Front Cover -- Introduction to Precise Numerical Methods -- Copyright Page -- Table of Contents -- Preface -- Acknowledgments -- 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 -- Software Exercises N -- Notes and References -- The Precise Numerical Methods Program PNM -- Index.