FFPACK: Finite Field Linear Algebra Package
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1 FFPACK: Finite Field Linear Algebra Package Jean-Guillaume Dumas, Pascal Giorgi and Clément Pernet {Jean.Guillaume.Dumas, P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 1
2 Introduction Motivation: Integer linear algebra P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 2
3 Introduction Motivation: Integer linear algebra Sparse or structured matrix: specific methods (Blackbox,...) Otherwise: Dense methods P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 2
4 Introduction Motivation: Integer linear algebra Sparse or structured matrix: specific methods (Blackbox,...) Otherwise: Dense methods Limit the growth of intermediate results: Several computations over distinct finite fields and reconstruction using Chinese Remaindering P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 2
5 Introduction Motivation: Integer linear algebra Sparse or structured matrix: specific methods (Blackbox,...) Otherwise: Dense methods Limit the growth of intermediate results: Several computations over distinct finite fields and reconstruction using Chinese Remaindering Applications: integer polynomial factorization, Gröbner basis computation, integer system solving,... P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 2
6 Exact Dense Linear Algebra Routines FFLAS Finite Field Linear Algebra Subroutines Based on a Matrix Multiplication kernel Using numerical BLAS through conversions Fast Matrix Multiplication algorithm P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 3
7 Exact Dense Linear Algebra Routines FFLAS Finite Field Linear Algebra Subroutines Based on a Matrix Multiplication kernel Using numerical BLAS through conversions Fast Matrix Multiplication algorithm Matrix matrix classic multiplication on a PIII 735 MHz ATLAS. (1) GF(19) Delayed with 32 bits. (2) GF(19) Prime field on top of Atlas. (3) GF(3 2 ) Galois field on top of Atlas. (4) 500 Speed (Mop/s) Matrix order P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 3
8 Exact Dense Linear Algebra Routines FFLAS Finite Field Linear Algebra Subroutines Based on a Matrix Multiplication kernel Using numerical BLAS through conversions Fast Matrix Multiplication algorithm Matrix matrix classic multiplication on a PIII 735 MHz ATLAS. (1) GF(19) Delayed with 32 bits. (2) GF(19) Prime field on top of Atlas. (3) GF(3 2 ) Galois field on top of Atlas. (4) 500 Speed (Mop/s) Matrix order FFPACK Finite Field Linear Algebra Package Higher Level (cf LAPACK) Based on matrix triangularization P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 3
9 Contents 1. Base field representations 2. Triangular System Solve (a) Three implementations (b) Two cascade algorithms and comparison 3. Triangularizations (a) Three implementations and comparison (b) Dealing with data locality 4. Conclusions and Perspectives P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 4
10 Base field representation Modular<double>: Based on machine double floating point representation Only using the mantissa Exact representation of integer up to 2 53 Avoids conversions and extra memory storage when using FFLAS P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 5
11 Base field representation Modular<double>: Based on machine double floating point representation Only using the mantissa Exact representation of integer up to 2 53 Avoids conversions and extra memory storage when using FFLAS Givaro-ZpZ: based on machine integer (16,32 or 64 bits) specialized dot-product (using delayed modulus) P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 5
12 Contents 1. Base field representations 2. Triangular System Solve (a) Three implementations (b) Two cascade algorithms and comparison 3. Triangularizations (a) Three implementations and comparison (b) Dealing with data locality 4. Conclusions and Perspectives P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 6
13 Triangular System Solve: trsm Compute a matrix X K m n, s.t. AX = B. P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 7
14 Triangular System Solve: trsm Compute a matrix X K m n, s.t. AX = B. Used for numerical computations (in the BLAS) P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 7
15 Triangular System Solve: trsm Compute a matrix X K m n, s.t. AX = B. Used for numerical computations (in the BLAS) Building block for triangularization block algorithms P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 7
16 Triangular System Solve: trsm Compute a matrix X K m n, s.t. AX = B. Used for numerical computations (in the BLAS) Building block for triangularization block algorithms Three different approaches for exact computation over a finite field : 1. A block recursive algorithm 2. A wrapping of the BLAS dtrsm 3. A matrix-vector based routine P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 7
17 Triangular System Solve: trsm Compute a matrix X K m n, s.t. AX = B. Used for numerical computations (in the BLAS) Building block for triangularization block algorithms Three different approaches for exact computation over a finite field : 1. A block recursive algorithm 2. A wrapping of the BLAS dtrsm 3. A matrix-vector based routine Cascade algorithms as solution P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 7
18 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
19 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
20 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). B 1 := B 1 A 2 X 2. P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
21 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). B 1 := B 1 A 2 X 2. X 1 := recursive call on (A 1,B 1 ). P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
22 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). B 1 := B 1 A 2 X 2. X 1 := recursive call on (A 1,B 1 ). Reduces to matrix multiplication P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
23 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). B 1 := B 1 A 2 X 2. X 1 := recursive call on (A 1,B 1 ). Reduces to matrix multiplication O(n ω ) algebraic time complexity P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
24 1. The block recursive algorithm [ ] [ ] [ ] A 1 A 2 0 A 3 X 1 X 2 = B 1 B 2 X 2 := recursive call on (A 3,B 2 ). B 1 := B 1 A 2 X 2. X 1 := recursive call on (A 1,B 1 ). Reduces to matrix multiplication O(n ω ) algebraic time complexity Efficiency of FFLAS P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 8
25 2. Wrapping the BLAS dtrsm Same approach as for the matrix multiplication in FFLAS: Conversion : Finite Field Real (double) Computation over the real (using BLAS dtrsm) Conversion : Real (double) Finite Field P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 9
26 2. Wrapping the BLAS dtrsm Same approach as for the matrix multiplication in FFLAS: Conversion : Finite Field Real (double) Computation over the real (using BLAS dtrsm) Conversion : Real (double) Finite Field Two constraints: No division must occur during BLAS computation No overflow P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 9
27 2. Wrapping the BLAS dtrsm First constraint: Divisions must be exact in x i = 1 a i,i b i n j=i+1 a i,j x j A must have a unit diagonal. Precondition A: diagonal of A. U = AD 1 A solve UY = B X = D 1 A Y where D A is the P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 10
28 2. Wrapping the BLAS dtrsm Second constraint: No overflow during the BLAS computation Must bound the growth of the coefficients: P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 11
29 2. Wrapping the BLAS dtrsm Second constraint: No overflow during the BLAS computation Must bound the growth of the coefficients: Naive:(p 1)p n 1 < 2 m P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 11
30 2. Wrapping the BLAS dtrsm Second constraint: No overflow during the BLAS computation Must bound the growth of the coefficients: Naive:(p 1)p n 1 < 2 m Using a classical prime field repr.: 0 x p 1: p 1 2 [ p n 1 + (p 2) n 1] < 2 m P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 11
31 2. Wrapping the BLAS dtrsm Second constraint: No overflow during the BLAS computation Must bound the growth of the coefficients: Naive:(p 1)p n 1 < 2 m Using a classical prime field repr.: 0 x p 1: p 1 2 [ p n 1 + (p 2) n 1] < 2 m Using a centered prime field repr.: p 1 2 x p 1 2 : p 1 2 ( ) n 1 p+1 2 < 2 m P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 11
32 2. Wrapping the BLAS dtrsm Limit the matrix order for a given prime p: P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 12
33 2. Wrapping the BLAS dtrsm Limit the matrix order for a given prime p: For m = 53: p = 2 n 55 p = 9739 n 4 p = n 2 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 12
34 2. Wrapping the BLAS dtrsm Limit the matrix order for a given prime p: For m = 53: Représentation positive Représentation centrée p = 2 n 55 p = 9739 n 4 p = n 2 n p P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 12
35 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
36 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: Using modular<double>: BLAS gemv and modulo P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
37 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: Using modular<double>: BLAS gemv and modulo Using integral representations: design of specialized dot-product routines [Dumas CASC 04] P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
38 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: Using modular<double>: BLAS gemv and modulo Using integral representations: design of specialized dot-product routines [Dumas CASC 04] Advantages: Delayed modulus (1 for each row of X) P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
39 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: Using modular<double>: BLAS gemv and modulo Using integral representations: design of specialized dot-product routines [Dumas CASC 04] Advantages: Delayed modulus (1 for each row of X) nicer bound: n(p 1) 2 < 2 m P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
40 3. Using Matrix-vector products A. Xi = X 1 Bi i = B i A.X i+1..n Matrix vector product Xi+1..n Different implementations: Using modular<double>: BLAS gemv and modulo Using integral representations: design of specialized dot-product routines [Dumas CASC 04] Advantages: Delayed modulus (1 for each row of X) nicer bound: n(p 1) 2 < 2 m Drawback: less efficient for large matrices (n 100) P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 13
41 Two cascade algorithms YES Idea: Or trsm rec n:=n/2 YES (n>n_blas)? NO trsm blas trsm rec n:=n/2 (n>n_delayed)? NO trsm delayed P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 14
42 Two cascade algorithms YES Idea: Or trsm rec n:=n/2 YES (n>n_blas)? NO trsm blas trsm rec n:=n/2 (n>n_delayed)? NO trsm delayed Timings of the cascade algorithm over Z/5Z using modular<double> on a P4 2.4Ghz Timings of the cascade algorithm over Z/5Z using Givaro ZpZ on a P4 2.4Ghz Speed (Mfops) Speed (Mfops) trsm rec trsm blas trsm delayed Matrix order trsm rec trsm blas trsm delayed Matrix order modular<double> p = 5 Givaro-ZpZ p = 5 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 14
43 Two cascade algorithms YES Idea: Or trsm rec n:=n/2 YES (n>n_blas)? NO trsm blas trsm rec n:=n/2 (n>n_delayed)? NO trsm delayed P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 14
44 Conclusion 1. trsm-blas highly depends on the prime trsm-delayed do not P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 15
45 Conclusion 1. trsm-blas highly depends on the prime trsm-delayed do not 2. If the field representation can be chosen Use Modular<double> with trsm-blas P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 15
46 Conclusion 1. trsm-blas highly depends on the prime trsm-delayed do not 2. If the field representation can be chosen Use Modular<double> with trsm-blas 3. Otherwise For some cases, a specialization of dot-product can slightly outperform trsm-blas P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 15
47 Contents 1. Base field representations 2. Triangular System Solve (a) Three implementations (b) Two cascade algorithms and comparison 3. Triangularizations (a) Three implementations and comparison (b) Dealing with data locality 4. Conclusions and Perspectives P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 16
48 Triangularization Specific issues: Have to deal with singular matrices Memory requirements P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 17
49 Triangularization Specific issues: Have to deal with singular matrices Memory requirements Provide a better analysis of the algebraic time complexity: improves the constant of the dominant term giving T = 2 3 n3 + O(n 2 ) in the nonsingular case with classic matrix multiplication P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 17
50 Triangularization Specific issues: Have to deal with singular matrices Memory requirements Provide a better analysis of the algebraic time complexity: improves the constant of the dominant term giving T = 2 3 n3 + O(n 2 ) in the nonsingular case with classic matrix multiplication We will compare 3 implementations: LSP: a block recursive algorithm [Ibara & Al.] LUdivine: LSP with lesser memory requirements LQUP : Fully in-place triangularization P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 17
51 LSP algorithms LSP [Ibara]: Split the row dimension P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 18
52 LSP algorithms LSP [Ibara]: Split the row dimension recursive call on A 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 18
53 LSP algorithms LSP [Ibara]: Split the row dimension recursive call on A 1 G A 21 U 1 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 18
54 LSP algorithms LSP [Ibara]: Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GV P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 18
55 LSP algorithms LSP [Ibara]: Split the row dimension recursive call on A 1 S S 1 S 2 G A 21 U 1 1 A 22 A 22 GV recursive call on A 22 L L1 G L 2 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 18
56 LSP algorithms LUdivine: result is in place Split the row dimension P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 19
57 LSP algorithms LUdivine: result is in place Split the row dimension recursive call on A 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 19
58 LSP algorithms LUdivine: result is in place Split the row dimension recursive call on A 1 G A 21 U 1 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 19
59 LSP algorithms LUdivine: result is in place Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GW P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 19
60 LSP algorithms LUdivine: result is in place Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GW recursive call on A 22 S L1 1 G S 2 L 2 V P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 19
61 LSP algorithms LQUP: fully in place Split the row dimension P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
62 LSP algorithms LQUP: fully in place Split the row dimension recursive call on A 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
63 LSP algorithms LQUP: fully in place Split the row dimension recursive call on A 1 G A 21 U 1 1 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
64 LSP algorithms LQUP: fully in place Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GV P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
65 LSP algorithms LQUP: fully in place Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GV recursive call on A 22 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
66 LSP algorithms LQUP: fully in place Split the row dimension recursive call on A 1 G A 21 U 1 1 A 22 A 22 GV recursive call on A 22 row permutations L U 1 1 G L U 2 2 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 20
67 Comparisons n LSP LUdivine LQUP Similar timings when matrix fit in the RAM LQUP is slightly faster LQUP is fully in-place no swap for n = 8000 P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 21
68 Dealing with data Locality Application: parallelism, out of core computations Use square recursive blocked data structure A triangularization algorithm: TURBO P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 22
69 Dealing with data Locality Application: parallelism, out of core computations Use square recursive blocked data structure A triangularization algorithm: TURBO 3000 TURBO vs LQUP for rank computation over Z101 on a P4 2.4Ghz, 512Mb RAM Mfops (2) LQUP using Givaro ZpZ Matrix order P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 22
70 Dealing with data Locality Application: parallelism, out of core computations Use square recursive blocked data structure A triangularization algorithm: TURBO 3000 TURBO vs LQUP for rank computation over Z101 on a P4 2.4Ghz, 512Mb RAM Mfops (1) TURBO using Givaro ZpZ (2) LQUP using Givaro ZpZ Matrix order P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 22
71 Conclusion Results: Approach the timings of numerical routines: 6.5s for a numeric LUP of a matrix 7.6s for a symbolic LQUP of a matrix P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 23
72 Conclusion Results: Approach the timings of numerical routines: 6.5s for a numeric LUP of a matrix 7.6s for a symbolic LQUP of a matrix Improved memory management of LSP factorization Further analysis of LSP time complexity. P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 23
73 Conclusion Results: Approach the timings of numerical routines: 6.5s for a numeric LUP of a matrix 7.6s for a symbolic LQUP of a matrix Improved memory management of LSP factorization Further analysis of LSP time complexity. Optimal bounds for the coefficient growth in trsm P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 23
74 Conclusion Results: Approach the timings of numerical routines: 6.5s for a numeric LUP of a matrix 7.6s for a symbolic LQUP of a matrix Improved memory management of LSP factorization Further analysis of LSP time complexity. Optimal bounds for the coefficient growth in trsm Part of the LinBox library [ P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 23
75 Conclusion Conclusion: Again: Wrapping numerical routines as much as possible appears to be the best choice When not possible (ex LSP) block recursive algorithms P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 24
76 Conclusion Conclusion: Again: Wrapping numerical routines as much as possible appears to be the best choice When not possible (ex LSP) block recursive algorithms BLAS No modulo Need to control the coefficient growth Bounds for correctness P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 24
77 Conclusion Conclusion: Again: Wrapping numerical routines as much as possible appears to be the best choice When not possible (ex LSP) block recursive algorithms BLAS No modulo Need to control the coefficient growth Bounds for correctness Cascade structure Switches between algorithms due to Correctness constraints (theoretical thresholds) Performance constraints (experimental thresholds) P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 24
78 Further developments Self adapting software: automatic setup of optimal experimental thresholds P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 25
79 Further developments Self adapting software: automatic setup of optimal experimental thresholds Apply of the factorization to other applications: characteristic polynomial, null space,... P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 25
80 FFPACK: Finite Field Linear Algebra Package Jean-Guillaume Dumas, Pascal Giorgi and Clément Pernet {Jean.Guillaume.Dumas, P. Giorgi, J-G. Dumas & C. Pernet - FFPACK: Finite Field Linear Algebra Package p. 26
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