On Computing the Resultant of Generic Bivariate Polynomials Gilles - - PowerPoint PPT Presentation

SLIDE 1

On Computing the Resultant of Generic Bivariate Polynomials

Gilles Villard ISSAC, New York, July 17th, 2018

SLIDE 2

New algorithm:

Outline of the talk

Appetizer The problem A key ingredient A central remark

SLIDE 3

New algorithm:

Outline of the talk

Appetizer The problem A key ingredient A central remark

SLIDE 4

Minimal polynomial in a field extension

Find the relation ? 픸 = 햪[y]/g(y) g(y) = y4 + 10 y3 + 3 y2 + 7 y + 4 mod 11

α(y) = y4 + 4 y2 + 8 y + 9 ∈ 픸 α(y)2 + α(y) + 4 = 0

SLIDE 5

Minimal polynomial in a field extension

μ(y) monic (and irreducible) such that

μ(α(y)) ≡ 0 mod g(y) 픸 = 햪[y]/g(y) deg g = n α(y) ∈ 픸 αn(y) + μn−1 αn−1(y) + … + μ1 α(y) + μ0 ≡ 0 mod g(y) ? Generic case:

μ(y) = χ(y)

SLIDE 6

Projection for saving operations: use a K-linear map:

π : 햪[y]/g(y) → 햪

SLIDE 7

Minimal polynomial in a field extension

• 1. Compute

π(1), π(α), π(α2), …, π(α2n−1)

/* The sequence is linearly generated over */

• 2. Compute the minimal polynomial of the sequence

햪

using e.g. a Euclidean scheme or the Berlekamp-Massey algorithm

• Nota. One uses the trace and Leverrier’s approach for the characteristic polynomial.

[Rifà, Borrell 1991] [Shoup 1994] [Ly 1989]

SLIDE 8

MinPoly / CharPoly

μ(α) ≡ 0 mod g

PowerProjections

π(1), π(α), π(α2), …, π(α2n−1)

π : 픸 → 햪

O˜(n)

SLIDE 9

PowerProjections

π(1), π(α), π(α2), …, π(α2n−1)

π : 픸 → 햪 Left matrix-vector product [π0 π1 π2 … πn−1] ⋅ 1 1 α0 α1 α2 … αn−1 1 1 픸 = 햪[y]/g(y) ≃ 햪n

SLIDE 10

PowerProjections

[π0 π1 π2 … πn−1] ⋅ 1 1 α0 α1 α2 … αn−1 1 1 ⋅ h0 h1 h2 ⋮ hn−1

π(1), π(α), π(α2), …, π(α2n−1)

π : 픸 → 햪 Left matrix-vector product

Modular Composition

Right matrix-vector product

For a polynomial

h(α) mod g ? h

픸 = 햪[y]/g(y) ≃ 햪n

SLIDE 11

PowerProjections

π(1), π(α), π(α2), …, π(α2n−1)

π : 픸 → 햪

For a polynomial

h(α) mod g ? h

Transposition principle

[Kaltofen 2000]

Modular Composition

[Canny, Kaltofen, Yagati 1989]

O(n)

[Shoup 94]

SLIDE 12

MinPoly / CharPoly

μ(α) ≡ 0 mod g

PowerProjections

π(1), π(α), π(α2), …, π(α2n−1)

π : 픸 → 햪

For a polynomial

h(α) mod g ? h

Modular Composition

O(n) O˜(n)

SLIDE 13

[Bostan, Flajolet, Salvy, Schost 2006] [Brent & Kung 1978] [Shoup 1994]

O˜(n2) ⟹ Modular Composition

Minimal polynomial in a field extension

• perations in

햪

MinPoly / CharPoly O˜(nω2/2)

( n × n) ⋅ ( n × n)

[Paterson, Stockmeyer 1973]

Baby steps / giant steps strategy

[Kaltofen 2000] [Huang, Pan 1998] [Le Gall, Urrutia 2018]

⟹ O(n1.626)

SLIDE 14

Improvement for generic polynomials

Particular case of the resultant approach

α(y) ∈ 픸

1 1 α0 α1 α2 … αn−1 1 1

A

n × n

A : u ↦ α u

Multiplication endomorphism:

1 1 α0 α1 α2 … αn−1 1 1

x I − A det

?

Characteristic polynomial

SLIDE 15

New algorithm:

Outline of the talk

Appetizer The problem A key ingredient A central remark

SLIDE 16

‘’structured’’ polynomial matrix

A(x) ∈ 햪[x]n × n det A(x) ?

Ex: Toeplitz, Sylvester matrix, Frobenius matrix algebra, etc.

SLIDE 17

Entries in K

p, q ∈ 햪[x]

S = pn qn pn−1 pn qn−1 qn ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ pn ⋮ ⋮ qn p0 ⋮ pn−1 q0 ⋮ qn−1 p0 ⋮ q0 ⋮ ⋱ ⋮ ⋱ ⋮ p0 q0

Res(p, q) = det S ∈ 햪 ?

Knuth-Schönhage-Moenck recursive polynomial gcd: operations

O˜(n)

deg p, q = n

∈ 햪2n × 2n

Sylvester matrix

[Bini, Pan 1994] [von zur Gathen, Gerhard 1999]

SLIDE 18

Cost over Rule of thumb:

⪯

햪[x] Cost over 햪 Output degree ×

(Evaluation-interpolation scheme)

SLIDE 19

Entries in K[x]

p, q ∈ 햪[x, y] det S(x) ?

Output degree:

degx = 1, degy = n

S(x) = pn(x) qn(x) pn−1(x) pn(x) qn−1(x) qn(x) ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ pn(x) ⋮ ⋮ qn(x) p0(x) ⋮ pn−1(x) q0(x) ⋮ qn−1(x) p0(x) ⋮ q0(x) ⋮ ⋱ ⋮ ⋱ ⋮ p0(x) q0(x)

∈ 햪[x]2n × 2n

2n

• perations

O˜(n × n) = O˜(n2)

⟹

points

2n

SLIDE 20

Cost over Rule of thumb:

⪯

햪[x] Cost over 햪 Output degree ×

?

(Evaluation-interpolation scheme)

SLIDE 21

Dense polynomial matrices

Determinant in O˜(nω d) ≪ O˜(nω × nd)

Output degree: nd

[Storjohann 2003-2005] [Labahn, Neiger, Zhou 2017]

Degree: d

A(x) ∈ 햪[x]n × n

• perations in 햪

SLIDE 22

Dense matrix fractions

• f degree d

Generic case

H(x) = R(x)Q(x)−1 ∈ 햪[x]n × n R, Q Matrix fraction reconstruction:

from terms of

O(d) H(x) = Σi Hi xi in O˜(nω d)

[Beckermann, Labahn 1994] [Giorgi et al. 2003]

• perations in 햪

SLIDE 23

The problem

det S(x) ?

degx = 1

S(x) = pn(x) qn(x) pn−1(x) pn(x) qn−1(x) qn(x) ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ ⋮ pn(x) ⋮ ⋮ qn(x) p0(x) ⋮ pn−1(x) q0(x) ⋮ qn−1(x) p0(x) ⋮ q0(x) ⋮ ⋱ ⋮ ⋱ ⋮ p0(x) q0(x)

∈ 햪[x]2n × 2n

To simplify:

Best known complexity bound Size of a system solution

O˜(n2)

( entries of degree )

2n

n

SLIDE 24

From 10.000 feet New algorithm

SLIDE 25

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A =

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69591193773 203713103035 97579672962 203713103035 284823690824 203713103035 29281306465 40742620607

− 187605083672

203713103035

− 7390918941

203713103035

− 39531524706

203713103035

− 28866179508

40742620607

− 19372027446

40742620607 35285114899 203713103035

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<latexit sha1_base64="RUgD2b3q7/u1CvE7LuDoUaY5zg8=">AFHiclZLPa9RAFMdf42r+qOtHr0MFsGLZWaSzGQuUvDizQquLWyWks3ObkOzSUiyQgn5O/xPvAmCeBOvInhV9L/wzTQL2lLsTsjum+97nzfvzcu4SJOqpvTbmnOtd/3G+sbN/q3bd+5ubm3fe13lizLWgzhP8/JwHFU6TI9qJM61YdFqaP5ONUH45Nnxn/wRpdVkmev6tNCj+bRLEumSRzVKB1tOy8ICVM9rYckHOtZkpEmKsvotG3iljThtIziRihfMaZcKd24dSVzGXUpa7ftv0wJGWR2ky5pwrifVSVK0HaekL5WQXAm+EscDL+CuUDTg3mqg4gFzqfCE3zYelR4XnAoqL8OedBwLpKA+DVws9moHLknpKqpYoDy2Gucq32U+9yQVq4E8CIRgUmG1q7WI4+OUS8TV+KWdfo8BnzAqXO1xnqbL8WPokLJPZcT0i/aOtHbpL7SIXDdYZO3v580PALCfb32FECaQwLmIOGDGq0U4igwmcIDCgUqI2gQa1EK7F+DS30kV1glMaICNUT/J3hbtipGe5NzsrSMZ6S4lsiSeARMjnGlWib04j1L2xmo16Wu7E5TW2n+D/ucs1RreEY1f9xy8ircqaXGqYQ2B4S7Kmwiuku7rIs7K2YyslfXdWYoUDN2BP0l2jHlzeM7FMZXs3dxtZ/y8baVSzj7vYBfw2VeKA2flxXjQGfFft0pc46KdwtjbgATyExzhOCXvwHPZhALHzvnu/HB+9t723vc+9j6dhTprHXMf/lm9L38AhI5RoA=</latexit><latexit sha1_base64="DYpr/QhONj1M8ZNLkLIiFH+wJj8=">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</latexit><latexit sha1_base64="DYpr/QhONj1M8ZNLkLIiFH+wJj8=">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</latexit>

A−1b =

<latexit sha1_base64="xMNE/iv+ELlrqEpNdEAIUHF8EqU=">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</latexit><latexit sha1_base64="3/HmoZev9EYliONyLQOkoJ6UOk=">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</latexit><latexit sha1_base64="3/HmoZev9EYliONyLQOkoJ6UOk=">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</latexit>

Determinant of ?

A

<latexit sha1_base64="mkpNzxqwrNH7ecnD7gVaTELXjbQ=">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</latexit><latexit sha1_base64="gybTDm0Xl4uNqkSbIxPJMjAVfjs=">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</latexit><latexit sha1_base64="gybTDm0Xl4uNqkSbIxPJMjAVfjs=">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</latexit>

Cramer’s rule: det A = −203713103035

<latexit sha1_base64="AxPSvWJ1Bf7T4r5/wkrfki58has=">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</latexit><latexit sha1_base64="dGVxexVt+RxG0NYgSu0BlPdbZxI=">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</latexit><latexit sha1_base64="dGVxexVt+RxG0NYgSu0BlPdbZxI=">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</latexit>

SLIDE 26

                         2 −5 −10 10 −10 10 −10 11 2 11 −5 −12 6 4 −11 2 −11 8 −9 11 −3 −2 −3 4 5 −2 −10 −1 8 −4 5 1 3 11 10 −6 11 8 10 −12 12 2 −2 8 2 8 1 7 −7 4 5 7 −10 −5 −2 −5 −11 3 12 −5 5 −2 8 −6 −5 4 −10 12 −3 −2 8 1 −6 6 −2 −9 10 −6 2 −1 12 10 −12 −5 −11 4 10 2 3 −5 6 1 −7 −12 −12                         

<latexit sha1_base64="BqYFxgli20j6lrlAnQ0aCrRPU=">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</latexit><latexit sha1_base64="rSIae5KuKSr0x7amipYJG7XGLh8=">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</latexit><latexit sha1_base64="rSIae5KuKSr0x7amipYJG7XGLh8=">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</latexit>

A =

<latexit sha1_base64="30Ovqz3cCT35aldzCyDN4Pz+jfs=">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</latexit><latexit sha1_base64="i/EJgSxANRyKQIA4Gtc+GReZ6w=">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</latexit><latexit sha1_base64="i/EJgSxANRyKQIA4Gtc+GReZ6w=">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</latexit>

                        

69591193773 203713103035 97579672962 203713103035 284823690824 203713103035 29281306465 40742620607

− 187605083672

203713103035

− 7390918941

203713103035

− 39531524706

203713103035

− 28866179508

40742620607

− 19372027446

40742620607 35285114899 203713103035

                        

<latexit sha1_base64="RUgD2b3q7/u1CvE7LuDoUaY5zg8=">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</latexit><latexit sha1_base64="DYpr/QhONj1M8ZNLkLIiFH+wJj8=">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</latexit><latexit sha1_base64="DYpr/QhONj1M8ZNLkLIiFH+wJj8=">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</latexit>

A−1b =

<latexit sha1_base64="xMNE/iv+ELlrqEpNdEAIUHF8EqU=">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</latexit><latexit sha1_base64="3/HmoZev9EYliONyLQOkoJ6UOk=">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</latexit><latexit sha1_base64="3/HmoZev9EYliONyLQOkoJ6UOk=">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</latexit>

Determinant of ?

A

<latexit sha1_base64="mkpNzxqwrNH7ecnD7gVaTELXjbQ=">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</latexit><latexit sha1_base64="gybTDm0Xl4uNqkSbIxPJMjAVfjs=">ACxHicjVHLTsJAFD3UF+ILdemkZi4IkUX6kYxLnSJRoQEiWmHARv6yszUhBD9Arf6Q36E8Q/0L7wzlEQlRm/T9sy595yZO9dLAl8qx3nLWVPTM7Nz+fnCwuLS8kpxde1KxqlgvM7iIBZNz5U8CNeV74KeDMR3A29gDe8/onON+64kH4cXapBwtuh24v8rs9cRdSFfXxTLDlx4Q9CSoZKB297JqoxcVXKODGAwpQnBEUIQDuJD0tFCBg4S4NobECUK+yXPco0DalKo4VbjE9unbo1UrYyNa09p1Ix2CegVpLSxRZqY6gRhvZt8qlx1uxv3kPjqc82oL+XeYXEKtwS+5duXPlfne5FoYt904NPSWG0d2xzCU1t6JPbn/pSpFDQpzGHcoLwswox/dsG40veu7dU3+3VRqVq9ZVpviQ5+SBlz5Oc5JUN8pH5Sdc6dUPcQo8tjAJrZpnHuo4gw1Mm6i0c84dk6tUJLWumo1MplmnV8C+vhE54mkW8=</latexit><latexit sha1_base64="gybTDm0Xl4uNqkSbIxPJMjAVfjs=">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</latexit>

Cramer’s rule: det A = −203713103035

<latexit sha1_base64="AxPSvWJ1Bf7T4r5/wkrfki58has=">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</latexit><latexit sha1_base64="dGVxexVt+RxG0NYgSu0BlPdbZxI=">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</latexit><latexit sha1_base64="dGVxexVt+RxG0NYgSu0BlPdbZxI=">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</latexit>

What if solving a linear system has prohibitive cost?

SLIDE 27

                         − 378816900

3134047739

− 20495829114

203713103035

− 67053094413

203713103035 9396074080 40742620607

− 58841813322

203713103035 8641632698 203713103035 1176300782 10721742265

− 305542579

3134047739

− 11872538116

203713103035

− 49238615442

203713103035 8998738354 40742620607

− 48926872543

203713103035 17364341402 203713103035 1603975448 10721742265

− 595667827

3134047739

− 34850589482

203713103035

− 93065264584

203713103035 16395446499 40742620607

− 99757356861

203713103035 26770440759 203713103035 2467702816 10721742265

− 74008954

3134047739

− 2169888633

40742620607

− 4053804427

40742620607 4874154765 40742620607

− 4349067980

40742620607 3634590188 40742620607 195062702 2144348453 239765981 3134047739 17661126586 203713103035 60948870672 203713103035

− 8711085182

40742620607 62978493878 203713103035

− 12358170402

203713103035

− 1419405818

10721742265 153069150 3134047739 4228351328 203713103035 32680318171 203713103035

− 4551698694

40742620607 28277713354 203713103035

− 20658574316

203713103035

− 564797499

10721742265 86854607 3134047739

−

64708557 203713103035 18276747571 203713103035

− 2331953340

40742620607 21325207739 203713103035

− 11479047526

203713103035

− 708058399

10721742265 84711069 3134047739 2041415721 40742620607 5533800772 40742620607

− 3120400038

40742620607 4453203683 40742620607

− 2490245417

40742620607

− 159017671

2144348453

− 115733957

3134047739

− 916054792

40742620607

− 1171020887

40742620607

− 309781915

40742620607

− 135778185

40742620607

− 1372889107

40742620607 7348981 2144348453

− 285726486

3134047739

− 15441589322

203713103035

− 55992257474

203713103035 8792979720 40742620607

− 51532000881

203713103035 15442993054 203713103035 1128412236 10721742265

<latexit sha1_base64="wbT8zj1F1taD2sgQ7I8T2ZpDdN4=">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</latexit><latexit sha1_base64="yvHA79mCPOWvOrZPEeAIfoVv+M0=">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</latexit><latexit sha1_base64="yvHA79mCPOWvOrZPEeAIfoVv+M0=">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</latexit>

                        

17807806326 203713103035

− 23405165014

203713103035 10100538629 203713103035 2562596724 203713103035

− 20657616816

203713103035 1957476176 203713103035 11632892698 203713103035

− 30848462707

203713103035 3042447147 203713103035

− 469567929

40742620607

− 926331297

40742620607

− 1221610838

40742620607

− 12553388579

203713103035 31097066916 203713103035

− 5825205336

203713103035 130624778 203713103035 18915310378 203713103035 286984762 203713103035

− 3859090862

203713103035 6143195823 203713103035 7665741127 203713103035 2076805993 40742620607 1591605402 40742620607

− 956914256

40742620607 809222740 40742620607 172501490 40742620607

− 716590790

40742620607 2784154348 203713103035

− 17109379672

203713103035

− 1441829408

203713103035

                        

<latexit sha1_base64="CphEb5jOhsSOGLIhPMmM9fGet2g=">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</latexit><latexit sha1_base64="PVpcQPztjn9W4E9Xj7+h3Y3eM=">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</latexit><latexit sha1_base64="PVpcQPztjn9W4E9Xj7+h3Y3eM=">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</latexit>

A−1 =

<latexit sha1_base64="/fT1/4ht4GYGVKY5tNGErp2+qZk=">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</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">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</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">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</latexit>

SLIDE 28

                         − 378816900

3134047739

− 20495829114

203713103035

− 67053094413

203713103035 9396074080 40742620607

− 58841813322

203713103035 8641632698 203713103035 1176300782 10721742265

− 305542579

3134047739

− 11872538116

203713103035

− 49238615442

203713103035 8998738354 40742620607

− 48926872543

203713103035 17364341402 203713103035 1603975448 10721742265

− 595667827

3134047739

− 34850589482

203713103035

− 93065264584

203713103035 16395446499 40742620607

− 99757356861

203713103035 26770440759 203713103035 2467702816 10721742265

− 74008954

3134047739

− 2169888633

40742620607

− 4053804427

40742620607 4874154765 40742620607

− 4349067980

40742620607 3634590188 40742620607 195062702 2144348453 239765981 3134047739 17661126586 203713103035 60948870672 203713103035

− 8711085182

40742620607 62978493878 203713103035

− 12358170402

203713103035

− 1419405818

10721742265 153069150 3134047739 4228351328 203713103035 32680318171 203713103035

− 4551698694

40742620607 28277713354 203713103035

− 20658574316

203713103035

− 564797499

10721742265 86854607 3134047739

−

64708557 203713103035 18276747571 203713103035

− 2331953340

40742620607 21325207739 203713103035

− 11479047526

203713103035

− 708058399

10721742265 84711069 3134047739 2041415721 40742620607 5533800772 40742620607

− 3120400038

40742620607 4453203683 40742620607

− 2490245417

40742620607

− 159017671

2144348453

− 115733957

3134047739

− 916054792

40742620607

− 1171020887

40742620607

− 309781915

40742620607

− 135778185

40742620607

− 1372889107

40742620607 7348981 2144348453

− 285726486

3134047739

− 15441589322

203713103035

− 55992257474

203713103035 8792979720 40742620607

− 51532000881

203713103035 15442993054 203713103035 1128412236 10721742265

<latexit 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                        

17807806326 203713103035

− 23405165014

203713103035 10100538629 203713103035 2562596724 203713103035

− 20657616816

203713103035 1957476176 203713103035 11632892698 203713103035

− 30848462707

203713103035 3042447147 203713103035

− 469567929

40742620607

− 926331297

40742620607

− 1221610838

40742620607

− 12553388579

203713103035 31097066916 203713103035

− 5825205336

203713103035 130624778 203713103035 18915310378 203713103035 286984762 203713103035

− 3859090862

203713103035 6143195823 203713103035 7665741127 203713103035 2076805993 40742620607 1591605402 40742620607

− 956914256

40742620607 809222740 40742620607 172501490 40742620607

− 716590790

40742620607 2784154348 203713103035

− 17109379672

203713103035

− 1441829408

203713103035

                        

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A−1 =

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SLIDE 29

                         − 378816900

3134047739

− 20495829114

203713103035

− 67053094413

203713103035 9396074080 40742620607

− 58841813322

203713103035 8641632698 203713103035 1176300782 10721742265

− 305542579

3134047739

− 11872538116

203713103035

− 49238615442

203713103035 8998738354 40742620607

− 48926872543

203713103035 17364341402 203713103035 1603975448 10721742265

− 595667827

3134047739

− 34850589482

203713103035

− 93065264584

203713103035 16395446499 40742620607

− 99757356861

203713103035 26770440759 203713103035 2467702816 10721742265

− 74008954

3134047739

− 2169888633

40742620607

− 4053804427

40742620607 4874154765 40742620607

− 4349067980

40742620607 3634590188 40742620607 195062702 2144348453 239765981 3134047739 17661126586 203713103035 60948870672 203713103035

− 8711085182

40742620607 62978493878 203713103035

− 12358170402

203713103035

− 1419405818

10721742265 153069150 3134047739 4228351328 203713103035 32680318171 203713103035

− 4551698694

40742620607 28277713354 203713103035

− 20658574316

203713103035

− 564797499

10721742265 86854607 3134047739

−

64708557 203713103035 18276747571 203713103035

− 2331953340

40742620607 21325207739 203713103035

− 11479047526

203713103035

− 708058399

10721742265 84711069 3134047739 2041415721 40742620607 5533800772 40742620607

− 3120400038

40742620607 4453203683 40742620607

− 2490245417

40742620607

− 159017671

2144348453

− 115733957

3134047739

− 916054792

40742620607

− 1171020887

40742620607

− 309781915

40742620607

− 135778185

40742620607

− 1372889107

40742620607 7348981 2144348453

− 285726486

3134047739

− 15441589322

203713103035

− 55992257474

203713103035 8792979720 40742620607

− 51532000881

203713103035 15442993054 203713103035 1128412236 10721742265

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                        

17807806326 203713103035

− 23405165014

203713103035 10100538629 203713103035 2562596724 203713103035

− 20657616816

203713103035 1957476176 203713103035 11632892698 203713103035

− 30848462707

203713103035 3042447147 203713103035

− 469567929

40742620607

− 926331297

40742620607

− 1221610838

40742620607

− 12553388579

203713103035 31097066916 203713103035

− 5825205336

203713103035 130624778 203713103035 18915310378 203713103035 286984762 203713103035

− 3859090862

203713103035 6143195823 203713103035 7665741127 203713103035 2076805993 40742620607 1591605402 40742620607

− 956914256

40742620607 809222740 40742620607 172501490 40742620607

− 716590790

40742620607 2784154348 203713103035

− 17109379672

203713103035

− 1441829408

203713103035

                        

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A−1 =

<latexit sha1_base64="/fT1/4ht4GYGVKY5tNGErp2+qZk=">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</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">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</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">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</latexit>

       64 47 −24 122 20 36 −36 140 44 66 −38 213 −13 18 −3 66               36 183 785 363 319 379 −41 −116 −299 672 −195 382 −387 344       

−1

<latexit sha1_base64="c+/h1+soTiUKEzeoTtaql0ESn3I=">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</latexit><latexit sha1_base64="iIZmBYQkw0knGAFHxshegtRWQ=">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</latexit><latexit sha1_base64="iIZmBYQkw0knGAFHxshegtRWQ=">AEH3icjVFNb9NAEB3XfBTz0RSOXFZUrLg48tomti+oEheORSJtRyK42zTVRzb8gdSZOXPcOMPcOfGCXFD/QXQGxw5IDG7dSgCbCW7dk372d2RnliSgryzrVNvRLl69c3bxmXL9x89ZWZ/v2fpnVRcz7cZkxeEoKnkiUt6vRJXw7zg0WyU8IPR9LHMH7ziRSmy9Fk1z/lwFk1ScSziqELoaFt7TsKEH1cDEo74RKSkiYoimi+aGNeC9FzqetS0XcpsOwxJmGZRIiZpE874uJyKfGFb1OlR08APc62VFNelPaQ4PrWZs5JhMocyHykGMkOejpdlkLAQk5NqSP5apSqB+Q71/Icr/Z2eQx0WUMPxAmq6bE0RDKu0g4D2PJuaLFj5duGbMajeKzrqz2RWOyhXHU2bG6lrkYsDaYGd3/+zNy7f+xlnU8QwhgyiKGXBIocI4gQhKfAbAwIcsSE0iBUYCZXnsADtTWyODIiRKf4neBu0KIp7qVnqdQxnpLgW6CSwH3UZMgrMJanEZWvlbNE13k3ylPWNsf/qPWaIVrBCaL/0i2Z/6uTvVRwDL7qQWBPuUJkd3HrUqtbkZWTX7q0CFHTMZjzBcYx0q5vGeiNKXqXd5tpPJfFOich+3BrOZJU4YPbnOC8GfbsbdK2nOhHcL424S7cgwc4Tg924QnsQR9i7Z32WfuqfdNf6+/1D/rHc+qG1mruwG9LP/0Ji20GLg=</latexit>

XT A−1Y = RQ−1 =

<latexit sha1_base64="fHwxC/ON7w6pKToCd/dNFiqD4k0=">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</latexit><latexit sha1_base64="6JdjiyIj6q1R8hwILwznWnXIVT4=">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</latexit><latexit sha1_base64="6JdjiyIj6q1R8hwILwznWnXIVT4=">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</latexit>

Step 1.

SLIDE 30

                         − 378816900

3134047739

− 20495829114

203713103035

− 67053094413

203713103035 9396074080 40742620607

− 58841813322

203713103035 8641632698 203713103035 1176300782 10721742265

− 305542579

3134047739

− 11872538116

203713103035

− 49238615442

203713103035 8998738354 40742620607

− 48926872543

203713103035 17364341402 203713103035 1603975448 10721742265

− 595667827

3134047739

− 34850589482

203713103035

− 93065264584

203713103035 16395446499 40742620607

− 99757356861

203713103035 26770440759 203713103035 2467702816 10721742265

− 74008954

3134047739

− 2169888633

40742620607

− 4053804427

40742620607 4874154765 40742620607

− 4349067980

40742620607 3634590188 40742620607 195062702 2144348453 239765981 3134047739 17661126586 203713103035 60948870672 203713103035

− 8711085182

40742620607 62978493878 203713103035

− 12358170402

203713103035

− 1419405818

10721742265 153069150 3134047739 4228351328 203713103035 32680318171 203713103035

− 4551698694

40742620607 28277713354 203713103035

− 20658574316

203713103035

− 564797499

10721742265 86854607 3134047739

−

64708557 203713103035 18276747571 203713103035

− 2331953340

40742620607 21325207739 203713103035

− 11479047526

203713103035

− 708058399

10721742265 84711069 3134047739 2041415721 40742620607 5533800772 40742620607

− 3120400038

40742620607 4453203683 40742620607

− 2490245417

40742620607

− 159017671

2144348453

− 115733957

3134047739

− 916054792

40742620607

− 1171020887

40742620607

− 309781915

40742620607

− 135778185

40742620607

− 1372889107

40742620607 7348981 2144348453

− 285726486

3134047739

− 15441589322

203713103035

− 55992257474

203713103035 8792979720 40742620607

− 51532000881

203713103035 15442993054 203713103035 1128412236 10721742265

<latexit 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                        

17807806326 203713103035

− 23405165014

203713103035 10100538629 203713103035 2562596724 203713103035

− 20657616816

203713103035 1957476176 203713103035 11632892698 203713103035

− 30848462707

203713103035 3042447147 203713103035

− 469567929

40742620607

− 926331297

40742620607

− 1221610838

40742620607

− 12553388579

203713103035 31097066916 203713103035

− 5825205336

203713103035 130624778 203713103035 18915310378 203713103035 286984762 203713103035

− 3859090862

203713103035 6143195823 203713103035 7665741127 203713103035 2076805993 40742620607 1591605402 40742620607

− 956914256

40742620607 809222740 40742620607 172501490 40742620607

− 716590790

40742620607 2784154348 203713103035

− 17109379672

203713103035

− 1441829408

203713103035

                        

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A−1 =

<latexit sha1_base64="/fT1/4ht4GYGVKY5tNGErp2+qZk=">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</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">ACyXicjVG7TsNAEJyYVwivACWNRYREQ2RDARAEA0FRZAwiQB2c4lWPFL5zMiRKn4Alr4HD4C8QfwF+xdHAmIEKxle252Zu/21ol9LxG8ZbTxsYnJqfy04WZ2bn5heLi0nkSpdxlhv5Ea87dsJ8L2SW8ITP6jFnduD4rOZ0jmS+dst4kXhmejGrBHY7dBrea4tiKodXvU2zP7edbFklA0V+igwM1A6eNlSUY2Kr7hExFcpAjAEIQ9mEjoecCJgzExDXQI4T8lSeoY8CeVNSMVLYxHbo26bVRcaGtJY1E+V2aRefXk5OHWvkiUjHCcvdJVPVWXJ/la7p2rKs3Xp72S1AmIFboj9yzdU/tcnexFoYUf14FPsWJkd25WJVW3Ik+uf+lKUIWYOImblOeEXeUc3rOuPInqXd6trfLvSilZuXYzbYoPeUoasPlznKPA2izvlo1To1TZxyDyWMEq1mc26jgGFVYqslHPOFZO9G4dqfdD6RaLvMs41toD5+wZ5Ny</latexit><latexit sha1_base64="7v14J+CzpeG4ysnLTn1EzQgdhJE=">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</latexit>

       64 47 −24 122 20 36 −36 140 44 66 −38 213 −13 18 −3 66               36 183 785 363 319 379 −41 −116 −299 672 −195 382 −387 344       

−1

<latexit sha1_base64="c+/h1+soTiUKEzeoTtaql0ESn3I=">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</latexit><latexit sha1_base64="iIZmBYQkw0knGAFHxshegtRWQ=">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</latexit><latexit sha1_base64="iIZmBYQkw0knGAFHxshegtRWQ=">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</latexit>

XT A−1Y = RQ−1 =

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det Q = det A = −203713103035

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Step 1. Step 2.

SLIDE 31

Compute a submatrix of the inverse without solving an entire system? Be sure that the matrix fraction is ‘’small’’?

SLIDE 32

New algorithm:

Outline of the talk

Appetizer The problem A key ingredient A central remark

SLIDE 33

Degrees of depending on ?

S(x)−1 =              1 1 1 1 1             

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We consider a north-western submatrix of the inverse:

S(x) =

               pn(x) qn(x) pn−1(x) pn(x) qn−1(x) qn(x) . . . . . . ... . . . . . . ... . . . . . . pn(x) . . . . . . qn(x) p0(x) . . . pn−1(x) q0(x) . . . qn−1(x) p0(x) . . . q0(x) . . . ... . . . ... . . . p0(x) q0(x)               

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H(x) = R(x) Q(x)−1

S(x)−1

m × m

R(x), Q(x)

m

SLIDE 34

Ex:

S(x) ∈ 햪[x]16×16 d = 1 p(x, y) = xy8 + y, q(x, y) = y8 + x

x 1 x 1 x 1 x 1 x 1 x 1 x 1 1 x 1 1 x 1 x 1 x 1 x 1 x 1 x 1 x x

S(x) =

SLIDE 35

Ex:

S(x) ∈ 햪[x]16×16 d = 1 p(x, y) = xy8 + ym, q(x, y) = y8 + x

Denominator degree:

16 m = 1

S(x)−1 =

2⌈n/m⌉ = 2⌈8/1⌉ = 16

H(x) = − 1 x(x15 + 1)

SLIDE 36

Denominator degree:

1 H(x) = I ⋅ S(x)−1

S(x)−1 =

S(x) ∈ 햪[x]16×16 d = 1

Ex: p(x, y) = xy8 + ym, q(x, y) = y8 + x

SLIDE 37

Denominator degree:

4 m = 5

S(x)−1 =

2⌈n/m⌉ = 2⌈8/5⌉ = 4

⇥ ¯ S(x)−1⇤

1,...,5;12,...,16 = −I ·

2 6 6 6 6 6 6 6 6 4 x x2 x x2 x4 x x4 x x4 x 3 7 7 7 7 7 7 7 7 5

−1

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H(x)

S(x) ∈ 햪[x]16×16 d = 1

Ex: p(x, y) = xy8 + ym, q(x, y) = y8 + x

SLIDE 38

⇥ ¯ S(x)−1⇤

1,...,3;14,...,16 = −I ·

2 6 6 4 x x4 x6 x x6 x 3 7 7 5

−1

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Denominator degree:

6 m = 3

S(x)−1 =

2⌈n/m⌉ = 2⌈8/3⌉ = 6 H(x)

S(x) ∈ 햪[x]16×16 d = 1

Ex: p(x, y) = xy8 + ym, q(x, y) = y8 + x

SLIDE 39

• Lemma. Generically,

with

Small size fraction

and

H(x) = R(x)Q(x)−1 ∈ 햪(x)m×m deg R, deg Q ∈ O(n/m) deg Q = det S O(n/m)

Expansion limited to order

See also [Kaltofen, Villard 2005]

SLIDE 40

System solution:

O(n)

entries expansion of order

n

O˜(n2)

Matrix fractions: entries

n × n = n O( n)

expansion of order

O˜(n n)

SLIDE 41

Compute a submatrix of the inverse without solving an entire system? Be sure that the matrix fraction is ‘’small’’?

SLIDE 42

New algorithm:

Outline of the talk

Appetizer The problem A key ingredient A central remark

SLIDE 43

Toeplitz-like matrices

(Widely used techniques)

픸 = 햪[y]/g(y) g(y) = y6 + 3 y5 + 4 y4 + 7 y3 + 6 y + 8 mod 11 α(y) = 7 y5 + 7 y4 + 3 y3 + 2 y2 + y + 9 ∈ 픸

Ex:

A = 9 10 2 7 6 9 1 6 10 6 10 2 1 6 10 6 3 8 2 3 7 8 9 9 5 2 7 8 6 2 3 7

SLIDE 44

Toeplitz-like matrices

픸 = 햪[y]/g(y) g(y) = y6 + 3 y5 + 4 y4 + 7 y3 + 6 y + 8 mod 11 α(y) = 7 y5 + 7 y4 + 3 y3 + 2 y2 + y + 9 ∈ 픸 A = 9 10 2 7 6 9 1 6 10 6 10 2 1 6 10 6 3 8 2 3 7 8 9 9 5 2 7 8 6 2 3 7

(Widely used techniques)

Ex:

A = 9 1 9 2 1 9 3 2 1 9 7 3 2 1 9 7 7 3 2 1 9 + 3 5 3 0 5 3 4 0 5 3 7 4 0 5 3 8 7 4 0 5 3 7 8 6 2 3 7 8 6 2 7 8 6 7 8 7

SLIDE 45

Toeplitz-like matrices

(Widely used techniques) [Kailath, Kung, Morf 1979] [Kaltofen 1994] [Labahn 1992] [Bini, Pan 1994]

Theory of displacement rank

ΣLU representation:

             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗              +              ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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T =

Ex: Toeplitz, Sylvester matrix, Frobenius matrix algebra, etc.

SLIDE 46

Toeplitz-like matrices

A−1 = 9 4 9 9 4 9 1 9 4 9 9 1 9 4 9 6 9 1 9 4 9 + 3 5 3 0 5 3 4 0 5 3 7 4 0 5 3 8 7 4 0 5 3 6 2 4 2 1 6 2 4 2 6 2 4 6 2 6

(Widely used techniques)

The structure is kept recursively during block Gaussian elimination à la Strassen

A−1 = 9 7 6 1 6 3 4 6 6 4 9 4 6 6 4 1 1 3 8 9 10 3 7 3 10 6 2 4 2 1

SLIDE 47

Compute a submatrix of the inverse without solving an entire system? Be sure that the matrix fraction is ‘’small’’?

SLIDE 48

             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗              +              ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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S(x)−1

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=

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ΣLU representation of the inverse O˜(n) operations on truncated power series (half-gcd)

[Kailath, Kung, Morf 1979] [Kaltofen 1994] [Labahn 1992] [Bini, Pan 1994]

SLIDE 49

             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗              +              ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

<latexit sha1_base64="4+2xaQhIadP72f3+f+um1fZ8zSw=">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</latexit><latexit sha1_base64="rzLoJ78joMuFs0AL7dl+VEwE6E=">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</latexit><latexit sha1_base64="rzLoJ78joMuFs0AL7dl+VEwE6E=">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</latexit>

             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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S(x)−1

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=

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SLIDE 50

Small polynomial Toeplitz matrix products

             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗              +              ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗                           ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗             

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S(x)−1

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=

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m m

SLIDE 51

• 2. Scalar fraction reconstruction
• 1. Expansion of S(x)−1 y

Determinant via (vector) lifting

• 3. Determinant from denominators

SLIDE 52

Algorithm ‘’Structured determinant’’ Output:

• 2. Reconstruct a fraction description

R(x)Q(x)−1 ∈ 햪(x)m×m

• 1. Compute an expansion of a submatrix of S(x)−1

det Q(x)

Input: a Toeplitz-like matrix

S(x)

Nota: The matrix reconstruction and the final determinant use dense matrix computations.

[Storjohann 2003-2005] [Labahn, Neiger, Zhou 2017] [Beckermann, Labahn 1994] [Giorgi et al. 2003]

SLIDE 53

Output:

• 2. Reconstruct a fraction description
• 1. Compute an expansion of a submatrix of S(x)−1

det Q(x)

Input: a Toeplitz-like matrix

S(x)

Nota: The matrix reconstruction and the final determinant use dense matrix computations.

[Storjohann 2003-2005] [Labahn, Neiger, Zhou 2017] [Beckermann, Labahn 1994] [Giorgi et al. 2003]

O˜(n ⋅ n m )

R(x)Q(x)−1 ∈ 햪(x)m×m

Algorithm ‘’Structured determinant’’

SLIDE 54

Output:

• 2. Reconstruct a fraction description
• 1. Compute an expansion of a submatrix of S(x)−1

det Q(x)

Input: a Toeplitz-like matrix

S(x)

Nota: The matrix reconstruction and the final determinant use dense matrix computations.

[Storjohann 2003-2005] [Labahn, Neiger, Zhou 2017] [Beckermann, Labahn 1994] [Giorgi et al. 2003]

O˜(n ⋅ n m ) O˜(mω ⋅ n m )

R(x)Q(x)−1 ∈ 햪(x)m×m

Algorithm ‘’Structured determinant’’

SLIDE 55

Conclusion

Block size or m = n1/ω

Generic resultant cost:

Determinant of a polynomial structured matrix

Expansion: O˜(n ⋅ n

m )

Dense linear algebra: O˜(mω ⋅ n m )

O˜(n2) O˜(n5/3) O(n1.58)

Here degree 1, analogous for degree d

See also [van der Hoeven and Larrieu 2018]

Application to Gröbner bases of generic ideals

m = n1/3

SLIDE 56

Conclusion bis

Generalization

CharPolyGen ≈ Modular CompositionGen <

?

Improvement of modular composition for generic polynomials Field extension CharPoly:

O(n1.63) O(n1.58)

{

CharPoly Modular Composition

better than composition ?

SLIDE 57

Thank you !