Built-in Mathematica Symbol

CoefficientArrays

CoefficientArrays[polys, vars]
gives the arrays of coefficients of the variables vars in the polynomials polys.

CoefficientArrays gives a list containing SparseArray objects, which can be converted to ordinary arrays using Normal.

If CoefficientArrays[polys, vars] gives {m₀, m₁, m₂, ...}, then polys can be reconstructed as m₀+m₁.vars+m₂.vars.vars+.... »

Any element of polys of the form lhsrhs is taken to correspond to the polynomial lhs-rhs. »

CoefficientArrays[polys, {form₁, form₂, ...}] takes all expressions in polys that match any of the form_i to be variables.

The length of the list CoefficientArrays[polys, vars] is one more than the total degree of polys.

For linear equations, the solution to Thread[polys0] is given by LinearSolve[m₁, -m₀].

For nonlinear equations, the m_i are not unique. CoefficientArrays by default assigns nonzero coefficients only to monomials where the variables appear in the same order as vars.

CoefficientArrays[polys, vars, Symmetric->True] makes all the m_i symmetric in all their indices. The resulting arrays will generally be less sparse.

Linear polynomial:

List of linear polynomials:

Nonlinear polynomials:

Polynomial equations:

The default coefficient of a quadratic form corresponding to y x is zero:

With "Symmetric"->True, CoefficientArrays gives symmetric arrays:

Convert linear equations to arrays suitable for LinearSolve:

The polynomial can be reconstructed by using Dot (.):

The symmetric coefficient arrays correspond to vector derivatives at the origin:

CoefficientArrays gives a list of arrays of polynomial coefficients ordered by total degrees:

CoefficientList gives a tensor of the coefficients:

Use Coefficient to find the coefficient of

:

In ca the position is a+b+1 concatenated with a 1s & b 2s (1 & 2 indicate the first & second variables):

In cl the coefficient of x^a y^b is the element at position a+1, b+1: