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Object-Oriented Modeling and Simulation
Published in Derek A. Linkens, CAD for Control Systems, 2020
Sven Erik Mattsson, Mats Andersson, Karl Johan Åström
The symbolic manipulation should not make the work for the numerical solver worse. Condition numbers and pivoting are important concepts when solving equation systems. By restricting manipulations to be local to each block, much is gained since it is, in general, a bad idea to use an equation to solve for a variable belonging to another diagonal block. When converting a problem into explicit-state space form, Gauss elimination is not a good approach to triangularize the block. It is impossible to make a proper pivoting when only having symbolic expressions. Furthermore, to get a well-conditioned system it may be necessary to change the pivoting along the simulation run. In the worst case it may happen that the triangularization causes division by zero. For small or sparse linear equation systems it is feasible to use Cramer’s rule and calculate the inverse by calculating the determinant and minors.
Linear Algebra
Published in Brian Vick, Applied Engineering Mathematics, 2020
Cramer’s rule is a solution technique that is best suited to small numbers of equations. Consider a 2-by-2 matrix [a11a12a21a22][x1x2]=[b1b2]
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
A standard application of determinants is the solution of a system of linear algebraic equations using Cramer’s Rule. As an example, we consider a simple system of two equations and two unknowns, x and y, in the form () ax+by=e,cx+dy=f.Cramer’s Rule for solving algebraic systems of equations.
Effect of sampling plan and trend removal on residual uncertainty
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2018
Gordon A. Fenton, Farzaneh Naghibi, Michael A. Hicks
whereand where are the coordinates of the centre of the jth sampled cell. The unknowns in the system of equations shown in Equation (13) can be explicitly solved using Cramer’s rule as follows:where is the matrix of coefficients shown in Equation (13) and is a matrix obtained by replacing the ith column of matrix with the right-hand side vector in Equation (13). Substituting the corresponding matrix determinants into Equation (15) and extracting the coefficients of gives the following components:from which the unknowns can be found using Equation (12).