{"version":9,"randomSeed":"027522d981741090eae18217c1e6feb0","graph":{"viewport":{"xmin":-2.4999999999999996,"ymin":-1.9954942150304038,"xmax":2.4999999999999996,"ymax":1.995494215030402}},"expressions":{"list":[{"type":"text","id":"34","text":"see https://www.youtube.com/watch?v=-RdOwhmqP5s"},{"type":"folder","id":"24","title":"utilities","collapsed":true},{"type":"text","id":"10","folderId":"24","text":"multiplication and division of complex numbers (as points)"},{"type":"expression","id":"2","folderId":"24","color":"#2d70b3","latex":"m\\left(w,z\\right)=\\left(w.x\\cdot z.x-w.y\\cdot z.y,w.x\\cdot z.y+w.y\\cdot z.x\\right)"},{"type":"text","id":"12","folderId":"24","text":"see https://en.wikipedia.org/wiki/Division_%28mathematics%29#Of_complex_numbers"},{"type":"expression","id":"5","folderId":"24","color":"#000000","latex":"d\\left(w,z\\right)=\\frac{\\left(w.x\\cdot z.x+w.y\\cdot z.y,w.y\\cdot z.x-w.x\\cdot z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"folder","id":"20","title":"setup","collapsed":true},{"type":"text","id":"14","folderId":"20","text":"function (z^3 - 1) and its derivative (3*z^2)"},{"type":"expression","id":"1","folderId":"20","color":"#c74440","latex":"F\\left(z\\right)=m\\left(z,m\\left(z,z\\right)\\right)-\\left(1,0\\right)"},{"type":"expression","id":"3","folderId":"20","color":"#388c46","latex":"f\\left(z\\right)=3\\cdot m\\left(z,z\\right)"},{"type":"text","id":"16","text":"Newton's method step"},{"type":"expression","id":"4","color":"#6042a6","latex":"N\\left(z\\right)=z-d\\left(F\\left(z\\right),f\\left(z\\right)\\right)"},{"type":"expression","id":"7","color":"#2d70b3","latex":"Q\\left(z\\right)=N\\left(N\\left(N\\left(N\\left(z\\right)\\right)\\right)\\right)"},{"type":"text","id":"26","text":"four iterations"},{"type":"expression","id":"27","color":"#388c46","latex":"Q\\left(\\left(x,y\\right)\\right).x\\ge0","hidden":true},{"type":"text","id":"29","text":"eight iterations"},{"type":"expression","id":"30","color":"#6042a6","latex":"Q\\left(Q\\left(\\left(x,y\\right)\\right)\\right).x\\ge0","lines":false,"fillOpacity":"1"},{"type":"expression","id":"31","color":"#388c46","latex":"Q\\left(Q\\left(\\left(x,y\\right)\\right)\\right).y\\ge0.1","lines":false,"fillOpacity":"1"},{"type":"expression","id":"32","color":"#2d70b3","latex":"Q\\left(Q\\left(\\left(x,y\\right)\\right)\\right).y\\le-0.1","lines":false,"fillOpacity":"1"}]}}