Numerical methods in fortran

Solving linear, nonlinear equations, integral, ordinary differential equations, … using numerical methods in fortran

1. Linear equations:

LU
PLU (TODO)
QR (TODO)

2. Non-uniform random number generator

Normal distribution
    - Box–Muller transform
    - Ratio-of-uniforms method

3. Nonlinear equations

Newton-Raphson
Fix point

4. Integration Methods

One-dimensional
    - Rectangle rule.
    - Trapezoidal rule.
    - Simpson's rule.   
    - Gauss-Hermite     (TODO)
    - Gauss-Laguerre    (TODO)
N-dimensional
    - Monte Carlo       (TODO)
    - Sparse grids      (TODO)
    - Bayesian Quadrature (TODO)

5. Ordinary differential equations (ODE)

Monostep
    - Euler explicit
    - Euler implicit (TODO)
    - Runge-Kutta fourth order method (classical)
Multistep
    - Adams-Bashforth
    - Adams-Moulton   (TODO)
    - Nyström         (TODO)
    - Mile-Simpson    (TODO)
    - Backward differentiation formula (TODO)

6. Stochastic Ordinary Differential Equations (SDE)

- Euler-Maruyama method
- Milstein Method (TODO)
- Strong Order 1.0 Runge-Kutta Method 
- Strong Order 1.5 Taylor Method (TODO)
- Weak Order 2 Taylor Method (TODO)
- Weak Order 2 Runge-Kutta Method

Examples

pgplot and dislin libraries are necessary to plot the examples:
pgplot, dislin

Example 1: ordinary differential equation

Example 2: Lotka–Volterra 1

Example 3: Lotka–Volterra 2

Example 4: isotherm transesterification reaction at 50 C (link)

Where:

• [TG] Triglycerides.


• [DG] Diglycerides.


• [MG] Monoglycerides.


• [GL] Glycerin.


• [A] Alcohol.


• [E] Ethylester.

Example 5: Bogdanov-Takens bifurcation

Example 6: pendulum

Example 7: Lorenz system

Example 8: Stochastic Ordinary Differential Equation (SDE)

Force Regulation by Nascent Adhesion Sites (Robijn Bruinsma)

Where:

• G(0,1) standard normal distribution.