Bài toán số là mối quan hệ giữa dữ liệu nhập(input data) biến độc lập trong bài toán và dữ liệu xuất (output data) kết quả cần tìm. Dữ liệu nhập và xuất gồm một số hữu hạn các đại lượng thực ( hoặc phức) và như vậy được biểu diễn bởi các véc tơ có kích thước hữu hạn.
[...]... i.e., the notion of the norm of a linear transformation that acts on a space of functions Such ideas are dealt with in functional analysis, and might profitably be used in a more sophisticated study of numerical methods We study briefly, in Section 2, the practical problem of the effect of rounding errors on the basic operations of arithmetic Except for calculalations involving only exact-integer arithmetic,... rounding errors are invariably present in any computation A most important feature of the later analysis of numerical methods is the incorporation of a treatment of the effects of such rounding errors Finally, in Section 3, we describe the computational problems that are "reasonable" in some general sense In effect, a numerical method which produces a solution insensitive to small changes in data or... of finite dimensional vectors and matrices This subject properly belongs to the field of linear algebra In later chapters, we may occasionally employ the notion of the norm of a function This is a straightforward extension of the notion of a vector norm to the infinite-dimensional case On the other hand, we shall not introduce the corresponding natural generalization, i.e., the notion of the norm of. .. matrix A of order n there exists a nonsingular matrix P, of order n, such that THEOREM = B P- 1AP is upper triangular and has the eigenvalues of A, say Aj == AlA), j = I, 2, , n, on the principal diagonal (i.e., any square matrix is equivalent to a triangular matrix) Proof We sketch the proof of this result The reader should have no difficulty in completing the proof in detail Let A be an eigenvalue of. .. Matrices 4 Iterative Methods 4.1 Jacobi or Simultaneous Iterations 4.2 Gauss-Seidel or Successive Iterations 4.3 Method of Residual Correction 4.4 Positive Definite Systems 4.5 Block Iterations 5 The Acceleration of Iterative Methods O 1 1 2 14 17 26 29 34 37 46 50 52 54 55 58 61 64 66 68 70 72 73 Xl Xli CONTENTS 6 5.1 Practical Application of Acceleration Methods 5.2 Generalizations of the Acceleration... Interpolation 273 3.4 Divergence of Sequences of Interpolation Polynomials 275 Calculus of Difference Operators 281 Numerical Differentiation 288 5.1 Differentiation Using Equidistant Points 292 Multivariate Interpolation 294 Chapter 7 O 1 2 Numerical Integration Introduction Interpolatory Quadrature 1.1 Newton-Cotes Formulae 1.2 Determination of the Coefficients Roundoff Errors and Uniform Coefficient... EulerCauchy Method 1.1 Improving the Accuracy of the Numerical Solution 1.2 Roundoff Errors 1.3 Centered Difference Method 1.4 A Divergent Method with Higher Order Truncation Error Multistep Methods Based on Quadrature Formulae 2.1 Error Estimates in Predictor-Corrector Methods 2.2 Change of Net Spacing One-Step Methods 3.1 Finite Taylor's Series 3.2 One-Step Methods Based on Quadrature Formulae Linear Difference... Consequences of Stability 5.2 The von Neumann Stability Test Bibliography Index XV 444 445 452 458 463 471 475 479 485 491 ~5 501 505 514 522 523 531 535 1 Norms, Arithmetic, and Well-Posed Computations O INTRODUCTION In this chapter, we treat three topics that are generally useful for the analysis of the various numerical methods studied throughout the book In Section I, we give the elements of the theory of. .. Theorem 2.4 of Chapter 4 (see Problem 2.13(b) of Chapter 4) We turn now to the basic content of this section, which is concerned with the generalization of the concept of distance in n-dimensional linear vector spaces The "distance" between a vector and the null vector, i.e., the origin, is a measure of the "size" or "length" of the vector This generalized notion of distance or size is called a norm In particular,... necessarily in its abstract setting, but at least with specific reference to finite-dimensional linear vector spaces over the field of complex scalars By "basic theory" we of course include: the theory of linear systems of equations, some elementary theory of determinants, and the theory of matrices or linear transformations to about the Jordan normal form We hardly employ the Jordan form in the present study . alt="" ANALYSIS OF NUMERICAL METHODS EUGENE ISAACSON Professor Emeritus of Mathematics Courant Institute of Mathematical Sciences New York University HERBERT BISHOP KELLER Profes~or of Applied. present in any computation. A most important feature of the later analysis of numerical methods is the incorporation of a treatment of the effects of such rounding errors. Finally, in Section 3, we describe. topics that are generally useful for the analysis of the various numerical methods studied throughout the book. In Section I, we give the elements of the theory of norms of finite dimen- sional vectors