AngouriMath
TrySolveOverPolynomials(AngouriMath.Entity[][],AngouriMath.Entity[],AngouriMath.Entity[]@)
Method (no overloads)
Summary
The system whose entries are polynomials over Q in some symbols, solved
exactly as such: fraction-free Gauss-Jordan elimination, so that every entry on
the way is a minor of the augmented matrix and every zero is a zero, and each
unknown comes out as one polynomial over the last pivot, in lowest terms.
Declined, for the elimination on entities, where an entry is not such a
polynomial -- a number outsideQ , a symbol under a root or below the bar
-- or where the system has no symbols at all and the entities are exact already.
exactly as such: fraction-free Gauss-Jordan elimination, so that every entry on
the way is a minor of the augmented matrix and every zero is a zero, and each
unknown comes out as one polynomial over the last pivot, in lowest terms.
Declined, for the elimination on entities, where an entry is not such a
polynomial -- a number outside
-- or where the system has no symbols at all and the entities are exact already.
Remarks
The elimination on entities does not collect terms, so with symbols in the entries
its zero test is numeric and its answers are what the arithmetic wrote:
1/((x + 1) sqrt(x^2 + x + b)) through the Euler substitution came back
correct in 24 KB, with partial-fraction coefficients like(2b - 2b) beside
terms that were zero, and1/((x + a) sqrt(x^2 + b x + c)) in 77 KB. Bareiss'
elimination keeps every intermediate a polynomial, divided exactly by the previous
pivot, and the Gauss-Jordan form of it (Nakos, Turner and Williams, 1997) reduces
above the pivot too, so at the end each pivot row readsd x_k = n_k with
d the last pivot -- one fraction per unknown, no back-substitution to
compound them.
its zero test is numeric and its answers are what the arithmetic wrote:
correct in 24 KB, with partial-fraction coefficients like
terms that were zero, and
elimination keeps every intermediate a polynomial, divided exactly by the previous
pivot, and the Gauss-Jordan form of it (Nakos, Turner and Williams, 1997) reduces
above the pivot too, so at the end each pivot row reads
compound them.
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