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New element matrices need not be generated and the singular numerical integration was avoided completely.
Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS).
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It avoids the singular integral and numerical interpolation problems encountered in the time-domain numerical method.
Appropriate sentences involving numerical singular terms are sometimes true; witness 'the number of planets is nine' and '1+1=2'.
However, for unstable or singular systems the numerical integration of the dynamic model may fail.
First, one might deny that sentences containing numerical singular terms are (literally) true; one might, for example, endorse a fictionalist account of mathematics (Field 1980).
It is backed up by the Fregean argument considered above that the left-hand side is ontologically committed to numbers in virtue of containing numerical singular terms.
For our purposes, we can sidestep this objection and focus on the other premise of the argument, that extensional sentences containing numerical singular terms, if true, refer to numbers which is to say, in other words, that such sentences, whether true or not, are ontologically committed to numbers.
The routine was able to find the solution with a computational time similar to the time used by a numerical singular value decomposition routine.
It is also important that such methods require the inversion of the Hessian matrix of the energy function, and in the case of reduced dimensionality the matrix is nearly singular, leading to numerical difficulties and loss of accuracy.
Techniques such as those from singular perturbation theory and numerical analysis have proven especially useful in this regard.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com