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Intro -- Preface -- Preface to the Second Edition -- Acknowledgments -- Preface to the First Edition -- Acknowledgments -- Contents -- Chapter 1 Introduction -- Background -- Pitfalls and Limitations -- Programming Languages -- The C Language -- Program Packages Matlab, Maple, Mathematica -- The Golden Middle Course: Python -- Part I Background -- Part II Core Languages -- Part III Commercial Computing Environments -- Part IV Distributed Computing -- Part V Specialized Programming Frameworks -- Part I Background -- Chapter 2 Mathematical Foundations of Programming -- 2.1 A Simple Machine Model -- Machine Language and Higher-Level Programming Languages -- Church's Thesis -- The Turing Machine -- Limits of Computability -- Repercussions in Mathematics -- Extended Number Ranges -- 2.2 Digital Computers -- Number Representation -- Natural Numbers -- Integer Numbers -- Real Numbers, Floating Point Numbers -- Higher-Level Programming Languages -- Part II Core Languages -- Chapter 3 Python, the Fundamentals -- 3.1 Python Interpreter -- A First Example -- Interactive Mode -- 3.2 Elementary Data Types -- Numbers and Arithmetic -- Integer Numbers -- Real Numbers, Floating Point Numbers -- Arithmetical Operations -- Error Propagation in float Computations -- The Types complex and bool -- Complex Numbers -- Boolean Operators, Comparisons -- 3.3 Variables and Value Assignments -- Basic Rule -- 3.4 Control Structures -- Loops -- While Loops -- For Loops -- Interlude: the range Function -- The break and continue Statements -- Conditional Statements -- Floating Point Precision Revisited: Machine Epsilon -- 3.5 Collection Types: Lists, Tuples, Dictionaries and Sets -- Lists -- list is a Reference Type -- Sieve of Eratosthenes -- Sublists -- Concatenation -- List Comprehensions -- Tuples -- Dictionaries -- Sets -- 3.6 Functions.
Anonymous Functions, Lambda Expressions -- Functions as Arguments -- Functions as Return Values -- Recursion -- The Quicksort Algorithm -- 3.7 String Interpolation and Formatting -- 3.8 Writing and Reading Files -- Writing Strings -- Interlude: Access Paths -- Reading -- Numbers -- Binary Files and pickle -- 3.9 Object-Oriented Programming and Classes -- Polynomials -- Inheritance -- Exercises -- Machine Epsilon -- Polynomial Class -- Linear Algebra -- Numerical Integration with the Trapezoidal Rule -- The Taylor Series -- The Newton Method -- Fermat's Last Theorem -- Chapter 4 Python in Scientific Computation -- 4.1 NumPy -- NumPy Arrays -- Vectors and Matrices -- Vectors -- Matrices -- Access to Matrix Components -- Flatten and Reshape -- Standard Matrices -- Advanced Construction Methods -- The Methods diag and kron, the Poisson Matrix -- 4.2 Conjugate Gradient Method -- Quadratic Form -- The Method -- 4.3 SciPy -- 4.4 Linear Algebra -- Matrix Algorithms -- Linear Equations -- LU Decomposition -- Cholesky Decomposition -- Application: Method of Least Squares -- QR Decomposition -- QR Decomposition in Rectangular Matrices -- The Gaussian Normal Equation -- Eigenvectors and Eigenvalues -- Sparse Matrices -- Sparse Poisson Matrix -- 4.5 Graphics with Matplotlib -- 4.6 Nonlinear Equations, Optimization -- Single Equations -- Equation Systems -- Minimization -- 4.7 Numerical Integration, Ordinary Differential Equations -- Integration -- Ordinary Differential Equations ODEs -- Initial Value Problems, solve_ivp -- The Solver solve_ivp -- Lotka-Volterra Equations -- Pendulum -- Boundary Value Problems BVPs, solve_ivp -- 4.8 Partial Differential Equations -- Discretization -- Solution Program -- Sparse Poisson Matrix -- Nonhomogeneous Boundary Conditions -- The SciPy Program -- Verification -- 4.9 Round off: Random Numbers.
Monte Carlo Method for the Computation of -- Exercises -- Linear Equations -- Nonlinear Equations -- Newton's Method in Rn -- Integration -- Optimization -- Ordinary Differential Equations, Initial Value Problems -- Ordinary Differential Equations, Boundary Value Problems -- Partial Differential Equations -- Chapter 5 Python in Computer Algebra -- 5.1 Symbolic Calculation, Numbers -- Number Types -- Symbolic Expressions -- Convention -- Value Assignments -- Advanced Transformations -- Functions -- 5.2 Equation Systems -- 5.3 Linear Algebra -- Matrices and Vectors -- Vector Space Bases -- Eigenvectors and Eigenvalues -- Matrices with ≥ 5 -- 5.4 Calculus -- Limits -- Differential Calculus -- Integration -- Series Expansion -- 5.5 Ordinary Differential Equations -- 5.6 Galerkin Method -- The Model Equation -- The Variational Problem -- The Method -- The SymPy Program -- Exact Solution -- Exercises -- Linear Equations -- Eigenvalues -- Nonlinear Equations -- Differential Equations -- Function Spaces -- Chapter 6 The C Language -- Integrated Development Environment -- Command Line Compiler -- 6.1 Basics -- Numbers and Arithmetic Operations -- Number Representations -- Arithmetic Operators -- Output -- Variables and Assignments -- 6.2 Control Structures: Branches, Loops -- Conditional Execution -- Loops -- 6.3 Functions -- Functions as Arguments -- 6.4 Arrays -- Arrays as Function Arguments -- Matrices -- 6.5 Pointers -- Arrays and Pointers -- Pointers as Arrays -- 6.6 Structures -- Named Structures -- Sparse Matrices -- 6.7 Files, Input and Output -- 6.8 Conclusion -- Chapter 7 The C++ Language -- 7.1 Transfer from C -- 7.2 Basics -- The Output Operator < -- < -- , Input/Output to Files -- Writing to Files -- Reading from Files -- Namespaces -- 7.3 Lambda Expressions -- 7.4 Data Type vector -- Sparse Vectors -- Matrices -- 7.5 Reference Operator.
7.6 Classes -- A Class Matrix -- Declaration Order -- 7.7 Header Files -- Compiling and Executing the Program -- 7.8 Subclassing -- The Subclass Parabola -- 7.9 Summary and Outlook -- Exercises -- Numerical Integration, Simpson's Rule -- Polynomial Interpolation -- Ordinary Differential Equations -- Classes -- Polynomials -- Quaternions -- Chapter 8 Julia -- 8.1 Basics -- Arithmetical Expressions -- Variables and Assignments -- Rational and Complex Numbers -- Mathematical Functions and Constants -- Boolean Operators, Comparisons -- 8.2 Control Structures: Branching, Loops -- Conditional Statements -- While Loop -- For Loop -- Interlude: Ranges -- The break and continue Statements -- 8.3 Functions -- Recursive Functions -- Functions as Arguments and Return Values -- Multiple Dispatch -- Interlude: The REPL and Just in Time Compilation -- 8.4 Collection Types -- Arrays and Vectors -- Vectors and Mutating Functions -- The filter Function -- Comprehension -- Tuples -- Dictionaries -- Sets -- 8.5 Composite Types -- Structures -- Interlude: String Interpolation -- Mutable Structures -- Structures vs. Classes -- 8.6 Linear Algebra -- Linear Equations -- Conjugate Gradient -- Special Matrices -- Submatrices -- Reshape -- The Standard Library Module Linear Algebra -- The Poisson Matrix -- Sparse Matrices -- 8.7 Ordinary Differential Equations -- Equation Systems -- Second Order ODEs -- Boundary Value Problems BVPs -- 8.8 Partial Differential Equations -- The Solution Program -- 8.9 Working with Files -- Julia Set Fractals -- Exercises -- Monte Carlo Method -- Continued Fractions -- Differential Equations -- Classes vs. Structs -- Spline Interpolation -- Part III Commercial Computing Environments -- Chapter 9 Matlab -- 9.1 Basics -- Numbers and Arithmetic -- Variables -- Boolean Operators, Comparisons -- 9.2 Vectors and Matrices -- Vectors.
The Colon Operator and linspace -- Dynamic Vector Extension -- Comparison Operators and Logical Arrays -- Matrices -- The Colon Operator for Matrices -- Standard Matrices -- Special Matrices -- Matrix Operations -- Division -- Elementwise Operations -- 9.3 Control Structures: Branching, Loops -- Conditional Statements -- Loops -- Loops vs. Vectors -- 9.4 Functions -- Scalar or Elementary Functions -- Vector Functions -- Matrix Functions -- 9.5 M-Files -- Script Files -- Function Files -- Subfunctions -- Functions as Arguments -- Anonymous Functions -- Save and Load Files -- 9.6 Linear Algebra -- Linear Equation Systems -- Matrix Decompositions -- LU Decomposition -- Cholesky Decomposition -- QR Decomposition -- 9.7 Ordinary Differential Equations -- The Matlab Function ode45 -- Initial Value Problems -- Second Order ODEs -- Boundary Value Problems -- The Bratu Equation -- 9.8 Partial Differential Equations -- The Matlab PDE Toolbox -- Exercises -- Matrix Computations -- ODE, Initial Value Problem -- ODE, Boundary Value Problem -- Partial Differential Equations PDEs -- Advanced Topic: The Mandelbrot Set in Complex Numbers -- Chapter 10 Maple -- 10.1 Basics -- 10.2 Functions -- User Defined Functions -- The Arrow Operator -- The unapply Command -- The proc Definition -- Visualization -- Equation Systems -- 10.3 Linear Algebra -- Function Values as Matrix Entries -- Vector Space Bases -- The LinearAlgebra Package -- Eigenvectors and Eigenvalues -- × Matrices with ≥ 5 -- 10.4 Calculus -- Derivation -- Integration -- Integral Norm -- 10.5 Interpolation with Spline Functions -- Polygonal Chains -- 10.6 Differential Equations -- Solution by Direct Integration -- The Euler Method as Maple Procedure -- 10.7 Galerkin Method -- Variational Problem -- The Maple Program -- Exact Solution -- 10.8 Finite Element Method -- The Maple Program -- Exercises.
Ordinary Differential Equations.
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Carl Adam Petri : Life and Science / / by Einar Smith
