{"version":10,"randomSeed":"ffc1e1e9ae02696bbed75593cdde764f","graph":{"viewport":{"xmin":-10,"ymin":-12.10877264841943,"xmax":10,"ymax":12.10877264841943}},"expressions":{"list":[{"type":"text","id":"50","text":"increment of sampling"},{"type":"expression","id":"4","color":"#6042a6","latex":"d=0.1","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"1"}},{"type":"text","id":"52","text":"sampling interval"},{"type":"expression","id":"78","color":"#388c46","latex":"a=-5"},{"type":"expression","id":"79","color":"#6042a6","latex":"b=5"},{"type":"expression","id":"5","color":"#000000","latex":"K=\\left[a,a+d,...b\\right]"},{"type":"text","id":"54","text":"given list representing function, evaluate function at point"},{"type":"expression","id":"9","color":"#6042a6","latex":"F\\left(L,x\\right)=L\\left[\\operatorname{floor}\\left(\\frac{x-a}{d}\\right)+1\\right]+\\left\\{\\operatorname{floor}\\left(\\frac{x-a}{d}\\right)<\\operatorname{length}\\left(L\\right)-1:\\left(\\frac{x-a}{d}-\\operatorname{floor}\\left(\\frac{x-a}{d}\\right)\\right)\\left(L\\left[\\operatorname{floor}\\left(\\frac{x-a}{d}\\right)+2\\right]-L\\left[\\operatorname{floor}\\left(\\frac{x-a}{d}\\right)+1\\right]\\right),0\\right\\}"},{"type":"text","id":"71","text":"given list representing function, differentiate function at point"},{"type":"expression","id":"69","color":"#000000","latex":"D\\left(L,x\\right)=\\frac{d}{dx}F\\left(L,x\\right)"},{"type":"text","id":"56","text":"y' = f(x, y)"},{"type":"expression","id":"35","color":"#c74440","latex":"f\\left(x,y\\right)=xy"},{"type":"text","id":"58","text":"where y(x0) = y0"},{"type":"expression","id":"43","color":"#6042a6","latex":"x_{0}=0"},{"type":"expression","id":"44","color":"#000000","latex":"y_{0}=-1"},{"type":"expression","id":"21","color":"#c74440","latex":"y\\to y_{0}+\\int_{x_{0}}^{x}f\\left(t,y\\left(t\\right)\\right)dt"},{"type":"text","id":"60","text":"transformation on function as list"},{"type":"expression","id":"14","color":"#6042a6","latex":"T\\left(L\\right)=\\left[\\left(y_{0}+\\left\\{k_{1}=x_{0}:0,k_{1}>x_{0}:\\operatorname{total}\\left(df\\left(\\left[x_{0},x_{0}+d,...k_{1}\\right],F\\left(L,\\left[x_{0},x_{0}+d,...k_{1}\\right]\\right)\\right)\\right),k_{1}<x_{0}:-\\operatorname{total}\\left(df\\left(\\left[k_{1},k_{1}+d,...x_{0}\\right],F\\left(L,\\left[k_{1},k_{1}+d,...x_{0}\\right]\\right)\\right)\\right)\\right\\}\\right)\\operatorname{for}k_{1}=K\\right]"},{"type":"text","id":"62","text":"initial approximation"},{"type":"expression","id":"82","color":"#2d70b3","latex":"y_{n}\\to0K+y_{0}"},{"type":"text","id":"86","text":"iterate"},{"type":"expression","id":"84","color":"#6042a6","latex":"y_{n}\\to T\\left(y_{n}\\right)"},{"type":"expression","id":"87","color":"#c74440","latex":"F\\left(y_{n},x\\right)"},{"type":"folder","id":"34","title":"maybe don't open","collapsed":true},{"type":"expression","id":"31","folderId":"34","color":"#c74440","latex":"y_{n}=\\left[-6517802.753657474,-3258901.376935698,-1662039.7022488746,-864260.6451706415,-458058.1419405637,-247351.3966479165,-136043.26815635516,-76184.230167559,-43425.01119550865,-25186.50649339499,-14860.038831103046,-8916.023298661827,-5438.774212183714,-3372.040011553906,-2124.3852072789605,-1359.606532658534,-883.7442462280471,-583.2712025105112,-390.7917056820422,-265.73835986378873,-183.3594683060143,-128.35162781421,-91.12965574808909,-65.61335213862415,-47.89774706119561,-35.444332825284754,-26.583249618963578,-20.20326971041232,-15.55651767701748,-12.134083788073633,-9.585926192578173,-7.668740954062539,-6.211680172790656,-5.093577741688336,-4.227669525601319,-3.5512424015051076,-3.0185560412793415,-2.5959581955002338,-2.2584836300852036,-1.987465594474979,-1.7688443790827313,-1.5919599411744583,-1.448683546468757,-1.3327888627512565,-1.2394936423586684,-1.1651240238171483,-1.106867822626291,-1.0625931097212393,-1.0307153164296021,-1.0101010101010102,-1,-1.0101010101010102,-1.0307153164296021,-1.0625931097212393,-1.106867822626291,-1.1651240238171483,-1.2394936423586684,-1.3327888627512565,-1.448683546468757,-1.5919599411744583,-1.7688443790827315,-1.987465594474979,-2.2584836300852036,-2.595958195500234,-3.018556041279342,-3.551242401505108,-4.2276695256013195,-5.093577741688336,-6.211680172790654,-7.668740954062535,-9.58592619257817,-12.134083788073632,-15.556517677017476,-20.203269710412307,-26.583249618963563,-35.44433282528475,-47.8977470611956,-65.61335213862411,-91.12965574808904,-128.3516278142099,-183.35946830601415,-265.7383598637886,-390.7917056820421,-583.2712025105106,-883.7442462280464,-1359.606532658533,-2124.3852072789578,-3372.0400115539014,-5438.774212183712,-8916.023298661821,-14860.038831103037,-25186.506493394976,-43425.011195508574,-76184.2301675589,-136043.268156355,-247351.3966479162,-458058.14194056334,-864260.6451706403,-1662039.7022488732,-3258901.3769356958,-6517802.753657473\\right]","hidden":true},{"type":"text","id":"68","text":"quality of solution (this should be zero)"},{"type":"expression","id":"72","color":"#6042a6","latex":"D\\left(y_{n},x\\right)-f\\left(x,F\\left(y_{n},x\\right)\\right)"}]}}