Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix Jean-Guillaume Dumas, Cl´ ement Pernet Universit´ e Grenoble Alpes, Laboratoire Jean Kuntzmann, UMR CNRS ISSAC’18, New York, USA July 17, 2017 Supported by OpenDreamKit Horizon 2020 European RI project (#676541) P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 1 / 16
Introduction Context Applications of symmetric Gaussian elimination Symmetric linear system solving Signature LLL: R factor of rational QR [Villard’12]) P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 2 / 16
Introduction Context Applications of symmetric Gaussian elimination Symmetric linear system solving Signature LLL: R factor of rational QR [Villard’12]) Compared to unsymmetric Gaussian elimination Save a factor of 2 in time complexity Invariants specific to symmetric matrices (signature) P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 2 / 16
Introduction Context Applications of symmetric Gaussian elimination Symmetric linear system solving Signature LLL: R factor of rational QR [Villard’12]) Compared to unsymmetric Gaussian elimination Save a factor of 2 in time complexity Invariants specific to symmetric matrices (signature) Motivation here fsytrf : finite field dense symmetric elimination in fflas-ffpack to be lifted for LinBox signature over Z reduction to matrix product: O ( n 2 r ω − 2 ) and BLAS3 investigate symmetric rank profile matrix and related pivoting P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 2 / 16
Introduction Outline State of the art on symmetric factorizations 1 Rank profile and pivoting 2 Algorithms 3 The characteristic 2 case 4 Performance 5 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 3 / 16
State of the art on symmetric factorizations Symmetric factorizations Symmetric Decomposition Exists for Field with sqrt & B T = A B Generic rank profile P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 4 / 16
State of the art on symmetric factorizations Symmetric factorizations Symmetric Decomposition Exists for Field with sqrt & B T = A B Generic rank profile L T Generic rank profile D A = L P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 4 / 16
State of the art on symmetric factorizations Symmetric factorizations Symmetric Decomposition Exists for Field with sqrt & B T = A B Generic rank profile L T Generic rank profile D A = L L T No [ 0 1 1 0 ]-like blocks T P D P A = L P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 4 / 16
State of the art on symmetric factorizations Symmetric factorizations Symmetric Decomposition Exists for Field with sqrt & B T = A B Generic rank profile L T Generic rank profile D A = L L T No [ 0 1 1 0 ]-like blocks T P D P A = L Any [Parlett-Reid 1970] L T T P T P = A L T tridiagonal P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 4 / 16
State of the art on symmetric factorizations Symmetric factorizations Symmetric Decomposition Exists for Field with sqrt & B T = A B Generic rank profile L T Generic rank profile D A = L L T No [ 0 1 1 0 ]-like blocks T P D P A = L Any [Parlett-Reid 1970] L T T P T P = A L T tridiagonal . . . Any [Bunch-Kaufmann 1977] L T T P Y P A = L Y with 1 × 1 and 2 × 2 blocks P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 4 / 16
State of the art on symmetric factorizations State of the art Form Properties [Parlett-Reid 1970] [Bunch-Parlett 1971] [Aasen 1971] Diagonal Pivoting Full pivoting Partial pivoting T Iterative Iterative Iterative 2 3 n 3 2 3 n 3 1 3 n 3 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 5 / 16
State of the art on symmetric factorizations State of the art Form Properties [Parlett-Reid 1970] [Bunch-Parlett 1971] [Aasen 1971] Diagonal Pivoting Full pivoting Partial pivoting T Iterative Iterative Iterative 3 n 3 2 2 3 n 3 1 3 n 3 [Bunch-Kaufmann 1977] Partial pivoting Y Iterative 1 3 n 3 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 5 / 16
State of the art on symmetric factorizations State of the art Form Properties [Parlett-Reid 1970] [Bunch-Parlett 1971] [Aasen 1971] Diagonal Pivoting Full pivoting Partial pivoting T Iterative Iterative Iterative 2 3 n 3 3 n 3 2 1 3 n 3 [Bunch-Kaufmann 1977] [Shklarski-Toledo 2011] Partial pivoting Partial pivoting Y Iterative Recursive (GRP hyp.) 1 3 n 3 1 3 n 3 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 5 / 16
State of the art on symmetric factorizations State of the art Form Properties [Parlett-Reid 1970] [Bunch-Parlett 1971] [Aasen 1971] Diagonal Pivoting Full pivoting Partial pivoting T Iterative Iterative Iterative 2 3 n 3 2 3 n 3 1 3 n 3 [Bunch-Kaufmann 1977] [Shklarski-Toledo 2011] [Yamazaki-Dongarra 2017] Partial pivoting Partial pivoting Partial pivoting Y Iterative Recursive (GRP hyp.) Block Iterative 1 3 n 3 1 3 n 3 1 3 n 3 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 5 / 16
State of the art on symmetric factorizations State of the art Form Properties [Parlett-Reid 1970] [Bunch-Parlett 1971] [Aasen 1971] Diagonal Pivoting Full pivoting Partial pivoting T Iterative Iterative Iterative 2 3 n 3 2 3 n 3 1 3 n 3 [Bunch-Kaufmann 1977] [Shklarski-Toledo 2011] [Yamazaki-Dongarra 2017] Partial pivoting Partial pivoting Partial pivoting Y Iterative Recursive (GRP hyp.) Block Iterative 1 3 n 3 1 3 n 3 1 3 n 3 Here Pivoting revealing the rank profile matrix ∆ Recursive for any matrix 3 n 3 when rank = n & ω =3) O ( n 2 r ω − 2 ) (gives 1 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 5 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal 0 0 0 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal 0 0 0 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal ⇒ L ∆ L T with ∆ block diagonal, 1 × 1 0 or 2 × 2 [ 0 1 1 0 ] blocks 0 0 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal ⇒ L ∆ L T with ∆ block diagonal, 1 × 1 0 or 2 × 2 [ 0 1 1 0 ] blocks 0 Off-diagonal pivoting with non-zero 0 diagonal 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal ⇒ L ∆ L T with ∆ block diagonal, 1 × 1 or 2 × 2 [ 0 1 1 0 ] blocks Off-diagonal pivoting with non-zero 0 diagonal 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting Symmetric pivoting Diagonal pivoting ⇒ LDL T with D diagonal Off-diagonal pivoting with zero diagonal ⇒ L ∆ L T with ∆ block diagonal, 1 × 1 or 2 × 2 [ 0 1 1 0 ] blocks Off-diagonal pivoting with non-zero 0 diagonal 0 ⇒ LDL T with D diagonal 0 ⇒ requires division by 2 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 6 / 16
Rank profile and pivoting The rank profile matrix Rank Profiles Given a matrix A of rank r : Example 0 0 0 2 3 0 4 0 A = 0 0 0 0 5 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 7 / 16
Rank profile and pivoting The rank profile matrix Rank Profiles Given a matrix A of rank r : RRP (Row Rank Profile): first r linearly independant rows Example 0 0 0 2 3 0 4 0 A = 0 0 0 0 5 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 7 / 16
Rank profile and pivoting The rank profile matrix Rank Profiles Given a matrix A of rank r : RRP (Row Rank Profile): first r linearly independant rows CRP (Column Rank Profile): first r linearly independant columns Example 0 0 0 2 3 0 4 0 A = 0 0 0 0 5 0 0 0 P · L · ∆ · LT · PT J-G. Dumas, C. Pernet (UGA) ISSAC’18, New York 7 / 16
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