Fractal Generation and Numerical Methods in MATLAB

Mandelbrot Set Iteration with Higher Powers

The Mandelbrot set calculation, specifically for z_n+1 = z_n⁶ + c, involves generating a grid of points in the specified complex plane and determining the associated c value (simply x + iy).

Initialization and Divergence Loops

Initialize matrices for the iterated value (z) and the number of iterations before blowing up (k). Everything will be done at once in a single while loop. This loop iterates the grid and checks for divergence, stopping once N reaches N_max.

Any k locations for which abs(w) > maxZ at this previous iteration are assigned the value of N, which indicates the rate of divergence and is used for color coding. If any k values are equal to 0, set them to the final iteration number. Write a single line of code that equals a binary sequence that bounces back and forth between low and high values.

• #define TRUE 1
• #define FALSE 0

Predator-Prey and Lotka-Volterra Equations

The following describes the background behind the 'atto-fox' problem:

• [tM, PM] = ode45(@(t, P) someFunc(t, P, k, L), tI, Pinit);
• [tM, PM] = ode45(@(t, P) someFunc(t, P, k, L), [0:stepsize:10], Pinit);

Analysis of ODE Solvers

Main Purpose of the Code

These lines of code are meant to numerically solve the Logistic equation: P’(t) = k * P(1 - P/L), where k and L are user inputs.

Equations and Variables

The equations are solved via ode45, but require an external function to define the ODE. Pinit represents the initial condition at tI(1), where tI = [0 10] defines the time boundaries. The stepsize is utilized for RK4 (Runge-Kutta) and Euler's method.

Differences Between Code Lines

The second line of code specifies a fixed step size for the output, whereas the first line allows the solver to determine the optimal time steps within the boundaries.

Interpolation Methods and Cubic Splines

This section compares Lagrange interpolation of data points with spline interpolation of the same data points. Figure 2 shows an example of the cubic spline curve through the same data points. In Figure 1, the jump in the height of data points near the middle has a strong effect on the interpolating polynomial curve near the ends. Cubic splines provide a good balance between simplicity and capturing curvature.

Tridiagonal Systems and Piecewise Polynomials

A tridiagonal linear system is solved for the information needed to describe the coefficients of the various cubic polynomials which make up the interpolating spline. The spline uses the functions ppval, mkpp, and unmkpp. These routines form a small suite of functions for working with piecewise polynomials.

Basic Algorithm for Spline Fitting

1. Fitting a cubic polynomial to a limited or centered group of data points.
2. Setting variable properties at the end points to guarantee smoothness (piecewise continuity).
3. Solving the resulting linear equations, which typically have a efficient matrix form.

Note: MATLAB also offers a separate 'Curve Fitting Toolbox' for advanced applications.