1. 1 Minimizing Communication in Numerical Linear Algebra www.cs.berkeley.edu/~demmel Case Study: Matrix Multiply Jim Demmel EECS & Math Departments, UC Berkeley demmel@cs.berkeley.edu

2. Summer School-Lecture 2 2 Why Matrix Multiplication? An important kernel in many problems Appears in many linear algebra algorithms Bottleneck for dense linear algebra Closely related to other algorithms, e.g., transitive closure on a graph using Floyd-Warshall Optimization ideas can be used in other problems The best case for optimization payoffs The most-studied algorithm in high performance computing

3. Summer School-Lecture 2 3 Matrix-multiply, optimized several ways Speed of n-by-n matrix multiply on Sun Ultra-1/170, peak = 330 MFlops

4. Summer School-Lecture 2 4 Note on Matrix Storage A matrix is a 2-D array of elements, but memory addresses are “1-D” Conventions for matrix layout by column, or “column major” (Fortran default); A(i,j) at A+i+j*n by row, or “row major” (C default) A(i,j) at A+i*n+j recursive (later) Column major (for now) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 4 8 12 16 1 5 9 13 17 2 6 10 14 18 3 7 11 15 19 Column major Row major cachelines Blue row of matrix is stored in red cachelines Figure source: Larry Carter, UCSD Column major matrix in memory

5. Summer School-Lecture 2 5 Computational Intensity: Key to algorithm efficiency Machine Balance: Key to machine efficiency Using a Simple(r) Model of Memory to Optimize Assume just 2 levels in the hierarchy, fast and slow All data initially in slow memory m = number of memory elements (words) moved between fast and slow memory t m = time per slow memory operation f = number of arithmetic operations t f = time per arithmetic operation << t m q = f / m average number of flops per slow memory access Minimum possible time = f* t f when all data in fast memory Actual time f * t f + m * t m = f * t f * (1 + t m /t f * 1/q) Larger q means time closer to minimum f * t f q  t m /t f needed to get at least half of peak speed

6. Summer School-Lecture 2 6 Warm up: Matrix-vector multiplication {implements y = y + A*x} for i = 1:n for j = 1:n y(i) = y(i) + A(i,j)*x(j) = + * y(i) A(i,:) x(:)

7. Summer School-Lecture 2 7 Warm up: Matrix-vector multiplication {read x(1:n) into fast memory} {read y(1:n) into fast memory} for i = 1:n {read row i of A into fast memory} for j = 1:n y(i) = y(i) + A(i,j)*x(j) {write y(1:n) back to slow memory} m = number of slow memory refs = 3n + n 2 f = number of arithmetic operations = 2n 2 q = f / m  2 Matrix-vector multiplication limited by slow memory speed

8. Summer School-Lecture 2 8 Modeling Matrix-Vector Multiplication Compute time for nxn = 1000x1000 matrix Time f * t f + m * t m = f * t f * (1 + t m /t f * 1/q) = 2*n 2 * t f * (1 + t m /t f * 1/2) For t f and t m, using data from R. Vuduc’s PhD (pp 351-3) http://bebop.cs.berkeley.edu/pubs/vuduc2003-dissertation.pdf For t m use minimum-memory-latency / words-per-cache-line machine balance (q must be at least this for ½ peak speed)

9. Summer School-Lecture 2 9 Simplifying Assumptions What simplifying assumptions did we make in this analysis? Ignored parallelism in processor between memory and arithmetic within the processor Sometimes drop arithmetic term in this type of analysis Assumed fast memory was large enough to hold three vectors Reasonable if we are talking about any level of cache Not if we are talking about registers (~32 words) Assumed the cost of a fast memory access is 0 Reasonable if we are talking about registers Not necessarily if we are talking about cache (1-2 cycles for L1) Memory latency is constant Could simplify even further by ignoring memory operations in X and Y vectors Mflop rate/element = 2 / (2* t f + t m )

10. Summer School-Lecture 2 10 Validating the Model How well does the model predict actual performance? Actual DGEMV: Most highly optimized code for the platform Model sufficient to compare across machines But under-predicting on most recent ones due to latency estimate

11. Summer School-Lecture 2 11 Naïve Matrix Multiply {implements C = C + A*B} for i = 1 to n for j = 1 to n for k = 1 to n C(i,j) = C(i,j) + A(i,k) * B(k,j) =+* C(i,j) A(i,:) B(:,j) Algorithm has 2*n 3 = O(n 3 ) Flops and operates on 3*n 2 words of memory q potentially as large as 2*n 3 / 3*n 2 = O(n)

12. Summer School-Lecture 2 12 Naïve Matrix Multiply {implements C = C + A*B} for i = 1 to n {read row i of A into fast memory} for j = 1 to n {read C(i,j) into fast memory} {read column j of B into fast memory} for k = 1 to n C(i,j) = C(i,j) + A(i,k) * B(k,j) {write C(i,j) back to slow memory} =+* C(i,j) A(i,:) B(:,j) C(i,j)

13. Summer School-Lecture 2 13 Naïve Matrix Multiply Number of slow memory references on unblocked matrix multiply m = n 3 to read each column of B n times + n 2 to read each row of A once + 2n 2 to read and write each element of C once = n 3 + 3n 2 So q = f / m = 2n 3 / ( n 3 + 3n 2 )  2 for large n, no improvement over matrix-vector multiply Inner two loops are just matrix-vector multiply, of row i of A times B Similar for any other order of 3 loops =+* C(i,j) A(i,:) B(:,j)

14. Summer School-Lecture 2 14 Matrix-multiply, optimized several ways Speed of n-by-n matrix multiply on Sun Ultra-1/170, peak = 330 MFlops

15. Summer School-Lecture 2 15 Naïve Matrix Multiply on RS/6000 T = N 4.7 O(N 3 ) performance would have constant cycles/flop Performance looks like O(N 4.7 ) Size 2000 took 5 days 12000 would take 1095 years Slide source: Larry Carter, UCSD

16. Summer School-Lecture 2 16 Naïve Matrix Multiply on RS/6000 Slide source: Larry Carter, UCSD Page miss every iteration TLB miss every iteration Cache miss every 16 iterations Page miss every 512 iterations

17. Summer School-Lecture 2 17 Blocked (Tiled) Matrix Multiply Consider A,B,C to be N-by-N matrices of b-by-b subblocks where b=n / N is called the block size for i = 1 to N for j = 1 to N {read block C(i,j) into fast memory} for k = 1 to N {read block A(i,k) into fast memory} {read block B(k,j) into fast memory} C(i,j) = C(i,j) + A(i,k) * B(k,j) {do a matrix multiply on blocks} {write block C(i,j) back to slow memory} =+* C(i,j) A(i,k) B(k,j)

18. Summer School-Lecture 2 18 Blocked (Tiled) Matrix Multiply Recall: m is amount memory traffic between slow and fast memory matrix has nxn elements, and NxN blocks each of size bxb f is number of floating point operations, 2n 3 for this problem q = f / m is our measure of algorithm efficiency in the memory system So: m = N*n 2 read each block of B N 3 times (N 3 * b 2 = N 3 * (n/N) 2 = N*n 2 ) + N*n 2 read each block of A N 3 times + 2n 2 read and write each block of C once = (2N + 2) * n 2 So computational intensity q = f / m = 2n 3 / ((2N + 2) * n 2 )  n / N = b for large n So we can improve performance by increasing the blocksize b Can be much faster than matrix-vector multiply (q=2)

19. Summer School-Lecture 2 19 Analyzing Machine Speed Limits The blocked algorithm has computational intensity q  b The larger the block size, the more efficient our algorithm will be Limit: All three blocks from A,B,C must fit in fast memory (cache), so we cannot make these blocks arbitrarily large Assume your fast memory has size M fast 3b 2  M fast, so q  b  (M fast /3) 1/2 To build a machine to run matrix multiply at 1/2 peak arithmetic speed of the machine, we need a fast memory of size M fast  3b 2  3q 2 = 3(t m /t f ) 2 This size is reasonable for L1 cache, but not for register sets Note: analysis assumes it is possible to schedule the instructions perfectly

20. Summer School-Lecture 2 20 Limits to Optimizing Matrix Multiply The blocked algorithm changes the order in which values are accumulated into each C[i,j] by applying commutativity and associativity Get slightly different answers from naïve code, because of roundoff - OK The previous analysis showed that the blocked algorithm has computational intensity: q  b  (M fast /3) 1/2 There is a lower bound result that says we cannot do any better than this (using only associativity) Theorem (Hong & Kung, 1981): Any reorganization of this algorithm (that uses only associativity) is limited to q = O( (M fast ) 1/2 ) Does not apply to algorithms like Strassen

21. Summer School-Lecture 2 Lower bound for all “direct” linear algebra Holds for sequential algorithms for Matmul, BLAS, LU, QR, eig, SVD, … Some whole programs (sequences of these operations, no matter how they are interleaved, eg computing A k ) Dense and sparse matrices (where #flops << n 3 ) Some graph-theoretic algorithms (eg Floyd-Warshall) Proof in Lecture 3 21 Let M = “fast” memory size #words_moved =  (#flops / M 1/2 ) #messages_sent =  (#flops / M 3/2 )

22. Summer School-Lecture 2 22 What if there are more than 2 levels of memory? Recall goal is to minimize communication between all levels The tiled algorithm requires finding a good block size Machine dependent Need to “block” b x b matrix multiply in inner most loop 1 level of memory  3 nested loops (naïve algorithm) 2 levels of memory  6 nested loops 3 levels of memory  9 nested loops … Cache Oblivious Algorithms offer an alternative Treat nxn matrix multiply as a set of smaller problems Eventually, these will fit in cache Will minimize # words moved between every level of memory hierarchy (between L1 and L2 cache, L2 and L3, L3 and main memory etc.) – at least asymptotically

23. Summer School-Lecture 2 Recursive Matrix Multiplication (RMM) (1/2) For simplicity: square matrices with n = 2 m C = = A · B = · · = True when each A ij etc 1x1 or n/2 x n/2 23 A 11 A 12 A 21 A 22 B 11 B 12 B 21 B 22 C 11 C 12 C 21 C 22 A 11 ·B 11 + A 12 ·B 21 A 11 ·B 12 + A 12 ·B 22 A 21 ·B 11 + A 22 ·B 21 A 21 ·B 12 + A 22 ·B 22 func C = RMM (A, B, n) if n = 1, C = A * B, else { C 11 = RMM (A 11, B 11, n/2) + RMM (A 12, B 21, n/2) C 12 = RMM (A 11, B 12, n/2) + RMM (A 12, B 22, n/2) C 21 = RMM (A 21, B 11, n/2) + RMM (A 22, B 21, n/2) C 22 = RMM (A 21, B 12, n/2) + RMM (A 22, B 22, n/2) } return

24. Summer School-Lecture 2 Recursive Matrix Multiplication (2/2) 24 func C = RMM (A, B, n) if n=1, C = A * B, else { C 11 = RMM (A 11, B 11, n/2) + RMM (A 12, B 21, n/2) C 12 = RMM (A 11, B 12, n/2) + RMM (A 12, B 22, n/2) C 21 = RMM (A 21, B 11, n/2) + RMM (A 22, B 21, n/2) C 22 = RMM (A 21, B 12, n/2) + RMM (A 22, B 22, n/2) } return A(n) = # arithmetic operations in RMM(.,., n) = 8 · A(n/2) + 4(n/2) 2 if n > 1, else 1 = 2n 3 … same operations as usual, in different order M(n) = # words moved between fast, slow memory by RMM(.,., n) = 8 · M(n/2) + 4(n/2) 2 if 3n 2 > M fast, else 3n 2 = O( n 3 / (M fast ) 1/2 + n 2 ) … same as blocked matmul

25. Summer School-Lecture 2 25 Recursion: Cache Oblivious Algorithms Recursion for general A (mxn) * B (nxp) Case1: m>= max{n,p}: split A horizontally: Case 2 : n>= max{m,p}: split A vertically and B horizontally Case 3: p>= max{m,n}: split B vertically Attains lower bound in O() sense Case 1 Case 3 Case 2 1 2

26. Summer School-Lecture 2 Experience with Cache-Oblivious Algorithms In practice, need to cut off recursion well before 1x1 blocks Call “Micro-kernel” for small blocks, eg 16 x 16 Implementing a high-performance Cache-Oblivious code is not easy Using fully recursive approach with highly optimized recursive micro-kernel, Pingali et al report that they never got more than 2/3 of peak. Issues with Cache Oblivious (recursive) approach Recursive Micro-Kernels yield less performance than iterative ones using same scheduling techniques Pre-fetching is needed to compete with best code: not well-understood in the context of Cache Oblivous codes Unpublished work, presented at LACSI 2006

27. Summer School-Lecture 2 Minimizing latency requires new data structures To minimize latency, need to load/store whole rectangular subblock of matrix with one “message” Incompatible with conventional columnwise (rowwise) storage Ex: Rows (columns) not in contiguous memory locations Blocked storage: store as matrix of bxb blocks, each block stored contiguously Ok for one level of memory hierarchy, what if more? Recursive blocked storage: store each block using subblocks Also known as “space filling curves”, “Morton ordering” 27

28. Summer School-Lecture 2 28 Comparing matrix-vector to matrix-matrix mult Data source: Jack Dongarra Matrix-matrix mult ≡ DGEMM, matrix-vector mult ≡ DGEMV

29. Summer School-Lecture 2 How hard is hand-tuning matmul, anyway? 29 Results of 22 student teams trying to tune matrix-multiply, in CS267 Spr09 Students given “blocked” code to start with Still hard to get close to vendor tuned performance (ACML) For more discussion, see www.cs.berkeley.edu/~volkov/cs267.sp09/hw1/results/

30. Summer School-Lecture 2 How hard is hand-tuning matmul, anyway? 30

31. Summer School-Lecture 2 31 What part of the Matmul Search Space Looks Like A 2-D slice of a 3-D register-tile search space. The dark blue region was pruned. (Platform: Sun Ultra-IIi, 333 MHz, 667 Mflop/s peak, Sun cc v5.0 compiler) Number of columns in register block Number of rows in register block Finding needle in haystack!

32. Summer School-Lecture 2 Automatic Performance Tuning Goal: Let machine do hard work of writing fast code What do tuning of Matmul, dense BLAS, FFTs, signal processing, have in common? Can do the tuning off-line: once per architecture, algorithm Can take as much time as necessary (hours, a week…) At run-time, algorithm choice may depend only on few parameters ( matrix dimensions, size of FFT, etc.) Examples: PHiPAC, ATLAS, FFTW, Spiral Can’t always do tuning off-line Algorithm and implementation may strongly depend on data only known at run-time Ex: Sparse matrix nonzero pattern determines both best data structure and implementation of Sparse-matrix-vector-multiplication (SpMV) Part of search for best algorithm just be done (very quickly!) at run-time Example: OSKI 32

33. Summer School-Lecture 2 33 Autotuning Matmul with ATLAS (n = 500) ATLAS is faster than all other portable BLAS implementations and it is comparable with machine-specific libraries provided by the vendor. ATLAS written by C. Whaley, inspired by PHiPAC, by Asanovic, Bilmes, Chin, D. Source: Jack Dongarra

34. Summer School-Lecture 2 Parallel matrix-matrix multiplication Consider distributed memory machines Each processor has its own private memory Communication by sending messages over a network Examples: MPI, UPC, Titanium First question: how is matrix initially distributed across different processors? 34

35. Summer School-Lecture 2 Different Parallel Data Layouts for Matrices (not all!) 0123012301230123 01230123 1) 1D Column Blocked Layout2) 1D Column Cyclic Layout 3) 1D Column Block Cyclic Layout 4) Row versions of the previous layouts Generalizes others 01010101 23232323 01010101 23232323 01010101 23232323 01010101 23232323 6) 2D Row and Column Block Cyclic Layout 0123 01 23 5) 2D Row and Column Blocked Layout b 35

36. Summer School-Lecture 2 36 Parallel Matrix-Vector Product Compute y = y + A*x, where A is a dense matrix Layout: 1D row blocked A(i) refers to the n by n/p block row that processor i owns x(i) and y(i) similarly refer to segments of x,y owned by i Algorithm: Foreach processor i Broadcast x(i) Compute y(i) = A(i)*x Algorithm uses the formula y(i) = y(i) + A(i)*x = y(i) +  j A(i,j)*x(j) x y P0 P1 P2 P3 P0 P1 P2 P3 A(0) A(1) A(2) A(3)

37. Summer School-Lecture 2 37 Matrix-Vector Product y = y + A*x A column layout of the matrix eliminates the broadcast of x But adds a reduction to update the destination y A 2D blocked layout uses a broadcast and reduction, both on a subset of processors sqrt(p) for square processor grid P0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15

38. Summer School-Lecture 2 38 Parallel Matrix Multiply Computing C=C+A*B Using basic algorithm: 2*n 3 Flops Depends on: Data layout Topology of machine Scheduling communication Use of simple performance model for algorithm design Message Time = “latency” + #words * time-per-word =  + #words *  Efficiency: serial time / (p * parallel time) perfect (linear) speedup  efficiency = 1

39. Summer School-Lecture 2 39 Matrix Multiply with 1D Column Layout Assume matrices are n x n and n is divisible by p A(i) refers to the n by n/p block column that processor i owns (similiarly for B(i) and C(i)) B(i,j) is the n/p by n/p sublock of B(i) in rows j*n/p through (j+1)*n/p Algorithm uses the formula C(i) = C(i) + A*B(i) = C(i) +  j A(j)*B(j,i) p0p1p2p3p5p4p6p7 May be a reasonable assumption for analysis, not for code

40. Summer School-Lecture 2 40 Matrix Multiply: 1D Layout on Bus or Ring Algorithm uses the formula C(i) = C(i) + A*B(i) = C(i) +  j A(j)*B(j,i) First consider a bus-connected machine without broadcast: only one pair of processors can communicate at a time (ethernet) Second consider a machine with processors on a ring: all processors may communicate with nearest neighbors simultaneously

41. Summer School-Lecture 2 41 MatMul: 1D layout on Bus without Broadcast Naïve algorithm: C(myproc) = C(myproc) + A(myproc)*B(myproc,myproc) for i = 0 to p-1 for j = 0 to p-1 except i if (myproc == i) send A(i) to processor j if (myproc == j) receive A(i) from processor i C(myproc) = C(myproc) + A(i)*B(i,myproc) barrier Cost of inner loop: computation: 2*n*(n/p) 2 = 2*n 3 /p 2 communication:  +  *n 2 /p

42. Summer School-Lecture 2 42 Naïve MatMul (continued) Cost of inner loop: computation: 2*n*(n/p) 2 = 2*n 3 /p 2 communication:  +  *n 2 /p … approximately Only 1 pair of processors (i and j) are active on any iteration, and of those, only i is doing computation => the algorithm is almost entirely serial Running time: = (p*(p-1) + 1)*computation + p*(p-1)*communication  2*n 3 + p 2 *  + p*n 2 *  This is worse than the serial time and grows with p. When might you still want to do this?

43. Summer School-Lecture 2 43 Matmul for 1D layout on a Processor Ring Pairs of adjacent processors can communicate simultaneously Copy A(myproc) into Tmp C(myproc) = C(myproc) + Tmp*B(myproc, myproc) for j = 1 to p-1 Send Tmp to processor myproc+1 mod p Receive Tmp from processor myproc-1 mod p C(myproc) = C(myproc) + Tmp*B( myproc-j mod p, myproc) Need to be careful about talking to neighboring processors May want double buffering in practice for overlap Ignoring deadlock details in code Time of inner loop = 2*(  +  *n 2 /p) + 2*n*(n/p) 2

44. Summer School-Lecture 2 44 Matmul for 1D layout on a Processor Ring Time of inner loop = 2*(  +  *n 2 /p) + 2*n*(n/p) 2 Total Time = 2*n* (n/p) 2 + (p-1) * Time of inner loop  2*n 3 /p + 2*p*  + 2*  *n 2 (Nearly) Optimal for 1D layout on Ring or Bus, even with Broadcast: Perfect speedup for arithmetic A(myproc) must move to each other processor, costs at least (p-1)*cost of sending n*(n/p) words Parallel Efficiency = 2*n 3 / (p * Total Time) = 1/(1 +  * p 2 /(2*n 3 ) +  * p/(2*n) ) = 1/ (1 + O(p/n)) Grows to 1 as n/p increases (or  and  shrink)

45. Summer School-Lecture 2 45 MatMul with 2D Layout Consider processors in 2D grid (physical or logical) Processors communicate with 4 nearest neighbors Assume p processors form square s x s grid, s = p 1/2 p(0,0) p(0,1) p(0,2) p(1,0) p(1,1) p(1,2) p(2,0) p(2,1) p(2,2) p(0,0) p(0,1) p(0,2) p(1,0) p(1,1) p(1,2) p(2,0) p(2,1) p(2,2) p(0,0) p(0,1) p(0,2) p(1,0) p(1,1) p(1,2) p(2,0) p(2,1) p(2,2) =*

46. Summer School-Lecture 2 46 Cannon’s Algorithm … C(i,j) = C(i,j) +  A(i,k)*B(k,j) … assume s = sqrt(p) is an integer forall i=0 to s-1 … “skew” A left-circular-shift row i of A by i … so that A(i,j) overwritten by A(i,(j+i)mod s) forall i=0 to s-1 … “skew” B up-circular-shift column i of B by i … so that B(i,j) overwritten by B((i+j)mod s), j) for k=0 to s-1 … sequential forall i=0 to s-1 and j=0 to s-1 … all processors in parallel C(i,j) = C(i,j) + A(i,j)*B(i,j) left-circular-shift each row of A by 1 up-circular-shift each column of B by 1 k

47. Summer School-Lecture 2 47 C(1,2) = A(1,0) * B(0,2) + A(1,1) * B(1,2) + A(1,2) * B(2,2) Cannon’s Matrix Multiplication

48. Summer School-Lecture 2 48 Initial Step to Skew Matrices in Cannon Initial blocked input After skewing before initial block multiplies A(1,0) A(2,0) A(0,1)A(0,2) A(1,1) A(2,1) A(1,2) A(2,2) A(0,0) B(0,1)B(0,2) B(1,0) B(2,0) B(1,1)B(1,2) B(2,1)B(2,2) B(0,0) A(1,0) A(2,0) A(0,1)A(0,2) A(1,1) A(2,1) A(1,2) A(2,2) A(0,0) B(0,1) B(0,2)B(1,0) B(2,0) B(1,1) B(1,2) B(2,1) B(2,2)B(0,0)

49. Summer School-Lecture 2 49 Skewing Steps in Cannon All blocks of A must multiply all like-colored blocks of B First step Second Third A(1,0) A(2,0) A(0,1)A(0,2) A(1,1) A(2,1) A(1,2) A(2,2) A(0,0) B(0,1) B(0,2)B(1,0) B(2,0) B(1,1) B(1,2) B(2,1) B(2,2)B(0,0) A(1,0) A(2,0) A(0,1) A(0,2) A(2,1) A(1,2) B(0,1) B(0,2)B(1,0) B(2,0) B(1,1) B(1,2) B(2,1) B(2,2)B(0,0) A(1,0) A(2,0) A(0,1) A(0,2) A(1,1) A(2,1) A(1,2) A(2,2) A(0,0) B(0,1) B(0,2)B(1,0) B(2,0) B(1,1) B(1,2) B(2,1) B(2,2)B(0,0) A(1,1) A(2,2) A(0,0)

50. Summer School-Lecture 2 50 Cost of Cannon’s Algorithm forall i=0 to s-1 … recall s = sqrt(p) left-circular-shift row i of A by i … cost ≤ s*(  +  *n 2 /p) forall i=0 to s-1 up-circular-shift column i of B by i … cost ≤ s*(  +  *n 2 /p) for k=0 to s-1 forall i=0 to s-1 and j=0 to s-1 C(i,j) = C(i,j) + A(i,j)*B(i,j) … cost = 2*(n/s) 3 = 2*n 3 /p 3/2 left-circular-shift each row of A by 1 … cost =  +  *n 2 /p up-circular-shift each column of B by 1 … cost =  +  *n 2 /p ° Total Time = 2*n 3 /p + 4 * s*  + 4*  *n 2 /s ° Parallel Efficiency = 2*n 3 / (p * Total Time) = 1/( 1 +  * 2*(s/n) 3 +  * 2*(s/n) ) = 1/(1 + O(sqrt(p)/n)) ° Grows to 1 as n/s = n/sqrt(p) = sqrt(data per processor) grows ° Better than 1D layout, which had Efficiency = 1/(1 + O(p/n))

51. Summer School-Lecture 2 Lower bound for all “direct” linear algebra Holds for parallel and sequential algorithms for Matmul (attained by Cannon), BLAS, LU, QR, eig, SVD, … Some whole programs (sequences of these operations, no matter how they are interleaved, eg computing A k ) Dense and sparse matrices (where #flops << n 3 ) Some graph-theoretic algorithms (eg Floyd-Warshall) Proof in Lecture 3 51 Let M = “fast” memory size = O(n 2 /p) per processor #words_moved (by at least one processor) =  (#flops / M 1/2 ) =  ((n 3 /p) / (n 2 /p) 1/2 ) =  (n 2 /p 1/2 ) #messages_sent (by at least one processor) =  (#flops / M 3/2 ) =  (p 1/2 )

52. Summer School-Lecture 2 52 Pros and Cons of Cannon So what if it’s “optimal”, is it fast? Yes: Local computation one call to (optimized) matrix-multiply Hard to generalize for p not a perfect square A and B not square Dimensions of A, B not perfectly divisible by s=sqrt(p) A and B not “aligned” in the way they are stored on processors block-cyclic layouts Memory hog (extra copies of local matrices)

53. Summer School-Lecture 2 53 SUMMA Algorithm SUMMA = Scalable Universal Matrix Multiply Slightly less efficient than Cannon, but simpler and easier to generalize Can accommodate any layout, dimensions, alignment Uses broadcast of submatrices instead of circular shifts Sends log p times as much data as Cannon Can use much less extra memory than Cannon, but send more messages Similar ideas appeared many times Used in practice in PBLAS = Parallel BLAS www.netlib.org/lapack/lawns/lawn{96,100}.ps

54. Summer School-Lecture 2 54 SUMMA * = i j A(i,k) k k B(k,j) i, j represent all rows, columns owned by a processor k is a block of b  1 rows or columns C(i,j) = C(i,j) +  k A(i,k)*B(k,j) Assume a p r by p c processor grid (p r = p c = 4 above) Need not be square C(i,j)

55. Summer School-Lecture 2 55 SUMMA For k=0 to n-1 … or n/b-1 where b is the block size … = # cols in A(i,k) and # rows in B(k,j) for all i = 1 to p r … in parallel owner of A(i,k) broadcasts it to whole processor row for all j = 1 to p c … in parallel owner of B(k,j) broadcasts it to whole processor column Receive A(i,k) into Acol Receive B(k,j) into Brow C_myproc = C_myproc + Acol * Brow * = i j A(i,k) k k B(k,j) C(i,j)

56. Summer School-Lecture 2 56 SUMMA performance For k=0 to n/b-1 for all i = 1 to s … s = sqrt(p) owner of A(i,k) broadcasts it to whole processor row … time = log s *(  +  * b*n/s), using a tree for all j = 1 to s owner of B(k,j) broadcasts it to whole processor column … time = log s *(  +  * b*n/s), using a tree Receive A(i,k) into Acol Receive B(k,j) into Brow C_myproc = C_myproc + Acol * Brow … time = 2*(n/s) 2 *b °Total time = 2*n 3 /p +  * log p * n/b +  * log p * n 2 /s °To simplify analysis only, assume s = sqrt(p)

57. Summer School-Lecture 2 57 SUMMA performance Total time = 2*n 3 /p +  * log p * n/b +  * log p * n 2 /s Parallel Efficiency = 1/(1 +  * log p * p / (2*b*n 2 ) +  * log p * s/(2*n) )  Same  term as Cannon, except for log p factor log p grows slowly so this is ok Latency (  ) term can be larger, depending on b When b=1, get  * log p * n As b grows to n/s, term shrinks to  * log p * s (log p times Cannon) Temporary storage grows like 2*b*n/s Can change b to tradeoff latency cost with memory

58. Summer School-Lecture 2 2/25/200958 PDGEMM = PBLAS routine for matrix multiply Observations: For fixed N, as P increases Mflops increases, but less than 100% efficiency For fixed P, as N increases, Mflops (efficiency) rises DGEMM = BLAS routine for matrix multiply Maximum speed for PDGEMM = # Procs * speed of DGEMM Observations (same as above): Efficiency always at least 48% For fixed N, as P increases, efficiency drops For fixed P, as N increases, efficiency increases

59. Summer School-Lecture 2 59 Summary of Parallel Matrix Multiplication 1D Layout Bus without broadcast - slower than serial Nearest neighbor communication on a ring (or bus with broadcast): Efficiency = 1/(1 + O(p/n)) 2D Layout Cannon Efficiency = 1/(1+O(  sqrt(p) /n    +  * sqrt(p) /n)) – optimal! Hard to generalize for general p, n, block cyclic, alignment SUMMA Efficiency = 1/(1 + O(  log p * p / (b*n 2 ) +  log p * sqrt(p) /n)) Very General b small => less memory, lower efficiency b large => more memory, high efficiency Used in practice (PBLAS)

60. Summer School-Lecture 2 Why so much about matrix multiplication? 60

61. Summer School-Lecture 2 A brief history of (Dense) Linear Algebra software (1/5) Libraries like EISPACK (for eigenvalue problems) Then the BLAS (1) were invented (1973-1977) Standard library of 15 operations (mostly) on vectors “AXPY” ( y = α·x + y ), dot product, scale (x = α·x ), etc Up to 4 versions of each (S/D/C/Z), 46 routines, 3300 LOC Goals Common “pattern” to ease programming, readability, self- documentation Robustness, via careful coding (avoiding over/underflow) Portability + Efficiency via machine specific implementations Why BLAS 1 ? They do O(n 1 ) ops on O(n 1 ) data Used in libraries like LINPACK (for linear systems) Source of the name “LINPACK Benchmark” (not the code!) CS267 Lecture 10 In the beginning was the do-loop…

62. Summer School-Lecture 2 A brief history of (Dense) Linear Algebra software (2/5) But the BLAS-1 weren’t enough Consider AXPY ( y = α·x + y ): 2n flops on 3n read/writes “Computational intensity” = #flops / #mem_refs = (2n)/(3n) = 2/3 Too low to run near peak speed (time for mem_refs dominates) Hard to vectorize (“SIMD’ize”) on supercomputers of the day (1980s) So the BLAS-2 were invented (1984-1986) Standard library of 25 operations (mostly) on matrix/vector pairs “GEMV”: y = α·A·x + β·x, “GER”: A = A + α·x·y T, “TRSV”: y = T -1 ·x Up to 4 versions of each (S/D/C/Z), 66 routines, 18K LOC Why BLAS 2 ? They do O(n 2 ) ops on O(n 2 ) data So computational intensity still just ~(2n 2 )/(n 2 ) = 2 OK for vector machines, but not for machine with caches CS267 Lecture 10

63. Summer School-Lecture 2 A brief history of (Dense) Linear Algebra software (3/5) The next step: BLAS-3 (1987-1988) Standard library of 9 operations (mostly) on matrix/matrix pairs “GEMM”: C = α·A·B + β·C, “SYRK”: C = α·A·A T + β·C, “TRSM”: C = T -1 ·B Up to 4 versions of each (S/D/C/Z), 30 routines, 10K LOC Why BLAS 3 ? They do O(n 3 ) ops on O(n 2 ) data So computational intensity (2n 3 )/(4n 2 ) = n/2 – big at last! Tuning opportunities machines with caches, other mem. hierarchy levels How much BLAS1/2/3 code so far (all at www.netlib.org/blas) Source: 142 routines, 31K LOC, Testing: 28K LOC Reference (unoptimized) implementation only Ex: 3 nested loops for GEMM Lots more optimized code Most computer vendors provide own optimized versions Motivates “automatic tuning” of the BLAS CS267 Lecture 10

64. Summer School-Lecture 2 A brief history of (Dense) Linear Algebra software (4/5) LAPACK – “Linear Algebra PACKage” - uses BLAS-3 (1989 – now) Ex: Obvious way to express Gaussian Elimination (GE) is adding multiples of one row to other rows – BLAS-1 How do we reorganize GE to use BLAS-3 ? (details later) Contents of LAPACK (summary) Algorithms we can turn into (nearly) 100% BLAS 3 –Linear Systems: solve Ax=b for x –Least Squares: choose x to minimize ||r|| 2   r i 2 where r=Ax-b Algorithms we can only make up to ~50% BLAS 3 (so far) –“Eigenproblems”: Find and x where Ax = x –Singular Value Decomposition (SVD): A T Ax=  2 x Error bounds for everything Lots of variants depending on A’s structure (banded, A=A T, etc) How much code? (Release 3.2, Nov 2008) (www.netlib.org/lapack) Source: 1582 routines, 490K LOC, Testing: 352K LOC Ongoing development (at UCB, UTK and elsewhere) CS267 Lecture 10

65. Summer School-Lecture 2 What could go into a linear algebra library? For all linear algebra problems For all matrix/problem structures For all data types For all programming interfaces Produce best algorithm(s) w.r.t. performance and accuracy (including condition estimates, etc) For all architectures and networks Need to prioritize, automate!

66. Summer School-Lecture 2 A brief history of (Dense) Linear Algebra software (5/5) Is LAPACK parallel? Only if the BLAS are parallel (possible in shared memory) ScaLAPACK – “Scalable LAPACK” (1995 – now) For distributed memory – uses MPI More complex data structures, algorithms than LAPACK Only (small) subset of LAPACK’s functionality available Details later (projects!) All at www.netlib.org/scalapack CS267 Lecture 10

67. Summer School-Lecture 2 67 Success Stories for Sca/LAPACK Cosmic Microwave Background Analysis, BOOMERanG collaboration, MADCAP code (Apr. 27, 2000). ScaLAPACK Widely used Adopted by Mathworks, Cray, Fujitsu, HP, IBM, IMSL, Intel, NAG, NEC, SGI, … >115M web hits(in 2010, 56M in 2006) @ Netlib (incl. CLAPACK, LAPACK95) New Science discovered through the solution of dense matrix systems Nature article on the flat universe used ScaLAPACK Other articles in Physics Review B that also use it 1998 Gordon Bell Prize www.nersc.gov/news/reports/new NERSCresults050703.pdf

68. Summer School-Lecture 2 So is there anything left to do? Yes! Performance of LAPACK and ScaLAPACK much lower than peak on new architectures Multicore (eg PLASMA) GPUs (eg MAGMA) Heterogeneous clusters of these (MAGMA) Cloud, Grid, … Major reason: almost no algorithms in Sca/LAPACK minimize communication Not enough to call BLAS3 in the inner loop of classical algorithms And there is still lots of missing functionality… 68

69. Summer School-Lecture 2 EXTRA SLIDES 69