1. Communication Avoiding Algorithms for Dense Linear Algebra Jim Demmel CS294, Lecture #4 Fall, 2011 Communication-Avoiding Algorithms www.cs.berkeley.edu/~odedsc/CS294 Based on: Bootcamp 2011, CS267, Summer-school 2010, many papers

2. 2 Outline for today Recall communication lower bounds What a “matching upper bound”, i.e. CA-algorithm, might have to do Summary of what is known so far Case Studies of CA algorithms Case study I: Matrix multiplication Case study II: LU, QR Case study III: Cholesky decomposition

3. 3 Communication lower bounds Applies to algorithms satisfying technical conditions discussed last time “smells like 3-nested loops” For each processor: M = size of fast memory of that processor G = #operations (multiplies) performed by that processor #words_moved to/from fast memory = Ω ( max ( G / M 1/2, #inputs + #outputs ) ) #messages to/from fast memory ≥ #words_moved max_message_size = Ω ( G / M 3/2 )

4. 4 Attaining these lower bounds Depends on what processor, memory refer to Sequential vs Parallel_distributed_memory vs Parallel_shared_memory Simple vs Hierarchical vs Messier… DRAM+cache vs multiple levels of cache Uniprocessors connected over network vs multicore processors connected over network vs multicore/multiboard/multirack/multisite Uniform communication in network vs slower if farther away Parallel machine with local memory hierarchies Register file vs shared memory on GPUs Homogeneous vs Heterogeneous All flop_rates/bandwidths/latencies/mem_sizes same vs different Use minimum fast_memory vs all available fast_memory Big design space (even just for matmul)

5. Summary of dense sequential algorithms attaining communication lower bounds 5 Algorithms shown minimizing # Messages use (recursive) block layout Not possible with columnwise or rowwise layouts Many references (see reports), only some shown, plus ours Cache-oblivious are underlined, Green are ours, ? is unknown/future work Algorithm2 Levels of MemoryMultiple Levels of Memory #Words Moved and # Messages#Words Moved and #Messages BLAS-3Usual blocked or recursive algorithms Usual blocked algorithms (nested), or recursive [Gustavson,97] Cholesky LAPACK (with b = M 1/2 ) [Gustavson 97] [BDHS09] [Gustavson,97] [Ahmed,Pingali,00] [BDHS09] (←same) LU with pivoting LAPACK (sometimes) [Toledo,97], [GDX 08] [GDX 08] not partial pivoting [Toledo, 97] [GDX 08]? QR Rank- revealing LAPACK (sometimes) [Elmroth,Gustavson,98] [DGHL08] [Frens,Wise,03] but 3x flops? [DGHL08] [Elmroth, Gustavson,98] [DGHL08] ? [Frens,Wise,03] [DGHL08] ? Eig, SVD Not LAPACK [BDD11] randomized, but more flops; [BDK11] [BDD11, BDK11?] [BDD11, BDK11?]

6. Summary of dense parallel algorithms attaining communication lower bounds 6 Assume nxn matrices on P processors Minimum Memory per processor = M = O(n 2 / P) Recall lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply Cholesky LU QR Sym Eig, SVD Nonsym Eig

7. Summary of dense parallel algorithms attaining communication lower bounds 7 Assume nxn matrices on P processors (conventional approach) Minimum Memory per processor = M = O(n 2 / P) Recall lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply[Cannon, 69]1 CholeskyScaLAPACKlog P LUScaLAPACKlog P QRScaLAPACKlog P Sym Eig, SVDScaLAPACKlog P Nonsym EigScaLAPACKP 1/2 log P

8. Summary of dense parallel algorithms attaining communication lower bounds 8 Assume nxn matrices on P processors (conventional approach) Minimum Memory per processor = M = O(n 2 / P) Recall lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply[Cannon, 69]11 CholeskyScaLAPACKlog P LUScaLAPACKlog Pn log P / P 1/2 QRScaLAPACKlog Pn log P / P 1/2 Sym Eig, SVDScaLAPACKlog Pn / P 1/2 Nonsym EigScaLAPACKP 1/2 log Pn log P

9. Summary of dense parallel algorithms attaining communication lower bounds 9 Assume nxn matrices on P processors (better) Minimum Memory per processor = M = O(n 2 / P) Recall lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply[Cannon, 69]11 CholeskyScaLAPACKlog P LU[GDX10]log P QR[DGHL08]log P log 3 P Sym Eig, SVD[BDD11]log Plog 3 P Nonsym Eig[BDD11]log P log 3 P

10. Can we do even better? 10 Assume nxn matrices on P processors Why just one copy of data: M = O(n 2 / P) per processor? Recall lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply[Cannon, 69]11 CholeskyScaLAPACKlog P LU[GDX10]log P QR[DGHL08]log P log 3 P Sym Eig, SVD[BDD11]log Plog 3 P Nonsym Eig[BDD11]log P log 3 P

11. Can we do even better? 11 Assume nxn matrices on P processors Why just one copy of data: M = O(n 2 / P) per processor? Increase M to reduce lower bounds: #words_moved =  ( (n 3 / P) / M 1/2 ) =  ( n 2 / P 1/2 ) #messages =  ( (n 3 / P) / M 3/2 ) =  ( P 1/2 ) AlgorithmReferenceFactor exceeding lower bound for #words_moved Factor exceeding lower bound for #messages Matrix Multiply[Cannon, 69]11 CholeskyScaLAPACKlog P LU[GDX10]log P QR[DGHL08]log P log 3 P Sym Eig, SVD[BDD11]log Plog 3 P Nonsym Eig[BDD11]log P log 3 P

12. 12 Other algorithms of interest Variations on pivoting to find “best” rows and/or columns Best = most independent PAP T = LDL T with symmetric pivoting A symmetric, P permutation, L lower triangular, D (block) diagonal Best row/column pairs first PAP T = LTL T with symmetric pivoting A symmetric, P permutation, L lower triangular, T tridiagonal Best row/column pairs first AP = QR with column pivoting Best columns first P r AP c T = LU with complete pivoting Best rows and columns first

13. Case Study I: Matrix Multiply

14. 14 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)

15. 15 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)

16. 16 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)

17. 17 Matrix-multiply, optimized several ways Speed of n-by-n matrix multiply on Sun Ultra-1/170, peak = 330 MFlops

18. 18 Recall: finding implementations for an algorithm that reduce communication Allowed: Reordering arithmetic operations, preserving the dependencies. Reordering summations, using associativity. Reordering multiplications, using commutativity. (irrelevant for matrix multiply and other bilinear algorithms). Not allowed (i.e., considered as different algorithms) Other reorderings even if by doing so we preserve the arithmetic correctness (e.g., using distributivity, as in Strassen’s algorithms).

19. 19 Blocked (Tiled) Matrix Multiply Consider A,B,C to be n/b-by-n/b matrices of b-by-b subblocks where b is called the block size for i = 1 to n/b for j = 1 to n/b {read block C(i,j) into fast memory} for k = 1 to n/b {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)

20. 20 Blocked (Tiled) Matrix Multiply Recall: m is #words_moved between slow and fast memory matrix has nxn elements, and n/b x n/b blocks each of size bxb So: m = n 3 /b read each block of B (n/b) 3 times ((n/b) 3 * b 2 = n 3 /b) + n 3 /b read each block of A (n/b) 3 times + 2n 2 read and write each block of C once = 2n 3 /b + 2n 2 So we can reduce communication by increasing the blocksize b Limit: 3b 2 ≤ M = fast memory size m ≥ 2 * 3 1/2 * n 3 / M 1/2

21. 21 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

22. 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 22 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

23. Recursive Matrix Multiplication (2/2) 23 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

24. 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

25. 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

26. How hard is hand-tuning matmul, anyway? 26 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/

27. How hard is hand-tuning matmul, anyway? 27

28. 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? 28

29. 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 29

30. 30 Matrix Multiply with 1D Column Block 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

31. 31 Matmul for 1D Column Block 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   latency,  latency,  time_per_flop

32. 32 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) =  /(  + (  * (p 2 /n 3 ) + (  * (p/n) ) = 1/ (1 + O(p/n)) Grows to 1 as n/p increases (or  and  shrink)

33. 33 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) =*

34. 34 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

35. 35 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

36. 36 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)

37. 37 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)

38. 38 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 …. attains lower bound! ° 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))

39. 39 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 Can you show optimal Cannon-like implementation (up to a constant), that can deal with items 1 and 3 above? Can you show optimal Cannon-like implementation (up to a constant), that can deal with item 2 above? (hint: what do you get if you use rectangle blocks with aspect ratios as the that of the input/output matrices?) Memory hog (extra copies of local matrices)

40. 40 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

41. 41 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)

42. 42 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)

43. 43 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)

44. 44 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

45. 45 Matrix multiplication – summary Cannon and SUMMA both (nearly) attain communication lower bound Assuming 1 copy of data is allowed (plus a constant amount of buffer space) Depend on assumptions about initial, final data layouts Called “2D algorithms” to reflect layout When more memory is available, get 2.5D algorithm Future talk (Edgar Solomonik) When processors heterogeneous, get another algorithm Future talk (Grey Ballard and Andrew Gearhart) Since Matmul naturally decomposes into smaller analogous problems, hierarchical machines can be accommodated

46. Why so much about matrix multiplication? 46

47. 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!) In the beginning was the do-loop…

48. 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

49. 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

50. 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)

51. 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!

52. 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 All at www.netlib.org/scalapack

53. 53 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, … >143M web hits(in 2011, 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

54. There is a lot left to do Do dense algorithms as implemented in LAPACK and ScaLAPACK attain communication lower bounds? Mostly not If not, are there other algorithms that do? Yes Do LAPACK or ScaLAPACK run (well or at all) on recent architectures? Not on Multicore (PLASMA) Not on GPUs, or heterogeneous clusters (MAGMA) Not Cloud, grid, … 54

55. Summer School Lecture 4 55 Gaussian Elimination (GE) for solving Ax=b Add multiples of each row to later rows to make A upper triangular Solve resulting triangular system Ux = c by substitution … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j tmp = A(j,i); for k = i to n A(j,k) = A(j,k) - (tmp/A(i,i)) * A(i,k) 0...00...0 0...00...0 0.00.0 0 0000 0...00...0 0...00...0 0.00.0 0...00...0 0...00...0 0...00...0 After i=1After i=2After i=3After i=n-1 …

56. 56 Refine GE Algorithm (1/5) Initial Version Remove computation of constant tmp/A(i,i) from inner loop. … for each column i … zero it out below the diagonal by adding multiples of row i to later rows for i = 1 to n-1 … for each row j below row i for j = i+1 to n … add a multiple of row i to row j tmp = A(j,i); for k = i to n A(j,k) = A(j,k) - (tmp/A(i,i)) * A(i,k) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i to n A(j,k) = A(j,k) - m * A(i,k) m i j

57. 57 Refine GE Algorithm (2/5) Last version Don’t compute what we already know: zeros below diagonal in column i for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - m * A(i,k) for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i to n A(j,k) = A(j,k) - m * A(i,k) Do not compute zeros m i j

58. 58 Refine GE Algorithm (3/5) Last version Store multipliers m below diagonal in zeroed entries for later use for i = 1 to n-1 for j = i+1 to n m = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - m * A(i,k) for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k) Store m here m i j

59. 59 Refine GE Algorithm (4/5) Last version for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k) Split Loop for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for j = i+1 to n for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k) Store all m’s here before updating rest of matrix i j

60. 60 Refine GE Algorithm (5/5) Last version Express using matrix operations (BLAS) for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) * ( 1 / A(i,i) ) … BLAS 1 (scale a vector) A(i+1:n,i+1:n) = A(i+1:n, i+1:n ) - A(i+1:n, i) * A(i, i+1:n) … BLAS 2 (rank-1 update) for i = 1 to n-1 for j = i+1 to n A(j,i) = A(j,i)/A(i,i) for j = i+1 to n for k = i+1 to n A(j,k) = A(j,k) - A(j,i) * A(i,k)

61. 61 What GE really computes Call the strictly lower triangular matrix of multipliers M, and let L = I+M Call the upper triangle of the final matrix U Lemma (LU Factorization): If the above algorithm terminates (does not divide by zero) then A = L*U Solving A*x=b using GE Factorize A = L*U using GE (cost = 2/3 n 3 flops) Solve L*y = b for y, using substitution (cost = n 2 flops) Solve U*x = y for x, using substitution (cost = n 2 flops) Thus A*x = (L*U)*x = L*(U*x) = L*y = b as desired for i = 1 to n-1 A(i+1:n,i) = A(i+1:n,i) / A(i,i) … BLAS 1 (scale a vector) A(i+1:n,i+1:n) = A(i+1:n, i+1:n ) - A(i+1:n, i) * A(i, i+1:n) … BLAS 2 (rank-1 update)

62. 62 Gaussian Elimination with Partial Pivoting (GEPP) Partial Pivoting: swap rows so that A(i,i) is largest in column for i = 1 to n-1 find and record k where |A(k,i)| = max {i  j  n} |A(j,i)| … i.e. largest entry in rest of column i if |A(k,i)| = 0 exit with a warning that A is singular, or nearly so elseif k ≠ i swap rows i and k of A end if A(i+1:n,i) = A(i+1:n,i) / A(i,i) … each |quotient| ≤ 1 A(i+1:n,i+1:n) = A(i+1:n, i+1:n ) - A(i+1:n, i) * A(i, i+1:n) Lemma: This algorithm computes A = P*L*U, where P is a permutation matrix. This algorithm is numerically stable in practice For details see LAPACK code at http://www.netlib.org/lapack/single/sgetf2.f Standard approach – but communication costs?

63. 63 Converting BLAS2 to BLAS3 in GEPP Blocking Used to optimize matrix-multiplication Harder here because of data dependencies in GEPP BIG IDEA: Delayed Updates Save updates to “trailing matrix” from several consecutive BLAS2 (rank-1) updates Apply many updates simultaneously in one BLAS3 (matmul) operation Same idea works for much of dense linear algebra One-sided factorization are ~100% BLAS3 Two-sided factorizations are ~50% BLAS3 First Approach: Need to choose a block size b Algorithm will save and apply b updates b should be small enough so that active submatrix consisting of b columns of A fits in cache b should be large enough to make BLAS3 (matmul) fast

64. 64 Blocked GEPP (www.netlib.org/lapack/single/sgetrf.f) for ib = 1 to n-1 step b … Process matrix b columns at a time end = ib + b-1 … Point to end of block of b columns apply BLAS2 version of GEPP to get A(ib:n, ib:end) = P’ * L’ * U’ … let LL denote the strict lower triangular part of A(ib:end, ib:end) + I A(ib:end, end+1:n) = LL -1 * A(ib:end, end+1:n) … update next b rows of U A(end+1:n, end+1:n ) = A(end+1:n, end+1:n ) - A(end+1:n, ib:end) * A(ib:end, end+1:n) … apply delayed updates with single matrix-multiply … with inner dimension b (For a correctness proof, see on-line notes from CS267 / 1996.)

65. Does GE Minimize Communication? (1/4) Model of communication costs with fast memory M BLAS2 version of GEPP costs O(n ·b) if panel fits in M: n·b  M O(n · b 2 ) (#flops) if panel does not fit in M: n·b > M Update of A(end+1:n, end+1:n ) by matmul costs O( max ( n·b·n / M 1/2, n 2 )) Triangular solve with LL bounded by above term Total # slow mem refs for GE = (n/b) · sum of above terms for ib = 1 to n-1 step b … Process matrix b columns at a time end = ib + b-1 … Point to end of block of b columns apply BLAS2 version of GEPP to get A(ib:n, ib:end) = P’ * L’ * U’ … let LL denote the strict lower triangular part of A(ib:end, ib:end) + I A(ib:end, end+1:n) = LL -1 * A(ib:end, end+1:n) … update next b rows of U A(end+1:n, end+1:n ) = A(end+1:n, end+1:n ) - A(end+1:n, ib:end) * A(ib:end, end+1:n) … apply delayed updates with single matrix-multiply … with inner dimension b

66. Does GE Minimize Communication? (2/4) Model of communication costs with fast memory M BLAS2 version of GEPP costs O(n ·b) if panel fits in M: n·b  M O(n · b 2 ) (#flops) if panel does not fit in M: n·b > M Update of A(end+1:n, end+1:n ) by matmul costs O( max ( n·b·n / M 1/2, n 2 )) Triangular solve with LL bounded by above term Total # slow mem refs for GE = (n/b) · sum of above terms Case 1: M < n (one column too large for fast mem) Total # slow mem refs for GE = (n/b)*O(max(n b 2, b n 2 / M 1/2, n 2 )) = O( max( n 2 b, n 3 / M 1/2, n 3 / b )) Minimize by choosing b = M 1/2 Get desired lower bound O(n 3 / M 1/2 ) 66

67. Does GE Minimize Communication? (3/4) Model of communication costs with fast memory M BLAS2 version of GEPP costs O(n ·b) if panel fits in M: n·b  M O(n · b 2 ) (#flops) if panel does not fit in M: n·b > M Update of A(end+1:n, end+1:n ) by matmul costs O( max ( n·b·n / M 1/2, n 2 )) Triangular solve with LL bounded by above term Total # slow mem refs for GE = (n/b) · sum of above terms Case 2: M 2/3 < n  M Total # slow mem refs for GE = (n/b)*O(max(n b 2, b n 2 / M 1/2, n 2 )) = O( max( n 2 b, n 3 / M 1/2, n 3 / b )) Minimize by choosing b = n 1/2 (panel does not fit in M) Get O(n 2.5 ) slow mem refs Exceeds lower bound O(n 3 / M 1/2 ) by factor (M/n) 1/2  M 1/6 67

68. Does GE Minimize Communication? (4/4) Model of communication costs with fast memory M BLAS2 version of GEPP costs O(n ·b) if panel fits in M: n·b  M O(n · b 2 ) (#flops) if panel does not fit in M: n·b > M Update of A(end+1:n, end+1:n ) by matmul costs O( max ( n·b·n / M 1/2, n 2 )) Triangular solve with LL bounded by above term Total # slow mem refs for GE = (n/b) · sum of above terms Case 3: M 1/2 < n  M 2/3 Total # slow mem refs for GE = (n/b)*O(max(n b, b n 2 / M 1/2, n 2 )) = O(max( n 2, n 3 / M 1/2, n 3 / b )) Minimize by choosing b = M/n (panel fits in M) Get O(n 4 /M) slow mem refs Exceeds lower bound O(n 3 / M 1/2 ) by factor n/M 1/2  M 1/6 Case 4: n  M 1/2 – whole matrix fits in fast mem 68

69. Alternative cache-oblivious GE formulation (1/2) Toledo (1997) Describe without pivoting for simplicity “Do left half of matrix, then right half” 69 function [L,U] = RLU (A) … assume A is m by n if (n=1) L = A/A(1,1), U = A(1,1) else [L1,U1] = RLU( A(1:m, 1:n/2)) … do left half of A … let L11 denote top n/2 rows of L1 A( 1:n/2, n/2+1 : n ) = L11 -1 * A( 1:n/2, n/2+1 : n ) … update top n/2 rows of right half of A A( n/2+1: m, n/2+1:n ) = A( n/2+1: m, n/2+1:n ) - A( n/2+1: m, 1:n/2 ) * A( 1:n/2, n/2+1 : n ) … update rest of right half of A [L2,U2] = RLU( A(n/2+1:m, n/2+1:n) ) … do right half of A return [ L1,[0;L2] ] and [U1, [ A(.,.) ; U2 ] ] A = L * U

70. Alternative cache-oblivious GE formulation (2/2) 70 function [L,U] = RLU (A) … assume A is m by n if (n=1) L = A/A(1,1), U = A(1,1) else [L1,U1] = RLU( A(1:m, 1:n/2)) … do left half of A … let L11 denote top n/2 rows of L1 A( 1:n/2, n/2+1 : n ) = L11 -1 * A( 1:n/2, n/2+1 : n ) … update top n/2 rows of right half of A A( n/2+1: m, n/2+1:n ) = A( n/2+1: m, n/2+1:n ) - A( n/2+1: m, 1:n/2 ) * A( 1:n/2, n/2+1 : n ) … update rest of right half of A [L2,U2] = RLU( A(n/2+1:m, n/2+1:n) ) … do right half of A return [ L1,[0;L2] ] and [U1, [ A(.,.) ; U2 ] ] Mem(m,n) = Mem(m,n/2) + O(max(m·n,m·n 2 /M 1/2 )) + Mem(m-n/2,n/2)  2 · Mem(m,n/2) + O(max(m·n,m·n 2 /M 1/2 )) = O(m·n 2 /M 1/2 + m·n·log M) = O(m·n 2 /M 1/2 ) if M 1/2 ·log M = O(n)

71. One-sided Factorizations (LU, QR), so far Classical Approach for i=1 to n update column i update trailing matrix #words_moved = O(n 3 ) 71 Blocked Approach (LAPACK) for i=1 to n/b update block i of b columns update trailing matrix #words moved = O(n 3 /M 1/3 ) Recursive Approach func factor(A) if A has 1 column, update it else factor(left half of A) update right half of A factor(right half of A) #words moved = O(n 3 /M 1/2 ) None of these approaches minimizes #messages or addresses parallelism Need another idea