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AuthorRutishauser, Heinz. author
TitleLectures on Numerical Mathematics [electronic resource] / by Heinz Rutishauser ; edited by Martin Gutknecht
ImprintBoston, MA : Birkhรคuser Boston, 1990
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Descript XVI, 546 p. online resource


The present book is an edition of the manuscripts to the courses "Numerical Methods I" and "Numerical Mathematics I and II" which Professor H. Rutishauser held at the E.T.H. in Zurich. The first-named course was newly conceived in the spring semester of 1970, and intended for beginners, while the two others were given repeatedly as elective courses in the sixties. For an understanding of most chapters the fundaยญ mentals of linear algebra and calculus suffice. In some places a little complex variable theory is used in addition. However, the reader can get by without any knowledge of functional analysis. The first seven chapters discuss the direct solution of systems of linear equations, the solution of nonlinear systems, least squares probยญ lems, interpolation by polynomials, numerical quadrature, and approximaยญ tion by Chebyshev series and by Remez' algorithm. The remaining chapters include the treatment of ordinary and partial differential equaยญ tions, the iterative solution of linear equations, and a discussion of eigenยญ value problems. In addition, there is an appendix dealing with the qdยญ algorithm and with an axiomatic treatment of computer arithmetic


1. An Outline of the Problems -- ยง 1.1. Reliability of programs -- ยง 1.2. The evolution of a program -- ยง 1.3. Difficulties -- Notes to Chapter 1 -- 2. Linear Equations and Inequalities -- ยง 2.1. The classical algorithm of Gauss -- ยง 2.2. The triangular decomposition -- ยง 2.3. Iterative refinement -- ยง 2.4. Pivoting strategies -- ยง 2.5. Questions of programming -- ยง 2.6. The exchange algorithm -- ยง 2.7. Questions of programming -- ยง 2.8. Linear inequalities (optimization) -- Notes to Chapter 2 -- 3. Systems of Equations With Positive Definite Symmetric Coefficient Matrix -- ยง 3.1. Positive definite matrices -- ยง 3.2. Criteria for positive definiteness -- ยง 3.3. The Cholesky decomposition -- ยง 3.4. Programming the Cholesky decomposition -- ยง 3.5. Solution of a linear system -- ยง 3.6. Influence of rounding errors -- ยง 3.7. Linear systems of equations as a minimum problem -- Notes to Chapter 3 -- 4. Nonlinear Equations -- ยง 4.1. The basic idea of linearization -- ยง 4.2. Newtonโ{128}{153}s method -- ยง 4.3. The regula falsi -- ยง 4.4. Algebraic equations -- ยง 4.5. Root squaring (Dandelin-Graeffe) -- ยง 4.6. Application of Newtonโ{128}{153}s method to algebraic equations -- Notes to Chapter 4 -- 5. Least Squares Problems -- ยง 5.1. Nonlinear least squares problems -- ยง 5.2. Linear least squares problems and their classical solution -- ยง 5.3. Unconstrained least squares approximation through orthogonalization -- ยง 5.4. Computational implementation of the orthogonalization -- ยง 5.5. Constrained least squares approximation through orthogonalization -- Notes to Chapter 5 -- 6. Interpolation -- ยง 6.1. The interpolation polynomial -- ยง 6.2. The barycentric formula -- ยง 6.3. Divided differences -- ยง 6.4. Newtonโ{128}{153}s interpolation formula -- ยง 6.5. Specialization to equidistant xi -- ยง 6.6. The problematic nature of Newton interpolation -- ยง 6.7. Hermite interpolation -- ยง 6.8. Spline interpolation -- ยง 6.9. Smoothing -- ยง 6.10.Approximate quadrature -- Notes to Chapter 6 -- 7. Approximation -- ยง 7.1. Critique of polynomial representation -- ยง 7.2. Definition and basic properties of Chebyshev polynomials -- ยง 7.3. Expansion in T-polynomials -- ยง 7.4. Numerical computation of the T-coefficients -- ยง 7.5. The use of T-expansions -- ยง 7.6. Best approximation in the sense of Chebyshev (T-approximation) -- ยง 7.7. The Remez algorithm -- Notes to Chapter 7 -- 8. Initial Value Problems for Ordinary Differential Equations -- ยง8.1. Statement of the problem -- ยง 8.2. The method of Euler -- ยง 8.3. The order of a method -- ยง 8.4. Methods of Runge-Kutta type -- ยง 8.5. Error considerations for the Runge-Kutta method when applied to linear systems of differential equations -- ยง 8.6. The trapezoidal rule -- ยง 8.7. General difference formulae -- ยง 8.8. The stability problem -- ยง 8.9. Special cases -- Notes to Chapter 8 -- 9. Boundary Value Problems For Ordinary Differential Equations -- ยง 9.1. The shooting method -- ยง 9.2. Linear boundary value problems -- ยง 9.3. The Floquet solutions of a periodic differential equation -- ยง 9.4. Treatment of boundary value problems with difference methods -- ยง 9.5. The energy method for discretizing continuous problems -- Notes to Chapter 9 -- 10. Elliptic Partial Differential Equations, Relaxation Methods -- ยง10.1. Discretization of the Dirichlet problem -- ยง10.2. The operator principle -- ยง10.3. The general principle of relaxation -- ยง10.4. The method of Gauss-Seidel, overtaxation -- ยง10.5. The method of conjugate gradients -- ยง10.6. Application to a more complicated problem -- ยง10.7. Remarks on norms and the condition of a matrix -- Notes to Chapter 10 -- 11. Parabolic and Hyperbolic Partial Differential Equations -- ยง11.1. One-dimensional heat conduction problems -- ยง11.2. Stability of the numerical solution -- ยง11.3. The one-dimensional wave equation -- ยง11.4. Remarks on two-dimensional heat conduction problems -- Notes to Chapter 11 -- 12. The Eigenvalue Problem For Symmetric Matrices -- ยง12.1. Introduction -- ยง12.2. Extremal properties of eigenvalues -- ยง12.3. The classical Jacobi method -- ยง12.4. Programming considerations -- ยง12.5. The cyclic Jacobi method -- ยง12.6. The LR transformation -- ยง12.7. The LR transformation with shifts -- ยง12.8. The Householder transformation -- ยง12.9. Determination of the eigenvalues of a tridiagonal matrix -- Notes to Chapter 12 -- 13. The Eigenvalue Problem For Arbitrary Matrices -- ยง13.1. Susceptibility to errors -- ยง13.2. Simple vector iteration -- Notes to Chapter 13 -- Appendix. An Axiomatic Theory of Numerical Computation with an Application to the Quotient-Difference Algorithm -- Editorโ{128}{153}s Foreword -- Al. Introduction -- ยงA1.1. The eigenvalues of a qd-row -- ยงA1.2. The progressive form of the qd-algorithm -- ยงA1.3. The generating function of a qd-row -- ยงA1.4. Positive qd-rows -- ยงA1.5. Speed of convergence of the qd-algorithm -- ยงA1.6. The qd-algorithm with shifts -- ยงA1.7. Deflation after the determination of an eigenvaluec -- A2. Choice of Shifts -- ยงA2.1. Effect of the shift v on Zโ{128}{153} -- ยงA2.2. Seropositive qd-rows -- ยง A2.4. A formal algorithm for the determination of eigenvalues -- A3. Finite Arithmetic -- ยงA3.1. The basic sets -- ยงA3.2. Properties of the arithmetic -- ยงA3.3. Monotonicity of the arithmetic -- ยงA3.4. Precision of the arithmetic -- ยงA3.5. Underflow and overflow control -- A4. Influence of Rounding Errors -- ยงA4.1. Persistent properties of the qd-algorithm -- ยงA4.2. Coincidence -- ยงA4.3. The differential form of the progressive qd-algorithm -- ยงA4.4. The influence of rounding errors on convergence -- A5. Stationary Form of the qd-Algorithm -- ยงA5.1. Development of the algorithm -- ยงA5.2. The differential form of the stationary qd-algorithm -- ยงA5.3. Properties of the stationary qd-algorithm -- ยงA5.4. Safe qd-steps -- Bibliography to the Appendix -- Author Index

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