By Robert Vichnevetsky, J. Vignes, R. Vichnevetsky
ISBN-10: 0444700676
ISBN-13: 9780444700674
Numerical arithmetic and functions
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Extra resources for Numerical Mathematics and Applications
Example text
For example, Φ can consist of step functions which have nodes and values which are floating-point numbers. An upward rounding operator Δ from the set R of real functions into Φ has the following properties: (i) Au = u if u 6 Φ, (ii) u < Au for u £ R, (iii) if u < v, then Au < Av. These conditions imply, among other things, that there is no element v of Φ such that u < v < Av [3], hence, this rounding is "optimal" in a certain sense. The downward rounding operator V is defined in a similar way, with the inequality signs reversed.
INCIDENCE OF ROUNDING ERRORS. where Θ. (x) + ε. inby the definition of |(2) n(3) . „ i > c i r bound ε. 3 - Proposition 5 (Lower bounds). (v such that K n keK. (x)|) X * X 1 > r . (x) / V X keK. 1 |Α·χν| X K Under assumptions (C) and (E_) one can obtain a mean value : Checking the Quality of the Solution of a Linear System (2) 1 k- ,2 = Π x (I k£K. (xJ )) Σ such that Θ. is n l close to 0 (less than 2 for instance), then there is an index k e K. such that the relative 1 value ε. 1 - The normed residue.
T))dt], (20) As an example of this type of computation, consider the integrand G(s,t,U(s),U(t)) = (19) = [g(s,t,û(s),u(t)), g(s,t,û(s),u(t))] will provide an interval inclusion of g(s,t,u(s),u(t)) for u € U. Since interval integration is inclusion monotone [1], integration of (19) will yield an interval inclusion G of the integral operator g. Methods using monotonicity properties of operators to obtain lower and upper bounds for solutions of functional equations have been investigated extensively by Collatz [2] and others.