{"version":9,"randomSeed":"09c6609cf86c1f674b4dc423d8eaeffe","graph":{"viewport":{"xmin":-0.3988862084038455,"ymin":-2.7917287673167963,"xmax":6.317089222114611,"ymax":2.55681905562454}},"expressions":{"list":[{"type":"folder","id":"11","title":"functions"},{"type":"text","id":"14","folderId":"11","text":"function of which to find a root"},{"type":"expression","id":"1","folderId":"11","color":"#c74440","latex":"f\\left(x\\right)=\\sin x"},{"type":"text","id":"46","folderId":"11","text":"linear function with points (a, f(a)) and (b, f(b))"},{"type":"expression","id":"47","folderId":"11","color":"#2d70b3","latex":"L\\left(a,b,x\\right)=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}\\cdot\\left(x-a\\right)+f\\left(a\\right)"},{"type":"text","id":"16","folderId":"11","text":"root of linear function with points (a, f(a)) and (b, f(b))"},{"type":"text","id":"38","folderId":"11","text":"see https://www.desmos.com/calculator/cwhcloetov"},{"type":"expression","id":"4","folderId":"11","color":"#6042a6","latex":"R\\left(a,b\\right)=-\\frac{b-a}{f\\left(b\\right)-f\\left(a\\right)}\\cdot f\\left(b\\right)+b"},{"type":"folder","id":"62","title":"commentary on R(a, b)"},{"type":"text","id":"53","folderId":"62","text":"we may rearrange this to"},{"type":"expression","id":"54","folderId":"62","color":"#388c46","latex":"R_{0}\\left(a,b\\right)=b-\\frac{f\\left(b\\right)}{\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}}"},{"type":"text","id":"56","folderId":"62","text":"notice that the denominator there is the secant approximation of f' (\"no calculus\" was a lie!)"},{"type":"text","id":"58","folderId":"62","text":"if we treat b as x_n, this matches Newton's method: R(a, b) = x_{n + 1}"},{"type":"text","id":"64","folderId":"62","text":"if we choose b as the preceding c and a as c - f(c) (close to c), we get"},{"type":"expression","id":"65","folderId":"62","color":"#2d70b3","latex":"R_{1}\\left(c\\right)=-\\frac{f\\left(c\\right)}{\\frac{f\\left(c\\right)-f\\left(c-f\\left(c\\right)\\right)}{c-\\left(c-f\\left(c\\right)\\right)}}+c","hidden":true},{"type":"text","id":"67","folderId":"62","text":"which simplifies to"},{"type":"expression","id":"68","folderId":"62","color":"#6042a6","latex":"R_{2}\\left(c\\right)=c-\\frac{f\\left(c\\right)^{2}}{f\\left(c\\right)-f\\left(c-f\\left(c\\right)\\right)}","hidden":true},{"type":"text","id":"70","folderId":"62","text":"this provides a method to find a better root given an imperfect root, requiring (with caching) two function evaluations"},{"type":"folder","id":"21","title":"step 0","collapsed":true},{"type":"expression","id":"2","folderId":"21","color":"#2d70b3","latex":"a_{0}=3"},{"type":"expression","id":"6","folderId":"21","color":"#c74440","latex":"\\left(a_{0},f\\left(a_{0}\\right)\\right)"},{"type":"expression","id":"3","folderId":"21","color":"#388c46","latex":"b_{0}=4"},{"type":"expression","id":"7","folderId":"21","color":"#2d70b3","latex":"\\left(b_{0},f\\left(b_{0}\\right)\\right)"},{"type":"expression","id":"48","folderId":"21","color":"#388c46","latex":"L\\left(a_{0},b_{0},x\\right)"},{"type":"expression","id":"5","folderId":"21","color":"#000000","latex":"c_{0}=R\\left(a_{0},b_{0}\\right)"},{"type":"expression","id":"28","folderId":"21","color":"#2d70b3","latex":"\\left(c_{0},f\\left(c_{0}\\right)\\right)"},{"type":"folder","id":"23","title":"step 1","collapsed":true},{"type":"expression","id":"24","folderId":"23","color":"#388c46","latex":"a_{1}=c_{0}"},{"type":"expression","id":"26","folderId":"23","color":"#000000","latex":"b_{1}=c_{0}-f\\left(c_{0}\\right)","hidden":true},{"type":"expression","id":"49","folderId":"23","color":"#6042a6","latex":"L\\left(a_{1},b_{1},x\\right)"},{"type":"expression","id":"29","folderId":"23","color":"#388c46","latex":"c_{1}=R\\left(a_{1},b_{1}\\right)"},{"type":"expression","id":"30","folderId":"23","color":"#6042a6","latex":"\\left(c_{1},f\\left(c_{1}\\right)\\right)"},{"type":"folder","id":"32","title":"step 2","collapsed":true},{"type":"expression","id":"33","folderId":"32","color":"#c74440","latex":"a_{2}=c_{1}"},{"type":"expression","id":"34","folderId":"32","color":"#2d70b3","latex":"b_{2}=c_{1}-f\\left(c_{1}\\right)"},{"type":"expression","id":"50","folderId":"32","color":"#000000","latex":"L\\left(a_{2},b_{2},x\\right)"},{"type":"expression","id":"35","folderId":"32","color":"#388c46","latex":"c_{2}=R\\left(a_{2},b_{2}\\right)"},{"type":"expression","id":"36","folderId":"32","color":"#6042a6","latex":"\\left(c_{2},f\\left(c_{2}\\right)\\right)"},{"type":"folder","id":"40","title":"step 3","collapsed":true},{"type":"expression","id":"41","folderId":"40","color":"#2d70b3","latex":"a_{3}=c_{2}"},{"type":"expression","id":"42","folderId":"40","color":"#388c46","latex":"b_{3}=c_{2}-f\\left(c_{2}\\right)"},{"type":"expression","id":"51","folderId":"40","color":"#c74440","latex":"L\\left(a_{3},b_{3},x\\right)"},{"type":"expression","id":"43","folderId":"40","color":"#6042a6","latex":"c_{3}=R\\left(a_{3},b_{3}\\right)"},{"type":"expression","id":"44","folderId":"40","color":"#000000","latex":"\\left(c_{3},f\\left(c_{3}\\right)\\right)"}]}}