Better comments

differentiation/derivative_backward_difference.m

@@ -4,11 +4,11 @@
     % All values in 'x' must be equally spaced.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
-    %     dy: array containing the first derivative values.
+    %     dy: an array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);

differentiation/derivative_five_point.m

@@ -4,11 +4,11 @@
     % All values in 'x' must be equally spaced.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
-    %     dy: array containing the first derivative values.
+    %     dy: an array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);

differentiation/derivative_three_point.m

@@ -4,11 +4,11 @@
     % All values in 'x' must be equally spaced.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
-    %     dy: array containing the first derivative values.
+    %     dy: an array containing the first derivative values.
 
     x_size = size(x, 2);
     y_size = size(y, 2);

integration/composite2_simpson.m

@@ -2,8 +2,8 @@
     % Calculate the integral from 1/3 Simpson's Rule.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
     %     xi: integral value.

integration/composite2_trapezoidal.m

@@ -1,9 +1,9 @@
 function [xi] = composite2_trapezoidal(x, y)
-    % Calculate the integral from Trapezoidal Rule.
+    % Calculate the integral from the Trapezoidal Rule.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
     %     xi: integral value.

integration/composite_simpson.m

@@ -3,8 +3,8 @@
     %
     % Args:
     %     f: function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %
     % Returns:

integration/composite_trapezoidal.m

@@ -1,10 +1,10 @@
 function [xi] = composite_trapezoidal(f, b, a, n)
-    % Calculate the integral from Trapezoidal Rule.
+    % Calculate the integral from the Trapezoidal Rule.
     %
     % Args:
     %     f: function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %
     % Returns:

interpolation/lagrange.m

@@ -1,9 +1,9 @@
 function [y_int] = lagrange(x, y, x_int)
-    % Interpolates a value using Lagrange polynomial.
+    % Interpolates a value using the 'Lagrange polynomial'.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %     x_int: value to interpolate.
     %
     % Returns:

interpolation/neville.m

@@ -1,9 +1,9 @@
 function [y_int, q] = neville(x, y, x_int)
-    % Interpolates a value using Neville polynomial.
+    % Interpolates a value using the 'Neville polynomial'.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %     x_int: value to interpolate.
     %
     % Returns:

linear_systems/backward_substitution.m

@@ -3,10 +3,10 @@
     %
     % Args:
     %     upper: upper triangular matrix.
-    %     d: array containing d values.
+    %     d: an array containing d values.
     %
     % Returns:
-    %     x: solution of linear system.
+    %     x: solution of linear the system.
 
     [n, m] = size(u);
 

linear_systems/forward_substitution.m

@@ -3,10 +3,10 @@
     %
     % Args:
     %     lower: lower triangular matrix.
-    %     c: array containing c values.
+    %     c: an array containing c values.
     %
     % Returns:
-    %     x: solution of linear system.
+    %     x: solution of linear the system.
 
     [n, m] = size(l);
 

linear_systems/gauss_elimination_pp.m

@@ -1,12 +1,12 @@
 function [a] = gauss_elimination_pp(a, b)
     % Gaussian Elimination with Partial Pivoting.
     %
-    % Calculate the upper triangular matrix from linear system Ax=b (do a row
+    % Calculate the upper triangular matrix from linear system Ax=b (make a row
     % reduction).
     %
     % Args:
     %     a: matrix A from system Ax=b.
-    %     b: array containing b values.
+    %     b: an array containing b values.
     %
     % Returns:
     %     a: augmented upper triangular matrix.
@@ -33,7 +33,7 @@
 
         end
 
-        % Cheking for nullity of the pivots
+        % Checking for nullity of the pivots
         while p <= n && a(p, i) == 0
             p = p + 1;
         end

linear_systems_iterative/gauss_seidel.m

@@ -3,13 +3,13 @@
     %
     % Args:
     %     a: matrix A from system Ax=b.
-    %     b: array containing b values.
-    %     x0: initial approximation of solution.
+    %     b: an array containing b values.
+    %     x0: initial approximation of the solution.
     %     tol: tolerance.
     %     iter_max: maximum number of iterations.
     %
     % Returns:
-    %     x: solution of linear system.
+    %     x: solution of linear the system.
     %     iter: used iterations.
 
     % L and U matrices

linear_systems_iterative/jacobi.m

@@ -3,13 +3,13 @@
     %
     % Args:
     %     a: matrix A from system Ax=b.
-    %     b: array containing b values.
-    %     x0: initial approximation of solution.
+    %     b: an array containing b values.
+    %     x0: initial approximation of the solution.
     %     tol: tolerance.
     %     iter_max: maximum number of iterations.
     %
     % Returns:
-    %     x: solution of linear system.
+    %     x: solution of linear the system.
     %     iter: used iterations.
 
     % D and M matrices

main.m

@@ -19,7 +19,7 @@
 % Bisection method (find roots of an equation)
 %   Pros:
 %       It is a reliable method with guaranteed convergence;
-%       It is a simple method that searches for the root employing a
+%       It is a simple method that searches for the root by employing a
 %           binary search;
 %       There is no need to calculate the derivative of the function.
 %   Cons:
@@ -58,7 +58,7 @@
 %   Cons:
 %       It may diverge if the function is not approximately linear in the
 %           range containing the root;
-%       It is necessary to give two points 'a' and 'b' where
+%       It is necessary to give two points, 'a' and 'b' where
 %           f(a)-f(b) must be nonzero.
 disp('> Run an example "Solutions: Secant method".')
 f = @(x) (4 * x^3 + x + cos(x) - 10);

ode/euler.m

@@ -1,18 +1,18 @@
 function [vx, vy] = euler(f, a, b, n, ya)
     % Calculate the solution of the initial-value problem (IVP).
     %
-    % Solve the IVP from Euler method.
+    % Solve the IVP from the Euler method.
     %
     % Args:
     %     f: function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %     ya: initial value.
     %
     % Returns:
-    %     vx: array containing x values.
-    %     vy: array containing y values (solution of IVP).
+    %     vx: an array containing x values.
+    %     vy: an array containing y values (solution of IVP).
 
     vx = zeros(1, n + 1);
     vy = zeros(1, n + 1);

ode/rk4.m

@@ -1,18 +1,18 @@
 function [vx, vy] = rk4(f, a, b, n, ya)
     % Calculate the solution of the initial-value problem (IVP).
     %
-    % Solve the IVP from Runge-Kutta (Order Four) method.
+    % Solve the IVP from the Runge-Kutta (Order Four) method.
     %
     % Args:
     %     f: function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %     ya: initial value.
     %
     % Returns:
-    %     vx: array containing x values.
-    %     vy: array containing y values (solution of IVP).
+    %     vx: an array containing x values.
+    %     vy: an array containing y values (solution of IVP).
 
     vx = zeros(1, n + 1);
     vy = zeros(1, n + 1);

ode/rk4_system.m

@@ -4,15 +4,15 @@
     % Solve from Runge-Kutta (Order Four) method.
     %
     % Args:
-    %     f: array of functions f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     f: an array of functions f(x).
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
-    %     ya: array of initial values.
+    %     ya: an array of initial values.
     %
     % Returns:
-    %     vx: array containing x values.
-    %     vy: array containing y values (solution of IVP).
+    %     vx: an array containing x values.
+    %     vy: an array containing y values (solution of IVP).
 
     m = size(f, 1);
 

ode/taylor2.m

@@ -1,19 +1,19 @@
 function [vx, vy] = taylor2(f, df1, a, b, n, ya)
     % Calculate the solution of the initial-value problem (IVP).
     %
-    % Solve the IVP from Taylor (Order Two) method.
+    % Solve the IVP from the Taylor (Order Two) method.
     %
     % Args:
     %     f: function f(x).
     %     df1: 1's derivative of function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %     ya: initial value.
     %
     % Returns:
-    %     vx: array containing x values.
-    %     vy: array containing y values (solution of IVP).
+    %     vx: an array containing x values.
+    %     vy: an array containing y values (solution of IVP).
 
     vx = zeros(1, n + 1);
     vy = zeros(1, n + 1);

ode/taylor4.m

@@ -1,21 +1,21 @@
 function [vx, vy] = taylor4(f, df1, df2, df3, a, b, n, ya)
     % Calculate the solution of the initial-value problem (IVP).
     %
-    % Solve the IVP from Taylor (Order Four) method.
+    % Solve the IVP from the Taylor (Order Four) method.
     %
     % Args:
     %     f: function f(x).
     %     df1: 1's derivative of function f(x).
     %     df2: 2's derivative of function f(x).
     %     df3: 3's derivative of function f(x).
-    %     a: initial point.
-    %     b: end point.
+    %     a: the initial point.
+    %     b: the final point.
     %     n: number of intervals.
     %     ya: initial value.
     %
     % Returns:
-    %     vx: array containing x values.
-    %     vy: array containing y values (solution of IVP).
+    %     vx: an array containing x values.
+    %     vy: an array containing y values (solution of IVP).
 
     vx = zeros(1, n + 1);
     vy = zeros(1, n + 1);

polynomials/briot_ruffini.m

@@ -1,14 +1,14 @@
 function [b, rest] = briot_ruffini(root, a)
-    % Divides a polynomial by another polynomial.
+    % Divide a polynomial by another polynomial.
     %
     % The format is: P(x) = Q(x) * (x-root) + rest.
     %
     % Args:
-    %     a: array containing the coefficients of the input polynomial.
+    %     a: an array containing the coefficients of the input polynomial.
     %     root: one of the polynomial roots.
     %
     % Returns:
-    %     b: array containing the coefficients of the output polynomial.
+    %     b: an array containing the coefficients of the output polynomial.
     %     rest: polynomial division Rest.
 
     n = size(a, 2) - 1;

polynomials/newton_divided_difference.m

@@ -1,14 +1,14 @@
 function [f] = newton_divided_difference(x, y)
     % Find the coefficients of Newton's divided difference.
     %
-    % Also findthe Newton's polynomial.
+    % Also, find Newton's polynomial.
     %
     % Args:
-    %     x: array containing x values.
-    %     y: array containing y values.
+    %     x: an array containing x values.
+    %     y: an array containing y values.
     %
     % Returns:
-    %     f: array containing Newton's divided difference coefficients.
+    %     f: an array containing Newton's divided difference coefficients.
 
     n = size(x, 2);
     q = zeros(n, n - 1);

solutions/bisection.m

@@ -1,5 +1,5 @@
 function [root, iter, converged] = bisection(f, a, b, tol, iter_max)
-    % Calculate the root of an equation by Bisection method.
+    % Calculate the root of an equation by the Bisection method.
     %
     % Args:
     %     f: function f(x).

solutions/newton.m

@@ -1,5 +1,5 @@
 function [root, iter, converged] = newton(f, df, x0, tol, iter_max)
-    % Calculate the root of an equation by Newton method.
+    % Calculate the root of an equation by the Newton method.
     %
     % Args:
     %     f: function f(x).