noexact

if provided, exact factorization of F will not be attempted

optimize

if given then a post-processing step is done on the output, using Optimization:-NLPSolve to return an approximate factorization with smaller backward error. Optionally, it can be given as optimize=list with a list of extra options to be passed to optimization.

After a series of initial preprocessing steps designed to handle exact and degenerate cases, numerical factors of F are found from the a low rank approximation of its RuppertMatrix.

This command works for univariate polynomials by calling factor which finds the real linear and quadratic factors from the roots.

$\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right)\:$

$F≔\mathrm{sort}\left(\mathrm{expand}\left(\left(x^2+y^2-1\right)\left(x^3-y^3+1\right)\right)\,\left[x\,y\right]\right)$

$F≔x^5+x^3{y}^2-x^2{y}^3-y^5-x^3+y^3+x^2+y^2-1$

$\mathrm{aF\_8}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-8}xy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_8}≔3.44406483319975\left(0.552477515351686-9.46729745577350\times {10}^{\mathrm{-11}}x-1.34863065022021\times {10}^{\mathrm{-9}}y-0.552477517557262x^2-2.03989792764670\times {10}^{\mathrm{-9}}xy-0.552477516011369y^2\right)\left(-0.525550037636230-6.82222545322290\times {10}^{\mathrm{-10}}x-3.43514241338219\times {10}^{\mathrm{-9}}y-7.62609004008427\times {10}^{\mathrm{-10}}{x}^2+6.10950533033342\times {10}^{\mathrm{-10}}xy+6.99437727875974\times {10}^{\mathrm{-10}}{y}^2-0.525550038352441x^3-3.70256462533906\times {10}^{\mathrm{-10}}y{x}^2-8.19581364011395\times {10}^{\mathrm{-10}}{y}^{2}x+0.525550036492460y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_8}\right)\right)\,\left[x\,y\right]\right)$

$1.x^5+0.9999999988x^3{y}^2-0.9999999958x^2{y}^3-0.9999999937y^5-0.9999999947x^3+0.9999999990y^3+0.9999999972x^2+0.9999999972y^2-0.9999999946$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_8}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-9}$

$\mathrm{aF\_4}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-4}xy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_4}≔3.47266293000094\left(-0.552885827441145-9.97108941000953\times {10}^{\mathrm{-6}}x+0.0000179791567479051y+0.552898569944418x^2+8.98886932467533\times {10}^{\mathrm{-6}}xy+0.552899121071388y^2\right)\left(0.520834024267511-5.33436675006424\times {10}^{\mathrm{-6}}x+0.0000175929390281242y+4.50859262542757\times {10}^{\mathrm{-6}}{x}^2-0.0000210937198292382xy+9.51805868057149\times {10}^{\mathrm{-6}}{y}^2+0.520828698460089x^3+2.48682307869211\times {10}^{\mathrm{-6}}y{x}^2-2.34026244961223\times {10}^{\mathrm{-6}}{y}^{2}x-0.520822689868150y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_4}\right)\,6\right)\,\left[x\,y\right]\right)$

$1.00001x^5+1.00000x^3{y}^2-0.999991x^2{y}^3-0.999996y^5-0.999994x^3+1.00001y^3+1.00001x^2+1.00000y^2-0.999994$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_4}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-5}$

$\mathrm{aF\_4I}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-4}\mathrm{I}xy\right)\,\left[x\,y\right]\right)$

$\mathrm{aF\_4I}≔\left(\mathrm{-3.33619010250857}-0.00138612068811785\mathrm{I}\right)\left(0.563243378342766-0.\mathrm{I}+\left(\mathrm{-3.11854963158353}\times {10}^{\mathrm{-10}}+0.0000272294190559388\mathrm{I}\right)x+\left(9.59484023866126\times {10}^{\mathrm{-11}}-0.0000398338053388040\mathrm{I}\right)y-\left(0.563243374460697+0.0000246224698940000\mathrm{I}\right)x^2+\left(8.66638205488023\times {10}^{\mathrm{-9}}+0.0000388463737235990\mathrm{I}\right)xy-\left(0.563243379994912+0.0000319121868867062\mathrm{I}\right)y^2\right)\left(0.532173243785897-0.000251787385887176\mathrm{I}+\left(\mathrm{-3.48434702385247}\times {10}^{\mathrm{-11}}+7.37230980399005\times {10}^{\mathrm{-6}}\mathrm{I}\right)x-\left(1.32880444269199\times {10}^{\mathrm{-8}}+0.0000153645724041642\mathrm{I}\right)y-\left(3.07790140400092\times {10}^{\mathrm{-9}}+4.12354147889653\times {10}^{\mathrm{-6}}\mathrm{I}\right)x^2+\left(7.05861311843272\times {10}^{\mathrm{-9}}+0.0000239417140052928\mathrm{I}\right)xy-\left(8.60791491508778\times {10}^{\mathrm{-10}}+0.0000123372414848047\mathrm{I}\right)y^2+\left(0.532173242984454-0.000244371634452508\mathrm{I}\right)x^3-\left(2.53274232099494\times {10}^{\mathrm{-9}}+2.35423155822770\times {10}^{\mathrm{-6}}\mathrm{I}\right)y{x}^2+\left(\mathrm{-3.21682849408933}\times {10}^{\mathrm{-10}}+4.05320747808123\times {10}^{\mathrm{-6}}\mathrm{I}\right)y^{2}x+\left(\mathrm{-0.532173249157492}+0.000240205547653828\mathrm{I}\right)y^3\right)$

$\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF\_4I}\right)\,6\right)\,\left[x\,y\right]\right)$

$1.x^5+1.00000x^3{y}^2-1.00000x^2{y}^3-1.00000y^5+0.000115711\mathrm{I}{x}^{3}y-1.00000x^3+1.00000y^3+1.00000x^2-0.000113958\mathrm{I}xy+1.00000y^2-1.00000+0.\mathrm{I}$

$\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF\_4I}\right)\,2\right)}{\mathrm{norm}\left(F\,2\right)}\right)$

$\mathrm{-5}$

Gao, S.; Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials via differential equations." Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004),  pp. 167-174. Ed. J. Guitierrez. ACM Press, 2004.

Kaltofen, E.; May, J.; Yang, Z.; and Zhi, L. "Approximate factorization of multivariate polynomials using singular value decomposition." Journal of Symbolic Computation Vol. 43(5), (2008): 359-376.

The PolynomialTools:-Approximate:-Factor command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.