By Wolfgang Bangerth
Textual content compiled from the fabric provided by way of the second one writer in a lecture sequence on the division of arithmetic of the ETH Zurich in the course of the summer season time period 2002. techniques of 'self-adaptivity' within the numerical answer of differential equations are mentioned, with emphasis on Galerkin finite aspect types. Softcover.
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Extra resources for Adaptive finite element methods for differential equations
Assume that floating-point numbers are represented in radix β, where β is a power of 2, and that their significands and exponents are stored on ws and we bits, respectively. 3, when a nonzero number x in the normal range is represented by the nearest floating-point number RN(x), a relative representation error x − RN(x) x is committed. We want to evaluate the maximum and average values of this relative error, for all x between the smallest positive normal floating-point number β emin and the largest one Ω = β emax · (β − β 1−p ).
These traps sometimes make the task of proving properties of arithmetic algorithms a difficult job when some operand can be very near a power of the radix. In the remainder of this book, ulp(x) will be GenGoldbergUlp(x). That is, we will follow Definition 5, not because it is the best (as we have seen, it is difficult to tell which one is the best), but because it is the most used. 6. 2 37 Errors in ulps and relative errors It is important to be able to establish links between errors expressed in ulps, and relative errors.
The bounds given here on the errors due to rounding will be used in particular in Chapter 6. 3 Exceptions In IEEE 754-1985 arithmetic (but also in other standards), an exception can be signaled along with the result of an operation. This can take the form of a status flag (which must be “sticky,” so that the user does not need to check it immediately, but after some sequence of operations, for instance at the end of a function) and/or some trap mechanism. Invalid: This exception is signaled when an input is invalid for the function.