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However, this is not to say that we should avoid the PBLAS layer. 1 and 4, as applied LU = P A factorization, are recent results obtained with Sid Chatterjee, Jim Sexton and mainly John Gunnels. 72 TFlops in the fall of 2004 that placed IBM number one in the TOP500 list. 2. References 1. R. C. Agarwal, F. G. Gustavson. A Parallel Implementation of Matrix Multiplication and LU factorization on the IBM 3090. 5 Working Group on Aspects of Computation on Asychronous Parallel Processors, book, Margaret Wright, ed.

TRSM Operation. First, we consider solving AX = C, where X overwrites C. A of size m × m is upper triangular, and C and X are m × n. Depending on m and n, there are several alternatives for doing a recursive splitting. Two of them are illustrated below. Case 1 (1 ≤ m ≤ n/2). Split C by columns only, A X1 X2 = C1 C2 , or, equivalently, AX1 = C1 , AX2 = C2 . Case 2 (1 ≤ n ≤ m/2). Split A, which is assumed to be upper triangular, by rows and columns. Since the number of right-hand sides n is much smaller than m, C is split by rows only, X1 C1 A11 A12 = , A22 X2 C2 or, equivalently, A11 X1 = C1 − A12 X2 , A22 X2 = C2 .

The TLB contains a finite set of pages. These pages are known as the current working set of New Generalized Data Structures for Matrices 15 the computation. If the computation addresses only memory in the TLB then there is no penalty. Otherwise, a TLB miss occurs resulting in a large performance penalty; see [2]. Cache blocking reduces traffic between the memory and cache. Analogously, register blocking reduces traffic between cache and the registers of the CPU. Cache and register blocking are further discussed in [2].