Eliminations and echelon forms in exact linear algebra
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1 Eliminations and echelon forms in exact linear algebra Clément PERNET, INRIA-MOAIS, Grenoble Université, France East Coast Computer Algebra Day, University of Waterloo, ON, Canada, April 9, 2011 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 1 / 59
2 Introduction Gaussian elimination in Computer Algebra Linear system solving: over Z p, Z, Q Polynomial system solving: Gröbner basis (Crypto, Number Theory) (Robotics, Crypto) Linear dependencies: rank, basis (of vector spaces, free modules, Krylov spaces,...) (Number Theory) Determinants: certificate of similarity (Graph Theory) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 2 / 59
3 Motivation Designing efficient dense gaussian elimination routines over an exact ring/field. Extensively studied for numerical computations Specificities of exact computations: No partial/full pivoting Rank profile matters size of coefficients (e.g. compressed in GF(2)) Study originating from, the design of the libraries FFLAS-FFPACK: word size finite fields M4RI: GF(2) asymmetry Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 3 / 59
4 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 4 / 59
5 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 5 / 59
6 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 6 / 59
7 LU decomposition A = L U L unit lower triangular, U non-sing upper triangular Exists for matrices having the generic rank profile (every leading principal minor is non zero) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 7 / 59
8 LUP, PLU decomposition A U = L P P a permutation matrix Exists for Any non-singular matrix Or any matrix with generic row rank profile Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 8 / 59
9 LUP, PLU decomposition A U = L P A = P L U P a permutation matrix Exists for Any non-singular matrix Or any matrix with generic row rank profile Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 8 / 59
10 LSP, LQUP, PLUQ decompositions A = L S P S: semi-upper triangular, Q permutation matrix Exists for any m n matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 9 / 59
11 LSP, LQUP, PLUQ decompositions A = L S P A = L Q U P S: semi-upper triangular, Q permutation matrix Exists for any m n matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 9 / 59
12 LSP, LQUP, PLUQ decompositions A = L S P A = L Q U P A = P L U Q S: semi-upper triangular, Q permutation matrix Exists for any m n matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 9 / 59
13 Echelon form decomposition Row Echelon Form XA = R X A = R X, Y : non-singular transformation matrices R, C: matrices in row/col echelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 10 / 59
14 Echelon form decomposition Row Echelon Form XA = R X A = R Column Echelon Form AY = C A Y = C X, Y : non-singular transformation matrices R, C: matrices in row/col echelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 10 / 59
15 Reduced echelon form decomposition Row Reduced Echelon Form XA = R X A = R X, Y : non-singular transformation matrices R, C: matrices in reduced row/col echelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 11 / 59
16 Reduced echelon form decomposition Row Reduced Echelon Form XA = R X A = R Column Reduced Echelon Form AY = C A Z = C X, Y : non-singular transformation matrices R, C: matrices in reduced row/col echelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 11 / 59
17 Turing factorization A = P L U R PLU decomposition of X 1, the inverse of the transformation matrix to REF Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 12 / 59
18 CUP and PLE decompositions A = C U P C: column echelon form E: row echelon form Exists for any m n matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 13 / 59
19 CUP and PLE decompositions A = C U P A = P L E C: column echelon form E: row echelon form Exists for any m n matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 13 / 59
20 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 14 / 59
21 Relations: up to permutations From LSP to LQUP S = QU A = L S P A = L Q U P Fact The first r = rank(a) values of the permutation Q are monotonically increasing. Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 15 / 59
22 Relations: up to permutations From LQUP to CUP A = L Q U P C = LQ A = L Q U P A = C U P And to PLE using transposition: PLE(A T ) = CUP(A) T Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 16 / 59
23 Relations: From LQUP to PLUQ A = L Q U P P Q, L = Q T LQ A = C U P A = P L U Q Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 17 / 59
24 Relations: From CUP to ColumnEchelon form A = C U P Y = P T [ U I n r ] 1 A = C U Id P A Y = Similarly, from PLE to RowEchelon form C Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 18 / 59
25 Relations: From Column Echelon form to Reduced Column Echelon form A Y = C M [ M Z = Y I n r ] 1 A Y = A Y = Q Q Id M Id A Z = C Similarly, from PLE to RowEchelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 19 / 59
26 Relations: From PLE to Turing A = P L E U = V, [ ] 1 V R = E I n r A = P L A = P L V V Q Id Q A = P L U R Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 20 / 59
27 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 21 / 59
28 Algorithms: main types Three ways to group operations: 1 simple iterative Apply the standard Gaussian elimination in dimension n Main loop for i=1 to n 2 block algorithms 1 block iterative (Tile) Apply Gaussian elimination in dimension n/k over blocks of size k Main loop: for i=1 to n/k 2 block recursive Apply Gaussian elimination in dimension 2 recursively on blocks of size n/2 i Main loop: for i=1 to 2 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 22 / 59
29 Type of algorithms Data locality: prefer block algorithms cache aware: block iterative cache oblivious: block recursive Base case efficieny: simple iterative Asymptotic time complexity: block recursive Parallelization: block iterative Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 23 / 59
30 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 24 / 59
31 Block recursive gaussian elimination Author Year Computation Requirement Strassen 69 Inverse gen. rank prof. Bunch, Hopcroft 74 LUP gen. row rank prof. Ibarra, Moran, Hui 82 LSP, LQUP none Schönage, Keller-Gerig 85 StepForm none Storjohann 00 Echelon, RedEch none here 11 CUP,PLE,PLUQ none Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 25 / 59
32 Block recursive gaussian elimination Author Year Computation Requirement Strassen 69 Inverse gen. rank prof. Bunch, Hopcroft 74 LUP gen. row rank prof. Ibarra, Moran, Hui 82 LSP, LQUP none Schönage, Keller-Gerig 85 StepForm none Storjohann 00 Echelon, RedEch none here 11 CUP,PLE,PLUQ none Comparison according to No requirement on the input matrix Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 25 / 59
33 Block recursive gaussian elimination Author Year Computation Requirement Strassen 69 Inverse gen. rank prof. Bunch, Hopcroft 74 LUP gen. row rank prof. Ibarra, Moran, Hui 82 LSP, LQUP none Schönage, Keller-Gerig 85 StepForm none Storjohann 00 Echelon, RedEch none here 11 CUP,PLE,PLUQ none Comparison according to No requirement on the input matrix Rank sensitive complexity Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 25 / 59
34 Block recursive gaussian elimination Author Year Computation Requirement Strassen 69 Inverse gen. rank prof. Bunch, Hopcroft 74 LUP gen. row rank prof. Ibarra, Moran, Hui 82 LSP, LQUP none Schönage, Keller-Gerig 85 StepForm none Storjohann 00 Echelon, RedEch none here 11 CUP,PLE,PLUQ none Comparison according to No requirement on the input matrix Rank sensitive complexity Memory allocations Constant factor in the time complexity Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 25 / 59
35 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 26 / 59
36 Memory requirements: Definition In place = output overrides the input and computation does not need extra memory (considering Matrix multiplication C C + AB as a black box) Remark: a unit lower triangular and an upper triangular matrix can be stored on the same m n storage! Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 27 / 59
37 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I ] [ ] D C 1 E Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
38 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I Compute F = C 1 E Compute G = D BF Compute [ H ] = A 1 G H Return F ] [ ] D C 1 E (Recursive call) (MM) (Recursive call) A B C D E Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
39 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I Compute F = C 1 E Compute G = D BF Compute [ H ] = A 1 G H Return F ] [ ] D C 1 E (Recursive call) (MM) (Recursive call) A B C D F Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
40 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I Compute F = C 1 E Compute G = D BF Compute [ H ] = A 1 G H Return F ] [ ] D C 1 E (Recursive call) (MM) (Recursive call) A B C G F Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
41 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I Compute F = C 1 E Compute G = D BF Compute [ H ] = A 1 G H Return F ] [ ] D C 1 E (Recursive call) (MM) (Recursive call) A B C H F Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
42 Preliminaries TRSM: TRiangular Solve with Matrix [ ] 1 [ ] [ ] [ ] [ A B D A 1 I B I = C E I I Compute F = C 1 E Compute G = D BF Compute [ H ] = A 1 G H Return F ] [ ] D C 1 E (Recursive call) (MM) (Recursive call) A B C H F O (n ω ) In place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 28 / 59
43 The LSP algorithm S A1 A2 1 Split A Row-wise L Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 29 / 59
44 The LSP algorithm S S 1 A21 V A 22 1 Split A Row-wise 2 Recursive call on A 1 L L1 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 29 / 59
45 The LSP algorithm S S 1 V A 22 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) U 1 L L1 G Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 29 / 59
46 The LSP algorithm S L S 1 L1 G V H 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MM) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 29 / 59
47 The LSP algorithm S L S 1 L1 G S 2 L 2 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MM) 5 Recursive call on H Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 29 / 59
48 The LQUP algorithm U A1 A2 1 Split A Row-wise L Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
49 The LQUP algorithm U U 1 A21 V A 22 1 Split A Row-wise 2 Recursive call on A 1 L L1 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
50 The LQUP algorithm U 1 U V A 22 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) L L1 G Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
51 The LQUP algorithm U L U 1 L1 G V H 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MMl) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
52 The LQUP algorithm L U 1 U 1 Split A Row-wise L1 G U 2 L 2 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MMl) 5 Recursive call on H Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
53 The LQUP algorithm U 1 U U 2 1 Split A Row-wise 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MMl) L L1 G L 2 5 Recursive call on H Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 30 / 59
54 The CUP decomposition A1 A2 1 Split A Row-wise Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
55 The CUP decomposition 1 Split A Row-wise C U 1 1 V A21 A22 2 Recursive call on A 1 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
56 The CUP decomposition 1 Split A Row-wise C U 1 1 G V A 22 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
57 The CUP decomposition 1 Split A Row-wise C U 1 1 G V H 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MM) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
58 The CUP decomposition 1 Split A Row-wise C U 1 1 U 2 G C 2 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MM) 5 Recursive call on H Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
59 The CUP decomposition 1 Split A Row-wise U1 C1 U 2 G C 2 2 Recursive call on A 1 3 G A 21 U 1 1 (trsm) 4 H A 22 G V (MM) 5 Recursive call on H 6 Row permutations Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 31 / 59
60 Memory: LSP vs LQUP vs PLUQ vs CUP Decomposition In place LSP N LQUP N PLUQ Y CUP Y Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 32 / 59
61 Echelon forms From CUP to ColumnEchelon form [ ] 1 Y = P T U1 U 2 I n r = P T [ U 1 1 U 1 1 U 2 I n r ] A = A = C C A Y = U U Id C P P Additional operations: U 1 U 2 trsm (triangular system solve) in-place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 33 / 59
62 Echelon forms From CUP to ColumnEchelon form [ ] 1 Y = P T U1 U 2 I n r = P T [ U 1 1 U 1 1 U 2 I n r ] A = A = C C A Y = U U Id C P P Additional operations: U 1 U 2 trsm (triangular system solve) in-place U 1 1 : trtri (triangular inverse) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 33 / 59
63 Echelon forms From CUP to ColumnEchelon form [ ] 1 Y = P T U1 U 2 I n r = P T [ U 1 1 U 1 1 U 2 I n r ] A = A = C C A Y = U U Id C P P Additional operations: U 1 U 2 trsm (triangular system solve) in-place U 1 1 : trtri (triangular inverse) in-place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 33 / 59
64 From LQUP to Column Echelon TRTRI: triangular inverse 1: if n = 1 then 2: U U 1 [ ] 1 [ U1 U 2 U 1 = 1 U 1 U 3 1 U 2U 1 ] 3 U 1 3 3: else 4: U 2 U 1 3 U 2 TRSM 5: U 2 U 2 U 1 3 TRSM 6: U 1 U 1 1 TRTRI 7: U 3 U 1 3 TRTRI 8: end if Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 34 / 59
65 Reduced Echelon forms From Column Echelon form to Reduced Column Echelon form A Y = C M [ M Z = Y I n r ] 1 A Y = A Y = Q Q Id M Id A Z = C Similarly, from PLE to RowEchelon form Again reduces to: U 1 X: TRSM, in-place U 1 : TRTRI, in-place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 35 / 59
66 Reduced Echelon forms From Column Echelon form to Reduced Column Echelon form A Y = C M [ M Z = Y I n r ] 1 A Y = A Y = Q Q Id M Id A Z = C Similarly, from PLE to RowEchelon form Again reduces to: U 1 X: TRSM, in-place U 1 : TRTRI, in-place UL: TRTRM, Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 35 / 59
67 Reduced Echelon forms From Column Echelon form to Reduced Column Echelon form A Y = C M [ M Z = Y I n r ] 1 A Y = A Y = Q Q Id M Id A Z = C Similarly, from PLE to RowEchelon form Again reduces to: U 1 X: TRSM, in-place U 1 : TRTRI, in-place UL: TRTRM, Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 35 / 59
68 Reduced Echelon forms From Column Echelon form to Reduced Column Echelon form A Y = C M [ M Z = Y I n r ] 1 A Y = A Y = Q Q Id M Id A Z = C Similarly, from PLE to RowEchelon form Again reduces to: U 1 X: TRSM, in-place U 1 : TRTRI, in-place UL: TRTRM, in-place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 35 / 59
69 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
70 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM U L 1 1 L L U 2 U Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
71 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM X 1 L L U 2 U Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
72 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM X 1 L L U 2 U Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
73 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM X 1 L L X 2 U Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
74 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM X 1 X L X 2 U Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
75 From Echelon to Reduced Echelon TRTRM: triangular triangular multiplication [ ] [ ] [ ] U1 U 2 L1 U1 L = 1 + U 2 L 2 U 2 L 3 U 3 L 2 L 3 U 3 L 2 U 3 L 3 1: X 1 U 1 L 1 TRTRM 2: X 1 X 1 + U 2 L 2 MM 3: X 2 U 2 L 3 TRMM 4: X 3 U 3 L 2 TRMM 5: X 4 U 3 L 3 TRTRM X 1 X 3 X 2 X 4 O (n ω ) In place Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 36 / 59
76 Example: in place matrix inversion A Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 37 / 59
77 Example: in place matrix inversion A L U CUP decomposition A = LU Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 37 / 59
78 Example: in place matrix inversion U A L L U 1 CUP decomposition Echelon AU 1 = L Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 37 / 59
79 Example: in place matrix inversion U A L L U 1 A 1 CUP decomposition Echelon Reduced Echelon A(U 1 L 1 ) = I Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 37 / 59
80 Experiments./test-invert A ,996,125,000 bytes x ms bytes 16M x804aa7e:_z10read_fieldi 14M 12M x8049a98:main 10M x804edc0:void%fflas::win 8M 6M x804edee:void%fflas::win 4M x804edd4:void%fflas::win 2M 0M ms./test-redechelon A ,663,687,500 bytes x ms bytes 8M x804a47e:_z10read_fieldi 6M x804b0ec:void%fflas::win 4M x804b11a:void%fflas::win 2M x804b100:void%fflas::win 0M ms Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 38 / 59
81 Direct computation of the Reduced Echelon form Strassen 69: inverse of generic matrices Storjohann 00: Gauss-Jordan generalization for any rank profile Matrix Inversion [Strassen 69] [ ] 1 [ A B A 1 = C D I ] [ I B I ] [ I (D CA 1 B) 1 ] [ I CA 1 I ] Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 39 / 59
82 Direct computation of the Reduced Echelon form Strassen 69: inverse of generic matrices Storjohann 00: Gauss-Jordan generalization for any rank profile Matrix Inversion [Strassen 69] [ ] 1 [ A B A 1 = C D I ] [ I B I ] [ I (D CA 1 B) 1 ] [ I CA 1 1: Compute E = A 1 (Recursive call) 2: Compute F = D CEB (MM) 3: Compute G = F 1 (Recursive call) 4: Compute H = EB (MM) 5: Compute J = HG (MM) 6: Compute K = CE (MM) 7: Compute L = E + JK (MM) 8: Compute [ M = GK ] (MM) E J 9: Return M G I ] Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 39 / 59
83 Strassen-Storjohann s Gauss-Jordan elimination Problem Needs to perform operations of the form A AB not doable in place by a usual matrix multiplication algorithm Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 40 / 59
84 Strassen-Storjohann s Gauss-Jordan elimination Problem Needs to perform operations of the form A AB not doable in place by a usual matrix multiplication algorithm Workaround [Storjohann]: 1 Decompose B = LU LU 2 A AL trmm 3 A AU trmm Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 40 / 59
85 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 41 / 59
86 Rank sensitive time complexity Fact Algorithms LSP, CUP, LQUP, PLUQ,... have a rank sensitive computation time: O ( mnr ω 2) n=3000, PIII 1.6Ghz, 512Mb RAM Block algorithm Iterative algorithm 100 Time (s) Rank Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 42 / 59
87 Rank sensitivity and Left/Right/Crout Looking variants Right looking / Left looking/ Crout variants does not affect 2x2 splitting for block iterative: always the same rank sensitive complexity: 2n 2 r 2nr 2 + 2/3r 3 Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 43 / 59
88 Time complexity: comparing constants O (n ω ) = C ω n 3 Algorithm Constant C ω C 3 C log2 7 in-place MM C ω 2 6 C TRSM ω ω 1 2 C TRTRI ω 1 8 (2 ω 1 2)(2 ω 1 1) 3 5 TRTRM, CUP C ω PLUQ LQUP, ω ω C Echelon ω ω ω 2 5 C RedEchelon ω(2 ω 1 +2) 44 2 (2 ω 1 2)(2 ω 1 1) 5 5C StepForm ω C ω ω 1 1 (2 ω 1 1)(2 ω 2 1) 5 GJ 2 8 Cω 2 ω 2 1 : GJ: GaussJordan alg of [Storjohann00] computing the reduced echelon form Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 44 / 59
89 Applications to standard linalg problems Problem Using C ω C 3 C log2 7 In place Rank RankProfile IsSingular Det Solve GJ CUP C ω 2 ω 2 1 C ω Cω 2 ω ω Inverse GJ CUP C ω 2 ω 2 1 C ω(2 ω 1 +2) (2 ω 1 2)(2 ω 1 1) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 45 / 59
90 Summary - Det - Rank - RankProfile - Solve - Echelon - Nullspace basis - Reduced Echelon form - Inverse A 2/3 L U U 1/3 L -1 1/3 L -1U-1 2/3 LUDecomp TRTRI TRTRI TRTRM Gauss 1 GaussJordan 2 LU decomp A=LU Echelon form decomp (TA=E) A -1 Reduced Echelon form decomp TA=R Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 46 / 59
91 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 47 / 59
92 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
93 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
94 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
95 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
96 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
97 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
98 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
99 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
100 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
101 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
102 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
103 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
104 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
105 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
106 Parallelization Using block iterative algorithm Parallelizing the matrix multiplication updates Always a critical path of n on the diagonal Scheduling tasks DAG, static scheduling,... Dynamic scheduling, work-stealing Advantage of exact computations: No need to update pivots (local pivoting) Multi-frontal approach made easier less ops Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 48 / 59
107 Dynamic scheduling with Kaapi hread0 hread1 hread2 hread3 hread4 hread5 hread6 hread7 hread8 hread9 hread10 hread11 hread12 hread13 hread14 hread15 hread16 hread17 hread18 hread19 hread20 hread21 hread22 hread23 hread24 hread25 hread26 hread27 hread28 hread29 hread30 hread31 hread32 hread33 hread34 hread35 hread36 hread37 hread38 hread39 hread40 hread41 hread42 hread43 hread44 hread45 hread46 hread47 ROOT Thread0 Thread1 Thread2 Thread3 Thread4 Thread5 Thread6 Thread7 Thread8 Thread9 Thread10 Thread11 Thread12 Thread13 Thread14 Thread15 Thread16 Thread17 Thread18 Thread19 Thread20 Thread21 Thread22 Thread23 Thread24 Thread25 Thread26 Thread27 Thread28 Thread29 Thread30 Thread31 Thread32 Thread33 Thread34 Thread35 Thread36 Thread37 Thread38 Thread39 Thread40 Thread41 Thread42 Thread43 Thread44 Thread45 Thread min: 0 max: min: 0 max: Thread0 Thread1 Thread2 Thread3 Thread4 Thread5 Thread6 Thread7 Thread8 Thread9 Thread10 Thread11 Thread12 Thread13 Thread14 Thread15 Thread16 Thread17 Thread18 Thread19 Thread20 Thread21 Thread22 Thread23 Thread24 Thread25 Thread26 Thread27 Thread28 Thread29 Thread30 Thread31 Thread32 Thread33 Thread34 Thread35 Thread36 Thread37 Thread38 Thread39 Thread40 Thread41 Thread42 Thread43 Thread44 Thread45 Thread46 Thread47 min: 0 max: Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 49 / 59
108 Outline 1 Decompositions and factorizations Gaussian elimination based matrix decompositions Relations between decompositions 2 Algorithms Block recursive gaussian elimination Memory allocations Time complexity 3 Parallelization 4 Algorithms into practice: the case of GF(2) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 50 / 59
109 Dedicated implementations over GF(2) The M4RI library [M Albrecht & Al.]: Dense basic linear algebra over GF(2) Packed representation: unsigned long long is a vector of 64 coefficients Matrix multiplication: Gray code table lookup (Methode of the 4 russians) O ( n 3 / log n ) Strassen on the coarse grain. O ( (n/k) ω k 3 / log k ) =O (n ω ) Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 51 / 59
110 Design of the gaussian elimination routine PLE vs CUP Row major storage easier to permute pivots along columns PLE rather than CUP PLE vs PLUQ PLUQ involves more back and forth pivoting Compact LAPACK representation of permutation: product of transpositions Not possible to maintain for P in O (1) PLUQ PLE rater than PLUQ Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 52 / 59
111 Design of the gaussian elimination routine Structure of the algorithm block recursive PLE on the coarse grain (O (n ω )) block iterative PLE with block size k = log n simple iterative PLE on cache fitting blocks Additional tricks: With gray code tables: inverting means reverse table look-up TRSM as efficient as MM... Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 53 / 59
112 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
113 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
114 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
115 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
116 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
117 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
118 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
119 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
120 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
121 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
122 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
123 Visualisation Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 54 / 59
124 Results: Reduced Row Echelon Form execution time t Magma Sage Magma/Sage Elimination log 2 relation Figure: 2.66 Ghz Intel i7, 4GB RAM Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 55 / 59
125 Results: Row Echelon Form Using one core we can compute the echelon form of a 500, , 000 dense random matrix over F 2 in seconds = 2.7 hours. Using four cores decomposition we can compute the echelon form of a random dense 500, , 000 matrix in seconds = 1.05 hours. Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 56 / 59
126 Work-in-Progress: Sensitivity to Sparsity M4RI M+P 0.15 PLS Magma execution time t nonzero elements per row on average Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 57 / 59
127 Work-in-Progress: Gröbner Basis Linear Algebra 64-bit Debian/GNU Linux, 2.6Ghz Opteron) Problem Matrix Density Magma M4RI PLS M+P 0.15 M+P 0.20 Dimension HFE 25 12, , s 3.28s 3.45s 3.03s 3.21s HFE 30 19, , s 23.72s 25.42s 23.84s 25.09s HFE 35 29, , s s s s s MXL 26, , s 23.03s 19.04s 17.91s 18.00s Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 58 / 59
128 Work-in-Progress: Multi-core Support M4RI BOpS & Speed-up PLE BOpS & Speed-up Clément Pernet (INRIA-MOAIS, Grenoble) Eliminations in exact linear algebra ECCAD 11, U of Waterloo 59 / 59
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