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By Hermann Stahl

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9) will be described in Sect. 1. 3 Admissible Partitions 25 Fig. 1 An admissible partition of I × I. algorithm for partitioning I × J with a minimum number of blocks is presented in Sect. 2. This partition is suitable for storing and multiplying A by a vector, but it does not allow to perform matrix operations such as the matrix product and the matrix inverse. If the latter operations are required, the partitioning from Sect. 5 has to be used instead. In that section partitions of I × J for arbitrary admissibility conditions will be constructed based on cluster trees for I and J.

N}. This kind of grid results, for instance, from adaptive refinement towards a singularity at 0; see Fig. 7. 0 ... 1 16 1 8 1 4 1 2 Fig. 7 Adaptive refinement. If we subdivide this set into t1 = {i ∈ t : zi < mt } and t2 = t \ t1 , where mt := 1 1 1 zi = ∑ 2−i = (1 − 2−n ), ∑ n i∈t n i∈t n then for i ∈ t1 it holds that n2−i < 1 − 2−n , which is satisfied for all i > log2 2n. The cluster tree will not be balanced, since t1 contains most of the indices while t2 has only few of them. This situation becomes even worse if we use the centroid mt := ∑ni=1 µ (Xi )zi ∑ni=1 µ (Xi ) with µ (Xi ) = 23 2−i .

If we subdivide this set into t1 = {i ∈ t : zi < mt } and t2 = t \ t1 , where mt := 1 1 1 zi = ∑ 2−i = (1 − 2−n ), ∑ n i∈t n i∈t n then for i ∈ t1 it holds that n2−i < 1 − 2−n , which is satisfied for all i > log2 2n. The cluster tree will not be balanced, since t1 contains most of the indices while t2 has only few of them. This situation becomes even worse if we use the centroid mt := ∑ni=1 µ (Xi )zi ∑ni=1 µ (Xi ) with µ (Xi ) = 23 2−i . From mt = 1 1 − 4−n ∑ni=1 4−i = n 3 1 − 2−n ∑i=1 2−i we see that i ∈ t1 is equivalent to 3 · 2−i (1 − 2−n ) < 1 − 4−n , which is satisfied for all i > log2 3.

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Abriss einer Theorie der algebraischen Funktionen einer Veraenderlichen by Hermann Stahl


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