 Login  |   Contact  |   About Us       Monday, August 8, 2022  ~ Laguerre Method ~ Hermite Method ~ Chebyshev Method ~ Bessel Method

 "Special Functions" Laguerre Polynomial Method
 x = -1.00, L20(x) = 709.931791843359 x = 0.00, L20(x) = 1 x = 1.00, L20(x) = -0.164258811827793 x = 2.00, L20(x) = 0.50072156756003 x = 3.00, L20(x) = -0.557509325217191 x = 4.00, L20(x) = -0.150791428102111 x = 5.00, L20(x) = 2.02022574447691

 Order of Laguerre Polynomial: Entries: [ Initial Order of Laguerre Polynomial: {20) ] [ Initial entries number: {7} ]

IMPLEMENTATION
Laguerre Polynomial Method

The Laguerre polynomial is the standard solution of the Laguerre equation:

 xy'' + ( 1-x )y' + ny = 0

Which is a second-order linear differential equation. This equation has non-singular solutions only if n is a positive integer.

The polynomials from the standard solutions of the above equation, usually denoted L0,,L1,2,... form a polynomial sequence which can be defined by the Rodrigues formula:

 Ln(x) = ex / n! dn / dxn(e-xxn)

Can easily derive the first few polynomials from the above equation:

 L0(x) = 1 L1(x) = -x + 1 L2(x) = 1/2( x2- 4x + 2)   Testing the Laguerre Polynomial Method

To test the Laguerre Polynomial method, a new static method has been added. The implementation of the type of polynomials are based on their recurrence relation. The TestLaguerrePolynomial() method has been written, added and executed:

static void TestLaGuerrePolynomial();
{
for (int i = 0; i < t2; i++)
{
double x = 1.0 * i - 1.0;
ListBox1.Items.Add(" x = " + x + ".00, " + "L" + t1 + "(x) = " + SpecialFunctions.Laguerre(x, t1).ToString());
}
}

 Other Implementations...

 Object-Oriented Implementation Graphics and Animation Sample Applications Ore Extraction Optimization Vectors and Matrices Complex Numbers and Functions Ordinary Differential Equations - Euler Method Ordinary Differential Equations 2nd-Order Runge-Kutta Ordinary Differential Equations 4th-Order Runge-Kutta Higher Order Differential Equations Nonlinear Systems Numerical Integration Numerical Differentiation Function Evaluation  Math, Analysis,
expertise..."

EIGENVALUE
SOLUTIONS...

> Rayleigh-Quotient Method

> Cubic Spline Method

 Applied Mathematical Algorithms ComplexFunctions NonLinear Differentiation Integration
 About Us KMP Software Engineering is an independent multidisciplinary engineering consulting company specializing in mathematical algorithms. Areas of Expertise SpecialFunctions VectorsMatrices OptimizationMethods ComplexNumbers Interpolation CurveFitting NonLinearSystems LinearEquations DistributionFunctions NumericalDifferentiation NumericalIntegration DifferentialEquations Smalltalk FiniteBoundary Eigenvalue Graphics UnderstandingMining MiningMastery MineralNews MineralCommodities MineralForum Crystallography Services NumericalModeling WebServices MainframeServices OutsourceServices LINKED IN KMP ARTICLES Brand Login Contact