Methods for Solving Systems of Polynomial Equations

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Methods for Solving Systems of Polynomial Equations. Zuzana Kúkelová. Motivation. Many problems in computer vision can be formulated using systems of polynomial equations 5 point relative pose problem, 6 point focal length problem

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### Methods for Solving Systems of Polynomial Equations

Zuzana Kúkelová

Motivation
• Many problems in computer vision can be formulated using systems of polynomial equations
• 5 point relative pose problem, 6 point focal length problem
• Problem of correcting radial distortion from point correspondences
• systems are not trivial => special algorithms have to be designed to achieve numerical robustness and computational efficiency

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

Polynomial equation in one unknown - Companion matrix
• Finding the roots of the polynomial in one unknown, is equivalent to determining the eigenvalues of so-called companion matrix
• The companion matrix of the monic polynomial in one unknown x

is a matrix defined as

• The characteristic polynomial of C(p)is equal to p
• The eigenvalues of some matrix A are precisely the roots of its characteristic polynomial

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

System of linear polynomial equations- Gauss elimination
• System of n linear equations in n unknowns can be written as
• We can represent this system in a matrix form

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

System of linear polynomial equations- Gauss elimination
• We can rewrite this
• Perform Gauss-elimination on the matrix A => reduces the matrix A to a triangular form
• Back-substitution to find the solution of the linear system

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

System of m polynomial equations in n unknowns (non-linear)
• A system of equations which are given by a set of m polynomials in nvariables with coefficients from
• Our goal is to solve this system
• Many different methods
• Groebner basis methods
• Resultant methods

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

System of m polynomial equations in n unknowns – An ideal
• An ideal generated by polynomials is the set of polynomials of the form:
• Contains all polynomials we can generate from F
• All polynomials in the ideal are zero on solutions of F
• Contains an infinite number of polynomials
• - generators of

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

Groebner basis method w.r.t. lexicographic ordering
• An ideal can be generated by many different sets of generators which all share the same solutions
• Groebner basis w.r.t. the lexicographic ordering which generates the ideal I = special set of generators which is easy to solve - one of the generators is a polynomial in one variable only

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

Groebner basis method w.r.t. lexicographic ordering
• A system of initial equations - initial generators of I
• This system of equations can be written in a matrix form
• M is the coefficient matrix
• X is the vector of all monomials
• - is a monomial

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

Groebner basis method w.r.t. lexicographic ordering
• Compute a Groebner basis w.r.t. lexicographic ordering
• Buchberger’s algorithm ~ Gauss elimination
• Generator - polynomial in one variable only
• Finding the roots of this polynomial using the companion matrix
• Back-substitution to find solutions of the whole system

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

An Analogy

Gauss elimination

Back-substitution

System of equations

in triangular form

One polynomial equation

in one unknown

Solving system of linear

equations

Solutions

Companion matrix +

Back-substitution

Buchberger’s algorithm

Groebner basis

w.r.t lexicographic ordering

- One polynomial equation

in one unknown

Solving system of

polynomial equations

Solutions

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

• For most problems - “Groebner basis method w.r.t. the lexicographic ordering” is not feasible (double exponential computational complexity in general)
• Therefore for some problems
• A Groebner basis G under another ordering, e.g. the graded reverse lexicographic ordering is constructed
• The properties of the quotient ring can be used
• The “action” matrix of the linear operator of the multiplication by a polynomial is constructed
• The solutions to the set of equations - read off directly from the eigenvalues and eigenvectors of this matrix

Polynomial division

Polynomial

Equations

Gröbner

Basis

Action

Matrix

Solutions

Buchberger’s algorithm

Eigenvectors

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz

An Analogy

Compute Companion

Matrix

Finding the Eigenvalues

of the Companion Matrix

Solving one Polynomial

Equation in one Unknown

Solutions

Requires a Groebner Basis

for Input Equations

Compute Action Matrix in Quotient Ring A

(Polynomials modulo the Groebner basis)

Finding the Eigenvalues

of the Action Matrix

Solving System of

Polynomial Equations

Solutions

Zuzana Kúkelovákukelova@cmp.felk.cvut.cz