The solution vector is dynamically updated by one of the Mathematica
built-in functions PseudoInverse
, or LinearSolve
. It is desirable to use PseudoInverse
when setting the matrix and vector elements. Once the desired matrix and rhs vector are fixed, you can compare the solutions given by all four built-in functions for real, interval, and uncertain numbers.
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 permits a vast spectrum of results to be readily understood and properly interpreted. The needs of numerous testing laboratories to calculate the measurement uncertainty on a large scale is supported by numerous software tools. Rasmussen  refers to 10 different software packages.
To facilitate the calculation of the combined standard uncertainty similar to complex and interval numbers, Aibe and Mikhailov introduced a new object
called an uncertain number  and developed in Mathematica
6 an Uncertain Calculus package transforming any functional relationship of uncertain numbers
into an uncertain number
, assuming that all arguments of the function
are independent. The rules manipulating uncertain numbers in this Demonstration are extracted from the Uncertain Calculus package .
This Demonstration solves a system of linear equations
is a vector of variables and
are a matrix and a vector with real, interval, and uncertain elements. The real and interval numbers are obtained from the uncertain numbers.
The elements of the initialization diagonal matrix and vector are uncertain numbers 1±0. The "size" of the system is limited to four equations in order to show the entire matrix dot multiplied by the solution vector and the rhs vector. The uncertainty of elements changes only the uncertainty of the solution.
When the matrix has a nonzero determinant, then the built-in functions Inverse
find the unique solution. However, if the determinant is zero (the matrix is singular) there may be either no vector or an infinite number of vectors satisfying the system. Nevertheless, for a singular matrix the built-in functions PseudoInverse
give solutions that minimize the sum of the squares of all entries in ·Inverse-
is the identity matrix. The Demonstration lets you compare the uncertainty introduced by all four of these built-in functions.
In the authors' experiments, Inverse
surprisingly introduces less uncertainty than LinearSolve
, while PseudoInverse
introduce the highest uncertainty.
 European Cooperation for Accreditation of Laboratories, Expression of the Uncertainty of Measurement in Calibration
, EAL-R2 and EAL-R2-S1, 1997.
 International Organization for Standardization, Guide to the Expression of Uncertainty in Measurement
, Geneva, Switzerland: ISO, 1995.
 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.
 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.