Applied Scientific Computing: With Python

✍ Scribed by Peter R. Turner, Thomas Arildsen, Kathleen Kavanagh

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 280
• Series: Texts in Computer Science
• Category: Library

✦ Synopsis

This book is a gentle and sympathetic introduction to many of the problems of scientific computing, and the wide variety of methods used for their solutions. It is ideal for students taking a first course in numerical mathematics who need a low level entry to the subject. It gives an appreciation of the need for numerical methods for the solution of different types of problem, and discusses basic approaches. For each of the problems, at least some mathematical justification and examples provide both practical evidence and motivations for the reader to follow. Practical justification of the methods is presented through computer examples and exercises. The book also includes an introduction to MATLAB, but the code used is not intended to exemplify sophisticated or robust pieces of software; it is purely illustrative of the methods under discussion.

✦ Table of Contents

Preface
Contents
1 Motivation and Background
1.1 Mathematical Modeling and Applications
1.2 Applied Scientific Computing
1.3 Python Programming
1.4 Background
1.4.1 Series Expansions
1.5 Modeling Errors Versus Errors
1.6 Conclusions and Connections: Motivation and Background
2 Number Representations and Errors
2.1 Introduction
2.2 Floating-Point Numbers
2.2.1 Python Number Representation
2.3 Sources of Errors
2.3.1 Rounding Errors
2.3.2 Truncation Error
2.3.3 Ill-Conditioning
2.4 Measures of Error and Precision
2.5 Floating-Point Arithmetic
2.6 Conclusions and Connections: Number Representation and Errors
3 Numerical Calculus
3.1 Introduction
3.2 Numerical Differentiation
3.3 Numerical Integration
3.4 Composite Formulas
3.5 Practical Numerical Integration
3.6 Conclusions and Connections: Numerical Calculus
3.7 Python Functions for Numerical Calculus
4 Linear Equations
4.1 Introduction
4.2 Gauss Elimination
4.2.1 Pivoting in Gauss Elimination
4.2.2 Tridiagonal Systems
4.3 LU Factorization and Applications
4.4 Iterative Methods
4.5 Linear Least Squares Approximation
4.6 Eigenvalues
4.7 Conclusions and Connections: Linear Equations
4.8 Python's Linear Algebra Functions
4.8.1 Linear Equations
4.8.2 Linear Least Squares
4.8.3 Eigenvalues
4.8.4 Basic Linear Algebra Functions
5 Iterative Solution of Nonlinear Equations
5.1 Introduction
5.1.1 Summary of Convergence of Sequences
5.2 The Bisection Method
5.3 Fixed Point Iteration
5.4 Newton's Method
5.5 The Secant Method
5.6 Newton's Method in Higher Dimensions
5.6.1 Newton's Method: Two Equations in Two Unknowns
5.7 Conclusions and Connections: Iterative Solution of Nonlinear Equations
5.8 Python Functions for Equation Solving
6 Interpolation
6.1 Introduction
6.2 Lagrange Interpolation
6.3 Difference Representations
6.3.1 Divided Difference Interpolation
6.4 Splines
6.5 Conclusions and Connections: Interpolation
6.6 Python Interpolation Functions
7 Differential Equations
7.1 Introduction and Euler's Method
7.2 Runge–Kutta Methods
7.3 Multistep Methods
7.4 Systems of Differential Equations
7.5 Boundary Value Problems: Shooting Methods
7.6 Conclusions and Connections: Differential Equations
7.7 Python Functions for Ordinary Differential Equations
Further Reading and Bibliography
Index

✦ Subjects

Computers, Programming, Algorithms, Computer Simulation, Mathematics, Applied, Computer Science, Discrete Mathematics, Information Technology, Desktop Applications, Design & Graphics