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Solving a Linear System with Uncertain Coefficients

This Demonstration shows the LinearSolve solution of linear algebraic systems with uncertain numbers of the form as entries. The rules are extracted from the authors' Uncertain Calculus package. The elements of the initialization diagonal matrix and right-hand side vector are . The "size" of the linear system can be set from 2×2 to 10×10.
A slider is used to set the "real" and "uncertain" parts of the matrix and the vector's uncertain entries for selected "row" and "column". The slider intervals for the real and uncertain parts of the uncertain number are and , respectively. These values are multiplied by , where ranges from -5 to 5. The slider jump is , where ranges from 1 to 6.
For the sizes up to 5, the entire matrix, the solution, and the right-hand side vector are all shown in the pane. For the sizes from 6 to 10, only the acting element of the matrix or vector is shown, but the setting "nothing" lets you inspect the values of the matrix and vector elements by using consecutively the corresponding "row" and "column" controller settings. When all elements of the matrix and vector have zero uncertainty, the solution coincides with the standard one. The uncertainty of elements changes the uncertainty of the solution.

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There is international consensus on the evaluation of standard uncertainty and combined standard uncertainty in measurement and computations [1, 2]. Just as the International System of Units (SI) has brought coherence to all scientific and technological measurements, the worldwide consensus on uncertainty permitted a vast spectrum of results to be readily understood and properly interpreted.The needs of numerous testing laboratories to calculate the measurement uncertainty in large scale are supported by numerous software tools. Rasmussen [3] reviewed 10 different software packages.
To facilitate the calculation of the combined standard uncertainty, similar to the complex and interval numbers, Aibe and Mikhailov introduced a new object, , called an uncertain number [4] and implemented the Uncertain Calculus in Mathematica 6. It transforms a functional relationship of uncertain numbers into an uncertain number , assuming that all arguments of the function are independent. It is related to interval arithmetic. The key difference is that this model of propagating uncertainty, under the standard arithmetic operations of plus, times, and power, is more realistic than the worst-case assumption underlying interval arithmetic's propagation of independent error. Specifically, an model of propagation is used, rather than the more conservative model of interval operations.
It is important to realize that uncertainty propagation, like error propagation from interval arithmetic, is defined operationally. This means, among other things, that a "black box" solver, such as the LinearSolve function of Mathematica, can give different results when using different methods. In particular, results will be heavily influenced by the internal pivot selection strategy. They will also be affected, though to a lesser extent, by whether row reduction is done in two separate stages (forward, then reverse), versus clearing both below and above pivots at the same time.
References:
1. European Cooperation for Accreditation of Laboratories, Expression of the Uncertainty of Measurement in Calibration, EAL-R2 and EAL-R2-S1, 1997.
2. International Organization for Standardization, Guide to the Expression of Uncertainty in Measurement, Geneva, Switzerland, 1993.
3. S. N. Rasmussen, "Software Tools for the Expression of Uncertainty in Measurement," MetroTrade Workshop on Traceability and Measurement Uncertainty in Testing, Berlin, January 30–31, 2003.
4. V. Y. Aibe and M. D. Mikhailov, "Uncertainty Calculus in Metrology," Proceedings of ENCIT 2008, 12th Brazilian Congress of Thermal Engineering and Sciences, Belo Horizonte, MG, Brazil, November 10–14, 2008.
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