{"version":9,"randomSeed":"e2c1f6ead0d70df7d0fe897dd0a7f79b","graph":{"viewport":{"xmin":0.2743379987282757,"ymin":-1.1396433872638267,"xmax":3.5189430053666215,"ymax":2.1852990639042447}},"expressions":{"list":[{"type":"text","id":"50","text":"cf https://www.desmos.com/calculator/xgr11df4rs"},{"type":"folder","id":"30","title":"functions"},{"type":"text","id":"32","folderId":"30","text":"function of which to find extremum"},{"type":"expression","id":"2","folderId":"30","color":"#2d70b3","latex":"f\\left(x\\right)=\\sin x"},{"type":"text","id":"34","folderId":"30","text":"secant approximation of derivative at (a + b) / 2"},{"type":"expression","id":"1","folderId":"30","color":"#c74440","latex":"D\\left(a,b\\right)=\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}"},{"type":"text","id":"36","folderId":"30","text":"double-secant approximation of second derivative around b"},{"type":"expression","id":"4","folderId":"30","color":"#6042a6","latex":"D_{2}\\left(a,b,c\\right)=\\frac{D\\left(b,c\\right)-D\\left(a,b\\right)}{\\frac{b+c}{2}-\\frac{a+b}{2}}"},{"type":"text","id":"38","folderId":"30","text":"expanded by definition of D(a, b)"},{"type":"expression","id":"5","folderId":"30","color":"#000000","latex":"D_{2e}\\left(a,b,c\\right)=2\\frac{\\frac{f\\left(c\\right)-f\\left(b\\right)}{c-b}-\\frac{f\\left(b\\right)-f\\left(a\\right)}{b-a}}{c-a}"},{"type":"text","id":"40","folderId":"30","text":"given three approximations to the extremum, find a better one"},{"type":"text","id":"44","folderId":"30","text":"x' = x - f'(x) / f''(x)"},{"type":"text","id":"42","folderId":"30","text":"https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization"},{"type":"expression","id":"3","folderId":"30","color":"#388c46","latex":"S\\left(a,b,c\\right)=\\frac{b+c}{2}-\\frac{D\\left(b,c\\right)}{D_{2e}\\left(a,b,c\\right)}"},{"type":"folder","id":"46","title":"initial approximations","collapsed":true},{"type":"expression","id":"6","folderId":"46","color":"#c74440","latex":"a_{0}=1"},{"type":"expression","id":"7","folderId":"46","color":"#2d70b3","latex":"a_{1}=2"},{"type":"expression","id":"8","folderId":"46","color":"#388c46","latex":"a_{2}=3"},{"type":"folder","id":"17","title":"iteration steps"},{"type":"expression","id":"9","folderId":"17","color":"#6042a6","latex":"a_{3}=S\\left(a_{0},a_{1},a_{2}\\right)"},{"type":"expression","id":"10","folderId":"17","color":"#000000","latex":"a_{4}=S\\left(a_{1},a_{2},a_{3}\\right)"},{"type":"expression","id":"11","folderId":"17","color":"#c74440","latex":"a_{5}=S\\left(a_{2},a_{3},a_{4}\\right)"},{"type":"expression","id":"15","folderId":"17","color":"#000000","latex":"a_{6}=S\\left(a_{3},a_{4},a_{5}\\right)"},{"type":"expression","id":"20","folderId":"17","color":"#6042a6","latex":"a_{7}=S\\left(a_{4},a_{5},a_{6}\\right)"},{"type":"expression","id":"21","folderId":"17","color":"#000000","latex":"a_{8}=S\\left(a_{5},a_{6},a_{7}\\right)"},{"type":"expression","id":"22","folderId":"17","color":"#c74440","latex":"a_{9}=S\\left(a_{6},a_{7},a_{8}\\right)"},{"type":"folder","id":"48","title":"display","collapsed":true},{"type":"expression","id":"24","folderId":"48","color":"#388c46","latex":"A=\\left[a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},a_{8},a_{9}\\right]"},{"type":"expression","id":"25","folderId":"48","color":"#6042a6","latex":"k=\\left[1...10\\right]"},{"type":"expression","id":"26","folderId":"48","color":"#000000","latex":"\\left(A,f\\left(A\\right)\\right)","showLabel":true,"label":"${k}"},{"type":"folder","id":"54","title":"convergence","hidden":true,"collapsed":true},{"type":"expression","id":"55","folderId":"54","color":"#2d70b3","latex":"q=1"},{"type":"expression","id":"57","folderId":"54","color":"#6042a6","latex":"s=\\frac{\\pi}{2}"},{"type":"expression","id":"56","folderId":"54","color":"#fa7e19","latex":"y_{1}=\\ln\\left(\\left|A\\left[k\\right]-s\\right|\\right)"},{"type":"expression","id":"59","folderId":"54","color":"#c74440","latex":"\\left(k,y_{1}\\right)"}]}}