Relaxed algorithm s, p-adic lifting and p olynom ial system solving ∗

Relaxed algorithm s, p -adic lifting and p olynom ial system solving ^∗

by Romain Lebreton ARITH Team, LIRMM Université Montpellier II

BIPOP-CASYS Seminar

Laboratoire Jean Kuntzmann, Grenoble March 7, 2013

∗. This document has been written using the GNU TEX^MACS text editor (see www.texmacs.org).

Computer algebra

1. Why is p-adic lifting interesting?

2. Relaxed p-adic lifting a. On-line algorithms b. Recursive p-adic

c. Application to:

i. Linear systems

ii. Linear q-differential systems iii. Polynomial root lifting

iv. Univariate representation lifting 3. Conclusion

Why modular computation?

P₁ = x⁵ y⁸+ (−x⁸+ 4x⁶)y⁷−x⁷y⁶ +(x¹⁰−4x⁸)y⁵+ y −x³+ 4x

P₂ = −x²y⁸+ (x⁵−4x³)y⁷+x⁴y⁴ +(−x⁷+ 4x⁵)y³+ y −x³+ 4x

Solutions of P₁= 0 Solutions of P₂= 0

Do they have a curve in common?

Why modular computation?

P₁ = x⁵ y⁸+ (−x⁸+ 4x⁶)y⁷−x⁷y⁶ +(x¹⁰−4x⁸)y⁵+ y −x³+ 4x

P₂ = −x²y⁸+ (x⁵−4x³)y⁷+x⁴y⁴ +(−x⁷+ 4x⁵)y³+ y −x³+ 4x

Solutions of P₁= 0 Solutions of P₂= 0

Do they have a curve in common?

YES !

Why modular computation?

P₁ = x⁵y⁸+ (−x⁸+ 4x⁶)y⁷−x⁷y⁶ +(x¹⁰−4x⁸)y⁵+ y −x³+ 4x

P₂ = −x²y⁸+ (x⁵−4x³)y⁷+x⁴y⁴ +(−x⁷+ 4x⁵)y³+ y −x³+ 4x

,

Euclid’s algorithm (in Q(x)[y])

,

R= y −x³+x Solutions of R= 0

Why modular computation?

P₁ = x⁵y⁸+ (−x⁸+ 4x⁶)y⁷−x⁷y⁶ +(x¹⁰−4x⁸)y⁵+ y −x³+ 4x

P₂ = −x²y⁸+ (x⁵−4x³)y⁷+x⁴y⁴ +(−x⁷+ 4x⁵)y³+ y −x³+ 4x

,

Euclid’s algorithm (in Q(x)[y])

,

R₁ = y²+ (−x³⁴+x³³+ 5x³²−5x³¹−5x³⁰+ 5x²⁹+x²⁸−x²⁷

−4x²³+ 4x²²+ 4x²¹−13x²⁰+10x¹⁹+12x¹⁸−18x¹⁷+ 2x¹⁶ +15x¹⁵−13x¹⁴−11x¹³+11x¹²−5x¹¹−11x¹⁰+ 8x⁹+ 2x⁸

−6x⁷+ 6x⁶+ 4x⁵−4x⁴+ 4x³+x²−x+ 1)/(x³¹−x³⁰−x²⁹ +x²⁸−x²²+x²¹+x²⁰−3x¹⁹+ 2x¹⁸+ 3x¹⁷−4x¹⁶+ 3x¹⁴

−3x¹³−3x¹²+ 2x¹¹−2x¹⁰−2x⁹+x⁸−x⁷) y+ (x³²−x³¹

−5x³⁰+ 5x²⁹+ 4x²⁸−3x²⁷−x²⁶−5x²⁵+ 7x²⁴+ 2x²³

−14x²²+10x²¹+ 8x²⁰−9x¹⁹+x¹⁸+ 4x¹⁷−5x¹⁶+x¹⁵+ 3x¹⁴

−8x¹³+ 8x¹²+12x¹¹−22x¹⁰+22x⁹+18x⁸−28x⁷+

R= y −x³+x Solutions of R= 0

High precision modular computation

Two different approaches:

multi-modular lifting Hensel lifting

Precision 1 Precision 1

High precision modular computation

Two different approaches:

multi-modular lifting Hensel lifting

Precision 2 Precision 2

High precision modular computation

Two different approaches:

multi-modular lifting Hensel lifting

Precision 3 Precision 3

High precision modular computation

Two different approaches:

multi-modular lifting Hensel lifting

Precision 4 Precision 4

Plan

1. Why is p-adic lifting interesting?

2. Relaxed p-adic lifting a. On-line algorithms b. Recursive p-adic

c. Application to:

i. Linear systems

ii. Linear q-differential systems iii. Polynomial root lifting

iv. Univariate representation lifting 3. Conclusion

Relaxed algorithms

Definition. (on-line or relaxed algorithm) [Hennie ’66]

: reading allowed

Relaxed algorithms

Definition. (on-line or relaxed algorithm) [Hennie ’66]

: reading allowed

Relaxed algorithms

Definition. (on-line or relaxed algorithm) [Hennie ’66]

: reading allowed

Relaxed algorithms

Definition. (on-line or relaxed algorithm) [Hennie ’66]

: reading allowed

Half-line or semi-relaxed algorithm : condition on one input.

Off-line or zealous algorithm : condition not met.

Relaxed algorithms

Definition. (on-line or relaxed algorithm) [Hennie ’66]

: reading allowed

Half-line or semi-relaxed algorithm : condition on one input.

Off-line or zealous algorithm : condition not met.

Example.

• The naive addition algorithm is on-line:

for i from 0 to N do cⁱ⁸ aⁱ+bⁱ

• The naive multiplication algorithm is on-line:

for i from 0 to N do cⁱ⁸ aⁱb₀+aⁱ₋₁b₁+ +a₀bⁱ Challenge. Find a quasi-optimal on-line multiplication algorithm

Relaxed multiplication c = a × b - step 0

b0

b1

b2

a0 a1 a2 a

b

Figure. What we must compute

b0

b1

b2

a0 a1 a2 a

b

Figure. Minimum knowledge on the input

0

Figure. What we compute

step 0: c = a₀b₀

Relaxed multiplication c = a × b - step 1

0 b

a b0

b₁

a0 a1

Figure. What we must compute

0 b

a

Figure. Minimum knowledge on the input

0 1

1

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

Relaxed multiplication c = a × b - step 2

0 1

1

Figure. What we must compute

0 1

1

Figure. Minimum knowledge on the input

0 1

1 2

2 2

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

step 2: c⁹ z²(a0b₂+a₂b₀+ (a1+a₂z) (b1+b₂z))

Relaxed multiplication c = a × b - step 3

0 1

1 2 2

2

Figure. What we must compute

0 1

1 2

2 2

Figure. Minimum knowledge on the input

0 1

1 2

2 2

3

3

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

step 2: c⁹ z²(a0b₂+a₂b₀+ (a1+a₂z) (b1+b₂z)) step 3: c⁹ z³(a₀b₃+a₃b₀)

Relaxed multiplication c = a × b - step 4

0 1

1 2

2 2

3

3

Figure. What we must compute

0 1

1 2

2 2

3

3

Figure. Minimum knowledge on the input

0 1

1 2

2 2

3

3

4 4

4 4

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

step 2: c⁹ z²(a0b₂+a₂b₀+ (a1+a₂z) (b1+b₂z)) step 3: c⁹ z³(a₀b₃+a₃b₀)

step 4: c⁹ z⁴(a0b₄+a₄b₀+ (a1+a₂z) (b3+b₄z) +(a₃+a₄z) (b₁+b₂z))

Relaxed multiplication c = a × b - step 5

0 1

1 2

2 2

3

3

4 4

4 4

Figure. What we must compute

0 1

1 2

2 2

3

3

4 4

4 4

Figure. Minimum knowledge on the input

0 1

1 2

2 2

3

3

4 4

4 4 5

5

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

step 2: c⁹ z²(a0b₂+a₂b₀+ (a1+a₂z) (b1+b₂z)) step 3: c⁹ z³(a₀b₃+a₃b₀)

step 4: c⁹ z⁴(a₀b₄+a₄b₀+ (a₁+a₂z) (b₃+b₄z) +(a₃+a₄z) (b₁+b₂z))

step 5: c⁹ z⁵(a₀b₅+a₅b₀)

Relaxed multiplication c = a × b - step 6

0 1

1 2

2 2

3

3

4 4

4 4 5

5

Figure. What we must compute

0 1

1 2

2 2

3

3

4 4

4 4 5

5

Figure. Minimum knowledge on the input

0 1

1 2

2 2

3

3

4 4

4 4 5

5 6

6 6

6

6

Figure. What we compute

step 0: c = a₀b₀

step 1: c⁹ z (a₀b₁+a₁b₀)

step 2: c⁹ z²(a0b₂+a₂b₀+ (a1+a₂z) (b1+b₂z)) step 3: c⁹ z³(a₀b₃+a₃b₀)

step 4: c⁹ z⁴(a₀b₄+a₄b₀+ (a₁+a₂z) (b₃+b₄z) +(a₃+a₄z) (b₁+b₂z))

step 5: c⁹ z⁵(a₀b₅+a₅b₀)

step 6: c⁹ z⁶(a0b₆+a₆b₀+ (a1+a₂z) (b5+b₆z) +(a₅+a₆z) (b₁+b₂z)

+(a3+ +a₆z³) (b3+ +b₆z³))

Relaxed multiplications

Theorem. [Fischer, Stockmeyer ’74], [Schröder ’97], [van der Hoeven ’97], [Berthomieu, van der Hoeven, Lecerf ’11], [L., Schost ’12]

M(N): cost of a×b in k[[x]] at precision N by an off-line algorithm.

R(N): cost of a×b in k[[x]] at precision N by an on-line algorithm. Then R(N) =O(M(N)logN) =O˜(N).

Relaxed multiplications

Theorem. [Fischer, Stockmeyer ’74], [Schröder ’97], [van der Hoeven ’97], [Berthomieu, van der Hoeven, Lecerf ’11], [L., Schost ’12]

M(N): cost of a×b in k[[x]] at precision N by an off-line algorithm.

R(N): cost of a×b in k[[x]] at precision N by an on-line algorithm. Then R(N) =O(M(N)logN) =O˜(N).

Theorem. [van der Hoeven ’07, ’12]

R(N) =M(N)log(N)^o(1). Seems not to be used yet in practice.

Relaxed multiplications

Relaxed multiplication [Fischer, Stockmeyer ’74]

Relaxed multiplications

Semi-relaxed multiplication [Fischer, Stockmeyer ’74] Relaxed multiplication [Fischer, Stockmeyer ’74]

Relaxed multiplications

Semi-relaxed multiplication [Fischer, Stockmeyer ’74] Relaxed multiplication [Fischer, Stockmeyer ’74]

?

Semi-relaxed multiplication [van der Hoeven ’03]

Relaxed multiplications

Semi-relaxed multiplication [Fischer, Stockmeyer ’74] Relaxed multiplication [Fischer, Stockmeyer ’74]

Semi-relaxed multiplication [van der Hoeven ’03] Relaxed multiplication [L., Schost ’12]

Timings of relaxed multiplications

0 2 4 6 8 10 12

2⁰ 2² 2⁴ 2⁶ 2⁸ 2¹⁰ 2¹² 2¹⁴ 2¹⁶ 2¹⁸

Ratio

precision N [Fischer, Stockmeyer ’74]

[L., Schost ’12]

Figure. Ratio R(N)/M(N) for different relaxed multiplication algorithms over F268435459[[x]].

Plan

1. Why is p-adic lifting interesting?

2. Relaxed p-adic lifting a. On-line algorithms b. Recursive p-adic

c. Application to:

i. Linear systems

ii. Linear q-differential systems iii. Polynomial root lifting

iv. Univariate representation lifting 3. Conclusion

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Definition. (Shifted algorithm)

: reading allowed

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Definition. (Shifted algorithm)

: reading allowed

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Definition. (Shifted algorithm)

: reading allowed

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Definition. (Shifted algorithm)

: reading allowed

Theorem. [Watt ’88], [van der Hoeven ’02]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y).

Relaxed recursive power series

Definition. A power series y∈k[[x]] is recursive if there exists Φ such that

• y= Φ(y)

• Φ(y)ⁿ only depends on y₀,, yn−1

Definition. (Shifted algorithm)

: reading allowed

Theorem. [Watt ’88], [van der Hoeven ’02], [Berthomieu, L. ISSAC ’12]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y) by a shifted algorithm.

Relaxed recursive power series

Theorem. (Fundamental theorem)

[Watt ’88], [van der Hoeven ’02], [Berthomieu, L. ISSAC ’12]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y) by a shifted algorithm.

Proof. y= Φ(y)

: reading allowed

?

?

Relaxed recursive power series

Theorem. (Fundamental theorem)

[Watt ’88], [van der Hoeven ’02], [Berthomieu, L. ISSAC ’12]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y) by a shifted algorithm.

Proof. y= Φ(y)⇒ ϕ₁= y₁

: reading allowed

?

Relaxed recursive power series

Theorem. (Fundamental theorem)

[Watt ’88], [van der Hoeven ’02], [Berthomieu, L. ISSAC ’12]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y) by a shifted algorithm.

Proof. y= Φ(y)

: reading allowed

?

Relaxed recursive power series

Theorem. (Fundamental theorem)

[Watt ’88], [van der Hoeven ’02], [Berthomieu, L. ISSAC ’12]

Let y∈k[[x]] such that y= Φ(y). Given y₀ and Φ, we can compute y at precision N in the time necessary to evaluate Φ(y) by a shifted algorithm.

Proof. y= Φ(y)⇒ ϕ₂= y₂

: reading allowed

Plan

1. Why is p-adic lifting interesting?

2. Relaxed p-adic lifting a. On-line algorithms b. Recursive p-adic

c. Application to:

i. Linear systems

ii. Linear q-differential systems iii. Polynomial root lifting

iv. Univariate representation lifting 3. Conclusion

Two paradigms

Lifting solutions methods:

Zealous algorithms Relaxed algorithms

Newton operator Fundamental theorem

implicit equations P(Y ) = 0 recursive equations Y =Φ(Y )

To do:

• Transform an implicit equations into recursive equations (+ shifted algorithm) P(Y ) =0 Y =Ψ(Y )

Linear systems

Zealous algorithms Relaxed algorithms

Linear systems

dense O˜(r^ωd+r^ω−1N) structured O˜(α²r d+α r N)

Example. Find a(x), b(x), c(x)∈k[[x]] such that







(1 +x)a(x) + (3−x²)b(x) + 2c(x) = 1 (x−x³)a(x) + (1 +x)b(x) + (3−x²)c(x) = x (4−2x)a(x) + (x−x³)b(x) + (1 +x)c(x) = x²

r number of unknowns

d degree of polynomials in input N precision of the output

α parameter associated to structured matrices α≪r ω exponent of matrix multiplication 26ω 63

Linear systems: [Moenck, Carter ’79], [Storjohann ’03]

Ideas

Example of system

A·Y =B⇔







(1 +x) (3−x²) 2 (x−x³) (1 +x) (3−x²) (4−2x) (x−x³) (1 +x)





·





a(x) b(x) c(x)



=



 1 x x²





Transformation of equation

Implicit equation 0 = f(Y )⁸ A·Y −B

Recursive equation Y = Φ(Y )⁸ A₀⁻¹·(B −(A−A₀)·Y )

Shifted algorithm

Ψ(Y )

A

·

B − x ×

h

0

x

·

Y i

Linear systems

Zealous algorithms Relaxed algorithms

Linear systems

dense O˜(r^ωd+r^ω−1N) O˜(r^ω+r²N)

structured O˜(α²rd+α r N) O˜(α²r+α r N)

[Berthomieu, L., ISSAC ’12], [L., Schost ’12]

Example. Find a(x), b(x), c(x)∈k[[x]] such that







(1 +x)a(x) + (3−x²)b(x) + 2c(x) = 1 (x−x³)a(x) + (1 +x)b(x) + (3−x²)c(x) = x (4−2x)a(x) + (x−x³)b(x) + (1 +x)c(x) = x²

r number of unknowns

d degree of polynomials in input N precision of the output

α parameter associated to structured matrices α≪r ω exponent of matrix multiplication 26ω 63

Linear systems: [Moenck, Carter ’79], [Storjohann ’03]

Plan

1. Why is p-adic lifting interesting?

2. Relaxed p-adic lifting a. On-line algorithms b. Recursive p-adic

c. Application to:

i. Linear systems

ii. Linear q-differential systems iii. Polynomial root lifting

iv. Univariate representation lifting 3. Conclusion

Differential systems

Zealous algorithms Relaxed algorithms

Regular differential

systems O(r^ω)M(N)^+O_N^{→ ∞(N)} O(r²)R(N)^+O_N^→∞(N) (q)-differential

singular systems

Example. Find a(x)∈k[[x]] such that

(1 +x)a(x) + (3−x²) (a(2x)−a(x)) = x

Applications:

• List-decoding of folded Reed-Solomon codes: solve Q(x, f(x), f (q x)) = 0

• Composition of power series f(g) with g(0) = 0 when f solution of a singular differential equation.

Differential systems: [Brent, Kung ’78], [van der Hoeven ’02 ’11], [Bostan, Chyzak et al. ’07]

Differential systems

Zealous algorithms Relaxed algorithms

Regular differential

systems O(r^ω)M(N)^+O_N^{→ ∞(N)} O(r²)R(N)^+O_N^→∞(N) (q)-differential

singular systems O(r^ω)M(N)+O_{N→ ∞}(N) O(r²)R(N)+O_N→∞(N)

[Bostan, Chowdhury, L., Salvy, Schost, ISSAC ’12]

Example. Find a(x)∈k[[x]] such that

(1 +x)a(x) + (3−x²) (a(2x)−a(x)) = x

r number of unknowns N precision of the output

Differential systems: [Brent, Kung ’78], [van der Hoeven ’02 ’11], [Bostan, Chyzak et al. ’07]

Polynomial systems

Zealous algorithms Relaxed algorithms

Polynomial

systems O(r L+r^ω)M(N)^+O_N^{→ ∞(N)} Univariate

representations O(r L+r^ω)M(N)M(d)+O_{N→ ∞}(N)

Example. Find a(x), b(x), c(x)∈k[[x]] such that







(1 +x) + (4−2x)a(x)²c(x)⁴+ 2b(x) = 0 (x−x³) +(3−x²)a(x)b(x)²+ (1 +x+x²)c(x) = 0 1 +b(x)⁴c(x)⁵+x a(x)³ = 0

r number of unknowns, N precision of the output, L complexity of evaluation

Polynomial systems: [Newton 1736], [Hensel 1918], [Lipson ’76]

[Berthomieu, van der Hoeven, Lecerf ’11], [van der Hoeven ’11]

Univariate representations: [Giusti, Lecerf, Salvy ’01], [Heintz, Matera, Waissbein ’01]

Polynomial systems

Zealous algorithms Relaxed algorithms

Polynomial

systems O(rL+r^ω)M(N)^+O_N^{→ ∞(N)} O(L)R(N)^+O_N^{→ ∞(N)} Univariate

representations O(rL+r^ω)M(N)M(d)+O_{N→ ∞}(N) O(L)R(N)M(d)+O_N→∞(N)

[Berthomieu, L., ISSAC ’12], [L. ’12]

Example. Find a(x), b(x), c(x)∈k[[x]] such that







(1 +x) + (4−2x)a(x)²c(x)⁴+ 2b(x) = 0 (x−x³) +(3−x²)a(x)b(x)²+ (1 +x+x²)c(x) = 0 1 +b(x)⁴c(x)⁵+x a(x)³ = 0

r number of unknowns, N precision of the output, L complexity of evaluation

Polynomial systems: [Newton 1736], [Hensel 1918], [Lipson ’76]

[Berthomieu, van der Hoeven, Lecerf ’11], [van der Hoeven ’11]

Univariate representations: [Giusti, Lecerf, Salvy ’01], [Heintz, Matera, Waissbein ’01]

Timings

1

2

3

4 0 200 400 600 800 1000 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

time Newton

Relaxed

r

N time

Figure. Timings with systems of singular (q)-differential equations

Timings

0 0.005 0.01 0.015 0.02

0 100 200 300 400 500 600 700 800 900 1000

time

N

Newton DAC Naive

Figure. Timings with a singular (q)-differential equation

Timings

1

2

3

4 0 200 400 600 800 1000 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

time Newton

Relaxed

r

N time

Figure. Timings with systems of singular (q)-differential equations

Timings

Katsura -3 Katsura -4 Katsura -5 Katsura -6 N zealous relaxed zealous relaxed zealous relaxed zealous relaxed

2 0.02 0.007 0.08 0.02 0.25 0.06 0.8 0.17

4 0.03 0.01 0.10 0.03 0.35 0.08 1.1 0.22

8 0.05 0.02 0.17 0.05 0.55 0.13 1.7 0.36

16 0.08 0.04 0.29 0.09 0.9 0.24 1.9 0.70

32 0.14 0.07 0.51 0.20 1.7 0.53 5.2 1.5

64 0.26 0.16 1.0 0.44 3.3 1.2 10 3.6

128 0.51 0.36 1.9 1 6.6 2.8 21 8.6

Table. Timings in seconds of lifting of univariate representations over F16411[[x]] for Katsura-n.

Conclusion

Two general paradigms:

Newton operator Relaxed algorithms

Solve implicit equations P(y) = 0 Solve recursive equations y= Φ(y) Faster for higher precision

Less on-line multiplications

Implementations:

Contributions to algebramix and geomsolvex packages of Mathemagix;

Perspectives:

• Lift a polynomial root under more general hypotheses (multiplicities);

• Link between relaxed and divide-and-conquer algorithms

• New horizons : Relaxed Gröbner bases?

Thank you for your attention ;-)