all other terms become 0. This ensures that the Lagrange interpolating polynomial passes exactly through all the data points

The expression for the cardinal function Li(x)L_i(x) is as follows:

Li(x)=∏k=0,k≠inx−xkxi−xk=(x−x0)(x−x1)...(x−xi−1)(x−xi+1)...(x−xn)(xi−x0)(xi−x1)...(xi−xi−1)(xi−xi+1)...(xi−xn) L_i(x) = \prod \limits_{k=0,k\neq i}^n \frac{x-x_k}{x_i-x_k} = \frac{(x-x_0)(x-x_1)\text{...}(x-x_{i-1})(x-x_{i+1})\text{...}(x-x_n)}{(x_i-x_0)(x_i-x_1)\text{...}(x_i-x_{i-1})(x_i-x_{i+1})\text{...}(x_i-x_n)}

Largrange interpolating polynomial of degree not more than n

pn(x)=∑i=0nf(xi)Li(x)=f(x0)L0(x)+f(x1)L1(x)+...+f(xn)Ln(x) p_n(x) = \sum\limits{i=0}^nf(x_i)L_i(x) = f(x_0)L_0(x) + f(x_1)L_1(x)+\text{...}+f(x_n)L_n(x)

Error bounds