Numerical Mathematics (Undergraduate Texts in Mathematics) [Paperback] [Jan 09, 1991] Hämmerlin, Günther; Hoffmann, Karl-Heinz and Schumaker, Larry L.

Hämmerlin, Günther; Hoffmann, Karl-Heinz

ISBN 10: 0387974946 / ISBN 13: 9780387974941

Editore: Springer, 1991

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Different representation of the cover. Signature to the title page, seems fine to read throughout.

Codice articolo 137192-1a

Dati bibliografici

• Titolo: Numerical Mathematics (Undergraduate Texts in Mathematics) [Paperback] [Jan 09, 1991] Hämmerlin, Günther; Hoffmann, Karl-Heinz and Schumaker, Larry L.
• Editore: Springer
• Data di pubblicazione: 1991
• Lingua: Inglese
• ISBN 10: 0387974946
• ISBN 13: 9780387974941
• Rilegatura: Brossura
• Condizione: Very Good

Descrizione articolo

Riassunto

This English translation of the highly successful German textbook Numerische Mathematik covers the usual classical topics of numerical analysis, and also includes an up-to-date treatment of both splines and linear optimization methods. The text is designed to be used in a first course in numerical analysis at the upper division undergraduate level or at the beginning graduate level. It features a careful balance between mathematical rigor and numerical insight and includes many worked out numerical examples. Each section concludes with an extensive set of exercises which instructors should find useful in helping students to master the material. Moreover, the authors have also provided carefully researched historical notes which will be of particular interest to experts as well as students.

Contenuti

1 Computing.- §1. Numbers and Their Representation.- 1.1 Representing numbers in arbitrary bases.- 1.2 Analog and digital computing machines.- 1.3 Binary arithmetic.- 1.4 Fixed-point arithmetic.- 1.5 Floating-point arithmetic.- 1.6 Problems.- §2. Floating Point Arithmetic.- 2.1 The roundoff rule.- 2.2 Combining floating point numbers.- 2.3 Numerically stable vs. unstable evaluation of formulae.- 2.4 Problems.- §3. Error Analysis.- 3.1 The condition of a problem.- 3.2 Forward error analysis.- 3.3 Backward error analysis.- 3.4 Interval arithmetic.- 3.5 Problems.- §4. Algorithms.- 4.1 The Euclidean algorithm.- 4.2 Evaluation of algorithms.- 4.3 Complexity of algorithms.- 4.4 The complexity of some algorithms.- 4.5 Divide and conquer.- 4.6 Fast matrix multiplication.- 4.7 Problems.- 2. Linear Systems of Equations.- §1. Gauss Elimination.- 1.1 Notation and statement of the problem.- 1.2 The elimination method.- 1.3 Triangular decomposition by Gauss elimination.- 1.4 Some special matrices.- 1.5 On pivoting.- 1.6 Complexity of Gauss elimination.- 1.7 Problems.- §2. The Cholesky Decomposition.- 2.1 Review of positive definite matrices.- 2.2 The Cholesky decomposition.- 2.3 Complexity of the Cholesky decomposition.- 2.4 Problems.- §3. The QR Decomposition of Householder.- 3.1 Householder matrices.- 3.2 The basic problem.- 3.3 The Householder algorithm.- 3.4 Complexity of the QR decomposition.- 3.5 Problems.- §4. Vector Norms and Norms of Matrices.- 4.1 Norms on vector spaces.- 4.2 The natural norm of a matrix.- 4.3 Special norms of matrices.- 4.4 Problems.- §5. Error Bounds.- 5.1 Condition of a matrix.- 5.2 An error bound for perturbed matrices.- 5.3 Acceptability of solutions.- 5.4 Problems.- §6. III-Conditioned Problems.- 6.1 The singular-value decomposion of a matrix.- 6.2 Pseudo-normal solutions of linear systems of equations.- 6.3 The pseudo-inverse of a matrix.- 6.4 More on linear systems of equations.- 6.5 Improving the condition and regularization of a linear system of equations.- 6.6 Problems.- 3. Eigenvalues.- §1. Reduction to Tridiagonal or Hessenberg Form.- 1.1 The Householder method.- 1.2 Computation of the eigenvalues of tridiagonal matrices.- 1.3 Computation of the eigenvalues of Hessenberg matrices.- 1.4 Problems.- §2. The Jacobi Rotation and Eigenvalue Estimates.- 2.1 The Jacobi method.- 2.2 Estimating eigenvalues.- 2.3 Problems.- §3. The Power Method.- 3.1 An iterative method.- 3.2 Computation of eigenvectors and further eigenvalues.- 3.3 The Rayleigh quotient.- 3.4 Problems.- §4. The QR Algorithm.- 4.1 Convergence of the QR algorithm.- 4.2 Remarks on the LR algorithm.- 4.3 Problems.- 4. Approximation.- §1. Preliminaries.- 1.1 Normed linear spaces.- 1.2 Banach spaces.- 1.3 Hilbert spaces and pre-Hilbert spaces.- 1.4 The spaces Lp[a, b].- 1.5 Linear operators.- 1.6 Problems.- §2. The Approximation Theorems of Weierstrass.- 2.1 Approximation by polynomials.- 2.2 The approximation theorem for continuous functions.- 2.3 The Korovkin approach.- 2.4 Applications of Theorem 2.3.- 2.5 Approximation error.- 2.6 Problems.- §3. The General Approximation Problem.- 3.1 Best approximations.- 3.2 Existence of a best approximation.- 3.3 Uniqueness of a best approximation.- 3.4 Linear approximation.- 3.5 Uniqueness in finite dimensional linear subspaces.- 3.6 Problems.- §4. Uniform Approximation.- 4.1 Approximation by polynomials.- 4.2 Haar spaces.- 4.3 The alternation theorem.- 4.4 Uniqueness.- 4.5 An error bound.- 4.6 Computation of the best approximation.- 4.7 Chebyshev polynomials of the first kind.- 4.8 Expansions in Chebyshev polynomials.- 4.9 Convergence of best approximations.- 4.10 Nonlinear approximation.- 4.11 Remarks on approximation in (C[a, b], || • ||1).- 4.12 Problems.- §5. Approximation in Pre-Hilbert Spaces.- 5.1 Characterization of the best approximation.- 5.2 The normal equations.- 5.3 Orthonormal systems.- 5.4 The Legendre polynomials.- 5.5 Properties of orthonormal polynomials.- 5.6 Convergence in C[a, b].- 5.7 Approximation of piecewise continuous functions.- 5.8 Trigonometric approximation.- 5.9 Problems.- §6. The Method of Least Squares.- 6.1 Discrete approximation.- 6.2 Solution of the normal equations.- 6.3 Fitting by polynomials.- 6.4 Coalescent data points.- 6.5 Discrete approximation by trigonometric functions.- 6.6 Problems.- 5. Interpolation.- §1. The Interpolation Problem.- 1.1 Interpolation in Haar spaces.- 1.2 Interpolation by polynomials.- 1.3 The remainder term.- 1.4 Error bounds.- 1.5 Problems.- §2. Interpolation Methods and Remainders.- 2.1 The method of Lagrange.- 2.2 The method of Newton.- 2.3 Divided differences.- 2.4 The general Peano remainder formula.- 2.5 A derivative-free error bound.- 2.6 Connection to analysis.- 2.7 Problems.- §3. Equidistant Interpolation Points.- 3.1 The difference table.- 3.2 Representations of interpolating polynomials.- 3.3 Numerical differentiation.- 3.4 Problems.- §4. Convergence of Interpolating Polynomials.- 4.1 Best interpolation.- 4.2 Convergence problems.- 4.3 Convergence results.- 4.4 Problems.- §5. More on Interpolation.- 5.1 Horner’s scheme.- 5.2 The Aitken-Neville algorithm.- 5.3 Hermite interpolation.- 5.4 Trigonometric interpolation.- 5.5 Complex interpolation.- 5.6 Problems.- §6. Multidimensional Interpolation.- 6.1 Various interpolation problems.- 6.2 Interpolation on rectangular grids.- 6.3 Bounding the interpolation error.- 6.4 Problems.- 6. Splines.- §1. Polynomial Splines.- 1.1 Spline spaces.- 1.2 A basis for the spline space.- 1.3 Best approximation in spline spaces.- 1.4 Problems.- §2. Interpolating Splines.- 2.1 Splines of odd degree.- 2.2 An extremal property of splines.- 2.3 Quadratic splines.- 2.4 Convergence.- 2.5 Problems.- §3. B-splines.- 3.1 Existence of B-splines.- 3.2 Local bases.- 3.3 Additional properties of B-splines.- 3.4 Linear B-splines.- 3.5 Quadratic B-splines.- 3.6 Cubic B-splines.- 3.7 Problems.- §4. Computing Interpolating Splines.- 4.1 Cubic splines.- 4.2 Quadratic splines.- 4.3 A general interpolation problem.- 4.4 Problems.- §5. Error Bounds and Spline Approximation.- 5.1 Error bounds for linear splines.- 5.2 On uniform approximation by linear splines.- 5.3 Least squares approximation by linear splines.- 5.4 Error bounds for splines of higher degree.- 5.5 Least squares splines of higher degree.- 5.6 Problems.- §6. Multidimensional Splines.- 6.1 Bilinear splines.- 6.2 Bicubic splines.- 6.3 Spline-blended functions.- 6.4 Problems.- 7. Integration.- §1. Interpolatory Quadrature.- 1.1 Rectangle rules.- 1.2 The trapezoidal rule.- 1.3 The Euler-MacLaurin expansion.- 1.4 Simpson’s rule.- 1.5 Newton-Cotes formulae.- 1.6 Unsymmetric quadrature formulae.- 1.7 Problems.- §2. Extrapolation.- 2.1 The Romberg method.- 2.2 Error analysis.- 2.3 Extrapolation.- 2.4 Convergence.- 2.5 Problems.- §3. Gauss Quadrature.- 3.1 The method of Gauss.- 3.2 Gauss quadrature as interpolation quadrature.- 3.3 Error formula.- 3.4 Modified Gauss quadrature.- 3.5 Improper integrals.- 3.6 Nodes and coefficients of Gauss quadrature formulae.- 3.7 Problems.- §4. Special Quadrature Methods.- 4.1 Integration over an infinite interval.- 4.2 Singular integrands.- 4.3 Periodic functions.- 4.4 Problems.- §5. Optimality and Convergence.- 5.1 Norm minimization.- 5.2 Minimizing random errors.- 5.3 Optimal quadrature formulae.- 5.4 Convergence of quadrature formulae.- 5.5 Quadrature operators.- 5.6 Problems.- §6. Multidimensional Integration.- 6.1 Tensor products.- 6.2 Integration over standard domains.- 6.3 The Monte-Carlo method.- 6.4 Problems.- 8. Iteration.- §1. The General Iteration Method.- 1.1 Examples of convergent iterations.- 1.2 Convergence of iterative methods.- 1.3 Lipschitz constants.- 1.4 Error bounds.- 1.5 Convergence.- 1.6 Problems.- §2. Newton’s Method.- 2.1 Accelerating the convergence of an iterative method.- 2.2 Geometric interpretation.- 2.3 Multiple zeros.- 2.4 The secant method.- 2.5 Newton’s method for m > 1..- 2.6 Roots of polynomials.- 2.7 Problems.- §3. Iterative Solution of Linear Systems of Equations.- 3.1 Sequences of iteration matrices.- 3.2 The Jacobi method.- 3.3 The Gauss-Seidel method.- 3.4 The theorem of Stein and Rosenberg.- 3.5 Problems.- §4. More on Convergence.- 4.1 Relaxation for the Jacobi method.- 4.2 Relaxation for the Gauss-Seidel method.- 4.3 Optimal relaxation parameters.- 4.4 Problems.- 9. Linear. Optimization.- §1. Introductory Examples and the General Problem.- 1.1 Optimal production planning.- 1.2 A semi-infinite optimization problem.- 1.3 A linear control problem.- 1.4 The general problem.- 1.5 Problems.- §2. Polyhedra.- 2.1 Characterization of vertices.- 2.2 Existence of vertices.- 2.3 The main result.- 2.4 An algebraic characterization of vertices.- 2.5 Problems.- §3. The Simplex Method.- 3.1 Introduction.- 3.2 The vertex exchange without degeneracy.- 3.3 Finding a starting vertex.- 3.4 Degenerate vertices.- 3.5 The two-phase method.- 3.6 The modified simplex method.- 3.7 Problems.- §4. Complexity Analysis.- 4.1 The examples of Klee and Minty.- 4.2 On the average behavior of the algorithm.- 4.3 Runtime of algorithms.- 4.4 Polynomial algorithms.- 4.5 Problems.- References.- Symbols.