• 1.5 vectorization_numpy.py Vectorization con numpy
• 1.6 plotting.py Gráfica con matplotlib
• 1.8 error.py document modules by adding a docstring

• 2.2 gaussElimin.py Gauss elimination
• 2.2 example2_4.py Gauss elimination example
• 2.3 LUdecomp.py LU decomposition and solution phases
• 2.3 choleski.py Choleski’s decomposition and solution phase
• 2.3 example2_7.py program that solves Ax = B with Doolittle’s decomposition method and computes|A|
• 2.3 example2_8.py Solve the equations Ax = b by Choleski’s decomposition
• 2.4 LUdecomp3.py LU decomposition and solution phases of a tridiagonal matrix
• 2.4 example2_11.py Use the functions LUdecmp3 and LUsolve3 to solve Ax = b
• 2.4 LUdecomp5.py LU decomposes a symmetric, pentadiagonal matrix A of the form A = [f\e\d\e\f] and find the solution (Ax = b) by LUsolve5
• 2.5 swap.py The function swapRows interchanges rows i and j of a matrix or vector v, whereas swapCols interchanges columns i and j of a matrix
• 2.5 gaussPivot.py The function gaussPivot performs Gauss elimination with row pivoting. The elimination and solution phases are identical to gaussElimin
• 2.5 LUpivot.py LU decomposition of matrix [A] using scaled row pivoting. Solves \(\left [ L \right ]\left [ U \right ]\left \{ x \right \}=\left \{ b \right \}\)
• 2.5 example2_13.py function that inverts a matrix using LU decomposition with pivoting and test
• 2.6 example2_14.py function that inverts a tridiagonal matrix
• 2.7 gaussSeidel.py The function gaussSeidel is an implementation of the Gauss-Seidel method with relaxation
• 2.7 conjGrad.py The function conjGrad implements the conjugate gradient algorithm
• 2.7 example2_17.py program to solve n simultaneous equations (tridiagonal) by the Gauss-Seidel method with relaxation
• 2.7 example2_18.py program to solve n simultaneous equations (tridiagonal) with the conjugate gradient method

• 3.2 newtonPoly.py This module contains the two functions required for interpolation by Newton’s method
• 3.2 neville.py Neville's method of polynomial interpolation
• 3.2 example3_4.py Interpolate the data \(f(x)=4.8cos\left ( \frac{\pi x}{20} \right )\) by Newton’s method at x = 0, 0.5, 1.0, . . . , 8.0, and compare with the “exact” values
• 3.2 rational.py Rational function interpolation
• 3.2 example3_6.py Interpolate data and plot the results. Use both the polynomial interpolation and the rational function interpolation
• 3.3 cubicSpline.py Interpolate by cubic spline and evaluate the interpolant at x
• 3.3 example3_9.py program that interpolates, using the module cubicSpline, between given data points with a natural cubic spline
• 3.4 polyFit.py Sets up and solves the normal equations for the coefficients of a polynomial of degree m. It returns the coefficients of the polynomial
• 3.4 plotPoly.py Function for plotting the data points and the fitting polynomial
• 3.4 example3_12.py fits a polynomial of arbitrary degree m to the data points. Use the program to determine m that best fits the data in the least-squares sense.

• 4.2 rootsearch.py Searches for a zero of the user-supplied function f(x) in the interval (a,b) in increments of \(\Delta\)x
• 4.2 example4_1.py compute the root of a polynomial function with four-digit accuracy
• 4.3 bisection.py Method of bisection to compute the root of f(x) = 0 that is known to lie in the interval (x1,x2)
• 4.3 example4_2.py use bisection to find the root of a polynomial function with four-digit accuracy
• 4.3 example4_3.py Find all the zeros of f(x) = x − tan x in the interval (0, 20) by the method of bisection
• 4.4 ridder.py Ridder’s method (modification of the false position method. Use if the derivative of f(x) is impossible or difficult to compute)
• 4.4 example4_5.py Compute the zero of the function \(f(x)=\frac{1}{(x-0.3)^{2}+0.01}- \frac{1}{(x-0.8)^{2}+0.04}\)
• 4.5 newtonRaphson.py Safe version of the Newton-Raphson method, assumes that the root to be computed is initially bracketed in (a,b)
• 4.5 example4_8.py Find the smallest positive zero of \(f(x)=x^{4}-6.4x^{3}+6.45x^{2}+20.538x-31.752\)
• 4.6 newtonRaphson2.py Newton-Raphson method for systems of equations, computes the Jacobian matrix using finite difference approximation and the simultaneous equations are solved by Gauss elimination using the function gaussPivot
• 4.6 example4_10.py Find a solution of a system of nonlinear equations
• 4.7 evalPoly.py Evaluates a polynomial and its derivatives
• 4.7 polyRoots.py Computes all the roots of \(P_{n}=0\) defined by its coefficient array \(a = \left [a_{0}, a_{1},...a_{n}\right ]\) using Laguerre's method
• 4.7 example4_12.py compute all the roots of \(x^{4}-5x^{3}-9x^{2}+155x-250=0\)

• 5.2 Finite Difference Approximations
• 5.4 Derivatives by Interpolation
• 5.4 example5_4.py given a set of data compute f'(2) and f''(2) using a natural cubic spline interpolant spanning all the data points

• 6.2 trapezoid.py Computes the integral evaluated with the composite trapezoidal rule
• 6.2 example6_4.py Use the recursive trapezoidal rule to evaluate \(\int_{0}^{\pi }\sqrt{x}\:\textup{cos}\,x\:dx\) to six decimal places
• 6.3 romberg.py Romberg integration
• 6.3 example6_7.py Use Romberg integration to evaluate \(\int_{0}^{\sqrt{\pi }}2x^{2}\,\textup{cos}\,x^{2}\:dx \)
• 6.4 gaussNodes.py Computes the nodal abscissas x_i and the corresponding weights A_i used in Gauss-Legendre quadrature over the "standard" interval
• 6.4 gaussQuad.py Uses gaussNodes to evaluate \(\int_{a}^{b}f(x)\:dx \) with Gauss- Legendre quadrature using m nodes
• 6.4 example6_11.py Determine how many nodes are required to evaluate \(\int_{a}^{\pi }\left ( \frac{\textup{sin}\,x}{x} \right )^{2}\:dx\) with Gauss-Legendre quadrature to six decimal places
• 6.5 gaussQuad2.py Computes \(\iint_{A}^{}f(x,y)\,dx\, dy \) over a quadrilateral element with Gauss-Legendre quadrature of integration order m
• 6.5 example6_15.py Use gaussQuad2 to evaluate \(\iint_{A}^{}f(x,y)\,dx\, dy \) over one quadrilateral where \(f(x)= (x-2)^{2}(y-2)^{2} \)
• 6.5 triangleQuad.py Computes \(\iint_{A}^{}f(x,y)\,dx\, dy \) over a triangular region using the cubic formula
• 6.5 example6_16.py evaluate \(\iint_{A}^{}f(x,y)\,dx\, dy \) over a equilateral triangle for a given function

Chapter 7. Initial value problems

Chapter 8. Two-point boundary value problems

Chapter 9. Symmetric matrix eigenvalue problems

Chapter 10. Introduction to optimization

From: Numerical methods in engineering with Python 3, Jaan Kiusalaas