Basic Numerical Mathematics: Numerical algebraThese two volumes have been designed especially for use as a curricular tool with students who have a strong grasp of the fundamentals of linear algebra (vector space, basis, matrix, determinant, characteristic values, and vectors) and calculus (particularly convergence and continuity), as this text approaches numerical analysis with a program of controlled computational experiments. Volume one is a tersely written introduction to numerical analysis which presents algorithms for root-finding, interpolation, numerical quadrature, and the solution of ordinary differential equations. Approximately one-third of the book is devoted to detailed solutions of selected exercises. Volume two introduces the handling of a variety of problems on programmable computers [graphing calculators]. Emphasis is placed on controlled computational experiments--comparison of the machine solution of a problem to the theoretical solution--as a means of gauging the efficiency of the programs used. Also provided are "bad examples" illustrating the difficulties inherent in the subject. |
Contents
Notations and Abbreviations | 7 |
Preface | 9 |
Manipulation of Vectors and Matrices | 13 |
Copyright | |
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A₁ a₁₁ A₂ algebra approximate assume axioms b₁ b₂ band matrix C₁ Chapter characteristic pair characteristic polynomial characteristic roots characteristic value problem characteristic values Chebyshev norm clearly column vector condition number convergence corresponding characteristic d₁ d₂ denote determine diag diagonal matrix discuss dominant characteristic value elements evaluate factorization Gaussian gives Hence hermitian Hilbert matrix inequality inverse inverse iteration iterative linear Math matrix norm multiplications non-singular normal nXn matrix obtained orthogonal matrix P₁ perturbation positive definite proof pseudo-inverse quadratic form Rayleigh quotient result S-¹AS satisfy scalar sin² singular value decomposition Solution Solve the system spectral radius submultiplicative symmetric matrix system Ax=b theorem triangular matrix triple diagonal matrix unitary upper triangular v₁ values and vectors vector norm W₁ Write a program x₁ zero