Handbook of Numerical Methods for the Solution of Algebraic by V. L. Zaguskin

By V. L. Zaguskin

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Extra resources for Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations

Example text

1. The error of a sum In the addition of two inequalities defining the limits of the poss­ ible spread of exact numbers a — Δα ^ a* ^ a -f- Δα, h _ Afe < fe* ^ fe - f Afe, we get the limits of their sum: a - f fe — (Δα - f Afe) ^ a * + * * < a - f * + (Δα - f Afe). Thus, the absolute error of the sum is equal to the sum of the absolute errors of the added components. If the abbreviated way of writing complex numbers is used and the numbers to be added have different absolute errors, then it is necess­ ary to round off again.

51) converges to the function f(x) the number a is a root of the function 26 HANDBOOK OF NUMERICAL METHODS We omit the proof. 50). If some of the roots being found from the sequence turn out to be close together, then they may be taken as an approxi­ mation to the value of a root of the given transcendental equation. With this method one usually succeeds in finding one or several roots with smaller moduli. In principle it makes no difference which method one uses to solve the algebraic equations which one obtains: however it is evidently most convenient to use the methods of Lobachevskii and Bernouilli.

However, the accuracy with which roots are determined is often very small: usually one succeeds in determining only the order of magnitude of the largest and the small­ est roots. 1. iy . . , xm of the poly­ nomial P(x) = a0xn + α,χ^ + a*xn~* + . . 4) with its coefficients: x n-lxn> — = ΧχΧι - } - ΛΓ1ΛΓ3 -f- X\$X\$ -\- . . -f- — — = ΧχΧ%Χζ -f" ^1-^2^4 - | - X{X^Xi - j - X^X^Xi -f" · · ■ "•~Txn-îxn-lxrr (— 1 ) -^— — X\X'i. . Xn^ì Χη-\ -f" Χ{ Χ·ι. . Χη^Χη χ χ *Ά -.. χη. 2. ^\xn\. (3#6) If 1 JC*!