CoefficientArrays

Status: Experimental

present and registered, but lightly documented and not yet covered by dedicated tests.

Description

CoefficientArrays[polys, vars]

gives the coefficient arrays of the polynomial system polys in the variables vars, grouped by total degree: element d + 1 holds every degree-d coefficient, so {b, m} = CoefficientArrays[eqs, vars] is the constant vector and coefficient matrix of a linear system. A degree-d coefficient sits at the index tuple naming its variables in non-decreasing order, every other permutation being 0; a list of polynomials carries the equation index as the leading axis. CoefficientArrays[poly, vars] treats a non-List first argument as a single polynomial. Option Modulus -> p reduces the coefficients. The arrays are dense Lists rather than SparseArrays.

Examples (2)

Every input below was run against the current Mathilda build and its output recorded.

Applications (2)

{constant vector, coefficient matrix}

In[1]:= CoefficientArrays[{x + 2 y - 3, 3 x - y + 1}, {x, y}]
Out[1]= {{-3, 1}, {{1, 2}, {3, -1}}}

Grouped by degree: {const, {linear}, {{quadratic}}}

In[2]:= CoefficientArrays[1 + 2 x + 3 x^2, x]
Out[2]= {1, {2}, {{3}}}

Implementation notes

Algorithm. builtin_coefficientarrays (src/poly/monomials.c) decomposes each polynomial into (exponent-vector, coefficient) terms (the shared monomial front end also used by MonomialList/CoefficientRules), reduces coefficients modulo an optional Modulus -> m, finds the maximum total degree D, and lays the terms into D + 1 nested dense arrays grouped by total degree: result element d + 1 holds every degree-d coefficient. A degree-d coefficient sits at the rank-d index tuple naming its variables in non-decreasing order, every other permutation of that tuple being 0. So {b, m} = CoefficientArrays[eqs, vars] is the constant vector b and the coefficient matrix m of a linear system. A List first argument is treated as a system — the equation index becomes the leading axis, raising each array's rank by one and forcing one shared column order across equations; a non-List argument is a single polynomial with no leading axis.

Data structures. The arrays are built as nested dense Lists of Expr (zero-filled by ca_make_zero, then overwritten by ca_set at a computed multi-index), not SparseArray objects — Normal is the identity on them. Any visible NDArray/packed-List argument is unpacked first, since this head is not on the AWARE list.

Complexity / limits. The dense layout costs O(nvars^d) entries at degree d, so a high degree in many variables blows up; the routine guards this with a CoefficientArrays ::dense message (routed through mth_message) and declines when the dense result would exceed its entry cap. FromCoefficientRules is the structural inverse of the CoefficientRules form rather than of these arrays.

Attributes: Protected.

References

• Source: src/poly/monomials.c
• Specification index: Mathilda_spec.md

Notes & additional examples

Notes

CoefficientArrays[polys, vars] returns the coefficient arrays of a polynomial system, grouped by total degree: element d + 1 holds every degree-d coefficient, with a degree-d coefficient placed at the index tuple naming its variables in non-decreasing order. For a linear system this means {b, m} = CoefficientArrays[eqs, vars] gives the constant vector b and coefficient matrix m directly — the first example returns {{-3, 1}, {{1, 2}, {3, -1}}}.

A List of polynomials is a system, carrying the equation index as the leading axis; a single polynomial (second example) carries no leading axis, so its degree-d slice is a rank-d array. The arrays are dense Lists, not SparseArray objects, and Modulus -> p reduces the coefficients. A high degree in many variables makes the dense result explode and is declined with a CoefficientArrays::dense message.