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This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer-based solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions using polynomials and construct accurate approximations for the solution of ordinary and partial differential equations. To make the format concrete and appealing, the programming environments Matlab and Octave are adopted as faithful companions. The book contains the solutions to several problems posed in exercises and examples, often originating from important applications. At the end of each chapter, a specific section is devoted to subjects which were not addressed in the book and contains bibliographical references for a more comprehensive treatment of the material.
What can’t be ignored
The MATLAB and Octave environments
Real numbers
Complex numbers
Matrices
Real functions
To err is not only human
The MATLAB language
What we haven’t told you
Exercises
Nonlinear equations
Some representative problems
The bisection method
The Newton method
The secant method
Systems of nonlinear equations
Fixed point iterations
Acceleration using Aitken method
Algebraic polynomials
What we haven’t told you
Exercises
Approximation of functions and data
Some representative problems
Approximation by Taylor’s polynomials
Interpolation
Piecewise linear interpolation
Approximation by spline functions
The least-squares method
What we haven’t told you
Exercises
Numerical differentiation and integration
Some representative problems
Approximation of function derivatives
Numerical integration
Interpolatory quadratures
Simpson adaptive formula
Monte Carlo Methods for Numerical Integration
What we haven’t told you
Exercises
Linear systems
Some representative problems
Linear system and complexity
The LU factorization method
The pivoting technique
How accurate is the solution of a linear system
How to solve a tridiagonal system
Overdetermined systems
What is hidden behind the MATLAB command
Iterative methods
Richardson and gradient methods
The conjugate gradient method
When should an iterative method be stopped
To wrap-up direct or iterative
What we haven’t told you
Exercises
Eigenvalues and eigenvectors
Some representative problems
The power method
Generalization of the power method
How to compute the shift
Computation of all the eigenvalues
What we haven’t told you
Exercises
Ordinary differential equations
Some representative problems
The Cauchy problem
Euler methods
The Crank-Nicolson method
Zero-stability
Stability on unbounded intervals
High order methods
The predictor-corrector methods
Systems of differential equations
Some examples
What we haven’t told you
Exercises
Numerical approximation of boundary-value problems
Some representative problems
Approximation of boundary-value problems
Hyperbolic equations a scalar pure advection problem
The wave equation
What we haven’t told you
Exercises
Solutions of the exercises