Factor - Maple Help

PolynomialTools[Approximate]

 Factor
 compute approximate factorization

 Calling Sequence Factor(F, vars) Factor(F, vars, options)

Parameters

 F - polynom({numeric,complex(numeric)}) vars - set or list of variables

Options

 • 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.

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{PolynomialTools}:-\mathrm{Approximate}\right):$
 > $F≔\mathrm{sort}\left(\mathrm{expand}\left(\left({x}^{2}+{y}^{2}-1\right)\left({x}^{3}-{y}^{3}+1\right)\right),\left[x,y\right]\right)$
 ${F}{≔}{{x}}^{{5}}{+}{{x}}^{{3}}{}{{y}}^{{2}}{-}{{x}}^{{2}}{}{{y}}^{{3}}{-}{{y}}^{{5}}{-}{{x}}^{{3}}{+}{{y}}^{{3}}{+}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{1}$ (1)
 > $\mathrm{aF_8}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-8}xy\right),\left[x,y\right]\right)$
 ${\mathrm{aF_8}}{≔}{-}{3.44406483319975}{}\left({-}{0.552477515351686}{+}{9.46726890028623}{×}{{10}}^{{-11}}{}{x}{+}{1.34863083369451}{×}{{10}}^{{-9}}{}{y}{+}{0.552477517557262}{}{{x}}^{{2}}{+}{2.03989782080192}{×}{{10}}^{{-9}}{}{x}{}{y}{+}{0.552477516011368}{}{{y}}^{{2}}\right){}\left({-}{0.525550037636229}{-}{6.82222680826386}{×}{{10}}^{{-10}}{}{x}{-}{3.43514242374310}{×}{{10}}^{{-9}}{}{y}{-}{7.62609054545801}{×}{{10}}^{{-10}}{}{{x}}^{{2}}{+}{6.10950662565009}{×}{{10}}^{{-10}}{}{x}{}{y}{+}{6.99437842035687}{×}{{10}}^{{-10}}{}{{y}}^{{2}}{-}{0.525550038352441}{}{{x}}^{{3}}{-}{3.70256461964700}{×}{{10}}^{{-10}}{}{y}{}{{x}}^{{2}}{-}{8.19581361585493}{×}{{10}}^{{-10}}{}{{y}}^{{2}}{}{x}{+}{0.525550036492460}{}{{y}}^{{3}}\right)$ (2)
 > $\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF_8}\right)\right),\left[x,y\right]\right)$
 ${1.}{}{{x}}^{{5}}{+}{0.9999999988}{}{{x}}^{{3}}{}{{y}}^{{2}}{-}{0.9999999958}{}{{x}}^{{2}}{}{{y}}^{{3}}{-}{0.9999999937}{}{{y}}^{{5}}{-}{0.9999999947}{}{{x}}^{{3}}{+}{0.9999999990}{}{{y}}^{{3}}{+}{0.9999999972}{}{{x}}^{{2}}{+}{0.9999999972}{}{{y}}^{{2}}{-}{0.9999999946}$ (3)
 > $\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF_8}\right),2\right)}{\mathrm{norm}\left(F,2\right)}\right)$
 ${-9}$ (4)
 > $\mathrm{aF_4}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-4}xy\right),\left[x,y\right]\right)$
 ${\mathrm{aF_4}}{≔}{3.47266293000095}{}\left({-}{0.552885827441145}{-}{9.97108941035032}{×}{{10}}^{{-6}}{}{x}{+}{0.0000179791567477263}{}{y}{+}{0.552898569944418}{}{{x}}^{{2}}{+}{8.98886932466931}{×}{{10}}^{{-6}}{}{x}{}{y}{+}{0.552899121071388}{}{{y}}^{{2}}\right){}\left({0.520834024267511}{-}{5.33436675048057}{×}{{10}}^{{-6}}{}{x}{+}{0.0000175929390281277}{}{y}{+}{4.50859262571895}{×}{{10}}^{{-6}}{}{{x}}^{{2}}{-}{0.0000210937198288496}{}{x}{}{y}{+}{9.51805868064093}{×}{{10}}^{{-6}}{}{{y}}^{{2}}{+}{0.520828698460088}{}{{x}}^{{3}}{+}{2.48682307894181}{×}{{10}}^{{-6}}{}{y}{}{{x}}^{{2}}{-}{2.34026244980658}{×}{{10}}^{{-6}}{}{{y}}^{{2}}{}{x}{-}{0.520822689868150}{}{{y}}^{{3}}\right)$ (5)
 > $\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF_4}\right),6\right),\left[x,y\right]\right)$
 ${1.00001}{}{{x}}^{{5}}{+}{1.00000}{}{{x}}^{{3}}{}{{y}}^{{2}}{-}{0.999991}{}{{x}}^{{2}}{}{{y}}^{{3}}{-}{0.999996}{}{{y}}^{{5}}{-}{0.999994}{}{{x}}^{{3}}{+}{1.00001}{}{{y}}^{{3}}{+}{1.00001}{}{{x}}^{{2}}{+}{1.00000}{}{{y}}^{{2}}{-}{0.999994}$ (6)
 > $\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF_4}\right),2\right)}{\mathrm{norm}\left(F,2\right)}\right)$
 ${-5}$ (7)
 > $\mathrm{aF_4I}≔\mathrm{Factor}\left(\mathrm{expand}\left(F+{10}^{-4}Ixy\right),\left[x,y\right]\right)$
 ${\mathrm{aF_4I}}{≔}\left({3.33619010286800}{+}{0.00138524690482115}{}{I}\right){}\left({-}{0.563243378344182}{+}{0.}{}{I}{+}\left({3.13353783832099}{×}{{10}}^{{-10}}{-}{0.0000272294178703771}{}{I}\right){}{x}{+}\left({-9.76499830597157}{×}{{10}}^{{-11}}{+}{0.0000398338050312695}{}{I}\right){}{y}{+}\left({0.563243374462419}{+}{0.0000246224700712112}{}{I}\right){}{{x}}^{{2}}{-}\left({8.65949653567788}{×}{{10}}^{{-9}}{+}{0.0000388463734635401}{}{I}\right){}{x}{}{y}{+}\left({0.563243379993721}{+}{0.0000319121864341227}{}{I}\right){}{{y}}^{{2}}\right){}\left({0.532173243848354}{-}{0.000251648003844275}{}{I}{+}\left({-3.44355626123618}{×}{{10}}^{{-11}}{+}{7.37230937854983}{×}{{10}}^{{-6}}{}{I}\right){}{x}{-}\left({1.32808396699356}{×}{{10}}^{{-8}}{+}{0.0000153645729853419}{}{I}\right){}{y}{-}\left({3.07623209457749}{×}{{10}}^{{-9}}{+}{4.12354158710398}{×}{{10}}^{{-6}}{}{I}\right){}{{x}}^{{2}}{+}\left({7.05525063405447}{×}{{10}}^{{-9}}{+}{0.0000239417134783695}{}{I}\right){}{x}{}{y}{-}\left({8.60740959257295}{×}{{10}}^{{-10}}{+}{0.0000123372409064711}{}{I}\right){}{{y}}^{{2}}{+}\left({0.532173243047371}{-}{0.000244232252846877}{}{I}\right){}{{x}}^{{3}}{-}\left({2.53189723598475}{×}{{10}}^{{-9}}{+}{2.35423160039879}{×}{{10}}^{{-6}}{}{I}\right){}{y}{}{{x}}^{{2}}{+}\left({-3.20721711648609}{×}{{10}}^{{-10}}{+}{4.05320710982624}{×}{{10}}^{{-6}}{}{I}\right){}{{y}}^{{2}}{}{x}{+}\left({-0.532173249216916}{+}{0.000240066165610370}{}{I}\right){}{{y}}^{{3}}\right)$ (8)
 > $\mathrm{sort}\left(\mathrm{fnormal}\left(\mathrm{expand}\left(\mathrm{aF_4I}\right),6\right),\left[x,y\right]\right)$
 ${1.00000}{}{{x}}^{{5}}{+}{1.00000}{}{{x}}^{{3}}{}{{y}}^{{2}}{-}{1.00000}{}{{x}}^{{2}}{}{{y}}^{{3}}{-}{1.00000}{}{{y}}^{{5}}{+}{0.000115711}{}{I}{}{{x}}^{{3}}{}{y}{-}{1.00000}{}{{x}}^{{3}}{+}{1.00000}{}{{y}}^{{3}}{+}{1.00000}{}{{x}}^{{2}}{-}{0.000113958}{}{I}{}{x}{}{y}{+}{1.00000}{}{{y}}^{{2}}{-}{1.00000}{+}{0.}{}{I}$ (9)
 > $\mathrm{ilog10}\left(\frac{\mathrm{norm}\left(\mathrm{expand}\left(F-\mathrm{aF_4I}\right),2\right)}{\mathrm{norm}\left(F,2\right)}\right)$
 ${-5}$ (10)

References

 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.

Compatibility

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