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Computing Popov and Hermite forms

• f 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

K a field Matrices over K[X]

  3X3 + X2 + 5X + 3 6X + 5 2X + 1 5 5X2 + 3X + 1 5X + 3 3X + 4 X3 + 4X + 1 4X2 + 3  

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/7Z)

Input matrix:

  3X3 + X2 + 5X + 3 6X + 5 2X + 1 5 5X2 + 3X + 1 5X + 3 3X + 4 X3 + 4X + 1 4X2 + 3  

Transform, via elementary row operations,

   rowi ← rowi + p(X)rowj rowi ↔ rowj rowi ← α rowi, α ∈ K\{0}

into Popov form [Popov ’72]

  X3 + 5X2 + 4X + 1 2X + 4 3X + 5 1 X2 + 2X + 3 X + 2 3X + 2 4X X2  

into Hermite form [Hermite 1851]

  X6 + 6X4 + X3 + X + 4 5X5 + 5X4 + 6X3 + 2X2 + 6X + 3 X 3X4 + 5X3 + 4X2 + 6X + 1 5 1  

3/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Context and contribution

Goal: fast algorithms

K a field Matrices over K[X]

  3X3 + X2 + 5X + 3 6X + 5 2X + 1 5 5X2 + 3X + 1 5X + 3 3X + 4 X3 + 4X + 1 4X2 + 3  

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

K a field Matrices over K[X]

  3X3 + X2 + 5X + 3 6X + 5 2X + 1 5 5X2 + 3X + 1 5X + 3 3X + 4 X3 + 4X + 1 4X2 + 3  

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

cost O˜(mωd)

m × m matrix with degree d

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˜(mω−1nd)

d the degree of the matrix size of Popov form is O(mnd)

previous fastest: O(rmnd2) [Mulders-Storjohann ’03] and O˜(mnωd) [⋆] Las Vegas: O˜(mω−1nd) 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˜(mω−1nδ)

δ md

δ = min(sum of row/col degrees)

size of Hermite form can be Θ(mnδ)

previous fastest: O˜(nω+1δ) [⋆] (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 Matrix K ∈ K[X]k×m such that    K has full row rank 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 Matrix B ∈ K[X]r×n such that B has full row rank 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˜(mω−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

    [6] [1] [2] [4] [2] [2] [1] [1] [3] [1] [3] [2] [5] [1] [2] [5]    

size O(m2d)

    [12] [11] [3] [11] [2] [0] [11] [2] [1]    

size O(m2d)

Popov

pivot of a row: rightmost entry of largest degree

Hermite

pivot of a row: rightmost nonzero entry

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

    [6] [1] [2] [4] [2] [2] [1] [1] [3] [1] [3] [2] [5] [1] [2] [5]    

size O(m2d)

    [6] [5] [5] [1] [5] [2] [4] [2] [2] [2] [2] [1] [1] [1] [3] [3] [3] [1] [3] [3] [2] [5] [5] [5] [1] [5] [2] [5]    

size O(mnd)

    [12] [11] [3] [11] [2] [0] [11] [2] [1]    

size O(m2d)

Popov

pivot of a row: rightmost entry of largest degree

Hermite

pivot of a row: rightmost nonzero entry

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

    [6] [1] [2] [4] [2] [2] [1] [1] [3] [1] [3] [2] [5] [1] [2] [5]    

size O(m2d)

    [6] [5] [5] [1] [5] [2] [4] [2] [2] [2] [2] [1] [1] [1] [3] [3] [3] [1] [3] [3] [2] [5] [5] [5] [1] [5] [2] [5]    

size O(mnd)

    [12] [11] [3] [11] [2] [0] [11] [2] [1]    

size O(m2d)

    [12] [11] [21] [3] [11] [37] [2] [193] [0] [11] [32] [2] [5] [243] [1]    

size O(mnδ)

Popov

pivot of a row: rightmost entry of largest degree

Hermite

pivot of a row: rightmost nonzero entry

8/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Algorithmic tools and general approach

Pivot support

From now, focus on Popov form

• full row rank m × n matrix A
• input/output size O(mnd)
• target cost O˜(mω−1nd)

    [4] [3] [5] [2] [1] [4] [5] [7] [9] [4] [4] [2] [9] [3] [5] [8] [5] [5] [0] [5] [9] [7] [4] [9] [3] [5] [5] [9]     = A     [6] [5] [5] [1] [5] [2] [4] [2] [2] [2] [2] [1] [1] [1] [3] [3] [3] [1] [3] [3] [2] [5] [5] [5] [1] [5] [2] [5]     = P unimodular

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

Overview of new algorithms

When the pivot support is known

    [4] [3] [5] [2] [1] [4] [5] [7] [9] [4] [4] [2] [9] [3] [5] [8] [5] [5] [0] [5] [9] [7] [4] [9] [3] [5] [5] [9]         [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?]     Input matrix Popov form unimodular     [4] [2] [4] [5] [7] [4] [9] [3] [5] [5] [5] [9] [7] [3] [5] [9]     Aπ =     [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?] [?]     = Pπ same unimodular square nonsingular case

Algorithm in O˜(mω−1nd) Step 1: Aπ ← submatrix of A with columns in pivot support Step 2: Pπ ← UAπ = Popov form of Aπ

// nonsingular case

Step 3: Pπ ← UAπ = (PπA−1

π )Aπ mod Xd+1

10/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support via saturation

Saturation of A = K[X]1×n ∩ K(X)1×mA = left kernel of any right kernel basis of A

same rank as A + contains the row space of A ⇒ same pivot support

Pivot Support in O˜(nωd) Step 1: K ← right kernel basis of A

// use [Zhou-Labahn-Storjohann ’12]

Step 2: B ← pivot support revealing left kernel basis of K Step 3: return the indices of pivots in B Ensuring efficiency in Step 2

• K has large average row degree (not suitable for [Zhou-Labahn-Storjohann ’12])
• K has low average column degree, and

we have deg(B) d since AK = 0

• btain B via fast Popov approximant basis [Jeannerod-Neiger-Schost-Villard ’16]

BK = 0 mod diag(Xδ1, . . . , Xδn−r)

11/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]   Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]   Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]     [4] [2] [4] [5] [4] [2] [9] [0] [3] [7] [4] [9] [3] [6] [3] [6] [7] [4] [5] [5] [5] [9] [6] [4] [6] [2] [1]  

1 4 6

Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]     [4] [2] [4] [5] [4] [2] [9] [0] [3] [7] [4] [9] [3] [6] [3] [6] [7] [4] [5] [5] [5] [9] [6] [4] [6] [2] [1]  

1 4 6

Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]     [4] [2] [4] [5] [4] [2] [9] [0] [3] [7] [4] [9] [3] [6] [3] [6] [7] [4] [5] [5] [5] [9] [6] [4] [6] [2] [1]  

1 4 6

  [4] [4] [5] [9] [0] [3] [7] [9] [3] [6] [7] [4] [5] [5] [9] [6] [2] [1]  

1 6 7

Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Find the pivot support of a wide matrix

  [4] [3] [5] [2] [1] [4] [5] [4] [2] [9] [0] [3] [7] [9] [4] [4] [2] [9] [3] [6] [3] [6] [7] [4] [5] [8] [5] [5] [0] [5] [9] [6] [4] [6] [2] [1]     [4] [2] [4] [5] [4] [2] [9] [0] [3] [7] [4] [9] [3] [6] [3] [6] [7] [4] [5] [5] [5] [9] [6] [4] [6] [2] [1]  

1 4 6

  [4] [4] [5] [9] [0] [3] [7] [9] [3] [6] [7] [4] [5] [5] [9] [6] [2] [1]   pivot support: (1, 7, 12)

1 6 7

Goal: when n ≫ m, lower the cost from O˜(nωd) to O˜(mω−1nd)

Strategy apply the previous algorithm iteratively to

• n/m overlapping submatrices of A
• each of dimensions m × 2m

12/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018

Overview of new algorithms

Conclusion and perspectives

Conclusion:

• fast algorithm to find pivot support for Popov form
• fast Popov form when pivot support known

Also in the paper:

• algorithms based on completing the matrix into a square one
• second item works for all shifted normal forms (including Hermite)

Perspectives:

• normal forms for arbitrary shifts
• kernel bases for arbitrary shifts
• cost bound sensitive to average row/column degrees

13/13 Vincent Neiger Computing Popov and Hermite forms of rectangular polynomial matrices ISSAC 2018