Inverting integer and polynomial matrices. Jo60. Arne Storjohann University of Waterloo

Transcription

1 Inverting integer and polynomial matrices Jo60 Arne Storjohann University of Waterloo

2 Integer Matrix Inverse Input: An n n matrix A filled with entries of size d digits. Output: The matrix inverse A 1. Example: 6 6 with d = = input A has total size O(n 2 d) words output A 1 has total size O (n 3 d) words 1

3 Polynomial Matrix Inverse Input: An n n matrix A K[x] of degree bounded by d. Output: The matrix inverse A 1. Example: 3 3 over Z 7 [x] with d = 3 6x 2 + 4x+2 2x 2 + 5x+3 2x 2 + x 5x 2 + 3x+2 6x 2 + 6x+4 4x x 2 + 3x 5x 2 + 4x+3 5x = 2x 4 +3x 2 +6x+2 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 6x 3 +4x 2 +3 x 5 +5x 4 +3x 3 +2x 2 +x+5 6x 4 +3x 3 +4x 2 +x+4 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 6x 3 +3x 2 +5x+6 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 4x 3 +3x 2 +5x+4 x 5 +5x 4 +3x 3 +2x 2 +x+5 3x 4 +3x 3 +2x+3 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 2x 4 +6x 3 +6x 2 +4x+4 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 6x 2 +2x+5 x 5 +5x 4 +3x 3 +2x 2 +x+5 x 4 +3x 3 +x 2 +6x+6 x 6 +4x 5 +5x 4 +6x 3 +6x 2 +4x+2 input A has total size O(n 2 d) field elements from K output A 1 has total size O(n 3 d) 2

4 Notation nonsingular input A has dimension n K[x]: size of entries is d dega count field operations fromk Z: size of entries is d log A count word operations ω is a feasible exponent for matrix multiplication: 2 ω 3 Classical inversion algorithms Hensel-Newton iteration: p-adic lifting Homomorphic imaging and Chinese remaindering Classical Goal O (n ω+1 d) O (n 3 d) 3

5 Outline: Three completely different approaches for inversion A1 Sparse inverse expansion useful representation of A 1 mod p n applicable for integer and polynomial inputs A2 Balanced kernel basis nearly optimal inversion algorithm for generic polynomials inputs [Jeannerod & Villard, 2005] A3 Outer product adjoint formula nearly optimal inversion algorithm for all polynomial inputs well conditioned integer inputs 4

6 Alternative representations for the inverse of A = [ 3 2 x 5 ] A 1 = = [ 2 6+2x x 6+2x 2 6+2x 4 6+2x ] [ 1+x+x 2 + x 3 + 1xx 2 x 3 + xx 2 x x+x 2 + x 3 + ] = [ ] + [ ] x+ [ ] x 2 +. A 1 over K[[x]] modulo x 2nd sufficient to reconstruct A 1 over K(x) compute expansion over K[[x]] using quadratic Newton iteration 5

7 The sparse inverse formula for A= [ 1cx ] Explicit inverse modulo x 32 (1cx) 1 mod x 32 1+cx+c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + +c 31 x 31 6

8 The sparse inverse formula for A= [ 1cx ] Explicit inverse modulo x 32 (1cx) 1 mod x 32 1+cx+c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + +c 31 x 31 (1+cx)(1+c 2 x 2 )(1+c 4 x 4 )(1+c 8 x 8 )(1+c 16 x 16 ) 7

9 The sparse inverse formula for A= [ 1cx ] Explicit inverse modulo x 32 (1cx) 1 mod x 32 = 32 coefficients {}}{ 1+cx+c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + +c 31 x 31 = (1+cx)(1+c 2 x 2 )(1+c 4 x 4 )(1+c 8 x 8 )(1+c 16 x 16 ) }{{} 5 coefficients R 4 {}}{ c 4 x 4 ) R {}}{ 1 {}}{ R 2 } (1+ c x)(1+ {{ c 2 x 2 )(1+ } (1+ (1cx) 1 mod x 8 R 8 {}}{ c 8 x 8 )(1+ R 16 {}}{ c 16 x 16 ) 8

10 Computing the residues for A=1cx R 2 x 2 = I A(1+cx) = c 2 x 2 { A 1 mod }} x 4 { R 4 x 4 = I A( 1+cx+c 2 x 2 + c 3 x 3 ) = c 4 x 4 { A 1 mod }} x 8 { R 8 x 8 = I A( 1+cx+c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + c 6 x 6 + c 7 x 7 ) = c 8 x 8 9

11 Computing the residues for A=1cx R 2 x 2 = I A(1+cx) = c 2 x 2 { A 1 mod }} x 4 { R 4 x 4 = I A( 1+cx+c 2 x 2 + c 3 x 3 ) = c 4 x 4 { A 1 mod }} x 8 { R 8 x 8 = I A( 1+cx+c 2 x 2 + c 3 x 3 + c 4 x 4 + c 5 x 5 + c 6 x 6 + c 7 x 7 ) = c 8 x 8 If dega=d then R 2,R 4,R 8, have degree d 1. Can use reverse short product need only top d coefficients of A 1 mod x 10

12 Standard quadratic lifting High-order component lifting

13 Sparse inverse formula for integer and polynomial matrices Example for 7 1 mod x 32, x= mod x 32 = = (((52(13x 2 )+2x 4 )(15x 4 )+4x 8 )(13x 8 )+2x 16 )(15x 16 )+4x 32 General case formula can be computed in time O (n ω d) sparse inverse formula for A 1 mod x n has size O(n 2 (logn)d) can be used to compute A 1 b in time O (n ω d) fast algorithms for linear algebra over Z and K[x] randomized Las Vegas algorithm for explicit inverse? 12

14 Outline: Three completely different approaches for inversion A1 Sparse inverse expansion useful representation of A 1 mod p n applicable for integer and polynomial inputs A2 Balanced kernel basis nearly optimal inversion algorithm for generic polynomials inputs [Jeannerod & Villard, 2005] A3 Outer product adjoint formula nearly optimal inversion algorithm for all polynomial inputs well conditioned integer inputs 13

15 Minimal kernel basis of generic polynomial matrices Example 1: A K[x] 2 1,K=Z 97 [ N [ ] A ] f g f = 0 g 1 1 Example 2: A K[x] 8 4,K=Z 97 N x x 45x+51 51x+44 33x x x+92 67x+90 83x+29 47x x x+64 42x+38 12x+67 67x x+78 31x+50 4x+52 93x+28 17x+52 A 60x+16 52x20 4x89 77x+69 64x+89 16x+59 69x46 33x x+36 91x22 51x27 31x27 65x+88 10x6 80x84 26x51 88x+97 67x+58 29x+37 9x91 81x+65 12x+78 5x63 42x+9 21x27 79x22 51x+16 95x+86 97x14 83x96 8x54 =0 4 4 generic A K[x] 2n n of degree d has kernel of degree d effective algorithms, e.g., [Giorgi, Jeannerod & Villard, ISSAC 03] 14

16 From generic kernel to generic inversion split input A K[x] n n of degree d into two halves: A= [ A L A R ] compute minimal kernel U for A L and U for A R [ U U ] [ AL A R ] = [ UAL UA R ] recurse on degree 2d matrices UA L and UA R Example: n=8 and d = 1 N 3 N 2 N 1 A [ AL A R ] =(deta)a 1 15

17 From generic kernel to generic inversion N 3 N 2 N 1 A cost to compute N 1,N 2,...,N log2 n is O (n ω d)) cost to compute (deta)a 1 = N logn N 2 N 1 is O (n 3 d) [ AL A R ] =(deta)a 1 Notes requires generic input seems to require n to be a power of 2 16

18 Outline: Three completely different approaches for inversion A1 Sparse inverse expansion useful representation of A 1 mod p n applicable for integer and polynomial inputs A2 Balanced kernel basis nearly optimal inversion algorithm for generic polynomials inputs [Jeannerod & Villard, 2005] A3 Outer product adjoint formula nearly optimal inversion algorithm for all polynomial inputs well conditioned integer inputs 17

19 Special case of the outer product adjoint formula Example input matrix A= , deta= The adjoint A adj = [ ] mod

20 General case of the output product adjoint formla suppose Smith form of A is Diag(s 1,s 2,...,s n ) A adj deta v n u n + deta v n1 u n1 + + deta v 1 u 1 (mod deta) s n s n1 s 1 Integer matrices only obtain inverse modulo deta inverse of A Z n n in time O (n 3 d+ n 3 logκ(a)) nearly optimal for well conditioned input randomized Las Vegas Polynomial matrices use randomization and reversion to extend to arbitrary input inverse of any A K[x] n n in time O (n 3 d) operations fromk randomized Las Vegas 19

21 Applications of the nearly optimal inversion formulas 20

22 Example: Linear solving over K: K=Z 7, n=5 A= b= deta=3 A 1 b= Note: no expression swell over Z 7 21

23 Example input over K[x]: n=5, d = 2 = expression swell A= 6x 2 + x+5 6x+1 2x 2 x+4 3x 2 + 6x+5 3x+5 2x 2 + x+2 6x 2 + 2x+3 3x 2 + 2x+1 x 2 + 2x 5x 2 + x+4 6x 2 + 3x+6 5x 2 + 2x 5x 2 + 6x+4 3x 2 + x 6x 2 + 6x+5 2x 2 + 3x+2 3x 2 + 6x 3x 2 + 2x+2 5x 2 + 5x+4 3x 2 + 2x+5 5x 2 + 3x+1 4x+1 5x 2 + 4x 2x 2 + 3x+2 b= x 2 + x+1 2x 2 + 2x+6 x 2 + 3x+2 2x 2 + x+4 x 2 + 4x+3 deta=6x x 9 + x 8 + 3x 6 + x 5 + x 4 + 4x 2 + 2x+3 Degree is n d where d is degrees in input matrix. x x 9 + 2x 8 + x 7 + 3x 6 + x 4 + 6x 3 + 3x 5x 9 + x 8 + 6x 7 + x 5 + 4x 4 + 5x 3 + x 2 + 3x+1 A 1 b= 6x 10 + x 9 + 3x 8 + 6x 7 + 3x 6 + 2x 5 + 2x 3 + 5x x x 9 + 4x 8 + 3x 7 + 2x 6 + 3x 5 + 5x 4 + 6x 3x x 9 + 4x 7 + 5x 6 + 2x 5 + x+6 (1/detA) 22

24 Nearly optimal linear system solving with precomputation Recall situation over K Input: Nonsingular A K n n Precompute: Inverse A 1 Compute: A 1 v for any given v K n 1 A b = A 1 b size of A 1 b is Ω(n) time to compute A 1 b is O(n 2 ) 23

25 Nearly optimal linear system solving with precomputation over K[x] Input: nonsingular A K[x] n n of degree d Precompute: adjoint decomposition N logn N 2 N 1 =(deta)a 1. Compute: A 1 v for any given v K n 1 of degree d. N 3 N 2 N 1 b [1] [1] [1] [1] [1] [1] [1] [1] =(deta)a 1 b size of A 1 b is Ω(n 2 d) cost to compute A 1 b is O (n 2 d) result for nongeneric input via outer product adjoint formula 24

26 Krylov basis and matrix powers Input: matrix A K n n and vector v K n 1 Output: v,av,a 2,A 3 v,...,a n1 v Standard algorithm for matrix vector interates compute Av,A(Av),A(A 2 v),a(a 3 v),...,a(a n2 v) requires n 1 matrix-vector products total cost is O(n 3 ) operations from K Fast algorithm [Keller-Gehrig, 1985] precompute A 2,(A 2 ) 2,(A 4 ) 2,..., compute iterates via matrix multiplication total cost is O(n w logn) operations fromk 25

27 Nearly optimal computation of matrix powers Input: matrix A K n n Output: I,A,A 2,A 3,A 4,...,A n1 consider matrix I xa K[x] compute(i xa) 1 mod x n = I+ xa+x 2 A 2 + x 3 A 3 + x n1 A n1 total size of output is Θ(n 3 ) elements of K total cost is O (n 3 ) using fast inversion algorithm 26