Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018 (New York, USA) Vincent Neiger, Johan Rosenkilde, Grigory Solomatov XLIM – University of Limoges, France Technical University of Denmark July 17, 2018
Outline • Context and contribution • Algorithmic tools and general approach • Overview of new algorithms 1/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Polynomial matrix computations 3 X 3 + X 2 + 5 X + 3 6 X + 5 2 X + 1 K a field 5 X 2 + 3 X + 1 5 5 X + 3 Matrices over K [ X ] X 3 + 4 X + 1 4 X 2 + 3 3 X + 4 Usual matrix operations • matrix multiplication • rank, determinant • system solving, inversion Transformations to normal forms • triangularization � Hermite form • row reduction � Popov form • diagonalization � Smith form 2/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Hermite and Popov forms ( K = Z / 7 Z ) 3 X 3 + X 2 + 5 X + 3 6 X + 5 2 X + 1 5 X 2 + 3 X + 1 Input matrix: 5 5 X + 3 X 3 + 4 X + 1 4 X 2 + 3 3 X + 4 row i ← row i + p ( X ) row j Transform, via elementary row operations, row i ↔ row j row i ← α row i , α ∈ K \{ 0 } � into Popov form [Popov ’72] X 3 + 5 X 2 + 4 X + 1 2 X + 4 3 X + 5 X 2 + 2 X + 3 1 X + 2 X 2 3 X + 2 4 X � into Hermite form [Hermite 1851] X 6 + 6 X 4 + X 3 + X + 4 0 0 5 X 5 + 5 X 4 + 6 X 3 + 2 X 2 + 6 X + 3 X 0 3 X 4 + 5 X 3 + 4 X 2 + 6 X + 1 5 1 3/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Goal: fast algorithms 3 X 3 + X 2 + 5 X + 3 6 X + 5 2 X + 1 K a field 5 X 2 + 3 X + 1 5 5 X + 3 Matrices over K [ X ] X 3 + 4 X + 1 4 X 2 + 3 3 X + 4 Usual matrix operations • matrix multiplication • rank, determinant • system solving, inversion Transformations to normal forms • triangularization � Hermite form [Gupta-Storjohann ’11] [Labahn-Neiger-Zhou ’17] • row reduction � Popov form [Gupta-Sarkar-Storjohann-Valeriote ’11 & ’12] • diagonalization � Smith form 4/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Goal: fast algorithms 3 X 3 + X 2 + 5 X + 3 6 X + 5 2 X + 1 K a field 5 X 2 + 3 X + 1 5 5 X + 3 Matrices over K [ X ] X 3 + 4 X + 1 4 X 2 + 3 3 X + 4 Usual matrix operations • matrix multiplication cost O ˜ ( m ω d ) • rank, determinant • system solving, inversion m × m matrix with degree d Transformations to normal forms • triangularization � Hermite form [Gupta-Storjohann ’11] [Labahn-Neiger-Zhou ’17] • row reduction � Popov form [Gupta-Sarkar-Storjohann-Valeriote ’11 & ’12] • diagonalization � Smith form 4/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Contribution: rectangular normal forms For a matrix in K [ X ] m × n with m � n : Popov form deterministic algorithm with cost O ˜ ( m ω − 1 nd ) d the degree of the matrix size of Popov form is O ( mnd ) previous fastest: O ( rmnd 2 ) [Mulders-Storjohann ’03] and O ˜ ( mn ω d ) [ ⋆ ] Las Vegas: O ˜ ( m ω − 1 nd ) assuming full row rank [Sarkar-Storjohann ’11] [ ⋆ ] = based on some kernel computation [Beckermann-Labahn-Villard ’06] 5/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Context and contribution Contribution: rectangular normal forms For a matrix in K [ X ] m × n with m � n : Hermite form deterministic algorithm with cost O ˜ ( m ω − 1 nδ ) δ � md δ = min ( sum of row/col degrees ) size of Hermite form can be Θ ( mnδ ) previous fastest: O ˜ ( n ω + 1 δ ) [ ⋆ ] (speed-up factor � n ) [ ⋆ ] = based on some kernel computation [Beckermann-Labahn-Villard ’06] 5/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Tool: kernel bases Left kernel basis of a matrix A ∈ K [ X ] m × n K has full row rank Matrix K ∈ K [ X ] k × m such that KA = 0 rows of K generate the kernel of A • core algorithmic tool: rank, inversion, determinant, Hermite form, . . . • kernel bases can now be computed fast in Popov form � combine [Zhou-Labahn-Storjohann ’12] + this work • shifted normal forms: UA = P (via shifted Popov approximant basis [Jeannerod-Neiger-Schost-Villard ’16]) � cost of this approach: unsatisfactory 6/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Reduction to the full row rank case Row basis of a matrix A ∈ K [ X ] m × n � B has full row rank Matrix B ∈ K [ X ] r × n such that B and A have the same row space Fast algorithm: [Zhou-Labahn ’13] Consequence: deal with m � n and rank-deficient matrices Normal form of arbitrary A Step 1: B ← row basis of A // use [Zhou-Labahn ’13] Step 2: P ← the normal form of B // full row rank case Step 3: Return P Step 1 costs O ˜ ( m ω − 1 ( m + n ) d ) What do pivots become in the full row rank case? 7/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Degree structure and pivots nonsingular, m × m pivot of a row: rightmost entry of largest degree [ 6 ] [ 1 ] [ 2 ] [ 4 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] Popov [ 3 ] [ 1 ] [ 3 ] [ 2 ] [ 5 ] [ 1 ] [ 2 ] [ 5 ] size O ( m 2 d ) pivot of a row: rightmost nonzero entry [ 12 ] [ 11 ] [ 3 ] Hermite [ 11 ] [ 2 ] [ 0 ] [ 11 ] [ 2 ] [ 1 ] size O ( m 2 d ) 8/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Degree structure and pivots nonsingular, m × m full row rank, m × n pivot of a row: rightmost entry of largest degree [ 6 ] [ 1 ] [ 2 ] [ 4 ] [ 6 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 4 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] [ 1 ] Popov [ 3 ] [ 1 ] [ 3 ] [ 2 ] [ 3 ] [ 3 ] [ 3 ] [ 1 ] [ 3 ] [ 3 ] [ 2 ] [ 5 ] [ 1 ] [ 2 ] [ 5 ] [ 5 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 5 ] size O ( m 2 d ) size O ( mnd ) pivot of a row: rightmost nonzero entry [ 12 ] [ 11 ] [ 3 ] Hermite [ 11 ] [ 2 ] [ 0 ] [ 11 ] [ 2 ] [ 1 ] size O ( m 2 d ) 8/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Degree structure and pivots nonsingular, m × m full row rank, m × n pivot of a row: rightmost entry of largest degree [ 6 ] [ 1 ] [ 2 ] [ 4 ] [ 6 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 4 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] [ 1 ] Popov [ 3 ] [ 1 ] [ 3 ] [ 2 ] [ 3 ] [ 3 ] [ 3 ] [ 1 ] [ 3 ] [ 3 ] [ 2 ] [ 5 ] [ 1 ] [ 2 ] [ 5 ] [ 5 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 5 ] size O ( m 2 d ) size O ( mnd ) pivot of a row: rightmost nonzero entry [ 12 ] [ 12 ] [ 11 ] [ 21 ] [ 3 ] [ 11 ] [ 3 ] Hermite [ 11 ] [ 37 ] [ 2 ] [ 193 ] [ 0 ] [ 11 ] [ 2 ] [ 0 ] [ 11 ] [ 32 ] [ 2 ] [ 5 ] [ 243 ] [ 1 ] [ 11 ] [ 2 ] [ 1 ] size O ( m 2 d ) size O ( mnδ ) 8/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
Algorithmic tools and general approach Pivot support [ 4 ] [ 3 ] [ 5 ] [ 2 ] [ 1 ] [ 4 ] [ 5 ] [ 7 ] [ 9 ] [ 4 ] [ 4 ] [ 2 ] [ 9 ] [ 3 ] From now, focus on Popov form = A [ 5 ] [ 8 ] [ 5 ] [ 5 ] [ 0 ] [ 5 ] [ 9 ] • full row rank m × n matrix A [ 7 ] [ 4 ] [ 9 ] [ 3 ] [ 5 ] [ 5 ] [ 9 ] • input/output size O ( mnd ) unimodular • target cost O ˜ ( m ω − 1 nd ) [ 6 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 4 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 1 ] [ 1 ] [ 1 ] = P [ 3 ] [ 3 ] [ 3 ] [ 1 ] [ 3 ] [ 3 ] [ 2 ] [ 5 ] [ 5 ] [ 5 ] [ 1 ] [ 5 ] [ 2 ] [ 5 ] pivot support of A : indices of the pivots in P � any multiple UA has its pivots in the pivot support Algorithmic approach • Find the pivot support of A • Use the pivot support to compute P 9/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018
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