State of the art A new algorithm The new algorithm into practice Faster algorithms for the characteristic polynomial Clément P ERNET and Arne S TORJOHANN Symbolic Computation Group University of Waterloo, Canada. ISSAC 2007, Waterloo, July 30 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Problem Compute the characteristic polynomial of a dense matrix over a field C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Problem Compute the characteristic polynomial of a dense matrix over a field Result Randomized Las-Vegas algorithm in O ( n ω ) field operations for large fields ( # F > 2 n 2 ). C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Problem Compute the characteristic polynomial of a dense matrix over a field Result Randomized Las-Vegas algorithm in O ( n ω ) field operations for large fields ( # F > 2 n 2 ). Improves previous complexity by a log n factor, Optimal reduction to Matrix multiplication. C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Problem Compute the characteristic polynomial of a dense matrix over a field Result Randomized Las-Vegas algorithm in O ( n ω ) field operations for large fields ( # F > 2 n 2 ). Improves previous complexity by a log n factor, Optimal reduction to Matrix multiplication. Practical efficiency. E.g. over Z 547 909 : n 500 5000 15 000 0.91s 4m44s 2h20m LinBox 1.27s 15m32s 7h28m magma-2.13 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Outline State of the art 1 A new algorithm 2 Shifted forms Principle of the new algorithm Complexity 3 The new algorithm into practice C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Outline State of the art 1 A new algorithm 2 Shifted forms Principle of the new algorithm Complexity 3 The new algorithm into practice C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Pre-Strassen age Leverrier 1840: trace of powers of A , and Newton’s formula improved/rediscovered by Souriau, Faddeev, Frame and Csanky n 4 � � O , based on Matrix multiplication Suited for parallel computation model C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Pre-Strassen age Leverrier 1840: trace of powers of A , and Newton’s formula improved/rediscovered by Souriau, Faddeev, Frame and Csanky n 4 � � O , based on Matrix multiplication Suited for parallel computation model Danilevskii 1937: elementary row/column operations � n 3 � ⇒O C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Pre-Strassen age Leverrier 1840: trace of powers of A , and Newton’s formula improved/rediscovered by Souriau, Faddeev, Frame and Csanky n 4 � � O , based on Matrix multiplication Suited for parallel computation model Danilevskii 1937: elementary row/column operations � n 3 � ⇒O Hessenberg 1942: transformation to quasi-upper triangular and determinant expansion formula. n 3 � � ⇒O C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Post-Strassen age Preparata & Sarwate 1978: Update Csanky with fast matrix multiplication n ω + 1 � � ⇒O C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Post-Strassen age Preparata & Sarwate 1978: Update Csanky with fast matrix multiplication n ω + 1 � � ⇒O Keller-Gehrig 1985, alg.1: computes ( A 2 i ) i = 1 ... log 2 n to form a Krylov basis. O ( n ω log n ) the best complexity up to now C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art A new algorithm The new algorithm into practice Post-Strassen age Preparata & Sarwate 1978: Update Csanky with fast matrix multiplication n ω + 1 � � ⇒O Keller-Gehrig 1985, alg.1: computes ( A 2 i ) i = 1 ... log 2 n to form a Krylov basis. O ( n ω log n ) the best complexity up to now Keller-Gehrig 1985, alg.2: inspired by Danilevskii, block operations O ( n ω ) but only valid with generic matrices C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Outline State of the art 1 A new algorithm 2 Shifted forms Principle of the new algorithm Complexity 3 The new algorithm into practice C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Definition (degree d Krylov matrix of one vector v ) A d − 1 v � � K = v Av . . . Property 2 3 0 ∗ 1 6 ∗ 7 A × K = K × 6 7 ... 6 7 4 5 ∗ 1 ∗ | {z } C PA , v min C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Definition (degree d Krylov matrix of one vector v ) A d − 1 v � � K = v Av . . . Property 2 3 0 ∗ 1 6 ∗ 7 A × K = K × 6 7 ... 6 7 4 5 ∗ 1 ∗ | {z } C PA , v min ⇒ if d = n , K − 1 AK = C P A car C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Definition (degree d Krylov matrix of one vector v ) A d − 1 v � � K = v Av . . . Property 2 3 0 ∗ 1 6 ∗ 7 A × K = K × 6 7 ... 6 7 4 5 ∗ 1 ∗ | {z } C PA , v min ⇒ if d = n , K − 1 AK = C P A car [Keller-Gehrig, alg. 2] : K − 1 AK in O ( n ω ) for A generic C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Definition (degree k Krylov matrix of several vectors v i ) � A k − 1 v 1 A k − 1 v 2 A k − 1 v l � K = v 1 . . . v 2 . . . . . . v l . . . Property k k ≤ k 0 1 1 0 1 1 A × K = K × 0 1 1 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Fact (Shift Hessenberg form) If ( d 1 , . . . d l ) is lexicographically maximal such that � A d 1 − 1 v 1 A d l − 1 v l � K = v 1 . . . . . . v l . . . is non-singular, then d 1 d 2 dl 0 1 1 0 1 A × K = K × 1 0 1 1 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Principle k -shifted form: k k ≤ k 0 1 1 0 1 1 0 1 1 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Principle k + 1-shifted form: k + 1 k + 1 d l 0 1 1 0 1 1 0 1 1 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Principle Compute iteratively from 1-shifted form to d 1 -shifted form C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Principle Compute iteratively from 1-shifted form to d 1 -shifted form each completed block appears in the increasing degree order C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Principle Compute iteratively from 1-shifted form to d 1 -shifted form each completed block appears in the increasing degree order until the shifted Hessenberg form is obtained: d 1 d 2 d l 0 1 1 0 1 1 0 1 1 C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
State of the art Shifted forms A new algorithm Principle of the new algorithm The new algorithm into practice Complexity Example C. P ERNET and A. S TORJOHANN Faster algorithms for the characteristic polynomial
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