By Erricos John Kontoghiorghes (auth.)
Parallel Algorithms for Linear Models offers a whole and specific account of the layout, research and implementation of parallel algorithms for fixing large-scale linear types. It investigates and provides effective, numerically sturdy algorithms for computing the least-squares estimators and different amounts of curiosity on hugely parallel structures.
The monograph is in components. the 1st half comprises 4 chapters and offers with the computational elements for fixing linear versions that experience applicability in different components. the rest chapters shape the second one half, which concentrates on numerical and computational equipment for fixing a number of difficulties linked to likely unrelated regression equations (SURE) and simultaneous equations types.
the sensible problems with the parallel algorithms and the theoretical elements of the numerical tools might be of curiosity to a huge variety of researchers operating within the components of numerical and computational equipment in records and econometrics, parallel numerical algorithms, parallel computing and numerical linear algebra. the purpose of this monograph is to advertise examine within the interface of econometrics, computational facts, numerical linear algebra and parallelism.
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Additional info for Parallel Algorithms for Linear Models: Numerical Methods and Estimation Problems
Example text
In order to simplify the complexity of the timing models the dimension of the data matrix Ai (i = 1, ... , G) is assumed to be a multiple of the size of the array processor and G::; eS2. That is, m = Mesl and n = Nes2, where M ~ N ~ G, eSI = 128 and eS2 = 64. Furthermore, the algorithms have been slightly modified in order to reduce the overheads arising in their straightforward implementation. These overheads mainly comprised the remapping of the affected subarrays into the DPU and were overcome by referencing a subarray only if it was using fewer memory layers than a previous extracted subarray.
10. 10 The Householder algorithm. 1: defHouseh_QRD(A,m,n,G) = 2: for i = 1, ... , n do 3: apply transform(Ai:,i:,: , m - i + 1, n - i + 1, G) 4: end for 5: end def 6: def transform(A, m, n, G) = 7: H := A,I,: 8: S:= sqrt(sum(H *H, 1)) 9: where (HI,: < 0) then S := -S 10: HI,: := HI,: +S 11: B:= HI,: *S 12: W := spread(H,2,n) 13: Z:= sum(W *A, l)jspread(B, l,n) 14: A:=A-W*spread(Z,I,m) 15: end def As in the Householder factorization algorithm, the Modified Gram-Schmidt (MGS) method generates the upper triangular factor Ri row by row, with the difference that it explicitly constructs the orthogonal matrices Qi, where Ai = QiRi (i = 1, ...
N do 2: for j = m, m - 1, ... 2 is given by: n I)m-i) = n(2m-n-1)/2. 2 is equivalent to A := . G~ljG~llA, see Fig. 2. , 15 d3,43)d2,32 )d3,42 )d1,21) x and a blank space denotes a possible non-zero ele'men't, an element annihilated by the Givens rotation and a zero element, respectively. 2. d2,32) - ... - G(1) 2,3 o •• •• G(1) \,2 I·· a·· •• G(3) o. 2, where m = 4 and n = 3. 3, where m = 10 and n = 6. A number i (1 ::; i ::; 39) at position (j, k) indicates where zeros are created by the ith Givens rotation (1 ::; k ::; n and k < j ::; m).