
Book

Author

Contents

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Graphics

Programs
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The book is devoted to the general field of numerical programming, with emphasis on methods specific to computational physics and engineering.
The book is addressed to advanced undergraduate and graduate students in natural sciences and engineering, with the aim of being suited as curriculum material for a one or twosemester course in numerical programming based on Python or C/C++. The book may also be used for independent study or as a reference material beyond academic studies. It may be useful, for instance, as an introductory text for researchers preparing to engage in scientific computing, or engineers needing effective numerical tools for applicative calculations. All the materials on this web site are released as 'open source', with the only request that the authorship and the book are mentioned. Please also visit the book's web site at CRC Press / Taylor & Francis Group. Feedback to the author is highly appreciated! 
I am Titus Adrian Beu, professor emeritus at the Faculty of Physics of the "BabeșBolyai" University from ClujNapoca, Romania.
I have been active in the broader field of computational physics for more than 40 years, both in research and academia. My research has been theoretical and computational in nature, covering a multitude of topics. These have evolved from Tokamak plasma and nuclear reactors in the 1980s, collision theory and molecular cluster spectroscopy in the 1990s, fullerenes and nanofluidics in the 2000s, to complex biomolecular systems of interest in drug delivery, in recent years. The development of complex computer codes has always been at the heart of all the research projects I have conducted. In parallel, I gave university lectures on general programming techniques and advanced numerical methods, computer simulation methods, and advanced molecular dynamics. The material covered in the book comes together from my research experience and the computational physics courses taught at the university. The core material has undergone a continuous process of distillation and reformulation over the years, following the evolution of modern programming concepts and technologies. I am happy to receive opinions, questions, requests,... Have fun reading the book and practicing numerical programming! titus,[email protected] 
1 Approximate Numbers
1.1 Sources of errors in numerical calculations 1.2 Absolute and relative errors 1.3 Representation of numbers 1.4 Significant digits 1.5 Errors of elementary operations 1.5.1 Error of a sum 1.5.2 Error of a difference 1.5.3 Error of a product 1.5.4 Error of a quotient References 2 Basic Programming Techniques 2.1 Programming concepts 2.2 Functions and parameters 2.3 Passing arguments to Python functions 2.4 Passing arguments to C/C++ functions 2.5 Arrays in Python 2.6 Dynamic array allocation in C/C++ 2.7 Basic matrix operations References 3 Elements of Scientific Graphics 3.1 The Tkinter package 3.2 The Canvas widget 3.2.1 Commonly used Canvas items 3.2.2 Methods for updating Canvas widgets 3.3 Simple Tkinter applications 3.4 Plotting functions of one variable 3.5 The graphics library graphlib.py 3.5.1 Auxiliary functions 3.5.2 Routines for plotting functions 3.5.3 Simple 3D rendering routines 3.5.4 Histograms 3.6 Creating plots in C++ using the library graphlib.py References 4 Sorting and Indexing 4.1 Introduction 4.2 Bubble sort 4.3 Insertion sort 4.4 Quicksort 4.5 Indexing and ranking 4.6 Implementations in C/C++ 4.7 Problems References 5 Evaluation of Functions 5.1 Evaluation of polynomials by Horner's scheme 5.2 Evaluation of analytic functions 5.3 Continued fractions 5.4 Orthogonal polynomials 5.5 Spherical harmonics. Associated Legendre functions 5.6 Spherical Bessel functions 5.7 Implementations in C/C++ 5.8 Problems References 6 Algebraic and Transcendental Equations 6.1 Root separation 6.2 Bisection method 6.3 Method of false position 6.4 Method of successive approximations 6.5 Newton's method 6.6 Secant method 6.7 BirgeVieta method 6.8 Newton's method for systems of nonlinear equations 6.9 Implementations in C/C++ 6.10 Problems References 7 Systems of Linear Equations 7.1 Introduction 7.2 Gaussian elimination with backward substitution 7.3 GaussJordan elimination 7.4 LU factorization 7.5 Inversion of triangular matrices 7.6 Cholesky factorization 7.7 Tridiagonal systems of linear equations 7.8 Block tridiagonal systems of linear equations 7.9 Complex matrix equations 7.10 Jacobi and GaussSeidel iterative methods 7.11 Implementations in C/C++ 7.12 Problems References 8 Eigenvalue problems 8.1 Introduction 8.2 Diagonalization of matrices by similarity transformations 8.3 Jacobi method 8.4 Generalized eigenvalue problems for symmetric matrices 8.5 Implementations in C/C++ 8.6 Problems References 
9 Modeling of Tabulated Functions
9.1 Interpolation and regression 9.2 Lagrange interpolation polynomial 9.3 Neville's interpolation method 9.4 Cubic spline interpolation 9.5 Linear regression 9.6 Multilinear regression models 9.7 Nonlinear regression. The LevenbergMarquardt method 9.8 Implementations in C/C++ 9.9 Problems References 10 Integration of Functions 10.1 Introduction 10.2 Trapezoidal Rule; A Heuristic Approach 10.3 The NewtonCotes Quadrature Formulas 10.4 Trapezoidal Rule 10.5 Simpson's Rule 10.6 Adaptive Quadrature Methods 10.7 Romberg's Method 10.8 Improper Integrals: Open Formulas 10.9 Midpoint Rule 10.10 Gaussian Quadratures 10.11 Multidimensional integration 10.12 Adaptive multidimensional integration 10.13 Implementations in C/C++ 10.14 Problems References 11 Monte Carlo Method 11.1 Introduction 11.2 Integration of functions 11.3 Importance sampling 11.4 Multidimensional integrals 11.5 Generation of random numbers 11.6 Implementations in C/C++ 11.7 Problems References 12 Ordinary Differential Equations 12.1 Introduction 12.2 Taylor series method 12.3 Euler's method 12.4 RungeKutta methods 12.5 Adaptive step size control 12.6 Methods for secondorder ODEs 12.7 Numerov's method 12.8 Shooting methods for twopoint problems 12.9 Finitedifference methods for linear twopoint problems 12.10 Implementations in C/C++ 12.11 Problems References 13 Partial Differential Equations 13.1 Introduction 13.2 Boundaryvalue problems for elliptic differential equations 13.3 Initialvalue problems for parabolic differential equations 13.3.1 Explicit finitedifference method 13.3.2 von Neumann stability analysis 13.3.3 Implicit finitedifference method 13.3.4 CrankNicolson method 13.3.5 Spatially variable diffusion coefficient 13.4 Timedependent Schrödinger equation 13.5 Initialvalue problems for hyperbolic differential equations 13.6 Implementations in C/C++ 13.7 Problems References A Dynamic array allocation in C/C++ B Basic operations with vectors and matrices C Embedding Python in C/C++ References D The numerical libraries numxlib.py and numxlib.h E The graphics library graphlib.py based on Tkinter F The C++ interface to the graphics library graphlib.py G List of programs by chapter Index 
The numerical libraries numxlib.py and numxlib.h
The partial numerical libraries developed along the chapters compose the general libraries numxlib.py and numxlib.h. The corresponding Python and C/C++ routines have identical names and functionalities. The only naming exception concerns the modules random1.py and random.h, and this is due to the existence of a standard Python module named random.py.
numxlib.h also includes memalloc.h, containing dynamic memory allocation routines for vectors and matrices in C/C+.
numxlib.h also includes memalloc.h, containing dynamic memory allocation routines for vectors and matrices in C/C+.


numxlib.py Library of numerical methods for Python.
numxlib.h Library of numerical methods for C.
Component modules
sort.py / sort.h
Contain routines for sorting, indexing, and ranking numeric sequences:
BubbleSort  Sorts a numeric sequence using bubble sort.
InsertSort  Sorts a numeric sequence by direct insertion.
QuickSort  Sorts a numeric sequence using Quicksort.
Index  Indexes a numeric sequence by direct insertion.
Rank  Ranks a numeric sequence based on the array of indexes.
Index2  Performs the indexing of two correlated arrays by direct insertion.
Contain routines for sorting, indexing, and ranking numeric sequences:
BubbleSort  Sorts a numeric sequence using bubble sort.
InsertSort  Sorts a numeric sequence by direct insertion.
QuickSort  Sorts a numeric sequence using Quicksort.
Index  Indexes a numeric sequence by direct insertion.
Rank  Ranks a numeric sequence based on the array of indexes.
Index2  Performs the indexing of two correlated arrays by direct insertion.
specfunc.py / specfunc.h
Contain routines for evaluating special functions:
Chebyshev  Evaluates Chebyshev polynomials of the first kind and their derivatives using the recurrence relation.
Legendre  Evaluates Legendre polynomials and their derivatives using the recurrence relation.
aLegendre  Evaluates associated Legendre functions.
Laguerre  Evaluates Laguerre polynomials and their derivatives using the recurrence relation.
aLaguerre  Evaluates associated Laguerre polynomials.
Hermite  Evaluates Hermite polynomials and their derivatives using the recurrence relation.
SpherY  Evaluates the real and imaginary parts of spherical harmonics.
SBessy  Evaluates spherical Neumann functions.
SBessj  Evaluates spherical Bessel functions.
Contain routines for evaluating special functions:
Chebyshev  Evaluates Chebyshev polynomials of the first kind and their derivatives using the recurrence relation.
Legendre  Evaluates Legendre polynomials and their derivatives using the recurrence relation.
aLegendre  Evaluates associated Legendre functions.
Laguerre  Evaluates Laguerre polynomials and their derivatives using the recurrence relation.
aLaguerre  Evaluates associated Laguerre polynomials.
Hermite  Evaluates Hermite polynomials and their derivatives using the recurrence relation.
SpherY  Evaluates the real and imaginary parts of spherical harmonics.
SBessy  Evaluates spherical Neumann functions.
SBessj  Evaluates spherical Bessel functions.
roots.py / roots.h
Contain routines for determining real roots of real functions:
Bisect  Determines a real root of a function by the bisection method.
FalsPos  Determines a real root of a function by the false position method.
Iter  Determines a real root of a function by the method of successive approximations.
Newton  Determines a real root of a function by the NewtonRaphson method using the analytical derivative.
NewtonNumDrv  Determines a real root of a function by the NewtonRaphson method using the numerical derivative.
Secant  Determines a real root of a function by the secant method.
BirgeVieta  Determines real roots of polynomials by the BirgeVieta method.
Jacobian  Calculates the Jacobian of a system of multidimensional functions using central finite differences.
NewtonSys  Solves a system of nonlinear equations by Newton's method.
Contain routines for determining real roots of real functions:
Bisect  Determines a real root of a function by the bisection method.
FalsPos  Determines a real root of a function by the false position method.
Iter  Determines a real root of a function by the method of successive approximations.
Newton  Determines a real root of a function by the NewtonRaphson method using the analytical derivative.
NewtonNumDrv  Determines a real root of a function by the NewtonRaphson method using the numerical derivative.
Secant  Determines a real root of a function by the secant method.
BirgeVieta  Determines real roots of polynomials by the BirgeVieta method.
Jacobian  Calculates the Jacobian of a system of multidimensional functions using central finite differences.
NewtonSys  Solves a system of nonlinear equations by Newton's method.
linsys.py / linsys.py
Contain routines for solving systems of linear equations:
Gauss  Solves a matrix equation by Gaussian elimination with partial pivoting.
GaussJordan0  Solves a matrix equation by GaussJordan elimination with partial pivoting.
GaussJordan1  Solves a matrix equation by GaussJordan elimination with partial pivoting and matrix inversion.
GaussJordan  Solves a matrix equation by GaussJordan elimination with complete pivoting and matrix inversion.
LUFactor  Performs LU factorization of a matrix by Doolittle's method.
LUSystem  Solves a linear system using the LU decomposition of its matrix.
MatInv  Performs matrix inversion using LU factorization.
MatTriInv  Inverts a triangular matrix.
Cholesky  Performs the Cholesky factorization of a symmetric positivedefinite matrix.
CholeskySys  Solves a positivedefinite linear system using the Cholesky decomposition of its matrix.
MatSymInv  Inverts a positivedefinite matrix using the Cholesky factorization.
TriDiagSys  Solves a linear system with tridiagonal matrix.
GaussSeidel  Solves a system of linear equations by the GaussSeidel method.
Contain routines for solving systems of linear equations:
Gauss  Solves a matrix equation by Gaussian elimination with partial pivoting.
GaussJordan0  Solves a matrix equation by GaussJordan elimination with partial pivoting.
GaussJordan1  Solves a matrix equation by GaussJordan elimination with partial pivoting and matrix inversion.
GaussJordan  Solves a matrix equation by GaussJordan elimination with complete pivoting and matrix inversion.
LUFactor  Performs LU factorization of a matrix by Doolittle's method.
LUSystem  Solves a linear system using the LU decomposition of its matrix.
MatInv  Performs matrix inversion using LU factorization.
MatTriInv  Inverts a triangular matrix.
Cholesky  Performs the Cholesky factorization of a symmetric positivedefinite matrix.
CholeskySys  Solves a positivedefinite linear system using the Cholesky decomposition of its matrix.
MatSymInv  Inverts a positivedefinite matrix using the Cholesky factorization.
TriDiagSys  Solves a linear system with tridiagonal matrix.
GaussSeidel  Solves a system of linear equations by the GaussSeidel method.
eigsys.p y / eigsys.h
Contain routines for solving eigenvalue problems:
Jacobi Solves an eigenvalue problem for a real symmetric matrix using the Jacobi method.
EigSym  Solves a generalized eigenvalue problem for real symmetric (positivedefinite) matrices.
EigSort  Sorts a set of eigenvalues and eigenvectors according to the eigenvalues.
Contain routines for solving eigenvalue problems:
Jacobi Solves an eigenvalue problem for a real symmetric matrix using the Jacobi method.
EigSym  Solves a generalized eigenvalue problem for real symmetric (positivedefinite) matrices.
EigSort  Sorts a set of eigenvalues and eigenvectors according to the eigenvalues.
modfunc.py / modfunc.h
Contain routines for modeling tabular dependences by interpolation or regression:
Lagrange  Evaluates the Lagrange interpolating polynomial for a set of data points.
Neville  Evaluates the Lagrange interpolating polynomial by Neville's method.
Spline  Performs cubic spline interpolation.
LinFit  Performs linear regression on a set of data points.
MultiFit  Performs multilinear regression on a set of data points.
PolFit  Performs polynomial regression on a set of data points.
MarqFit  Performs nonlinear regression based on the LevenbergMarquardt method.
Contain routines for modeling tabular dependences by interpolation or regression:
Lagrange  Evaluates the Lagrange interpolating polynomial for a set of data points.
Neville  Evaluates the Lagrange interpolating polynomial by Neville's method.
Spline  Performs cubic spline interpolation.
LinFit  Performs linear regression on a set of data points.
MultiFit  Performs multilinear regression on a set of data points.
PolFit  Performs polynomial regression on a set of data points.
MarqFit  Performs nonlinear regression based on the LevenbergMarquardt method.
integral.py / integral.h
Contain 1D and 3D integrators for real functions with real variables:
qTrapz  Function integrator based on the trapezoidal rule.
qSimpson  Function integrator based on Simpson's rule.
qTrapzCtrl  Adaptive integrator based on the trapezoidal rule.
qSimpsonCtrl  Adaptive integrator based on Simpson's rule.
qRomberg  Adaptive integrator based on Romberg's method.
qImprop1  Adaptive integrator for improper integrals of the first kind (with infinite limits).
qImprop2  Adaptive integrator for improper integrals of the second kind (with singular integrable limits).
qMidPoint  Adaptive integrator based on the midpoint rule.
xGaussLeg  Generates abscissas and weights for GaussLegendre quadratures.
qGaussLeg  GaussLegendre integrator.
xGaussLag  Generates abscissas and weights for GaussLaguerre quadratures.
qGaussLag  GaussLaguerre integrator.
qTrapz3D  3D integrator based on the trapezoidal rule.
xSimpson  Generates integration points and weights for Simpson's rule.
qSimpson3D  3D integrator based on Simpson's rule.
qGaussLeg3D  3D integrator based on Gaussian quadratures.
qSimpsonAng  Integrator in spherical angular coordinates based on Simpson's rule.
qSimpsonSph  Integrator in spherical coordinates based on Simpson's rule.
qGaussSph  Integrator in spherical coordinates based on Gaussian quadratures.
qSimpsonCyl  Integrator in cylindrical coordinates based on Simpson's rule.
Contain 1D and 3D integrators for real functions with real variables:
qTrapz  Function integrator based on the trapezoidal rule.
qSimpson  Function integrator based on Simpson's rule.
qTrapzCtrl  Adaptive integrator based on the trapezoidal rule.
qSimpsonCtrl  Adaptive integrator based on Simpson's rule.
qRomberg  Adaptive integrator based on Romberg's method.
qImprop1  Adaptive integrator for improper integrals of the first kind (with infinite limits).
qImprop2  Adaptive integrator for improper integrals of the second kind (with singular integrable limits).
qMidPoint  Adaptive integrator based on the midpoint rule.
xGaussLeg  Generates abscissas and weights for GaussLegendre quadratures.
qGaussLeg  GaussLegendre integrator.
xGaussLag  Generates abscissas and weights for GaussLaguerre quadratures.
qGaussLag  GaussLaguerre integrator.
qTrapz3D  3D integrator based on the trapezoidal rule.
xSimpson  Generates integration points and weights for Simpson's rule.
qSimpson3D  3D integrator based on Simpson's rule.
qGaussLeg3D  3D integrator based on Gaussian quadratures.
qSimpsonAng  Integrator in spherical angular coordinates based on Simpson's rule.
qSimpsonSph  Integrator in spherical coordinates based on Simpson's rule.
qGaussSph  Integrator in spherical coordinates based on Gaussian quadratures.
qSimpsonCyl  Integrator in cylindrical coordinates based on Simpson's rule.
random1.py / random.h
Contain routines for generating pseudorandom numbers:
randLCG1  Linear congruential random number generator based on specifications from Press et al. 2002.
randLCG2  Linear congruential random number generator based on specifications from Rapaport 2004.
randMCG  MultiplywithCarry random number generator based on specifications of George Marsaglia.
randNrm  Generates random numbers with normal distribution based on the central limit theorem.
randNrm2  Returns two random numbers with normal distribution.
randMet  Random number generator based on the Metropolis method.
Contain routines for generating pseudorandom numbers:
randLCG1  Linear congruential random number generator based on specifications from Press et al. 2002.
randLCG2  Linear congruential random number generator based on specifications from Rapaport 2004.
randMCG  MultiplywithCarry random number generator based on specifications of George Marsaglia.
randNrm  Generates random numbers with normal distribution based on the central limit theorem.
randNrm2  Returns two random numbers with normal distribution.
randMet  Random number generator based on the Metropolis method.
ode.py / ode.h
Contain routines for solving systems of ordinary differential equations:
Euler  Euler propagator for systems of firstorder ODEs.
EulerPC  Euler predictorcorrector propagator for systems of firstorder ODEs.
RungeKutta  Fourthorder RungeKutta solver for Cauchy problems for systems of firstorder ODEs.
RKadapt  Adaptive ODE solver using stephalving and the RungeKutta method.
RKFehlberg  Adaptive ODE solver based on the RungeKuttaFehlberg method.
Euler1  Euler propagator for a secondorder ODE.
EulerCromer1  EulerCromer propagator for a secondorder ODE.
Verlet1  Verlet propagator for a secondorder ODE.
Euler2  Euler propagator for a system of secondorder ODEs.
EulerCromer  EulerCromer propagator for a system of secondorder ODEs.
Verlet2  2D velocity Verlet propagator for a single particle.
Verlet  Velocity Verlet propagator for a system of particles in interaction.
EulerCromerQM  EulerCromer integrator for the 1D Schrödinger equation.
Numerov  Numerov integrator for the 1D Schrödinger equation.
Propag  Propagates the solution of a secondorder ODE on a regular mesh using the EulerCromer method.
Shoot  Solves a twopoint boundaryvalue problem for a secondorder ODE using the shooting method.
ShootQM  Solves the 1D Schrödinger equation using the shooting method.
Bilocal  Linear bilocal problem solver based on the finitedifference method.
Contain routines for solving systems of ordinary differential equations:
Euler  Euler propagator for systems of firstorder ODEs.
EulerPC  Euler predictorcorrector propagator for systems of firstorder ODEs.
RungeKutta  Fourthorder RungeKutta solver for Cauchy problems for systems of firstorder ODEs.
RKadapt  Adaptive ODE solver using stephalving and the RungeKutta method.
RKFehlberg  Adaptive ODE solver based on the RungeKuttaFehlberg method.
Euler1  Euler propagator for a secondorder ODE.
EulerCromer1  EulerCromer propagator for a secondorder ODE.
Verlet1  Verlet propagator for a secondorder ODE.
Euler2  Euler propagator for a system of secondorder ODEs.
EulerCromer  EulerCromer propagator for a system of secondorder ODEs.
Verlet2  2D velocity Verlet propagator for a single particle.
Verlet  Velocity Verlet propagator for a system of particles in interaction.
EulerCromerQM  EulerCromer integrator for the 1D Schrödinger equation.
Numerov  Numerov integrator for the 1D Schrödinger equation.
Propag  Propagates the solution of a secondorder ODE on a regular mesh using the EulerCromer method.
Shoot  Solves a twopoint boundaryvalue problem for a secondorder ODE using the shooting method.
ShootQM  Solves the 1D Schrödinger equation using the shooting method.
Bilocal  Linear bilocal problem solver based on the finitedifference method.
pde.py / pde.h
Contain routines for solving partial differential equations:
Poisson0  Solver for the 2D Poisson equation in Cartesian coordinates with Dirichlet boundary conditions.
PoissonXY  Solver for the 2D Poisson equation in Cartesian coordinates with arbitrary boundary conditions.
Poisson2D  Solver for the 2D Poisson equation in Cartesian coordinates on an irregular domain.
PropagFTCS  Explicit forwardtime centralspace propagator for the diffusion equation.
PropagBTCS  Implicit backwardtime centralspace propagator for the diffusion equation.
PropagCN  CrankNicolson propagator for the diffusion equation.
PropagDiff  CrankNicolson propagator for the diffusion equation with variable diffusion coefficient.
PropagQTD  CrankNicolson propagator for the Schrödinger equation based on tridiagonal solver.
PropagQGS  CrankNicolson propagator for the Schrödinger equation using the GaussSeidel method.
PropagWave  Explicit forwardtime centralspace propagator for the classical wave equation.
Contain routines for solving partial differential equations:
Poisson0  Solver for the 2D Poisson equation in Cartesian coordinates with Dirichlet boundary conditions.
PoissonXY  Solver for the 2D Poisson equation in Cartesian coordinates with arbitrary boundary conditions.
Poisson2D  Solver for the 2D Poisson equation in Cartesian coordinates on an irregular domain.
PropagFTCS  Explicit forwardtime centralspace propagator for the diffusion equation.
PropagBTCS  Implicit backwardtime centralspace propagator for the diffusion equation.
PropagCN  CrankNicolson propagator for the diffusion equation.
PropagDiff  CrankNicolson propagator for the diffusion equation with variable diffusion coefficient.
PropagQTD  CrankNicolson propagator for the Schrödinger equation based on tridiagonal solver.
PropagQGS  CrankNicolson propagator for the Schrödinger equation using the GaussSeidel method.
PropagWave  Explicit forwardtime centralspace propagator for the classical wave equation.
utils.py / utils.h
Contain utility routines:
Nint  Returns the nearest integer to its realtype argument.
Sign  Transfers the sign of its second argument onto the absolute value of the first argument.
Magn  Returns the order of magnitude of its argument as a power of 10.
Index  Indexes a numeric sequence by direct insertion
Contain utility routines:
Nint  Returns the nearest integer to its realtype argument.
Sign  Transfers the sign of its second argument onto the absolute value of the first argument.
Magn  Returns the order of magnitude of its argument as a power of 10.
Index  Indexes a numeric sequence by direct insertion
Auxiliary libraries
memalloc.h
Contains functions for dynamic memory allocation of vectors and matrices in C:
Vector, FreeVector  Allocate/deallocate a double vector with indices in the range [imin,imax].
Matrix, FreeMatrix  Allocate/deallocate a double matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
IVector, FreeIVector  Allocate/deallocate an int vector with indices in the range [imin,imax].
IMatrix, FreeIMatrix  Allocate/deallocate an int matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
CVector, FreeCVector  Allocate/deallocate a complex vector with indices in the range [imin,imax].
CMatrix, FreeCMatrix  Allocate/deallocate a complex matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
Contains functions for dynamic memory allocation of vectors and matrices in C:
Vector, FreeVector  Allocate/deallocate a double vector with indices in the range [imin,imax].
Matrix, FreeMatrix  Allocate/deallocate a double matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
IVector, FreeIVector  Allocate/deallocate an int vector with indices in the range [imin,imax].
IMatrix, FreeIMatrix  Allocate/deallocate an int matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
CVector, FreeCVector  Allocate/deallocate a complex vector with indices in the range [imin,imax].
CMatrix, FreeCMatrix  Allocate/deallocate a complex matrix, with row and column indices in the range [imin,imax] x [jmin,jmax].
coords.py / coords.h
Contain routines for transforming the coordinates of systems of particles:
MovetoCM  Moves a system of n particles to the CM system.
PrincipalAxes  Rotates a set of n particles to the system of principal axes.
Contain routines for transforming the coordinates of systems of particles:
MovetoCM  Moves a system of n particles to the CM system.
PrincipalAxes  Rotates a set of n particles to the system of principal axes.
elemfunc.py / elemfunc.h
Contain routines for evaluating elementary functions:
Poly  Evaluates a real polynomial using Horner's scheme.
PolyDerive  Returns the coefficients of the derivative of a real polynomial.
PolyDivide  Returns the coefficients of the division of a real polynomial by a binomial (xx0).
Contain routines for evaluating elementary functions:
Poly  Evaluates a real polynomial using Horner's scheme.
PolyDerive  Returns the coefficients of the derivative of a real polynomial.
PolyDivide  Returns the coefficients of the division of a real polynomial by a binomial (xx0).
matutil.py / matutil.h
Contain utility routines for basic operations with vectors and matrices:
MatRead  Reads from the keyboard the elements of matrix.
MatPrint  Prints the elements of a matrix on the display.
MatPrintTrans  Prints the elements of a transposed matrix on the display.
MatZero  Zeros the elements of a matrix.
MatCopy  Copies a matrix into another matrix.
MatTrans  Replaces a square matrix by its transpose.
MatDiff  Returns the difference of two matrices.
MatProd  Returns the product of two matrices.
MatPow  Returns a power of a square matrix.
MatNorm  Returns the max norm of a matrix.
VecPrint  Prints the elements of a vector on the display.
VecZero  Zeros the elements of a vector.
VecCopy  Copies a vector into another vector.
VecDiff  Returns the difference of two vectors.
VecNorm  Returns the 2norm of a vector.
MatVecProd  Returns the product of matrix and a vector.
Contain utility routines for basic operations with vectors and matrices:
MatRead  Reads from the keyboard the elements of matrix.
MatPrint  Prints the elements of a matrix on the display.
MatPrintTrans  Prints the elements of a transposed matrix on the display.
MatZero  Zeros the elements of a matrix.
MatCopy  Copies a matrix into another matrix.
MatTrans  Replaces a square matrix by its transpose.
MatDiff  Returns the difference of two matrices.
MatProd  Returns the product of two matrices.
MatPow  Returns a power of a square matrix.
MatNorm  Returns the max norm of a matrix.
VecPrint  Prints the elements of a vector on the display.
VecZero  Zeros the elements of a vector.
VecCopy  Copies a vector into another vector.
VecDiff  Returns the difference of two vectors.
VecNorm  Returns the 2norm of a vector.
MatVecProd  Returns the product of matrix and a vector.
The graphics library grafxlib.py
The graphics library graphlib.py, accompanying the book was renamed grafxlib.py to avoid confusions with a recent standard module of Python. grafxlib.py is used throughout to produce runtime graphics both from Python and C/C++. The C/C++ applications can access grafxlib.py via the interface grafxlib.h.
grafxlib.py 
Graphics library for Python based on Tkinter. 
grafxlib.h 
C/C++ interface to the library grafxlib.py. 


Sample plots
Simple example  plotting a function of one variable:
Python code:
C code:
To see all graphics applications, go to section Programs.
Structure of grafxlib.py
Auxiliary functions
Limits extends an arbitrary real interval so as to include an integer number of equal subintervals of length expressible as d x 10p, with d = 2, 5, or 10 and p integer. Output: xmin and xmaxadjusted values of, scalescale factor (10p), nsigdnumber of significant digits required to optimally represent the limits, nintvresulting integer number of subintervals. Limits is typically called by Plot and MultiPlot to adjust the x and y user intervals for optimal labeling of the axes, by Contour to resize contour plots, and by ColorLegend to resize color legends.
FormStr determines the optimal labeling format for a given value x on the basis of previous output from Limits (scale factor scale and number of significant digits nsigd), and separates the mantissa in mant[] and the exponent of 10 in expn[].
ColorLegend and RGBcolors are used to display a color legend alongside a contour plot.
FormStr determines the optimal labeling format for a given value x on the basis of previous output from Limits (scale factor scale and number of significant digits nsigd), and separates the mantissa in mant[] and the exponent of 10 in expn[].
ColorLegend and RGBcolors are used to display a color legend alongside a contour plot.
Routines for plotting functions
Plot plots a real function of one variable. The arrays x[] and y[] should contain the coordinates pairs of the representative points. The plot color is specified by the parameter col, and the corresponding arguments can be RGB colors or standard color names. The plotstyle parameter sty can take one of the integer values: 0scatter plot, 1line plot, 2polar plot, 3drop lines, 4histogram. fxmin, fxmax, fymin, fymax are the fractional limits of the plotting domain on the canvas. xtext, ytext, and title are the titles of the axes and of the whole graph. "None" as axis title inhibits the labeling of the axis altogether. The independent coordinate x[] is not supposed to vary monotonically, and this also enables the representation of cyclic functions.
MultiPlot plots several functions of one variable on a single graph. The representative points of the nplot plots are contiguously packed in sequenced sets in the same vectors x[] and y[], with an additional array, n[], specifying the ending index for each set. sig[] is only used when error bars (sty[i]=4) are employed for at least one of the plots. col[] contains the colors of the individual plots and sty[] specifies the corresponding style parameters: 0scatter plot, 1line plot, 1line plot with dashed line, 2polar plot, 2polar plot with dashed line, 3drop lines, 4error bars, 4error bars and continuous line. For ioptx set to 0, the xaxis is resized automatically, while for nonzero ioptx, the axis is resized based on the provided user interval [xminp,xmaxp]. The yaxis is resized analogously, in accordance with the option iopty.
Contour creates the contour plot of a real function of two variables, defined in the domain [xmin,xmax] x [ymin,ymax] and specified by its values z[i][j] at the nx x ny nodes of a regular grid. The representation is confined to zvalues between zmin and zmax.
MultiPlot plots several functions of one variable on a single graph. The representative points of the nplot plots are contiguously packed in sequenced sets in the same vectors x[] and y[], with an additional array, n[], specifying the ending index for each set. sig[] is only used when error bars (sty[i]=4) are employed for at least one of the plots. col[] contains the colors of the individual plots and sty[] specifies the corresponding style parameters: 0scatter plot, 1line plot, 1line plot with dashed line, 2polar plot, 2polar plot with dashed line, 3drop lines, 4error bars, 4error bars and continuous line. For ioptx set to 0, the xaxis is resized automatically, while for nonzero ioptx, the axis is resized based on the provided user interval [xminp,xmaxp]. The yaxis is resized analogously, in accordance with the option iopty.
Contour creates the contour plot of a real function of two variables, defined in the domain [xmin,xmax] x [ymin,ymax] and specified by its values z[i][j] at the nx x ny nodes of a regular grid. The representation is confined to zvalues between zmin and zmax.
Simple 3D rendering routines
PlotParticles depicts systems of particles as spheres, optionally connected by cylindrical bonds. The input consists of the coordinate vectors x[], y[], and z[] of the centers of the n particles, their radii r[], and colors col[]. dmax is a cutoff distance beyond which the particles are no longer linked by bonds (with dmax set to 0, no bonds at all are depicted). The ioptx and iopty parameters control whether the x and yaxis, respectively, are to be sized automatically in order to reveal the entire structure (ioptx=0 and iopty=0), or, instead, the input user coordinates (xminp,xmaxp) and (yminp,ymaxp) are to be considered to crop the image.
PlotStruct is a simple 3D rendering routine for structures specified by nodes and triangular surfaces defined between them. The input to PlotStruct consists of the coordinate vectors x[], y[], and z[] of the n nodes and the indexes ind1[], ind2[], and ind3[] of the nodes defining each of the n3 triangles.
PlotStruct is a simple 3D rendering routine for structures specified by nodes and triangular surfaces defined between them. The input to PlotStruct consists of the coordinate vectors x[], y[], and z[] of the n nodes and the indexes ind1[], ind2[], and ind3[] of the nodes defining each of the n3 triangles.
Histograms
Histograms can be plotted using Plot with the style parameter sty=4. Before plotting a histogram, the data need to be binned. HistoBin is normally called in three successive modes:
 Initializing the histogramdefining the boundaries of the bins and zeroing the corresponding frequencies.
 Binning new observationsassigning them to the appropriate bins and incrementing the corresponding occurrence frequencies.
 Normalizing the histogramdividing the frequencies by the total number of observations.
Creating plots in C/C++ using the library grafxlib.py
The C/C++ programs including the interface file grafxlib.h can seamlessly call with similar syntax the Python graphics functions defined in the module grafxlib.py. The proper functioning of grafxlib.h requires access to grafxlib.py (and to the therein imported module utils.py)) either by search path settings or by simply placing the three files in the same folder.
List of programs by chapter
The file names of the application programs contain the complete information on their location within the folder structure descending from the root folder /INP/. The general format of the file names is Pnnname.py or Pnnname.cpp, whereby nn indicates the chapter folder /INP/Chnn/ and the extension indicates the respective subfolder, that is, /INP/Chnn/Python/ or /INP/Chnn/C/.
A trailing "~" in their names, indicates updated versions of the files.
A trailing "~" in their names, indicates updated versions of the files.
2 Basic Programming Techniques
Listing  File  Output  

2.1 2.2 
P02Fact_global.py P02Fact_global.cpp 
Factorial with global variables.  
2.3 2.4 
P02Fact_comb.py P02Fact_comb.cpp 
Combinations using a factorial function.  
P02Fact_recur.py P02Fact_recur.cpp 
Factorial based on recursivity.  
2.7 2.9 2.10 
P02Swap~.py P02Swap.c P02Swap.cpp 
Returning swapped arguments from a function.  
2.8  P02Swap_list~.py  Returning swapped list elements from a function.  
2.18 2.19 
P02MatOp.py P02MatOp.cpp 
Check of identity (A B)^{T} = B^{T} A^{T} for random arrays A and B. 
p02.zip  
File Size:  4 kb 
File Type:  zip 
3 Elements of Scientific Graphics
p03.zip  
File Size:  12 kb 
File Type:  zip 
4 Sorting and Indexing
p04.zip  
File Size:  10 kb 
File Type:  zip 
5 Evaluation of Functions
p05.zip  
File Size:  10 kb 
File Type:  zip 
6 Algebraic and Transcendental Equations
p06.zip  
File Size:  14 kb 
File Type:  zip 
7 Systems of Linear Equations
Listing  File  Output  

7.2 
P07Gauss.py P07Gauss.cpp 
Solution of a linear system by Gaussian elimination.  
7.6 
P07GaussJordan.py P07GaussJordan.cpp 
Solution of a matrix equation by GaussJordan elimination.  
7.11 
P07MatInv.py P07MatInv.cpp 
Check of matrix inversion based on LU factorization.  
7.18 
P07TriDiagSys.py P07TriDiagSys.cpp 
Solution of linear system with tridiagonal matrix by LU factorization.  
P07GaussSeidel.py P07GaussSeidel.cpp 
Solution of linear system by the GaussSeidel method.  
7.21 7.22 
P07GaussJordan1.py P07GaussJordan1.cpp 
Solution of a matrix equation by GaussJordan elimination.  
7.23 7.24 
P07LUFactor.py P07LUFactor.cpp 
Solution of multiple linear systems by LU factorization.  
7.25 
P07TriDiagSys1.py P07TriDiagSys1.cpp 
Solution of system with tridiagonal matrix by LU factorization.  
7.26 
P07MatDetIdent.py P07MatDetIdent.cpp 
Check of identity for determinant of matrix product.  
7.27 
P07MatInvIdent.py P07MatInvIdent.cpp 
Check of identity for inverse of matrix product.  
7.28 
P07MatSymInv.py P07MatSymInv.cpp 
Inversion of symmetric positivedefinite matrix by Cholesky factorization.  
7.29 
P07Wavelet.py P07Wavelet.cpp 
Solution of linear system for Daubechies D4 wavelet. 
p07.zip  
File Size:  14 kb 
File Type:  zip 
8 Eigenvalue problems
p08.zip  
File Size:  11 kb 
File Type:  zip 
9 Modeling of Tabulated Functions
p09.zip  
File Size:  40 kb 
File Type:  zip 
10 Integration of Functions
p10.zip  
File Size:  17 kb 
File Type:  zip 
11 Monte Carlo Method
p11.zip  
File Size:  18 kb 
File Type:  zip 
12 Ordinary Differential Equations
p12.zip  
File Size:  52 kb 
File Type:  zip 
13 Partial Differential Equations
p13.zip  
File Size:  42 kb 
File Type:  zip 
Appendix C: Embedding Python in C/C++
Listing  File  Output  

C.1  operations.py  Module operations.py containing functions called from the C/C++ programs C.2, C.3, and C.4.  
C.2  PyEmbed1.cpp  Calling Python functions with scalar arguments from C++ code.  
C.3  PyEmbed2.cpp  Calling Python functions with vector arguments from C++ code.  
C.4  PyEmbed3.cpp  Calling Python functions via C++ wrapper classes. 
pyembed.zip  
File Size:  2 kb 
File Type:  zip 