(Ebook PDF) Introduction to precise numerical methods 2nd Edition by Oliver Aberth ISBN 9780080471204 008047120X full chapters

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ISBN 10:008047120X

ISBN 13:9780080471204

Author: Oliver Aberth

Precise numerical analysis may be defined as the study of computer methods for solving mathematical problems either exactly or to prescribed accuracy. This book explains how precise numerical analysis is constructed. The book also provides exercises which illustrate points from the text and references for the methods presented.

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Table of Contents:

• Chapter 1 Introduction
• 1.1 Open-source software
• 1.2 Calling up a program
• 1.3 Log files and print files
• 1.4 More on log files
• 1.5 The tilde notation for printed answers
• Chapter 2 Computer Arithmetics
• 2.1 Floating-point arithmetic
• 2.2 Variable precision floating-point arithmetic
• 2.3 Interval arithmetic
• 2.4 Range arithmetic
• 2.5 Practical range arithmetic
• 2.6 Interval arithmetic notation
• 2.7 Computing standard functions in range arithmetic
• 2.8 Rational arithmetic
• Software Exercises A
• Notes and References
• Chapter 3 Classification of Numerical Computation Problems
• 3.1 A knotty problem
• 3.2 The impossibility of untying the knot
• 3.3 Repercussions from nonsolvable problem 3.1
• 3.4 Some solvable and nonsolvable decimal place problems
• 3.5 The solvable problems handled by calc
• 3.6 Another nonsolvable problem
• 3.7 The trouble with discontinuous functions
• Notes and References
• Chapter 4 Real-Valued Functions
• 4.1 Elementary functions
• Software Exercises B
• Chapter 5 Computing Derivatives
• 5.1 Power series of elementary functions
• 5.2 An example of series evaluation
• 5.3 Power series for elementary functions of several variables
• 5.4 A more general method of generating power series
• 5.5 The demo program deriv
• Software Exercises C
• Notes and References
• Chapter 6 Computing Integrals
• 6.1 Computing a definite integral
• 6.2 Formal interval arithmetic
• 6.3 The demo program integ for computing ordinary definite integrals
• 6.4 Taylor’s remainder formula generalized
• 6.5 The demo program mulint for higher dimensional integrals
• 6.6 The demo program impint for computing improper integrals
• Software Exercises D
• Notes and References
• Chapter 7 Finding Where a Function f(x) is Zero
• 7.1 Obtaining a solvable problem
• 7.2 Using interval arithmetic for the problem
• 7.3 Newton’s method
• 7.4 Order of convergence
• Software Exercises E
• Chapter 8 Finding Roots of Polynomials
• 8.1 Polynomials
• 8.2 A bound for the roots of a polynomial
• 8.3 The Bairstow method for finding roots of a real polynomial
• 8.4 Bounding the error of a rational polynomial’s root approximations
• 8.5 Finding accurate roots for a rational or a real polynomial
• 8.6 The demo program roots
• Software Exercises F
• Notes and References
• Chapter 9 Solving n Linear Equations in n Unknowns
• 9.1 Notation
• 9.2 Computation problems
• 9.3 A method for solving linear equations
• 9.4 Computing determinants
• 9.5 Finding the inverse of a square matrix
• 9.6 The demo programs equat, r_equat, and c_equat
• Software Exercises G
• Notes and References
• Chapter 10 Eigenvalue and Eigenvector Problems
• 10.1 Finding a solution to Ax=0 when det A=0
• 10.2 Eigenvalues and eigenvectors
• 10.3 Companion matrices and Vandermonde matrices
• 10.4 Finding eigenvalues and eigenvectors by Danilevsky’s method
• 10.5 Error bounds for Danilevsky’s method
• 10.6 Rational matrices
• 10.7 The demo programs eigen, c_eigen, and r_eigen
• Software Exercises H
• Chapter 11 Problems of Linear Programming
• 11.1 Linear algebra using rational arithmetic
• 11.2 A more efficient method for solving rational linear equations
• 11.3 Introduction to linear programming
• 11.4 Making the simplex process foolproof
• 11.5 Solving n linear interval equations in n unknowns
• 11.6 Solving linear interval equations via linear programming
• 11.7 The program linpro for linear programming problems
• 11.8 The program i_equat for interval linear equations
• Software Exercises I
• Notes and References
• Chapter 12 Finding Where Several Functions are Zero
• 12.1 The general problem for real elementary functions
• 12.2 Finding a suitable solvable problem
• 12.3 Extending the f(x) solution method to the general problem
• 12.4 The crossing parity
• 12.5 The crossing number and the topological degree
• 12.6 Properties of the crossing number
• 12.7 Computation of the crossing number
• 12.8 Newton’s method for the general problem
• 12.9 Searching a more general region for zeros
• Software Exercises J
• Notes and References
• Chapter 13 Optimization Problems
• 13.1 Finding a function’s extreme values
• 13.2 Finding where a function’s gradient is zero
• 13.3 The demo program extrema
• Software Exercises K
• Notes and References
• Chapter 14 Ordinary Differential Equations
• 14.1 Introduction
• 14.2 Two standard problems of ordinary differential equations
• 14.3 Difficulties with the initial value problem
• 14.4 Linear differential equations
• 14.5 Solving the initial value problem by power series
• 14.6 Degree 1 interval arithmetic
• 14.7 An improved global error
• 14.8 Solvable two-point boundary-value problems
• 14.9 Solving the boundary-value problem by power series
• 14.10 The linear boundary-value problem
• Software Exercises L
• Notes and References
• Chapter 15 Partial Differential Equations
• 15.1 Partial differential equation terminology
• 15.2 ODE and PDE initial value problems
• 15.3 A power series method for the ODE problem
• 15.4 The first PDE solution method
• 15.5 A simple PDE problem as an example
• 15.6 A defect of the first PDE method
• 15.7 The revised PDE method with comparison computation
• 15.8 Higher dimensional spaces
• 15.9 Satisfying boundary conditions
• Software Exercises M
• Notes and References
• Chapter 16 Numerical Methods with Complex Functions
• 16.1 Elementary complex functions
• 16.2 The demo program c_deriv
• 16.3 Computing line integrals in the complex plane
• 16.4 Computing the roots of a complex polynomial
• 16.5 Finding a zero of an elementary complex function f(z)
• 16.6 The general zero problem for elementary complex functions

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Oliver Aberth,Introduction,Precise Numerical Methods