Data Shackling Locality enhancement of dense numerical linear algebra codes Traversals along co-ordinate axes Data-centric reference for each statement No array copying

Workloads Basic Linear Algebra Subroutines Dot Product : xTy Sapxy : α * x + y Matrix vector : y = A * x Triangular Solve : Lx = b Matrix Multiplication : C = C + A*B Matrix Factorizations Cholesky LU QR

Why is BLAS approach not good enough? What is BLAS? Machine specific code for each BLAS Matrix factorizations are blocked Exposes a BLAS like interface BLAS not portable but matrix codes are  Automating the production of block codes not easy Too difficult an approach for compilers

What is a data shackle? One array in the program Divided into blocks Using parallel equally spaced cutting planes Order for visiting blocks of data One reference of that array is selected for each statement

Intuitively…A Data Shackle Specifies the order for touching blocks Data centric reference Which iterations of each statement are performed When that block is touched Code is generated to group those iterations together

Example – Matrix Multiplication Obtain the data shackle on C Divide C into 25 x 25 blocks Vertical and horizontal cutting planes Visited : L-R T-B C(i,j) is data-centric reference

Example Contd…

Discussion on example More optimzed code can be produced by folding the loop bounds Within a block order of instructions preserved Cannot say the same for program Not the real blocked code Low locality for A and B Compose shackles.. later

Are all shackles legal? Definitely not Program instructions reordered