Code refactor

cfgnunes · Mar 2, 2022 · 7166421 · 7166421
1 parent efe6bec
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diff --git a/README.md b/README.md
@@ -2,7 +2,7 @@
 
 Numerical methods implementation in MATLAB.
 
-For the "Python version", see [this repository](https://github.com/cfgnunes/numerical-methods-python).
+For the implementation in Python, see [this repository](https://github.com/cfgnunes/numerical-methods-python).
 
 ## Getting Started
 

diff --git a/differentiation/derivative_backward_difference.m b/differentiation/derivative_backward_difference.m
@@ -1,11 +1,14 @@
 function [dy] = derivative_backward_difference(x, y)
-% Calculate the first derivative
-% All values in 'x' must be equally spaced
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-% Outputs:
-%        dy: Array containing the first derivative values
+    % Calculate the first derivative.
+    %
+    % All values in 'x' must be equally spaced.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %
+    % Returns:
+    %     dy: array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);
@@ -22,13 +25,17 @@
 
     n = x_size;
     dy = zeros(1, n);
+
     for i = 1:n
+
         if i == n
             h = x(i) - x(i - 1);
             dy(i) = dy_difference(- h, y(i), y(i - 1));
         else
             h = x(i + 1) - x(i);
             dy(i) = dy_difference(h, y(i), y(i + 1));
         end
+
     end
+
 end
diff --git a/differentiation/derivative_five_point.m b/differentiation/derivative_five_point.m
@@ -1,11 +1,14 @@
 function [dy] = derivative_five_point(x, y)
-% Calculate the first derivative
-% All values in 'x' must be equally spaced
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-% Outputs:
-%        dy: Array containing the first derivative values
+    % Calculate the first derivative.
+    %
+    % All values in 'x' must be equally spaced.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %
+    % Returns:
+    %     dy: array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);
@@ -24,13 +27,17 @@
     h = x(2) - x(1);
     n = x_size;
     dy = zeros(1, n);
+
     for i = 1:n
+
         if i == 1 || i == 2
             dy(i) = dy_end(h, y(i), y(i + 1), y(i + 2), y(i + 3), y(i + 4));
         elseif i == n || i == n - 1
             dy(i) = dy_end(- h, y(i), y(i - 1), y(i - 2), y(i - 3), y(i - 4));
         else
             dy(i) = dy_mid(h, y(i - 2), y(i - 1), y(i + 1), y(i + 2));
         end
+
     end
+
 end
diff --git a/differentiation/derivative_three_point.m b/differentiation/derivative_three_point.m
@@ -1,11 +1,14 @@
 function [dy] = derivative_three_point(x, y)
-% Calculate the first derivative
-% All values in 'x' must be equally spaced
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-% Outputs:
-%        dy: Array containing the first derivative values
+    % Calculate the first derivative.
+    %
+    % All values in 'x' must be equally spaced.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %
+    % Returns:
+    %     dy: array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);
@@ -24,13 +27,17 @@
     h = x(2) - x(1);
     n = x_size;
     dy = zeros(1, n);
+
     for i = 1:n
+
         if i == 1
             dy(i) = dy_end(h, y(i), y(i + 1), y(i + 2));
         elseif i == n
             dy(i) = dy_end(- h, y(i), y(i - 1), y(i - 2));
         else
             dy(i) = dy_mid(h, y(i - 1), y(i + 1));
         end
+
     end
+
 end
diff --git a/integration/composite2_simpson.m b/integration/composite2_simpson.m
@@ -1,10 +1,12 @@
 function [xi] = composite2_simpson(x, y)
-% Calculate the integral from 1/3 Simpson's Rule
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-% Outputs:
-%        xi: Integral value
+    % Calculate the integral from 1/3 Simpson's Rule.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %
+    % Returns:
+    %     xi: integral value.
 
     x_size = size(x, 2);
     y_size = size(y, 2);
@@ -20,11 +22,13 @@
     sum_even = 0;
 
     for i = 2:(n - 1)
+
         if mod(i, 2) == 0
             sum_even = sum_even + y(i);
         else
             sum_odd = sum_odd + y(i);
         end
+
     end
 
     xi = h / 3 * (y(1) + 2 * sum_even + 4 * sum_odd + y(n));

diff --git a/integration/composite2_trapezoidal.m b/integration/composite2_trapezoidal.m
@@ -1,10 +1,12 @@
 function [xi] = composite2_trapezoidal(x, y)
-% Calculate the integral from Trapezoidal Rule
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-% Outputs:
-%        xi: Integral value
+    % Calculate the integral from Trapezoidal Rule.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %
+    % Returns:
+    %     xi: integral value.
 
     x_size = size(x, 2);
     y_size = size(y, 2);

diff --git a/integration/composite_simpson.m b/integration/composite_simpson.m
@@ -1,12 +1,14 @@
 function [xi] = composite_simpson(f, b, a, n)
-% Calculate the integral from 1/3 Simpson's Rule
-% Inputs:
-%         f: Function f(x)
-%         a: Initial point
-%         b: End point
-%         n: Number of intervals
-% Outputs:
-%        xi: Integral value
+    % Calculate the integral from 1/3 Simpson's Rule.
+    %
+    % Args:
+    %     f: function f(x).
+    %     a: initial point.
+    %     b: end point.
+    %     n: number of intervals.
+    %
+    % Returns:
+    %     xi: integral value.
 
     h = (b - a) / n;
 
@@ -15,11 +17,13 @@
 
     for i = 1:(n - 1)
         x = a + i * h;
+
         if mod(i, 2) == 0
             sum_even = sum_even + f(x);
         else
             sum_odd = sum_odd + f(x);
         end
+
     end
 
     xi = h / 3 * (f(a) + 2 * sum_even + 4 * sum_odd + f(b));

diff --git a/integration/composite_trapezoidal.m b/integration/composite_trapezoidal.m
@@ -1,12 +1,14 @@
 function [xi] = composite_trapezoidal(f, b, a, n)
-% Calculate the integral from Trapezoidal Rule
-% Inputs:
-%         f: Function f(x)
-%         a: Initial point
-%         b: End point
-%         n: Number of intervals
-% Outputs:
-%        xi: Integral value
+    % Calculate the integral from Trapezoidal Rule.
+    %
+    % Args:
+    %     f: function f(x).
+    %     a: initial point.
+    %     b: end point.
+    %     n: number of intervals.
+    %
+    % Returns:
+    %     xi: integral value.
 
     h = (b - a) / n;
 

diff --git a/interpolation/lagrange.m b/interpolation/lagrange.m
@@ -1,22 +1,30 @@
 function [y_int] = lagrange(x, y, x_int)
-% Interpolates a value using Lagrange polynomial
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-%     x_int: Value to interpolate
-% Outputs:
-%     y_int: Interpolated value
+    % Interpolates a value using Lagrange polynomial.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %     x_int: value to interpolate.
+    %
+    % Returns:
+    %     y_int: interpolated value.
 
     n = size(x, 2);
 
     y_int = 0;
+
     for i = 1:n
         p = y(i);
+
         for j = 1:n
+
             if i ~= j
                 p = p * ((x_int - x(j)) / (x(i) - x(j)));
             end
+
         end
+
         y_int = y_int + p;
     end
+
 end
diff --git a/interpolation/neville.m b/interpolation/neville.m
@@ -1,22 +1,26 @@
 function [y_int, q] = neville(x, y, x_int)
-% Interpolates a value using Neville polynomial
-% Inputs:
-%         x: Array containing x values
-%         y: Array containing y values
-%     x_int: Value to interpolate
-% Outputs:
-%     y_int: Interpolated value
-%         q: Coefficients matrix
+    % Interpolates a value using Neville polynomial.
+    %
+    % Args:
+    %     x: array containing x values.
+    %     y: array containing y values.
+    %     x_int: value to interpolate.
+    %
+    % Returns:
+    %     y_int: interpolated value.
+    %     q: coefficients matrix.
 
     n = size(x, 2);
     q = zeros(n, n - 1);
-    q = [y' q];  % Insert 'y' in the first column of the matrix 'q'
+    q = [y' q]; % Insert 'y' in the first column of the matrix 'q'
 
     for i = 2:n
+
         for j = 2:i
             q(i, j) = (x_int - x(i - j + 1)) * q(i, j - 1) - (x_int - x(i)) * q(i - 1, j - 1);
             q(i, j) = q(i, j) / (x(i) - x(i - j + 1));
         end
+
     end
 
     y_int = q(n, n);

diff --git a/linear_systems/backward_substitution.m b/linear_systems/backward_substitution.m
@@ -1,10 +1,12 @@
 function [x] = backward_substitution(u, d)
-% Solve the upper linear system ux=d
-% Inputs:
-%         u: Upper triangular matrix
-%         d: Array containing d values
-% Outputs:
-%         x: Solution of linear system
+    % Solve the upper linear system ux=d.
+    %
+    % Args:
+    %     upper: upper triangular matrix.
+    %     d: array containing d values.
+    %
+    % Returns:
+    %     x: solution of linear system.
 
     [n, m] = size(u);
 
@@ -15,11 +17,13 @@
     x = zeros(1, n);
 
     for i = n:-1:1
+
         if (u(i, i) == 0)
             error('Error: Matrix "u" is singular.')
         end
 
         x(i) = d(i) / u(i, i);
         d(1:i - 1) = d(1:i - 1) - u(1:i - 1, i) * x(i);
     end
+
 end
diff --git a/linear_systems/forward_substitution.m b/linear_systems/forward_substitution.m
@@ -1,10 +1,12 @@
 function [x] = forward_substitution(l, c)
-% Solve the lower linear system lx=c
-% Inputs:
-%         l: Lower triangular matrix
-%         c: Array containing c values
-% Outputs:
-%         x: Solution of linear system
+    % Solve the lower linear system lx=c.
+    %
+    % Args:
+    %     lower: lower triangular matrix.
+    %     c: array containing c values.
+    %
+    % Returns:
+    %     x: solution of linear system.
 
     [n, m] = size(l);
 
@@ -15,11 +17,13 @@
     x = zeros(1, n);
 
     for i = 1:n
+
         if (l(i, i) == 0)
             error('Error: Matrix "l" is singular.')
         end
 
         x(i) = c(i) / l(i, i);
         c(i + 1:n) = c(i + 1:n) - l(i + 1:n, i) * x(i);
     end
+
 end