By Hermann Stahl

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Additional resources for Abriss einer Theorie der algebraischen Funktionen einer Veraenderlichen

Example text

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.