Contents
- Explaining the parts of the lppl function
- Hyperbolic part
- Cosinus part
- Both parts together
- variation of first parameter
- variation of second parameter
- variation of third parameter
- variation of forth parameter
- variation of fifth parameter
- variation of sixth parameter
- variation of seventh parameter
- variation of eighth parameter
Explaining the parts of the lppl function
The lppl function consists of two parts: an hyperbolic function that increases to infinity at a singularity, and an oscillation of increasing frequency.
% set up realistic parameters params = [50.9057703950095; -39.2228401917215; 1348.19815338505; 0.0142416695966602; -0.00177914571357913; -6.22638522525240; 0.00974118509923082; 0.915241664436276]; % plot hyperbolic part grid = (1:0.2:(params(3)/params(8)-1)); % init anonymous function to get function values evalLppl = @(x, params) ... params(1) + params(2)*(params(3)-params(8)*x).^params(4).*... (1+params(5)*cos(params(6).*... log(params(3)-params(8)*x)+params(7))); vals = evalLppl(grid, params); plot(grid, vals)

Hyperbolic part
If we set the fifth parameter equal to zero, the oscillating part of the function gets removed.
paramsHyperb = params; paramsHyperb(5) = 0; valsHyperb = evalLppl(grid, paramsHyperb); plot(grid, valsHyperb)

Cosinus part
If we set parameters one and four to zero, and two and five to one, we get oscillating part up to rescaling
paramsOsc = params; paramsOsc([1, 4]) = 0; paramsOsc(2) = 1; paramsOsc(5) = 1; valsOs = evalLppl(grid, paramsOsc); plot(grid, valsOs);

Both parts together
plot(grid, valsOs + valsHyperb)

variation of first parameter
variations = params(1)*[0.8 0.9 1.1 1.2]; plotVaryLpplParameters(params, 1, variations); % The first parameter shifts the function additively up and down. % Higher values increase the level of the function.

variation of second parameter
clf variations = params(2)*[0.2 0.9 1.1 1.9]; plotVaryLpplParameters(params, 2, variations); % The second parameter linearly scales the function up and down. % Higher values increase the level of the function.

variation of third parameter
clf variations = params(3)*[0.8 0.9 1.1 1.2]; plotVaryLpplParameters(params, 3, variations); % The third parameter additively shifts the function left and right. % Higher values shift the function to the right.

variation of forth parameter
clf variations = params(4)*[0.2 0.9 1.1 1.5]; plotVaryLpplParameters(params, 4, variations); % The forth parameter determines the shape of the function. Hence, it % determines how long the period of curvature persists, and hence how % long the period of super-exponential persists.

variation of fifth parameter
clf variations = params(5)*[0.01 0.5 5 20]; plotVaryLpplParameters(params, 5, variations); % The fifth parameter determines the amplitude of the cosinus. Large % values increase the amplitude.

variation of sixth parameter
clf variations = params(6)*[0.01 0.5 5 20]; plotVaryLpplParameters(params, 6, variations); % The sixth parameter determines the frequency of the cosinus. Large % values increase the frequency.

variation of seventh parameter
clf variations = params(7)*[-1000 -200 200 1000]; plotVaryLpplParameters(params, 7, variations); % The seventh parameter shifts the cosinus additively left and right. % Positive values shift to the right.

variation of eighth parameter
clf variations = params(8)*[0.8 0.9 1.1 1.2]; plotVaryLpplParameters(params, 8, variations);
