The block WZ factorization

Optimized Cramer’s Rule in WZ Factorization and Applications

“…and we then proceed similarly for the central submatrices of size (n − 2k) and so on, where k = 1, 2, ..., n 2 , i, j = k + 1, ..., n − k and z (k) i, j ∈ R, see [9]. We can now re-write Equation ( 12) in matrix form as…”

“…The determinant of Equation ( 19) is nonsingular because Z 2,2 − Z 2,1 Z −1 1,1 Z 1,2 is a lower triangular invertible matrix (see [9]) and…”

“…The factorization has been applied in scientific computing -especially in science and engineering -see also [10,19,21,25,39]. Other variations of W Z factorization are detailed in [15,20,23,36,38], but block W Z factorization (or its Z system ) is discussed in [7,9,26]. The newest and alternative form of W Z factorization with applications is the W H factorization, see [4,6].…”

“…Therefore, W Z factorization has the adaptability to solve linear systems on Single Instruction, Multiple Data (SIMD) or Multiple Instruction, Multiple Data (MIMD) shared memory parallel computers or mesh multiprocessors, see [3,27,30] and the references therein. The efficiency of W Z factorization depends on an efficacious use of the memory echelon because computational cost often relies not only on the total number of arithmetic operations used but also the data transferring time between different memory levels [9].…”

The Parallel Tiled WZ Factorization Algorithm for Multicore Architectures

“…The block WZ factorization is described by Bylina (2018). We present the design of the tiled WZ factorization algorithm, dividing the matrix only into 4 × 4 = 16 tiles, and this construction can be easily generalized for r 2 tiles.…”