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Amplitude equations are often used to describe marginally unstable phenomena in physical systems. Allowing for spatial modulation, the canonical forms for these equations are the (in gneeral, coupled) Ginzburg-Landau equations. The form of these equations is universal but the sets of coefficients vary. To treat a specific problem, it is necessary to calculate the values of the coefficients appearing in the amplitude equations, and it is desirable to have an efficient and reliable way to do this.
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In this paper we describe a purely algebraic approach to the problem of finding coupling coefficients, which we have used and described previously, but without details. We show that under the assumption that the Q-R factorization or the Singular Value Decomposition (SVD) of matrices of linear problems is available, steps can be replaced with one matrix-vector multiplication and one division of two complex numbers. There is no need for the explicit calculation of adjoint eigenfunctions of the linear problem.
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Our method provides substantial simplification of the standard computational procedure and reduces the number of operations. All steps of the method are purely algebraic, and numerical quadrature, which adds additional programming complexity and can degrade accuracy, is avoided.
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