(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 217141, 4704] NotebookOptionsPosition[ 203128, 4457] NotebookOutlinePosition[ 203703, 4479] CellTagsIndexPosition[ 203660, 4476] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Scientific Programming 5", "Title", CellChangeTimes->{{3.724943992364118*^9, 3.724944007146431*^9}, 3.725524593957123*^9, 3.726336972840088*^9, {3.821347032422962*^9, 3.8213470388273163`*^9}, 3.821955106714554*^9, 3.8225584238953733`*^9},ExpressionUUID->"28513503-cae1-403e-8849-\ 3300f1c8ba08"], Cell[BoxData[ RowBox[{ RowBox[{"SetOptions", "[", RowBox[{ RowBox[{"EvaluationNotebook", "[", "]"}], ",", " ", RowBox[{"Background", " ", "\[Rule]", " ", "LightBlue"}], ",", RowBox[{"CellBracketOptions", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"\"\\"", "\[Rule]", "Blue"}], ",", RowBox[{"\"\\"", "\[Rule]", RowBox[{"{", RowBox[{"20", ",", "20"}], "}"}]}], ",", RowBox[{"\"\\"", "\[Rule]", "3.5"}]}], "}"}]}], ",", RowBox[{"FontColor", "\[Rule]", "Black"}]}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.8203427063672333`*^9, 3.820342710366617*^9}, { 3.820342935130333*^9, 3.820342954631484*^9}, {3.8203429969742928`*^9, 3.820343001387989*^9}, {3.820343450595797*^9, 3.8203434962099524`*^9}, { 3.82133799284402*^9, 3.821337995589003*^9}, 3.821338037042634*^9, { 3.821338104274653*^9, 3.821338118749104*^9}, 3.8213382124094257`*^9, { 3.821338261882592*^9, 3.821338269905648*^9}, 3.821338306335579*^9, { 3.821339170488145*^9, 3.821339195906999*^9}}, CellLabel->"In[1]:=",ExpressionUUID->"ea488723-95d0-4f83-bdb4-9046c033fe06"] }, Open ]], Cell[CellGroupData[{ Cell["Accelerated Convergence", "Title", CellChangeTimes->{{3.759060056578732*^9, 3.7590600688772907`*^9}},ExpressionUUID->"e9931230-8976-4e0a-a11e-\ d1ba4998089d"], Cell[CellGroupData[{ Cell["Richardson Extrapolation", "Section", CellChangeTimes->{{3.822594159158592*^9, 3.8225941855066032`*^9}},ExpressionUUID->"d0cc536c-aa87-4084-80f3-\ 4efe06191168"], Cell[TextData[{ "Suppose that you have to evaluate a sum like \n", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["\[Sum]", RowBox[{"k", "=", "1"}]], "\[Infinity]"], TraditionalForm]], ExpressionUUID->"ec0ee5f6-2c9e-486c-a8b9-9c2c6638837c"], " ", Cell[BoxData[ FormBox[ FractionBox["1", SuperscriptBox["k", "2"]], TraditionalForm]],ExpressionUUID-> "502ffeea-a6aa-45e3-9d79-8d9fed05b9c3"], "\nWe forget that we know that the sum is ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Pi]", "2"], "/", "6"}], TraditionalForm]], ExpressionUUID->"e0ddf877-0ab0-4dd2-9bd2-fa2de8947168"], " . Let ", Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "72e3ee65-e23b-4344-9f0f-39e7bd4fdf3d"], " denote the partial sum \n", Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "2b99c1cb-fed5-438d-b594-1b63e89bdd11"], " = ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["\[Sum]", RowBox[{"k", "=", "1"}]], "n"], TraditionalForm]],ExpressionUUID-> "38c62092-e46a-4706-a073-292d62f0f9f2"], " ", Cell[BoxData[ FormBox[ FractionBox["1", SuperscriptBox["k", "2"]], TraditionalForm]],ExpressionUUID-> "8bb82331-73ae-4b30-bada-1fefb4d1c40b"], " .\nThen \n", Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "a2c421b7-d874-40e2-833f-c84d9a9a4fa4"], " = ", Cell[BoxData[ FormBox[ SubscriptBox["s", RowBox[{"n", "-", "1"}]], TraditionalForm]],ExpressionUUID-> "8f198ca6-5698-455c-b27f-1a6fd57fb94c"], "+ ", Cell[BoxData[ FormBox[ FractionBox["1", SuperscriptBox["n", "2"]], TraditionalForm]],ExpressionUUID-> "d229bd12-5643-4173-b75f-b289b042d60c"], " ; ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["s", "0"], " ", "=", " ", RowBox[{"0", " ", "."}]}], TraditionalForm]],ExpressionUUID-> "52b3c485-e5d1-48fa-9443-c4d7c39d4274"], "\nThe traditional way to compute the sum is to iterate the recurrence \ relation and stop when, say, \n", Cell[BoxData[ FormBox[ RowBox[{"Abs", "[", " ", SubscriptBox["s", RowBox[{"n", " "}]]}], TraditionalForm]],ExpressionUUID-> "3b3c9641-eaa7-4810-bb54-c4fb118573bd"], "- ", Cell[BoxData[ FormBox[ SubscriptBox["s", RowBox[{"n", "-", "1", " "}]], TraditionalForm]],ExpressionUUID-> "fbe5076f-58da-450b-9259-78cef5fb2eac"], "] < ", Cell[BoxData[ FormBox[ RowBox[{" ", SuperscriptBox["10", RowBox[{"-", "8"}]]}], TraditionalForm]],ExpressionUUID-> "e32890a3-ecb0-4f21-a11c-217e19e25880"], " , so when n = ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "4"], TraditionalForm]],ExpressionUUID-> "ba8fc94c-dfd6-4bf9-a875-28a70d48f801"], ". \nThis is very bad since the error after n terms is\n", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Epsilon]", "n"], "~"}], TraditionalForm]],ExpressionUUID-> "8132d140-49f7-446d-863f-58bfd42cec58"], " ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["\[Sum]", RowBox[{"k", "=", RowBox[{"n", "+", "1"}]}]], "\[Infinity]"], TraditionalForm]], ExpressionUUID->"dbb5ebca-5ae8-4600-81a6-423075e2d0ef"], " ", Cell[BoxData[ FormBox[ FractionBox["1", SuperscriptBox["k", "2"]], TraditionalForm]],ExpressionUUID-> "1291a865-9ebd-40df-bbc0-833bf0374aef"], " ~ ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", RowBox[{"n", "+", "1"}]], "\[Infinity]"], " ", FractionBox[ RowBox[{"\[DifferentialD]", "k"}], SuperscriptBox["k", "2"]]}], TraditionalForm]],ExpressionUUID-> "125cfa08-bc54-49b3-8e3d-e108784f2f31"], " ~ ", Cell[BoxData[ FormBox[ FractionBox["1", "n"], TraditionalForm]],ExpressionUUID-> "623e643e-3c5a-43ed-90bf-f6bcc3d59454"], "\nso the error is ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["10", RowBox[{"-", "4"}]], " ", "not", " ", SuperscriptBox["10", RowBox[{"-", "8"}]]}], TraditionalForm]],ExpressionUUID-> "a3d7532c-32fe-4c07-b412-dd118660845d"], ". This is an example of a slowly convergent series." }], "Text", CellChangeTimes->{{3.8225942441234617`*^9, 3.822594351652936*^9}, { 3.8225943888651447`*^9, 3.8225944200582857`*^9}, {3.8225944798155947`*^9, 3.822594481482918*^9}, {3.822594522193264*^9, 3.8225945362104893`*^9}, { 3.822594575021371*^9, 3.822594740836803*^9}, {3.8225947716371593`*^9, 3.82259483197294*^9}, {3.822594888857998*^9, 3.82259523644738*^9}, { 3.822595481724609*^9, 3.822595697899536*^9}, {3.822596476950296*^9, 3.8225964919719877`*^9}},ExpressionUUID->"604dd131-d176-4bfe-87ad-\ 63f6ca430477"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"N", "[", RowBox[{"1", "/", SuperscriptBox["k", "2"]}], "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", SuperscriptBox["10", "4"]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.8225965461060266`*^9, 3.8225965861176147`*^9}, { 3.822596631409012*^9, 3.822596636882573*^9}}, CellLabel->"In[2]:=",ExpressionUUID->"1b6bea4b-4f00-4d34-8d5c-d3cbbdb41b8b"], Cell[BoxData["1.6448340718480652`"], "Output", CellChangeTimes->{3.8540165752195168`*^9}, CellLabel->"Out[2]=",ExpressionUUID->"dfddc673-d508-4836-9cc8-56a1dd17f973"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{ SuperscriptBox["\[Pi]", "2"], "/", "6"}], "]"}]], "Input", CellChangeTimes->{{3.8225966500186234`*^9, 3.822596668401388*^9}}, CellLabel->"In[3]:=",ExpressionUUID->"0f3d8da6-4af7-4e6d-8b36-890d5f03cc00"], Cell[BoxData["1.6449340668482262`"], "Output", CellChangeTimes->{3.854016587610921*^9}, CellLabel->"Out[3]=",ExpressionUUID->"91c1044d-ca54-4720-b3d5-7a93b35b2c7b"] }, Open ]], Cell[TextData[{ "The problem here is that\n\n", Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "487271fb-f1e6-45c1-81cf-675686228821"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Alpha]", "0"], " ", "+", " ", FractionBox[ SubscriptBox["\[Alpha]", "1"], "n"]}], TraditionalForm]],ExpressionUUID-> "8e0f7a2b-b122-429f-9397-a0cf140e37c3"], " + ...\n\nWe want to evaluate ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", "0"], TraditionalForm]],ExpressionUUID-> "5bde52e4-1354-4086-bcc5-e23e50dec7ab"], ". To start truncate the \[Alpha] - series after ", Cell[BoxData[ FormBox[ SubscriptBox["\[Alpha]", RowBox[{"1", " "}]], TraditionalForm]],ExpressionUUID-> "08a94ddf-69df-4105-ad33-b6e33c392f92"], "and write\n\nn ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["s", "n"], " ", "=", " ", "n"}], TraditionalForm]],ExpressionUUID->"c1f9adec-6240-4b75-8710-fa2d694666d3"], " ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Alpha]", "0"], " "}]],ExpressionUUID-> "5fafc1cd-c4af-4dd2-b8a5-573845b0c832"], "+ ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Alpha]", "1"], " "}]],ExpressionUUID-> "394f525c-01dd-4d24-bb4a-cfd8b65102d2"], "\n(n-1) ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["s", RowBox[{"n", "-", "1"}]], " ", "=", " ", RowBox[{"(", "n"}]}], TraditionalForm]],ExpressionUUID-> "9c8ea5fd-35e2-489d-a209-ea2c3b2f3099"], "-1) ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Alpha]", "0"], " "}]],ExpressionUUID-> "af8ca8e2-1c6b-4484-990b-f4fda8e26fc6"], "+ ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Alpha]", 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"Text", CellChangeTimes->{{3.82259724278689*^9, 3.8225972926437674`*^9}, { 3.822597339830559*^9, 3.82259741683107*^9}, 3.82259747486524*^9},ExpressionUUID->"09b42c45-56f7-4397-bcf6-\ 7240d59b502b"], Cell[TextData[{ "and the error here is of order ", Cell[BoxData[ FormBox[ SuperscriptBox["n", RowBox[{"-", "2"}]], TraditionalForm]],ExpressionUUID-> "6b1a6409-1a97-4bc8-9c1b-248281bbec55"], "." }], "Text", CellChangeTimes->{{3.822597420757943*^9, 3.8225974518417683`*^9}},ExpressionUUID->"a90b2037-181a-4f88-ad22-\ fbbb6a6de05b"], Cell[TextData[{ "\nLet us seek to go further, write\n\n", Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "3988474d-ff3d-4905-aa2a-069648ac545b"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Alpha]", "0"], " ", "+"}], TraditionalForm]], ExpressionUUID->"592370e8-f52c-48f0-b401-83e8b22a0033"], " ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "1"], "n"], TraditionalForm]],ExpressionUUID-> "4812baae-957e-4fdb-9a41-86927f584765"], " + ..... + ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "k"], SuperscriptBox["n", "k"]], TraditionalForm]],ExpressionUUID-> "42efcd3e-1024-448f-ad67-7a2a784b0a99"], "\n", Cell[BoxData[ FormBox[ SubscriptBox["s", RowBox[{"n", "-", "1"}]], TraditionalForm]],ExpressionUUID-> "2ad6385a-1c54-4ef7-8d9e-144443af0313"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Alpha]", "0"], "+"}], TraditionalForm]],ExpressionUUID-> "ee32856a-4d3d-46b3-b702-a1bc47806ed5"], " ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "1"], RowBox[{"n", "-", "1"}]], TraditionalForm]],ExpressionUUID-> "a089307c-c757-4217-b1e2-3de1308121fd"], " + ..... + ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "k"], SuperscriptBox[ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "k"]], TraditionalForm]],ExpressionUUID-> "2215e1b5-8add-40c7-aecc-ac3f0bf2978f"], "\n.........\n", Cell[BoxData[ FormBox[ SubscriptBox["s", RowBox[{"n", "-", "k"}]], TraditionalForm]],ExpressionUUID-> "fea29b1a-105c-4488-ad7e-e2c32a9e56b5"], " = ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Alpha]", "0"], "+"}], TraditionalForm]],ExpressionUUID-> "2f04f83f-0b43-4b98-8437-a87926996557"], " ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "1"], RowBox[{"n", "-", "k"}]], TraditionalForm]],ExpressionUUID-> "3833f0fb-4052-4f4d-856f-c0af3d3f9e35"], " + ..... + ", Cell[BoxData[ FormBox[ FractionBox[ SubscriptBox["\[Alpha]", "k"], SuperscriptBox[ RowBox[{"(", RowBox[{"n", "-", "k"}], ")"}], "k"]], TraditionalForm]],ExpressionUUID-> "71818d0b-a3eb-4c28-9ee1-1333f9650afc"], "\n\nmultiply up\n\n ", Cell[BoxData[ FormBox[ SuperscriptBox["n", "k"], TraditionalForm]],ExpressionUUID-> "de2c76ee-f250-43e4-88d7-730695576762"], Cell[BoxData[ FormBox[ SubscriptBox["s", "n"], TraditionalForm]],ExpressionUUID-> "6bb8a678-571b-49a5-b0f1-67072c73ba33"], " = ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{ SuperscriptBox["n", "k"], 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