e-pdiv-trivial
▸ function ePdivTrivial
(a: number[][], b: number[][], positiveMultiplier?: boolean): object
Defined in euclidean-division-related/expansion/e-pdiv-trivial.ts:52
Performs a trivial pseudo-division and returns the quotient and remainder
of the pseudo division of a/b (a, b both being polynomials) in such a way
that all intermediate calculations and the final result are done in ℤ, i.e.
performs Euclidean (i.e. long) division on the two given polynomials, a/b,
and returns a scaled r and q in the formula a = bq + r, where
degree(r) < degree(b). q is called the quotient and r the remainder.
precondition: the coefficients must integers (and also Shewchuk floating point expansions); if they are not they can easily be scaled from floating point numbers to Shewchuk expansions by calling scaleFloatsToInts or similar before calling this function (recall that all floating point numbers are rational).
Intermediate calculations (and the input coefficients) are done in infinite precision up to overlow (meaning integers can be represented exactly up to
2^1024 === 1797...(300 more digits)...37216) and may thus not be applicable to very high degree polynomials (in which case it is better to use bPdivTrivial)precondition: b !== [], i.e. unequal to the zero polynomial.
see also Polynomial long division
see The subresultant polynomial remainder sequence algorithm by Ruiyuan (Ronnie) Chen, p.10
Parameters:
| Name | Type | Default value | Description |
|---|---|---|---|
a | number[][] | - | the polynomial a in the formula a = bq + r; the polynomial is given with coefficients as a dense array of integer Shewchuk expansions from highest to lowest power, e.g. [[5],[-3],[0]] represents the polynomial 5x^2 - 3x |
b | number[][] | - | the polynomial b in the formula a = bq + r |
positiveMultiplier | boolean | false | defaults to false - if set to true then the multiplier (of the coefficients of the dividend) leadingCoeff(b)^(deg(a)-deg(b)+1) will be modified to abs(leadingCoeff(b)^(deg(a)-deg(b)+1)) |
Returns: object