Introduction to Numerical AnalysisThis book is based on a one-year introductory course on numerical analysis given by the authors at several universities in Germany and the United States. The authors concentrate on methods which can be worked out on a digital computer. For important topics, algorithmic descriptions (given more or less formally in ALGOL 60), as well as thorough but concise treatments of their theoretical founda tions, are provided. Where several methods for solving a problem are presented, comparisons of their applicability and limitations are offered. Each comparison is based on operation counts, theoretical properties such as convergence rates, and, more importantly, the intrinsic numerical properties that account for the reliability or unreliability of an algorithm. Within this context, the introductory chapter on error analysis plays a special role because it precisely describes basic concepts, such as the numerical stability of algorithms, that are indispensable in the thorough treatment of numerical questions. The remaining seven chapters are devoted to describing numerical methods in various contexts. In addition to covering standard topics, these chapters encom pass some special subjects not usually found in introductions to numerical analysis. Chapter 2, which discusses interpolation, gives an account of modem fast Fourier transform methods. In Chapter 3, extrapolation techniques for spe~d ing up the convergence of discretization methods in connection with Romberg integration are explained at length. |
Contents
Interpolation | 37 |
Topics in Integration | 117 |
Systems of Linear Equations | 159 |
Finding Zeros and Minimum Points by Iterative | 244 |
Eigenvalue Problems | 314 |
The Method of Hyman | 353 |
Ordinary Differential Equations | 404 |
BoundaryValue Problems | 476 |
Shooting Method | 487 |
Iterative Methods for the Solution of Large Systems | 536 |
| 597 | |
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Common terms and phrases
a₁ algorithm arbitrary arithmetic b₁ boundary-value problem characteristic polynomial coefficients column computed condition convergence D₁ decomposition defined determined Dh(x diagonal differential equations eigenvalues eigenvector Euler's method exact solution example exists floating-point formula Gauss-Seidel method given H₁ Hermitian Hessenberg matrix initial-value problem integration interval Jacobi method Jordan normal form linear equations lub(A method of order multiple multistep method n x n Newton's method nonsingular norm numerically stable obtained orthogonal P₁ P₁(x parameters positive definite PROOF QR method R₁ recursion roots roundoff errors satisfies Section sequence shooting method solution y(x solving spline function step stepsize system of equations t₁ Theorem tion tridiagonal tridiagonal matrix u₁ unitary unitary matrix V₁ vector x₁ Xi+1 y₁ zero