1. October 14-15, 2005Conformal Computing 20051 Geometry of Arrays: Mathematics of Arrays and  calculus Lenore R. Mullin Computer Science Department College of Computing and Information University at Albany, State University of New York

2. October 14-15, 2005Conformal Computing 20052 Overview Mathematics of Arrays (MoA): an algebraic system for dealing with arbitrary multi- dimensional arrays  calculus: reducing a complicated set of array operations to direct indexing of the original array Elimination of temporary arrays: significant enhancement in speed and reduction of memory requirements Example: block decomposition “lifts” (raises) the dimension of the a array

3. October 14-15, 2005Conformal Computing 20053 “Reshape” Operator 2-dimensional Becomes 4-dimensional

4. October 14-15, 2005Conformal Computing 20054 Block Decomposition 2-dimensional Viewed as 4-dimensional

5. October 14-15, 2005Conformal Computing 20055 Array “shapes” Shape operator:  returns a vector containing the lengths of each dimension First vector (two-dimensional): Second vector (four-dimensional):

6. October 14-15, 2005Conformal Computing 20056 “Transpose” operator Transpose operator  permutes the dimensions

7. October 14-15, 2005Conformal Computing 20057 “  operator” The  operator extracts a component of the array Full index extracts an element: Partial index extracts a sub array:

8. October 14-15, 2005Conformal Computing 20058 “Reshape” operator The process of “lifting” the dimension is carried out with the “reshape” operator We write: And:

9. October 14-15, 2005Conformal Computing 20059 “Hypercube” Representation The arrays and are examples of “hypercubes” Often array operations simplify in the hypercube representation In a hypercube: all dimensions have length 2 For an arbitrary array B we write:

10. October 14-15, 2005Conformal Computing 200510 “Product” operator The product operator  returns an integer equal to the product of the elements of a vector The total number of elements in A is: Number of hypercube dimensions:

11. October 14-15, 2005Conformal Computing 200511 Composing operations The hypercube representation of A is thus written: Likewise: Lastly:

12. October 14-15, 2005Conformal Computing 200512 Big Picture:  calculus Multiple operations acting on original array (previous slide) Each operation defined in terms of its action on the indices of the array By applying all operator definitions we “  reduce” the composite operations Final result: prescription (operational normal form (ONF)) for indexing operations on original array

13. October 14-15, 2005Conformal Computing 200513  reduction By selecting the ‘th component of the two- dimensional array: We obtain the ONF: for all 0<=i<2,0<=j<2 (2^0)+(2^0)+(2^0) (2^0)=0, (2^0)+(2^0)+(2^0) (2^1)=1, (2^0)+(2^1)+(2^0) (2^0)=4, (2^0)+(2^1)+(2^0) (2^1)=5 NOTE: a is the row major layout of A.

14. October 14-15, 2005Conformal Computing 200514 Temporal Index Composition Usual methods: Pipeline multiple temporal operations LU followed by QR for example Decompose LU Map to Processors Execute Start the pipeline Set up for QR Decompose Map to processors Execute Feed pipeline

15. October 14-15, 2005Conformal Computing 200515 Temporal Index Composition LU Decomposition 3 3 2 2 1 1 T=0 T=1T=2 3 3 2 2 1

16. October 14-15, 2005Conformal Computing 200516 LU to QR(or anything else) Think of it as indexing Temporal index = 0 1 2 Then again with QR: temporal index = 0 1 2 Then again with something else: index = 0 1 2 Formulate temporal index as an array index like processors or anything else Consequently temporal indices can be Transposed to produce 0 0 0 1 1 1 2 2 2

17. October 14-15, 2005Conformal Computing 200517 3 3 T=0 3 3 2 2 2 2 T=1 1 1 1 1 T=2

18. October 14-15, 2005Conformal Computing 200518 Temporal Index Composition Without transpose: Data moves around connecting networks. Multiple decompositions and communication Filling and emptying pipeline causes delays With transpose: One decomposition Mimimal data flow Highest Performance

19. October 14-15, 2005Conformal Computing 200519 Impact: efficiency and programming ease Efficiency: through direct indexing access we eliminate temporary arrays allowing BIGGER PROBLEMS Minimizing operations: significant speed enhancement Programming ease: ONF directly translated into code (no thinking or hackery) Recent FFT cache optimization: with design (ONF) in hand…code built in ONE DAY and it WORKED!!

20. October 14-15, 2005Conformal Computing 200520 Mathematical Reasoning Same mathematics used to describe the problem as is used to describe the organization of the machine Machine: decomposition over levels of memory, processors, networks, FPGA’s in terms of array dimensions Provability: two designs leading to the same ONF are equivalent!

21. October 14-15, 2005Conformal Computing 200521 Success to date This approach applied to many operations ubiquitous across science and engineering disciplines –FFT, QR decomposition, LU decomposition –Digital signal processing: RADAR (MIT Lincoln Labs) Most recent: FFT cache optimization factor of FOUR speedup over previous records (next talk)

22. October 14-15, 2005Conformal Computing 200522 Ongoing Research General-radix, multiple-dimension FFT optimized over cache and processors Application-specific hardware: programming hybrid software/hardware (FPGA) calculations Grid computing Large-scale materials simulations with density- functional theory Quantum computing and quantum algorithms (Density Matrix naturally expressed as a hypercube and qubits as indices(talk follows).