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The relation between local and global truncation errors is slightly different from in the simpler setting of one-step methods. For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors.
Quantization, in mathematics and digital signal processing, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements. Rounding and truncation are typical examples of quantization processes. Quantization is involved to some degree in ...
One advantage of the local methods is that it is possible to make a matrix to represent all the sensitivities in a system, thus providing an overview that cannot be achieved with global methods if there is a large number of input and output variables. [26]
Example A: Find the truncation in calculating the first derivative of () = at = using a step size of = Solution: The first derivative of () = is ′ =, and at = ...
The condition number of a problem is the ratio of the relative change in the solution to the relative change in the input. [3] A problem is well-conditioned if small relative changes in input result in small relative changes in the solution.
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The backward Euler method is a variant of the (forward) Euler method.Other variants are the semi-implicit Euler method and the exponential Euler method.. The backward Euler method can be seen as a Runge–Kutta method with one stage, described by the Butcher tableau:
In simple cases like in the example above, programmers may manually eliminate the duplicate expressions while writing the code. The greatest source of CSEs are intermediate code sequences generated by the compiler, such as for array indexing calculations, where it is not possible for the developer to manually intervene. In some cases language ...