A Gentle Introduction to Scientific Computing
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Scientific Computation has established itself as a stand-alone area of knowledge at the borderline between computer science and applied mathematics. Nonetheless, its interdisciplinary character cannot be denied: its methodologies are increasingly used in a wide variety of branches of science and engineering.
A Gentle Introduction to Scientific Computing intends to serve a very broad audience of college students across a variety of disciplines. It aims to expose its readers to some of the basic tools and techniques used in computational science, with a view to helping them understand what happens "behind the scenes" when simple tools such as solving equations, plotting and interpolation are used.
To make the book as practical as possible, the authors explore their subject both from a theoretical, mathematical perspective and from an implementation-driven, programming perspective.
- Middle-ground approach between theory and implementation.
- Suitable reading for a broad range of students in STEM disciplines. Could be used as the primary text for a first course in scientific computing.
- Introduces mathematics majors, without any prior computer science exposure, to numerical methods.
- All mathematical knowledge needed beyond Calculus (together with the most widely used Calculus notation and concepts) is introduced in the text to make it self-contained.
The erratum document for A Gentle Introduction to Scientific Computing can be accessed here.
1. Introduction. 1.1. Scientific Computing. 1.2. MATLAB: what and why? 1.3. A Word of Caution. 1.4. Additional Resources. 2. Vectors and Matrices. 2.1. Unidimensional Arrays: Vectors. 2.2. Bidimensional Arrays: Matrices. 2.3. Matrix Operations. 2.4. Systems of Linear Equations. 2.5. Eigenvalues and Eigenvectors. 2.6. Operation Counts. 2.7. Exercises. 3. Basics of MATLAB. 3.1. Defining and Using Scalar Variables. 3.2. Saving and Reloading the Workspace. 3.3. Defining and Using Arrays. 3.4. Operations on Vectors and Matrices. 3.5. More on Plotting Functions of One Variable. 3.6. Loops and Logical Operators. 3.7. Working with indices and arrays. 3.8. Organizing Your Outputs. 3.9. Number representation. 3.10. Machine epsilon. 3.11. Exercises. 4. Solving Nonlinear Equations. 4.1. The Bisection Method for Root-Finding. 4.2. Convergence Criteria and Efficiency. 4.3. Scripts and Function Files. 4.4. The False Position Method. 4.5. The Newton—Raphson Method for Root-Finding. 4.6 Fixed Point Iteration. 4.7. MATLAB built-in functions. 4.8. Exercises. 5. Systems of Equations. 5.1. Linear Systems. 5.2. Newton’s Method for Nonlinear Systems. 5.3. MATLAB built-in functions. 5.4. Exercises. 6. Approximation of Functions. 6.1. A hypothetical example. 6.2. Global Polynomial Interpolation. 6.3. Spline Interpolation. 6.4. Approximation with Trigonometric Functions. 6.5. MATLAB built-in functions. 6.6. Exercises. 7. Numerical Differentiation. 7.1. Basic Derivative Formulae. 7.2. Derivative Formulae Using Taylor Series. 7.3. Derivative Formulae Using Interpolants. 7.4. Errors in Numerical Differentiation. 7.5. Richardson Extrapolation. 7.6. MATLAB built-in functions. 7.7. Exercises. 8. Numerical Optimization. 8.1. The need for optimization methods. 8.2. Line Search Methods. 8.3. Successive Parabolic Interpolation. 8.4. Optimization Using Derivatives. 8.5. Linear programming. 8.6. Constrained nonlinear optimization. 8.7. MATLAB built-in functions. 8.8. Exercises. 9. Numerical Quadrature. 9.1. Basic Quadrature Formulae. 9.2. Gauss Quadrature. 9.3. Extrapolation Methods: Romberg Quadrature. 9.4. Higher-Dimensional Integrals. 9.5. Monte Carlo Integration. 9.6. MATLAB built-in functions. 9.7. Exercises. 10. Numerical Solution of Differential Equations. 10.1. First-order Models. 10.2. Second-order Models. 10.3. Basic Numerical Methods. 10.4. Global error and the order of accuracy. 10.5. Consistency, Stability and Convergence. 10.6. Explicit vs. Implicit Methods. 10.7. Multistep Methods. 10.8. Higher-Order Initial Value Problems. 10.9. Boundary Value Problems. 10.10. MATLAB built-in functions. 10.11. Exercises. Appendix A. Calculus Refresher. A.1. Taylor Series. A.2. Riemann Integrals. A.3. Other Important Results. Appendix B. Introduction to Octave. B.1. The Problem of Choice. B.2. Octave Basics. B.3. Octave Code Examples. Appendix C. Introduction to Python. C.1. The problem of choice. C.2. Python Basics. C.3. Installing Python. C.4. Python Code Examples. Appendix D. Introduction to Julia. D.1. The problem of choice. D.2. Julia Basics. D.3. Julia Code Examples. Appendix E. Hints and Answers for Selected Exercises. Bibliography. Index.
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