(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 232876, 4729]*) (*NotebookOutlinePosition[ 233584, 4753]*) (* CellTagsIndexPosition[ 233540, 4749]*) (*WindowFrame->Normal*) Notebook[{ Cell["Math 221 - Numerical PDE's, Lab 1", "Subtitle"], Cell[TextData[{ "Several software packages are very useful in developing and analyzing \ numerical methods for PDE's. Today we'll use ", StyleBox["Mathematica", FontSlant->"Italic"], " to derive and test out finite difference approximations of derivatives" }], "Text"], Cell[BoxData[ \(\(Off[General::spell1];\)\)], "Input"], Cell[CellGroupData[{ Cell["1. Many ways to find finite difference formulas", "Section"], Cell[CellGroupData[{ Cell["Using geometrical reasoning", "Subsection"], Cell["The three formulas inspired by graphical observation are:", "Text"], Cell[BoxData[ \(\(u'\)[t] = \(u[t + k] - u[t]\)\/k, \ forward\ difference\ formula\)], "DisplayFormula"], Cell[BoxData[ \(\(u'\)[t] = \(u[t] - u[t - k]\)\/k, \ backward\ difference\ formula\)], "DisplayFormula"], Cell[BoxData[ \(\(u'\)[t] = \(u[t + k] - u[t - k]\)\/\(2 k\), \ centered\ difference\ formula\)], "DisplayFormula"], Cell["\<\ We can see how each approximates the tangent slope by defining a \ function u(t) and drawing the corresponding lines\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(u[t_] = t\^3;\)\), "\[IndentingNewLine]", \(k = 0.5; \ t0 = 1.0;\), "\[IndentingNewLine]", \(\(pleft = {t0 - k, u[t0 - k]};\)\), "\[IndentingNewLine]", \(\(p0 = {t0, u[t0]};\)\), "\[IndentingNewLine]", \(\(pright = {t0 + k, u[t0 + k]};\)\), "\[IndentingNewLine]", \(\(ptanl = {t0 - 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o`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?oo o`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0oooo00<000000?oo o`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00@0oooo00<000000?oo o`3oool0103oool000h0ooooo`00000=000000H0oooo0000\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {0.416134, -0.901648, \ 0.00394232, 0.0247178}}] }, Open ]], Cell["\<\ The centered approximation obviously has a better chance of \ approximating the correct slope, i.e. to be parallel to the tangent. The left \ and right approximations will always have an error since they intersect the \ tangent at (t0,u(t0)).\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Method of undetermined coefficients", "Subsection"], Cell["We seek a formula of the form", "Text"], Cell[BoxData[ \(\(u'\)[ t] = \(1\/k\) \((a\ u[t + 2 k] + b\ u[t + k] + c\ u[t - k]\ + \ d\ u[t - 2 k])\)\)], "DisplayFormula"], Cell["\<\ Define a Mathematica expression for the general form of the above \ approximation\ \>", "Text"], Cell[BoxData[{ \(\(Clear[u, k];\)\), "\[IndentingNewLine]", \(\(f[t_, k_, n1_, n2_] := \(1\/k\) Sum[a[j]\ u[t + j\ k], {j, n1, n2}];\)\)}], "Input"], Cell["\<\ Examples of the above definition: 1. The formula asked for in this problem\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t, k, \(-2\), 2]\ /. \ a[0] \[Rule] 0\)], "Input"], Cell[BoxData[ \(\(a[\(-2\)]\ u[\(-2\)\ k + t] + a[\(-1\)]\ u[\(-k\) + t] + a[1]\ u[k + \ t] + a[2]\ u[2\ k + t]\)\/k\)], "Output"] }, Open ]], Cell["\<\ 2. The centered, second order formula we derived by graphical \ reasoning\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f[t, k, \(-1\), 1]\ /. \ a[0] \[Rule] 0\)], "Input"], Cell[BoxData[ \(\(a[\(-1\)]\ u[\(-k\) + t] + a[1]\ u[k + t]\)\/k\)], "Output"] }, Open ]], Cell["\<\ To find the coefficients a[j] we must carry out: 1. A Taylor series expansion\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(expansion = Simplify[Series[ f[t, k, \(-2\), 2]\ /. \ a[0] \[Rule] 0, {k, 0, 4}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{\(\(\((a[\(-2\)] + a[\(-1\)] + a[1] + a[2])\)\ u[t]\)\/k\), "+", RowBox[{\((\(-2\)\ a[\(-2\)] - a[\(-1\)] + a[1] + 2\ a[2])\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "t", "]"}]}], "+", RowBox[{\(1\/2\), " ", \((4\ a[\(-2\)] + a[\(-1\)] + a[1] + 4\ a[2])\), " ", RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}], " ", "k"}], "+", RowBox[{\(1\/6\), " ", \((\(-8\)\ a[\(-2\)] - a[\(-1\)] + a[1] + 8\ a[2])\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((3)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^2\)}], "+", RowBox[{\(1\/24\), " ", \((16\ a[\(-2\)] + a[\(-1\)] + a[1] + 16\ a[2])\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^3\)}], "+", RowBox[{\(1\/120\), " ", \((\(-32\)\ a[\(-2\)] - a[\(-1\)] + a[1] + 32\ a[2])\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^4\)}], "+", InterpretationBox[\(O[k]\^5\), SeriesData[ k, 0, {}, -1, 5, 1]]}], SeriesData[ k, 0, { Times[ Plus[ a[ -2], a[ -1], a[ 1], a[ 2]], u[ t]], Times[ Plus[ Times[ -2, a[ -2]], Times[ -1, a[ -1]], a[ 1], Times[ 2, a[ 2]]], Derivative[ 1][ u][ t]], Times[ Rational[ 1, 2], Plus[ Times[ 4, a[ -2]], a[ -1], a[ 1], Times[ 4, a[ 2]]], Derivative[ 2][ u][ t]], Times[ Rational[ 1, 6], Plus[ Times[ -8, a[ -2]], Times[ -1, a[ -1]], a[ 1], Times[ 8, a[ 2]]], Derivative[ 3][ u][ t]], Times[ Rational[ 1, 24], Plus[ Times[ 16, a[ -2]], a[ -1], a[ 1], Times[ 16, a[ 2]]], Derivative[ 4][ u][ t]], Times[ Rational[ 1, 120], Plus[ Times[ -32, a[ -2]], Times[ -1, a[ -1]], a[ 1], Times[ 32, a[ 2]]], Derivative[ 5][ u][ t]]}, -1, 5, 1]]], "Output"] }, Open ]], Cell["\<\ 2. Get the coefficients of the successive derivatives of \ u[t]\ \>", "Text"], Cell[BoxData[ \(c[j_] := Coefficient[expansion, D[u[t], {t, j}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(coefs = Table[c[j], {j, 0, 3}]\)], "Input"], Cell[BoxData[ \({\(a[\(-2\)] + a[\(-1\)] + a[1] + a[2]\)\/k, \(-2\)\ a[\(-2\)] - a[\(-1\)] + a[1] + 2\ a[2], 1\/2\ k\ \((4\ a[\(-2\)] + a[\(-1\)] + a[1] + 4\ a[2])\), 1\/6\ k\^2\ \((\(-8\)\ a[\(-2\)] - a[\(-1\)] + a[1] + 8\ a[2])\)}\)], "Output"] }, Open ]], Cell["\<\ 3. Set the coefficient of u'[t]=1 and all the others as 0 to get a \ linear system for the a[j]'s. Note that we only have four coefficients at our \ disposal so we can only impose four conditions.\ \>", "Text"], Cell[BoxData[ \(\(rhs = {0, 1, 0, 0};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(sol = Solve[coefs \[Equal] rhs, {a[\(-2\)], a[\(-1\)], a[1], a[2]}]\)], "Input"], Cell[BoxData[ \({{a[\(-2\)] \[Rule] 1\/12, a[\(-1\)] \[Rule] \(-\(2\/3\)\), a[1] \[Rule] 2\/3, a[2] \[Rule] \(-\(1\/12\)\)}}\)], "Output"] }, Open ]], Cell[TextData[{ "Note that even though we did not impose the condition that the O(", Cell[BoxData[ \(TraditionalForm\`k\^3\)]], ") term be zero, it turns out that it is null" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(c[3]\ /. \ sol\)], "Input"], Cell[BoxData[ \({0}\)], "Output"] }, Open ]], Cell["\<\ This is a benefit of the centered formula: we get one more order of \ precision than we would expect from the number of coefficients at our \ disposal. The same thing happened when we looked at the geometrical \ approximations previously.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Polynomial interpolation", "Subsection"], Cell["\<\ Let's define our forward and backward finite difference operators\ \ \>", "Text"], Cell[BoxData[{ \(dforward[u_[t_]] := u[t + k] - u[t]\), "\[IndentingNewLine]", \(dbackward[u_[t_]] := u[t] - u[t - k]\)}], "Input"], Cell["\<\ We'll also define these operators with a second argument denoting \ the number of times it gets applied\ \>", "Text"], Cell[BoxData[{ \(dforward[u_[t_], n_] := If[n \[Equal] 1, \ dforward[u[t]], \ dforward[u[t + k] - u[t], n - 1]]\), "\[IndentingNewLine]", \(\(dforward[a_\ u_[t_], n_]\ = \ a\ dforward[u[t], n];\)\ \), "\[IndentingNewLine]", \(\(dforward[a_\ + \ b_, n_] = dforward[a, n] + dforward[b, n];\)\), "\[IndentingNewLine]", \(dbackward[u_[t_], n_] := If[n \[Equal] 1, \ dbackward[u[t]], \ dbackward[u[t] - u[t - k], n - 1]]\), "\[IndentingNewLine]", \(\(dbackward[a_\ u_[t_], n_] = a\ dbackward[u[t], n];\)\), "\[IndentingNewLine]", \(\(dbackward[a_\ + \ b_, n_] = dbackward[a, n] + dbackward[b, n];\)\)}], "Input"], Cell["We can check that the operators give the expected answers", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(dforward[u[t], 4]\)], "Input"], Cell[BoxData[ \(u[t] - 4\ u[k + t] + 6\ u[2\ k + t] - 4\ u[3\ k + t] + u[4\ k + t]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(dbackward[u[t], 5]\)], "Input"], Cell[BoxData[ \(u[t] - u[\(-5\)\ k + t] + 5\ u[\(-4\)\ k + t] - 10\ u[\(-3\)\ k + t] + 10\ u[\(-2\)\ k + t] - 5\ u[\(-k\) + t]\)], "Output"] }, Open ]], Cell["The operator series approximating the first derivatives are", "Text"], Cell[BoxData[{ \(s1forward[u_[t_], n_] := Simplify[Sum[dforward[u[t], j]\ \((\(-1\))\)\^\(j + 1\)/j, {j, 1, n}]/ k]\), "\[IndentingNewLine]", \(s1backward[u_[t_], n_] := Simplify[Sum[dbackward[u[t], j]\ /j, {j, 1, n}]/k]\)}], "Input"], Cell["And those approximating the second derivatives are", "Text"], Cell[BoxData[{ \(s2forwardcoef[l_] := Coefficient[Sum[\(\((\(-1\))\)\^\(j + 1\)\) x\^j/j, {j, 1, l}]^2, x\^l]\), "\[IndentingNewLine]", \(s2backwardcoef[l_] := Coefficient[Sum[x\^j/j, {j, 1, l}]^2, x\^l]\), "\[IndentingNewLine]", \(s2forward[u_[t_], n_] := Simplify[Sum[\ s2forwardcoef[j]\ dforward[u[t], j], {j, 2, n + 1}]/ k\^2]\), "\[IndentingNewLine]", \(s2backward[u_[t_], n_] := Simplify[Sum[\ s2backwardcoef[j]\ dbackward[u[t], j], {j, 2, n + 1}]/ k\^2]\)}], "Input"], Cell["The formulas we're after are", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f31 = s1forward[u[t], 3]\)], "Input"], Cell[BoxData[ \(\(\(-11\)\ u[t] + 18\ u[k + t] - 9\ u[2\ k + t] + 2\ u[3\ k + t]\)\/\(6\ \ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b31 = s1backward[u[t], 3]\)], "Input"], Cell[BoxData[ \(\(11\ u[t] - 2\ u[\(-3\)\ k + t] + 9\ u[\(-2\)\ k + t] - 18\ u[\(-k\) + \ t]\)\/\(6\ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f32 = s2forward[u[t], 3]\)], "Input"], Cell[BoxData[ \(\(35\ u[t] - 104\ u[k + t] + 114\ u[2\ k + t] - 56\ u[3\ k + t] + 11\ \ u[4\ k + t]\)\/\(12\ k\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b32 = s2backward[u[t], 3]\)], "Input"], Cell[BoxData[ \(\(35\ u[t] + 11\ u[\(-4\)\ k + t] - 56\ u[\(-3\)\ k + t] + 114\ \ u[\(-2\)\ k + t] - 104\ u[\(-k\) + t]\)\/\(12\ k\^2\)\)], "Output"] }, Open ]], Cell["\<\ We can check that the formulas are correct by carrying out Taylor \ series expansions\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[f31, {k, 0, 3}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{\(1\/4\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^3\)}], "+", InterpretationBox[\(O[k]\^4\), SeriesData[ k, 0, {}, 0, 4, 1]]}], SeriesData[ k, 0, { Derivative[ 1][ u][ t], 0, 0, Times[ Rational[ 1, 4], Derivative[ 4][ u][ t]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[b31, {k, 0, 3}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "-", RowBox[{\(1\/4\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((4)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^3\)}], "+", InterpretationBox[\(O[k]\^4\), SeriesData[ k, 0, {}, 0, 4, 1]]}], SeriesData[ k, 0, { Derivative[ 1][ u][ t], 0, 0, Times[ Rational[ -1, 4], Derivative[ 4][ u][ t]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[f32, {k, 0, 3}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{\(5\/6\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^3\)}], "+", InterpretationBox[\(O[k]\^4\), SeriesData[ k, 0, {}, 0, 4, 1]]}], SeriesData[ k, 0, { Derivative[ 2][ u][ t], 0, 0, Times[ Rational[ 5, 6], Derivative[ 5][ u][ t]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Series[b32, {k, 0, 3}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]\[Prime]", MultilineFunction->None], "[", "t", "]"}], "-", RowBox[{\(5\/6\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((5)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^3\)}], "+", InterpretationBox[\(O[k]\^4\), SeriesData[ k, 0, {}, 0, 4, 1]]}], SeriesData[ k, 0, { Derivative[ 2][ u][ t], 0, 0, Times[ Rational[ -5, 6], Derivative[ 5][ u][ t]]}, 0, 4, 1]]], "Output"] }, Open ]], Cell[BoxData[ \(\(\(Note\)\(\ \)\(that\)\(\ \)\(in\)\(\ \)\(order\)\(\ \)\(to\)\(\ \ \)\(get\)\(\ \)\(O\) \((k\^n)\)\(\ \)\(precision\)\(\ \)\(in\)\(\ \)\(the\)\(\ \ \ \)\(formulas\)\(\ \)\(for\)\(\ \)\(the\)\(\ \)\(second\)\(\ \ \)\(derivative\)\(\ \)\(we\)\(\ \)\(must\)\(\ \)\(include\)\(\ \)\(n\)\(\ \ \)\(terms\)\(\ \)\(of\)\(\ \)\(the\)\(\ \)\(expansion\)\(\ \)\((as\ opposed\ \ to\ truncating\ the\ expansion\ at\ the\ power\ n\ of\ the\ delta\ \ operator)\)\(\ \)\)\)], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2. Using symbolic packages to determine high order formulas", "Section"], Cell["\<\ The operator technique is more amenable to programming on a \ computer. For computation by hand the choice is a matter of personal \ preference. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f6 = Simplify[\((s1forward[u[t], 5] + s1backward[u[t], 5])\)/2]\)], "Input"], Cell[BoxData[ \(\(\(1\/\(120\ k\)\)\((\(-12\)\ u[\(-5\)\ k + t] + 75\ u[\(-4\)\ k + t] - 200\ u[\(-3\)\ k + t] + 300\ u[\(-2\)\ k + t] - 300\ u[\(-k\) + t] + 300\ u[k + t] - 300\ u[2\ k + t] + 200\ u[3\ k + t] - 75\ u[4\ k + t] + 12\ u[5\ k + t])\)\)\)], "Output"] }, Open ]], Cell["\<\ In the above, I've taken advantage of the fact that the centered \ appproximation gives an added order of precision, so if we're looking for a \ sixth order formula the expansion may be carried out only to five terms. The \ central approximation is the average of the forward and backward \ approximations. We can check the order of precision achieved.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Series[f6, {k, 0, 6}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ RowBox[{ SuperscriptBox["u", "\[Prime]", MultilineFunction->None], "[", "t", "]"}], "+", RowBox[{\(5\/14\), " ", RowBox[{ SuperscriptBox["u", TagBox[\((7)\), Derivative], MultilineFunction->None], "[", "t", "]"}], " ", \(k\^6\)}], "+", InterpretationBox[\(O[k]\^7\), SeriesData[ k, 0, {}, 0, 7, 1]]}], SeriesData[ k, 0, { Derivative[ 1][ u][ t], 0, 0, 0, 0, 0, Times[ Rational[ 5, 14], Derivative[ 7][ u][ t]]}, 0, 7, 1]]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["3. Testing convergence", "Section"], Cell["Some more formulas", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f11 = s1forward[u[t], 1]\)], "Input"], Cell[BoxData[ \(\(\(-u[t]\) + u[k + t]\)\/k\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b11 = s1backward[u[t], 1]\)], "Input"], Cell[BoxData[ \(\(u[t] - u[\(-k\) + t]\)\/k\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(c11 = Simplify[\((f11 + b11)\)/2]\)], "Input"], Cell[BoxData[ \(\(\(-u[\(-k\) + t]\) + u[k + t]\)\/\(2\ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(f21 = s1forward[u[t], 2]\)], "Input"], Cell[BoxData[ \(\(-\(\(3\ u[t] - 4\ u[k + t] + u[2\ k + t]\)\/\(2\ k\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b21 = s1backward[u[t], 2]\)], "Input"], Cell[BoxData[ \(\(3\ u[t] + u[\(-2\)\ k + t] - 4\ u[\(-k\) + t]\)\/\(2\ k\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(u[t_] = Sin[t^Cos[t]];\)\), "\[IndentingNewLine]", \(upexact = 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", StyleBox["Mathematica", FontSlant->"Italic"], " allows us however to increase the numerical precision with which we're \ working. Let's make all computations have 30 decimal digits of precision and \ redo the computation." }], "Text"], Cell[BoxData[{ \(eps[f_] := N[Log[10, Abs[\((N[f, 30] - upexact)\)/upexact]], 30]\), "\[IndentingNewLine]", \(\(relerrf11 = Table[{\(-p\), eps[f11\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\), "\[IndentingNewLine]", \(\(relerrb11 = Table[{\(-p\), eps[b11\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\), "\[IndentingNewLine]", \(\(relerrc11 = Table[{\(-p\), eps[c11\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\), "\[IndentingNewLine]", \(\(relerrf31 = Table[{\(-p\), eps[f31\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\), "\[IndentingNewLine]", \(\(relerrb31 = Table[{\(-p\), eps[b31\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\), "\[IndentingNewLine]", \(\(relerrf6 = Table[{\(-p\), eps[f6\ /. \ {t \[Rule] 2, k \[Rule] 10\^\(-p\)}]}, {p, 1, 10}];\)\)}], "Input"], Cell[BoxData[{ \(\(f11plot = 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Cell["\<\ We see that the relative error now decreases smoothly. If we were \ to go to even smaller k values the resolution again would not be enough and \ we would get an increase in the error. Let's also verify the predicted order \ of convergence by carrying out a linear regression analysis on the \ data.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Fit[relerrf11, {1, logk}, logk]\), "\[IndentingNewLine]", \(Fit[relerrb11, {1, logk}, logk]\)}], "Input"], Cell[BoxData[ \(\(-0.8207107904589627`\) + 0.9892479297227588`\ logk\)], "Output"], Cell[BoxData[ \(\(-0.6835728005935584`\) + 1.007834547892648`\ logk\)], "Output"] }, Open ]], Cell["\<\ The first order formulas indeed have the coefficient of log(k) \ close to 1\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Fit[Take[relerrc11, 9], {1, logk}, logk]\)], "Input"], Cell[BoxData[ \(\(-0.19740677924080452`\) + 1.9998489612349353`\ logk\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(Fit[Take[relerrf31, 6], {1, logk}, logk]\), "\[IndentingNewLine]", \(Fit[Take[relerrb31, 6], {1, logk}, logk]\)}], "Input"], Cell[BoxData[ \(\(-0.008795354962615036`\) + 3.0878801020565896`\ logk\)], "Output"], Cell[BoxData[ \(\(-0.2569444607616427`\) + 3.0397593284563555`\ logk\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Fit[Take[relerrf6, 3], {1, logk}, logk]\)], "Input"], Cell[BoxData[ \(\(\(2.4294155069926577`\)\(\[InvisibleSpace]\)\) + 5.979573743408687`\ logk\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["4. 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