Numerical View of Polynomial Systems · 3Choose two polynomials in G and let S ij = (a ij=g i)f i (a ij=g j)f j. 4Reduce (completely) S ij, with the multivariate division algorithm

Numerical View of Polynomial Systems

Abhishek BhardwajSupervisor: Professor Markus Hegland

Australian National University

June 13, 2018

Abhishek Bhardwaj (ANU) Numerical Algebra June 13, 2018 :

Polynomials in Geometry & Algebra

Given a set of polynomials {p1, . . . , pn} ⊆ R[x1, . . . , xk ], consider theproblem of finding x = (x1, . . . , xk) such that,

p1(x) = 0,

...

pn(x) = 0.

There are many descriptions to the solution of this problem.



The solutions x form a Variety

V ({p1, . . . , pn}) = {x | p1(x) = · · · = pn(x) = 0}

Example

Taking p1(x1, x2) = x21 + x22 − 2, and p2(x1, x2) = x1 + x2, gives us

V ({p1, p2}) = {(−1, 1), (1,−1)}.



The solutions x form a Variety

V ({p1, . . . , pn}) = {x | p1(x) = · · · = pn(x) = 0}

Example

Taking p1(x1, x2) = x21 + x22 − 2, and p2(x1, x2) = x1 + x2, gives us

V ({p1, p2}) = {(−1, 1), (1,−1)}.



A more generic description is that of the Ideal I ⊆ R[x], generated by{p1, . . . , pn};

I = 〈p1, . . . , pn〉 =

{n∑1

qipi | q1, . . . , qn ∈ R[x]

}

Question

Is {p1, . . . , pn} the simplest set of generators of I?



A more generic description is that of the Ideal I ⊆ R[x], generated by{p1, . . . , pn};

I = 〈p1, . . . , pn〉 =

{n∑1

qipi | q1, . . . , qn ∈ R[x]

}

Question

Is {p1, . . . , pn} the simplest set of generators of I?


Grobner Basis

There are in fact many ”simpler” generators of I .

Grobner Basis

A Grobner basis for an ideal I is a set G , such that for every p ∈ I , thereexist g1, . . . , gk ∈ G and h1, . . . , hk ∈ R[x] such that

p = h1g1 + · · ·+ hkgk .

How does this help?


Grobner Basis

There are in fact many ”simpler” generators of I .

Grobner Basis

A Grobner basis for an ideal I is a set G , such that for every p ∈ I , thereexist g1, . . . , gk ∈ G and h1, . . . , hk ∈ R[x] such that

p = h1g1 + · · ·+ hkgk .

How does this help?


Buchberger’s Algorithm

Input: A set of polynomials F which generate the ideal I .Output: A Grobner Basis G.

1 Set G := F .

2 For every fi , fj ∈ G , denote by gi the leading term of fi with respect tothe given ordering, and by aij the least common multiple of gi and gj .

3 Choose two polynomials in G and let Sij = (aij/gi )fi − (aij/gj)fj .

4 Reduce (completely) Sij , with the multivariate division algorithm relativeto the set G . If the result is non-zero, add it to G .

5 Repeat steps 1-4 until all possible pairs are considered, including thoseinvolving the new polynomials added in step 4.

6 Output G




1 Set G := F .





6 Output G




1 Set G := F .





6 Output G




1 Set G := F .





6 Output G




1 Set G := F .





6 Output G




1 Set G := F .





6 Output G


The Matricial Representation

Rewrite the polynomials in matrix form

C =

xd1 . . . xdk . . . xd1 . . . xk−1 xdk . . . xk 1

p1 ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗...

......

......

......

......

...pn−1 ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗pn ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗

.

Letxk =

(xd1 . . . xdk . . . x21 . . . xdk . . . 1

),

then p1(x) = 0...

pn(x) = 0

⇔ Cxk = 0



Rewrite the polynomials in matrix form

C =

xd1 . . . xdk . . . xd1 . . . xk−1 xdk . . . xk 1

p1 ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗...

......

......

......

......

...pn−1 ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗pn ∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗

.

Letxk =

(xd1 . . . xdk . . . x21 . . . xdk . . . 1

),

then p1(x) = 0...

pn(x) = 0

⇔ Cxk = 0



Buchberger’s algorithm takes the following form. Take two polynomials inthe system;

c12 c1 c2( )pi 0 0 . . . 1 . . . ∗ . . . ∗ ∗pj 0 0 . . . 0 . . . 1 . . . ∗ ∗

Compute the relevant LCM of the leading terms, and modify the rows toobtain

c12 c1 c2( )pi 0 1 . . . ∗ . . . ∗ . . . ∗ ∗pj 0 1 . . . ∗ . . . ∗ . . . ∗ ∗

,

the S-polynomial, Sij is the difference pi − pj .



Buchberger’s algorithm takes the following form. Take two polynomials inthe system;

c12 c1 c2( )pi 0 0 . . . 1 . . . ∗ . . . ∗ ∗pj 0 0 . . . 0 . . . 1 . . . ∗ ∗

Compute the relevant LCM of the leading terms, and modify the rows toobtain

c12 c1 c2( )pi 0 1 . . . ∗ . . . ∗ . . . ∗ ∗pj 0 1 . . . ∗ . . . ∗ . . . ∗ ∗

,

the S-polynomial, Sij is the difference pi − pj .



Update the system to

C =

xd1 . . . xdk . . . xd1 . . . xdk . . . xk 1

p1 ∗ . . . ∗ . . . ∗ . . . ∗ ∗...

......

......

......

......

pn ∗ . . . ∗ . . . ∗ . . . ∗ ∗Si ,j ∗ . . . ∗ . . . ∗ . . . ∗ ∗

.

Repeat until

C =

xd1 . . . xdk . . . xd1 . . . xk−1 xdk . . . xk 1

∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗...

......

......

......

......

∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗0 . . . 0 . . . 0 ∗ . . . ∗ ∗

.



Update the system to

C =

xd1 . . . xdk . . . xd1 . . . xdk . . . xk 1

p1 ∗ . . . ∗ . . . ∗ . . . ∗ ∗...

......

......

......

......

pn ∗ . . . ∗ . . . ∗ . . . ∗ ∗Si ,j ∗ . . . ∗ . . . ∗ . . . ∗ ∗

.

Repeat until

C =

xd1 . . . xdk . . . xd1 . . . xk−1 xdk . . . xk 1

∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗...

......

......

......

......

∗ . . . ∗ . . . ∗ ∗ . . . ∗ ∗0 . . . 0 . . . 0 ∗ . . . ∗ ∗

.


Updating C

Notice that we may write

xk =

xdk 0 . . . 0...

......

...x0k 0 . . . 00 xdk . . . 0...

......

...0 x0k . . . 00 0 . . . xdk...

......

......

......

...0 0 . . . x0k

xk−1


Updating C

Which leads to

Cxk = C

xdk 0 . . . 0...

......

...x0k 0 . . . 00 xdk . . . 0...

......

...0 x0k . . . 00 0 . . . xdk...

......

......

......

...0 0 . . . x0k

xk−1


Solving for x

Numerically solving for the last variable at each step, we can iterate theprocess to obtain

Cxk = CkCk−1 . . .C2x1.


Initial Results

Let p = 106 and consider the system of equations

f1 = xp + yp,

f2 = xp.

The Grobner Basis is clearly {xp, yp}.Our software takes 3.58 ms and less than 0.01 MB of memory to solvethis. On the other hand SymPy takes 6.04 s and 168.7 MB.


Initial Results

Let q = 104 and consider

f1 = x2qzq + xqyqzq + xqyq,

f2 = x2qy2qz2q + x2qxq + xqyq,

f3 = xqyqzq + xz2q + zq

For this kind of high degree system, our software performs much bettertaking only 326 ms and less than 0.01 MB memory, while SymPy takes625 ms and 5.74 MB.


Gaussian Quadrature

Consider the quadrature equations with constant weightsx1 x2 x3 . . . xk a1...

......

......

...

xk1 xk2 xk3... xkk ak

.

Instead of using the S-polynomial, we can eliminate x1 from the higherorder equations by using the first one.


Gaussian Quadrature

Consider the quadrature equations with constant weightsx1 x2 x3 . . . xk a1...

......

......

...

xk1 xk2 xk3... xkk ak

.

Instead of using the S-polynomial, we can eliminate x1 from the higherorder equations by using the first one.


Gaussian Quadrature

Question

Can Buchberger’s algorithm be simplified for problems with specialstructure?

Conjecture

If we use pi (x) to eliminate xi from pj(x) (with i < j), then the leadingterm of pj(x) is divisible by the leading term of pi (x) at every step of theelimination.


Future Work

1 Parallelizing procedures in the algorithms

2 Ensuring the sparsity of C3 Symmetric systems

i Higher dimensional quadratureii Orthogonal polynomialsiii Simplifying the algorithm

4 Generalizations

i Tensors and tensor calculusii Non-commutative polynomial systemsiii Semi-algebraic systems


Thank you.


Numerical View of Polynomial Systems · 3Choose two polynomials in G and let S ij = (a ij=g i)f i (a ij=g j)f j. 4Reduce (completely) S ij, with the multivariate division algorithm

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