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Numerical Modeling and Simulation
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
Gaussian elimination is used in a wide variety of engineering problems. The method reduces the matrix (A) to an upper triangular matrix so that the bottom row has only one element, as shown in Figure 4.2(a). Then the equation corresponding to this bottom row is a linear equation with only one unknown, which is easily determined. The remaining unknowns are obtained by back-substitution, going from the bottom row to the top while considering each row in turn, so that each has only one unknown. For the numerical solution, an augmented matrix is formed by placing the column vector (B) at the end of matrix (A). The first step involves eliminating the element in the first column of all the rows below the first row, which is termed the pivot row and the element in the first column the pivot element. In the second step, the second row becomes the pivot row and the element in the second column the pivot element. Again, all the elements in the second column for rows below the second row are eliminated. The process is repeated until the matrix (A) is replaced by an upper triangular matrix.
Fundamentals of Systems of Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
We note that the (1, 1) entry of the augmented matrix (–2 in this case) needed to be nonzero in order to eliminate the entries in the same column, that lie below it. We call −2 a pivot and observe that we will always need to have a nonzero pivot in order to proceed. In the event that we obtain a 0 for a pivot element, we may interchange rows if it will help us proceed; if not, Gaussian elimination fails and the system has dependent rows. Continuing with our previous situation (the next pivot element is 2): () R3−2R2→R3gives[−2042024200−2−4].
Mathematical programming
Published in Dušan Teodorović, Miloš Nikolić, Quantitative Methods in Transportation, 2020
Dušan Teodorović, Miloš Nikolić
The pivot element is the element at the intersection of the pivot row and the pivot column. After selecting the pivot element, we generate the new table in the following way: All elements in the pivot column are equal to zero, except the element which was the pivot element. The value of that element is 1.The new values of the elements in the pivot row are equal to the old values divided by the pivot number.The other elements are determined in the following way. Suppose that element aij is a pivot number, and that we want to determine the new value of the element ars. We should also note elements ais and arj, that are the elements at the intersection of the pivot row and the column s, and at the intersection of the row r and the pivot column. The element at the intersection of row r and column s will have the new value ars−ais⋅arjaij.
Optimal Settings for Adaptive Overcurrent Relay Coordination in Grid-Connected Wind Farms
Published in Electric Power Components and Systems, 2020
Step 5: The current basic variable in the equation corresponding to the smallest nonnegative ratio from step 4 is chosen as the variable to leave the set of basic variable in the current basic feasible solution. Element corresponding to entering variable column and leaving variable row is the pivot element.