Programming Numerical Methods in Python

Explore the simple iterations method for solving higher-degree equations, including polynomials and transcendental cases, by rearranging the equation, selecting an initial guess, iterating, and checking convergence.

Demonstrates implementing the simple iterations method in Python with a for loop, using an initial guess, a degree of accuracy with absolute difference, and comparing results to analytical roots.

Code the bisection method in Python to solve 2x^2 -5x +3 = 0. Track sign differences and midpoints with a degree of accuracy to converge on roots 1 and 1.5.

Explore using SciPy's optimize module for root finding in Python, applying Newton, bisect, fsolve, and root to solve polynomial equations from initial guesses and intervals.

Explore linear interpolation between two data points in a time-temperature dataset, estimating values with a straight-line slope and Python examples using lambda and a user-defined function.

Build and code Newton's interpolation in Python by constructing the divided differences table with NumPy and implementing the substitution stage to evaluate the interpolation polynomial.

Apply the least squares method to fit a straight line to data using linear regression, deriving coefficients A and B from summations, x-bar, y-bar, and n.

Code the linear regression method using numpy arrays to compute the slope and intercept without loops. Import numpy, define arrays, and use mean and sum for efficient, vectorized calculations.

Learn to code polynomial curve fitting in python with numpy, build data points, form the system of equations, and compute coefficients for quadratic and cubic fits.

Learn to use SciPy's interpolation functions for 1D and multi-dimensional data, including interp1d and Lagrange, to build linear, quadratic, and cubic interpolants from X and Y data in Python.

Explore scipy curve fitting and linear regression with linregress and curve_fit in Python, using models for quadratic and cubic polynomials to extract coefficients and fit data.

Code the forward finite differences method to estimate first and second derivatives of a polynomial at x=0.1, compare with analytical values, and explore how smaller step sizes improve accuracy.

Explore using scipy.misc.derivative to compute first and second derivatives via central differences, with customizable step size and order, and compare results with plotting in Python.

Apply the trapezoidal rule to approximate the area under a curve, a basic numerical integration method. Divide [A, B] into equal-width trapezoids and sum their areas; more divisions reduce error.

Explore Simpson's one third rule for numerical integration, using weighting factors to improve accuracy with even numbers of strips, contrasting it with the trapezoidal rule and outlining its summation-based formula.

Explore the Simpson's 3/8 rule, using four points and three strips with weights on f(x1) and f(x2); derive the integral formula and coding approach.

Learn to perform double integration using Simpson's one-third and three-eighth rules with an extended form, applying outer and inner loops and combining their factors.

Gauss elimination code to handle zero pivots by skipping operations when pivot is zero and swapping rows, preventing division by zero and NaNs while solving linear systems.

Explore Jacobi's method, a basic iterative algorithm for solving linear systems by rearranging equations, using an initial guess and updating all x_i simultaneously until the whole system converges.

Code Jacobi's method in Python using numpy arrays, shape, full, zeros, and empty; set tolerance and iterations, implement i and j loops, convergence check, and copy to avoid aliasing.

Explore the Gauss-Seidel method as an iterative algorithm, applying new x values immediately within each equation, and compare its convergence and efficiency to Jacobi's method.

Learn to solve linear systems in numpy and scipy via direct solve and inverse-based methods, with Python examples, and compare elimination with Gauss-Seidel and Jacobi iterative schemes requiring diagonal dominance.

Explore Euler's method for solving initial value problems and implement it in Python, highlighting step size effects and comparing numerical results to the analytical solution.

Explore the fourth order Runge-Kutta method to solve y' = 2y + 3 on [2,4] with h=0.1, compare to the exact solution, and visualize results with numpy and matplotlib.