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Finding all solutions of piecewise-linear resistive circuits using linear programming

Finding all solutions of piecewise-linear resistive circuits using linear programming,10.1109/81.669067,IEEE Transactions on Circuits and Systems I-re

Finding all solutions of piecewise-linear resistive circuits using linear programming   (Citations: 31)
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An efficient algorithm is proposed for finding all solutions of piecewise-linear resistive circuits. This algorithm is based on a new test for nonexistence of a solution to a system of piecewise-linear equations fi(x)=0(i=1.2,···,n) in a super-region. Unlike the conventional sign test, which checks whether the solution surfaces of the single piecewise-linear equations exist or not in a super-region, the new test checks whether they intersect or not in the super-region. Such a test can be performed by using linear programming. It is shown that the simplex method can be performed very efficiently by exploiting the adjacency of super-regions in each step. The proposed algorithm is much more efficient than the conventional sign test algorithms and can find all solutions of large scale circuits very efficiently. Moreover, it can find all characteristic curves of piecewise-linear resistive circuits
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    • ...For example, by using the piecewise linear approximation to represent the nonlinear characteristic of circuit elements, then the problem can be handled by checking each piece of linear sub-problem for the existence of solutions[1]...

    Sermsak Uatrongjit. A box-splitting strategy of interval Newton method for finding all DC ...

    • ...Recently, other LP tests have been considered by other authors[12]–[16]...

    Stefano Pastore. Fast and Efficient Search for All DC Solutions of PWL Circuits by Mean...

    • ...(1) Some of them [2]-[3] can efficiently find all the solutions of piecewise-linear resistive circuits, where the nonlinear elements are modeled by the piecewise-linear segments...

    Seizo Haginoet al. Tracing Bifurcation Branches of High-Dimensional Nonlinear Algebraic E...

    • ...Therefore, there have been published many papers [1]-[5] for finding the multiple and/or all the solutions of the following nonlinear equation:...
    • ...Some of them [1]-[2] can efficiently find all the solutions of piecewise-linear resistive circuits, where the nonlinear elements are modeled by the piecewise-linear segments...

    Seizo Haginoet al. Finding Complex Conjugate Solutions of Nonlinear Algebraic Equations

    • ...Finding all solutions of piecewise-linear (PWL) resistive circuits is an important problem in circuit simulation, and many efficient algorithms have been proposed for this problem (for example, see [1]‐[12] and also the references cited in [10] and [12])...
    • ...In 1998, an efficient algorithm was proposed for finding all solutions of PWL resistive circuits using a new type of nonexistence test, termed the linear programming (LP) test [10]...
    • ...Namely, PWL resistive circuits (where each PWL resistor is controlled by a single voltage or current) are generally described by systems of PWL equations of the form , but the algorithm proposed in [10] can be applied only when is an identity matrix...
    • ...However, this algorithm is less efficient than the algorithm in [10]...
    • ...This is because the LP test in [12] is less powerful than that in [10] in most cases...
    • ...Another reason is that the LP test in [10] uses special techniques for improving the computational efficiency, but they cannot be applied to the LP test in [12]...
    • ...In this brief, we propose an improved version of the algorithm proposed in [10] that is not only very efficient but also applicable to the case where is not an identity matrix...
    • ...In this section, we first review the algorithm proposed in [10] briefly...
    • ...In the LP test proposed in [10], we apply the simplex method1 to the LP problem...
    • ...In the algorithm proposed in [10], the initial region is divided as shown in Fig. 1, and the LP test is performed on each box...
    • ...In Section IV of [10], two techniques (termed techniques 1 and 2) are proposed for improving the computational efficiency of the algorithm...
    • ...Furthermore, it is easily seen that techniques 1 and 2 proposed in [10] can be applied with minor and trivial modifications...
    • ...We assume that the reader has read Section IV of [10]...
    • ...If we add the row describing the equality constraint to the tableau and do not delete the row describing the inequality constraint (as done in [10]), then the tableau becomes larger as increases...
    • ...Then, we perform Phase I of the simplex method using as the objective function, and then minimize by Phase II (see Technique 2 of [10])...
    • ...Since is an identity matrix in this example, we can also compared the proposed algorithm with the algorithm in [10]...
    • ...The computation time of the algorithm in [10] when is 66 941 s, which is about 25 times larger than that of the proposed algorithm...
    • ...Note that the algorithm proposed in [10] cannot be applied to this system because is not an identity matrix...
    • ...In this brief, an improved version of the algorithm in [10] has been proposed for finding all solutions of PWL resistive circuits...
    • ...The proposed algorithm is not only applicable to a broader class of circuits than the algorithm in [10] but also much more efficient than the algorithms in [10] and [12]...
    • ...The proposed algorithm is not only applicable to a broader class of circuits than the algorithm in [10] but also much more efficient than the algorithms in [10] and [12]...

    Kiyotaka Yamamuraet al. Finding all solutions of piecewise-linear resistive circuits using the...

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