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== Numerical Solutions of Polynomial Systems with PHCpack == by Marshall Hampton; requires phcpack optional package (PHCpack written by Jan Verschelde). The example below is a two-parameter deformation of the cyclic-6 problem. Solution paths are tracked through the parameter homotopy. {{{ zringA.<z0,z1,z2,z3,z4,z5,a,b> = PolynomialRing(QQ,8) cyclic6 = [z0 + z1 + z2 + z3 + z4 + z5+a, z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0, z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1, z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 + z5*z0*z1*z2, z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z0 + z3*z4*z5*z0*z1 + z4*z5*z0*z1*z2 + z5*z0*z1*z2*z3, z0*z1*z2*z3*z4*z5 - b] zring.<z0,z1,z2,z3,z4,z5> = PolynomialRing(QQ,6) z1 = [zring(x.subs({a:1/10, b:1/10})) for x in cyclic6] s1 = phc.blackbox(z1,zring) s1sas = s1.save_as_start(start_filename = DATA + 's1phc') cstate = [open(DATA + 's1phc').read()] def def_cyclic(ain, bin): eqs = [zring(x.subs({a:ain, b:bin})) for x in cyclic6] return eqs slines2d = [] mpts = [] @interact def tbp_tracker(show_eqs = checkbox(False),a = slider(-1,1,1/100,1/100), b = slider(-1,1,1/100,1/100), h_c_skew = slider(0,.1,.001,0.0, label='Homotopy skew'), scale = slider([2.0^x for x in srange(.1,4,.025)],default = 2^1.6)): z_pt = phc._path_track_file(start_filename_or_string = cstate[-1], polys = def_cyclic(a,b), input_ring = zring, c_skew = h_c_skew) cstate.append(open(z_pt).read()) z_pp = phc._parse_path_file(z_pt) hue_v = len(cstate)/(len(cstate)+1) znames = ['z0','z1','z2','z3','z4','z5'] for a_sol in z_pp: for z in znames: mpts.append(point([a_sol[0][z].real(), a_sol[0][z].imag()], hue=hue_v,pointsize=3)) mpts.append(point([a_sol[-1][z].real(), a_sol[-1][z].imag()], hue=hue_v,pointsize=3)) for a_sol in z_pp: zlines = [[] for q in znames] for data in a_sol: for i in range(len(znames)): zn = znames[i] zlines[i].append([data[zn].real(), data[zn].imag()]) for zl in zlines: slines2d.append(line(zl, thickness = .5)) show(sum(slines2d)+sum(mpts), figsize = [5,5], xmin = -scale, xmax=scale, ymin=-scale,ymax=scale, axes = false) if show_eqs: pols = def_cyclic(a,b) for i in len(pols): show(pols[i]) }}} attachment:pathtrack.png |
Sage Interactions - Algebra
goto [:interact:interact main page]
Groebner fan of an ideal
by Marshall Hampton; (needs sage-2.11 or higher, with gfan-0.3 interface)
@interact def gfan_browse(p1 = input_box('x^3+y^2',type = str, label='polynomial 1: '), p2 = input_box('y^3+z^2',type = str, label='polynomial 2: '), p3 = input_box('z^3+x^2',type = str, label='polynomial 3: ')): R.<x,y,z> = PolynomialRing(QQ,3) i1 = ideal(R(p1),R(p2),R(p3)) gf1 = i1.groebner_fan() testr = gf1.render() html('Groebner fan of the ideal generated by: ' + str(p1) + ', ' + str(p2) + ', ' + str(p3)) show(testr, axes = False, figsize=[8,8*(3^(.5))/2])
attachment:gfan_interact.png
Numerical Solutions of Polynomial Systems with PHCpack
by Marshall Hampton; requires phcpack optional package (PHCpack written by Jan Verschelde). The example below is a two-parameter deformation of the cyclic-6 problem. Solution paths are tracked through the parameter homotopy.
zringA.<z0,z1,z2,z3,z4,z5,a,b> = PolynomialRing(QQ,8) cyclic6 = [z0 + z1 + z2 + z3 + z4 + z5+a, z0*z1 + z1*z2 + z2*z3 + z3*z4 + z4*z5 + z5*z0, z0*z1*z2 + z1*z2*z3 + z2*z3*z4 + z3*z4*z5 + z4*z5*z0 + z5*z0*z1, z0*z1*z2*z3 + z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z0 + z4*z5*z0*z1 + z5*z0*z1*z2, z0*z1*z2*z3*z4 + z1*z2*z3*z4*z5 + z2*z3*z4*z5*z0 + z3*z4*z5*z0*z1 + z4*z5*z0*z1*z2 + z5*z0*z1*z2*z3, z0*z1*z2*z3*z4*z5 - b] zring.<z0,z1,z2,z3,z4,z5> = PolynomialRing(QQ,6) z1 = [zring(x.subs({a:1/10, b:1/10})) for x in cyclic6] s1 = phc.blackbox(z1,zring) s1sas = s1.save_as_start(start_filename = DATA + 's1phc') cstate = [open(DATA + 's1phc').read()] def def_cyclic(ain, bin): eqs = [zring(x.subs({a:ain, b:bin})) for x in cyclic6] return eqs slines2d = [] mpts = [] @interact def tbp_tracker(show_eqs = checkbox(False),a = slider(-1,1,1/100,1/100), b = slider(-1,1,1/100,1/100), h_c_skew = slider(0,.1,.001,0.0, label='Homotopy skew'), scale = slider([2.0^x for x in srange(.1,4,.025)],default = 2^1.6)): z_pt = phc._path_track_file(start_filename_or_string = cstate[-1], polys = def_cyclic(a,b), input_ring = zring, c_skew = h_c_skew) cstate.append(open(z_pt).read()) z_pp = phc._parse_path_file(z_pt) hue_v = len(cstate)/(len(cstate)+1) znames = ['z0','z1','z2','z3','z4','z5'] for a_sol in z_pp: for z in znames: mpts.append(point([a_sol[0][z].real(), a_sol[0][z].imag()], hue=hue_v,pointsize=3)) mpts.append(point([a_sol[-1][z].real(), a_sol[-1][z].imag()], hue=hue_v,pointsize=3)) for a_sol in z_pp: zlines = [[] for q in znames] for data in a_sol: for i in range(len(znames)): zn = znames[i] zlines[i].append([data[zn].real(), data[zn].imag()]) for zl in zlines: slines2d.append(line(zl, thickness = .5)) show(sum(slines2d)+sum(mpts), figsize = [5,5], xmin = -scale, xmax=scale, ymin=-scale,ymax=scale, axes = false) if show_eqs: pols = def_cyclic(a,b) for i in len(pols): show(pols[i])
attachment:pathtrack.png