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res/approximation_scripts/Ln.m

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+%% Padé approximation of natural logarithms
+% This file contains the code to approximate the natural logarithm ($\ln x$) 
+% using Padé approximation. The |pade| function provides approximation for many 
+% functions.
+%% 
+% *Preparation*: Clear everything and set formatting.
+
+clc
+format loose
+
+syms x
+f = log(x); % f is the natural logarithm of x
+%% 
+% *Approximation*: Try to find the best approximant of the natural logarithm.
+% 
+% _Note_:
+%% 
+% * Drag the slider to adjust the center of the Padé approximant.
+% * 1 seems to produce a clear and simple output.
+
+a =1;
+
+% 1 is the best for now...
+% error: 0.0105
+
+g = pade(f, x, a, 'Order', [5 5])
+e = abs(f - g);
+%% 
+% *Evaluation*: Find the errors of the approximation.
+
+x_values = linspace(0, 1, 1e2);
+err = double(subs(e, x, x_values));
+mean(err, 'omitnan')
+%% 
+% *Visualization*: Graph the errors and see the covergence.
+
+figure
+hold off
+hold on
+fplot(g, [-1 1])
+fplot(f, [-1 1])
+plot(err)
+
+title('Error of Padé approximants of ln(x)')
+legend('g (approximated)', 'f', 'error')
+xlabel('x')
+ylabel('error')

res/approximation_scripts/Trig.m

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+%% Padé approximation of cosine
+% This file contains the code to approximate cosine using Padé approximation. 
+% The |pade| function provides approximation for many functions.
+%% 
+% *Preparation*: Clear everything and set formatting.
+
+clc
+format loose
+format short
+
+syms x
+f = cos(x); % f is the cosine value of x
+%% 
+% *Approximation*: Try to find the best approximant of cosine.
+% 
+% _Note_:
+%% 
+% * Drag the slider to adjust the center of the Padé approximant.
+% * 0 seems to produce a clear and simple output.
+
+a =0;
+
+% 1 is the best for now...
+% error: 0.0105
+
+g = pade(f, x, a, 'Order', [6 5])
+e = abs(f - g);
+%% 
+% *Evaluation*: Find the errors of the approximation.
+
+x_values = linspace(0, 1, 1e2);
+err = double(subs(e, x, x_values));
+mean(err, 'omitnan')
+%% 
+% *Visualization*: Graph the errors and see the covergence.
+
+figure
+hold off
+hold on
+fplot(g, [-1 1])
+fplot(f, [-1 1])
+plot(err)
+
+title('Error of Padé approximants of ln(x)')
+legend('g (approximated)', 'f', 'error')
+xlabel('x')
+ylabel('error')