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MiS Preprint
18/2000
The efficient computation of scalar products of certain antisymmetric functions
Wolfgang Hackbusch
Abstract
The solution of Schrödinger's equation leads to a high number N of independent variables. Furthermore, the restriction to (anti)symmetric functions implies some complications. We propose a sparse-grid approximation which leads to a set of non-orthogonal basis. Due to the antisymmetry, scalar products are expressed by sums of $ N \times N $-determinants. More precisely, we have to determine $$ det_K (A) := \sum\limits_{1\leq i_1 , i_2 ,...,i_K \leq N} det (\alpha^{(\beta)}_{i_\alpha , i_\beta })_{\alpha, \beta =1,...,K` }$$ where $ \alpha^{(\beta)}_{i_\alpha ,i_\beta}$ are entries of the K matrices in $A:=(A^{(1)},...,A^{K}$ We propose a method to evaluate this expression such that the computational cost amounts to $O(N^3)$ for fixed K, while the storage requirements are $O(N^2)$.
