Eugene Isaacson, Herbert Bishop Keller's Analysis of Numerical Methods PDF

By Eugene Isaacson, Herbert Bishop Keller

ISBN-10: 0486680290

ISBN-13: 9780486680293

First-class advanced-undergraduate and graduate textual content covers norms, numerical resolution of linear platforms and matrix factoring, iterative ideas of nonlinear equations, eigenvalues and eigenvectors, polynomial approximation and extra. cautious research and pressure on suggestions for constructing new equipment. Examples and difficulties. 1966 version. Bibliography.

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Extra resources for Analysis of Numerical Methods

Example text

Again by (iv) we have IIAmll::; IIAllm • so that condition (b) of Theorem 4 holds. Another important characterization and property of convergent matrices is contained in THEOREM 5. (a) The geometric series I converges iff A + A + A2 + A3 + ... , is convergent. (b) If A is convergent, then I - A is non-singular and (/- A)-l = I + A + A2 + A3 + .... , that A be convergent. The sufficiency will follow from part (b). Let A be convergent, whence by Theorem 4 we know that peA) < 1. Since the eigenvalues of I - A are 1 - A(A), it follows that det (I - A) #- 0 and hence this matrix is non-singular.

In these algebraic cases, the accuracy of the solution x, as measured in (5), is seen to depend on the well-posed ness of the original mathematical problem. In algorithms, 24 NORMS, ARITHMETIC, AND WELL-POSED COMPUTATIONS [Ch. 1] (1), that arise from differential equation problems, other techniques are developed to estimate the accuracy of the approximation. For differential equation problems the well-posed ness of the resulting algorithms (1) is referred to as the stability of the finite difference schemes (see Chapters 8 and 9).

I'iJIf = A +E where E == (eiJ) satisfies (24) lei < IJ - {Uj2aG(n)101-t, - 1)2aG(n)101- t, for i ~ j; for i > j. Here G(n) is any bound satisfying (22). 2] NUMERICAL SOLUTION OF LINEAR SYSTEMS where from Lemma 1 and (22) it follows that i~k+l,j~k+I. (25b) Similarly, multiplying the last line of (2Ic) by I + j; where again we find that i > j. (26b) Upon recalling (21) we have f£' == (Si;) , so that (f£'qj)ij ~ Sikbk~) = k=l min (I,n "" L, Sik b(k) kj' k=l Now let i > j and sum (25a) over k = I, 2, ' .

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Analysis of Numerical Methods by Eugene Isaacson, Herbert Bishop Keller

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