Abstract
The problem of computing the periodic steady-state response can be formulated as solving a nonlinear equation of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z = F(z)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(z)</tex> Is the solution vector for the nonlinear network after one period of integration from the initial vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> . The convergence of the sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y_0 , y_1 , \cdots</tex> generated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_{r+1} = F(y_r)</tex> can be accelerated by extrapolation methods. This paper presents a unified analysis of three extrapolation methods: the scalar and vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon</tex> -algorithms and the minimum polynomial extrapolation algorithm. The main result of the paper is the theorem giving conditions for quadratic convergence of the extrapolation methods. To obtain this result the methods are studied for linear problems (where F is a linear function) and the error propagation properties are investigated. For autonomous systems a function called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</tex> similar to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> can be defined. In order to obtain quadratic convergence from the extrapolation methods, the derivatives of F and G must be Lipschitz continuous. The appendixes give sufficient conditions for the Lipschitz continuity. A discussion of practical problems related to the implementation of the extrapolation methods is based on the convergence theorem and the error analysis. The performance of the extrapolation methods is demonstrated and compared with other methods for steady-state analysis by four examples, two autonomous and two nonautonomous. Extrapolation methods are very easy to implement, and they are efficient for the steady-state analysis of nonlinear circuits with few reactive elements giving rise to slowly decaying transients.
Published Version
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