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Cell["Seminar Programovani v systemu Mathematica", "Subsubtitle",
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Cell["Praha, CVUT, zari - rijen 2003 ", "Text",
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Cell["\<\
Pozor na podstatny rozdil mezi \"1.0\" a \"1\". Ve vstupnich datech \
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\>", "Text"]
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Cell["Slozene vyrazy", "Subsubsection"],

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Derivative[1] je funkce Derivative v bode 1, jejimz argumentem je funkce f, \
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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Mistroke
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.78 .58908 L
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Mistroke
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Mfstroke
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.145 .5744 L
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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Mistroke
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.70438 .08773 L
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Mistroke
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Mistroke
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.96297 .15395 L
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.97211 .42135 L
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Cell["Priklad - definice logaritmu ", "Text"],

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Cell[TextData[{
  "Zde je pricina, proc neni rozvinut i ",
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Cell["Pridame tedy dalsi pravidlo:", "Text"],

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Cell["Zatim nase funkce log neni \"listovatelna\":", "Text"],

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Cell["Ale tento atribut muzeme snadno pridat:", "Text"],

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Priklad definice funkce se stejnou hlavickou a ruznym poctem argumentu\
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Cell["Programovaci styly", "Subsection"],

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Cell["Uloha: vytvorit proceduru pro druhe mocniny listu", "Subsubsection"],

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Cell["Ryzi funkce ", "Text"],

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Cell["Ve zkracenem zapisu", "Text"],

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Cell["Zakladni funkce Map", "Text"],

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Cell["Ryzi funkci je samozrejme mozne pojmenovat:", "Text"],

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Alternativni vyuziti listovatelnosti (1. element 1. listu umocnen na 1. \
element 2. listu ...)\
\>", "Text"],

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Cell["Rekursivni styl ", "Subsubsection"],

Cell["\<\
Kazdy list je bud prazdny nebo ma prvni element a zbytek - ten zbytek muze \
byt prazdny list.\
\>", "Text"],

Cell["Klicova funkce pro tuto proceduru:", "Text"],

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Predlohy: x_ prave jeden argument, x__ jeden a vice argumentu, x___ nula a \
vice argumentu\
\>", "Text"],

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_  nebo ekvivalentne Blank[] ... predloha, ktere vyhovuje vse (any expression \
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Tvar: predloha/; podminka. Nejdrive se vyhledaji vyrazy vyhovujici predloze a \
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          {GridBox[{
                {\(fibB[1] = 1\)},
                {" "},
                {\(fibB[2] = 1\)},
                {" "},
                {\(fibB[n_Integer?Positive] := \(fibB[n] = 
                      fibB[n - 1] + fibB[n - 2]\)\)}
                },
              GridBaseline->{Baseline, {1, 1}},
              ColumnWidths->0.999,
              ColumnAlignments->{Left}]}
          },
        GridBaseline->{Baseline, {1, 1}},
        ColumnAlignments->{Left}],
      Definition[ "fibB"],
      Editable->False]], "Print",
  CellTags->"Info3274076305-8162128"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[fibB[n], {n, 5}]\)], "Input"],

Cell[BoxData[
    \({1, 1, 2, 3, 5}\)], "Output"]
}, Open  ]],

Cell["\<\
Narocnost na pamet je zanedbatelna. Vyzkousejte sina vice hodnot a pote \
zkuste\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(MaxMemoryUsed[]\)], "Input"],

Cell[BoxData[
    \(3302312\)], "Output"]
}, Open  ]],

Cell["Vsimneme si nyni na zapamatovanych hodnot:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?fibB\)\)], "Input"],

Cell["Global`fibB", "Print",
  CellTags->"Info3274076359-7528684"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {GridBox[{
                {\(fibB[1] = 1\)},
                {" "},
                {\(fibB[2] = 1\)},
                {" "},
                {\(fibB[3] = 2\)},
                {" "},
                {\(fibB[4] = 3\)},
                {" "},
                {\(fibB[5] = 5\)},
                {" "},
                {\(fibB[n_Integer?Positive] := \(fibB[n] = 
                      fibB[n - 1] + fibB[n - 2]\)\)}
                },
              GridBaseline->{Baseline, {1, 1}},
              ColumnWidths->0.999,
              ColumnAlignments->{Left}]}
          },
        GridBaseline->{Baseline, {1, 1}},
        ColumnAlignments->{Left}],
      Definition[ "fibB"],
      Editable->False]], "Print",
  CellTags->"Info3274076359-7528684"]
}, Open  ]],

Cell["\<\
Priklad 4 (Programovani zalozene na pravidlech, vlastni definice linearni \
funkce)\
\>", "Text"],

Cell[BoxData[{
    \(\(Clear[f];\)\), "\[IndentingNewLine]", 
    \(f[c_?NumericQ*x_] := c*f[x]\), "\[IndentingNewLine]", 
    \(f[x_ + y_] := f[x] + f[y]\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[\[Pi] + \(\@2\) x + 3*a*x\^2]\)], "Input"],

Cell[BoxData[
    \(f[\[Pi]] + \@2\ f[x] + 3\ f[a\ x\^2]\)], "Output"]
}, Open  ]],

Cell["To je v poradku, ale  v poradku neni ", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(f[2*\[Pi]]\)], "Input"],

Cell[BoxData[
    \(\[Pi]\ f[2]\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "Odstranime tim, ze za pivotni hodnotu budeme uvazovat ",
  Cell[BoxData[
      \(TraditionalForm\`f[1]\)]],
  "."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(f[c_?NumericQ] /; c =!= 1 := c*f[1]\), "\[IndentingNewLine]", 
    \({f[1], f[2*\[Pi]]}\)}], "Input"],

Cell[BoxData[
    \({f[1], 2\ \[Pi]\ f[1]}\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Uzitecne funkce Thread a MapThread", "Subsubsection"],

Cell[BoxData[
    \(\(Clear[a, b, c];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Thread[h[a + b + c], Plus]\), "\[IndentingNewLine]", 
    \(Thread[h[a*b*c], Times]\)}], "Input"],

Cell[BoxData[
    \(h[a] + h[b] + h[c]\)], "Output"],

Cell[BoxData[
    \(h[a]\ h[b]\ h[c]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Thread[{a, b, c} \[Rule] {1, 2, 3}]\)], "Input"],

Cell[BoxData[
    \({a \[Rule] 1, b \[Rule] 2, c \[Rule] 3}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(MapThread[Equal, {{x, y, z}, {1, 2, 3}}]\)], "Input"],

Cell[BoxData[
    \({x \[Equal] 1, y \[Equal] 2, False}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Clear[z];\)\), "\[IndentingNewLine]", 
    \(MapThread[Equal, {{x, y, z}, {1, 2, 3}}]\)}], "Input"],

Cell[BoxData[
    \({x \[Equal] 1, y \[Equal] 2, z \[Equal] 3}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Solve[%, {x, y, z}]\)], "Input"],

Cell[BoxData[
    \({{x \[Rule] 1, y \[Rule] 2, z \[Rule] 3}}\)], "Output"]
}, Open  ]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Funkcionalni programovani", "Subsection"],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Opakovani ryzich funkci (pure functions)", \
"Subsubsection"]], "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[Sqrt, {a, b, c}]\)], "Input"],

Cell[BoxData[
    \({\@a, \@b, \@c}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[Function[x, x\^2], {a, b, c}]\)], "Input"],

Cell[BoxData[
    \({a\^2, b\^2, c\^2}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(#^2 &\)[3]\)], "Input"],

Cell[BoxData[
    \(9\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\((#^2 &)\)'\)\(\ \ \)\( (*\ 
      Derivace\ ryzi\ funkce\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(2\ #1 &\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(\((#1^2 + #2^2 &)\)[x, y]\)\(\ \)\( (*\ 
      Ryzi\ funkce\ dvou\ promennych\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(x\^2 + y\^2\)], "Output"]
}, Open  ]],

Cell["Funkce Map a Apply", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Map[f, {a, b, c}]\), "\[IndentingNewLine]", 
    \(Apply[f, {a, b, c}]\)}], "Input"],

Cell[BoxData[
    \({f[a], f[b], f[c]}\)], "Output"],

Cell[BoxData[
    \(f[a, b, c]\)], "Output"]
}, Open  ]],

Cell["\<\
Prvni argument obou funkci je funkce. Druhy argument je obvykle list, ale \
neni to pravidlem, viz nasledujici prikaz. Rozdil mezi obema funkcemi je ze \
Map ponecha hlavicku druheho argumentu a aplikuje prvni argument na elementy \
druheho argumentu zatimco Apply vymeni hlavicku druheho argumentu.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \({Map[f, g[a]], Apply[f, g[a]]}\)], "Input"],

Cell[BoxData[
    \({g[f[a]], f[a]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[Function[x, 1/x], e10 = {1, 2, 1/3, I}]\)], "Input"],

Cell[BoxData[
    \({1, 1\/2, 3, \(-\[ImaginaryI]\)}\)], "Output"]
}, Open  ]],

Cell["Ponekud mene prehledne, ale funkcne ekvivalentni zapis:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(1/# &\)\  /@ \ e10\)], "Input"],

Cell[BoxData[
    \({1, 1\/2, 3, \(-\[ImaginaryI]\)}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "\[WarningSign] Vyhnete se uziti funkce Sum (",
  Cell[BoxData[
      \(TraditionalForm\`\[CapitalSigma]\)]],
  ") vsude, kde je to jen mozne. V nasledujicim pripade to mozne neni:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \({\[Sum]\+\(i = 1\)\%\[Infinity] 1\/i\^2, \[Sum]\+\(i = 1\)\%\[Infinity] \
1\/i\^4}\)], "Input"],

Cell[BoxData[
    \({\[Pi]\^2\/6, \[Pi]\^4\/90}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Apply[Plus, Range[5]]\)], "Input"],

Cell[BoxData[
    \(15\)], "Output"]
}, Open  ]],

Cell["Ukazka vyuziti pri vypoctu aritmetickeho prumeru a rozptylu", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    StyleBox[\(Clear[prumer];\),
      FontColor->RGBColor[1, 0, 
      0]], "\[IndentingNewLine]", \(prumer[data_?VectorQ] := 
      Apply[Plus, data]/Length[data] /; 
        Length[data] > 0\), "\[IndentingNewLine]", \(prumer[
      Range[5]]\)}], "Input"],

Cell[BoxData[
    \(3\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(mean = prumer;\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(rozptyl[data_?VectorQ] := 
      mean[\((data - mean[data])\)\^2]\), "\[IndentingNewLine]", 
    \(rozptyl[Range[5]]\)}], "Input"],

Cell[BoxData[
    \(2\)], "Output"]
}, Open  ]],

Cell["Samozrejme funguje i pro symbolicke promenne.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(prumer[{a, b, c}]\)], "Input"],

Cell[BoxData[
    \(1\/3\ \((a + b + c)\)\)], "Output"]
}, Open  ]],

Cell["Vsimnete si zkratkoveho zapisu funkce Apply (@@):", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?prumer\)\)], "Input"],

Cell["Global`prumer", "Print",
  CellTags->"Info3274077878-2704486"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {GridBox[{
                {\(prumer[data_?VectorQ] := 
                    Plus @@ data\/Length[data] /; Length[data] > 0\)}
                },
              GridBaseline->{Baseline, {1, 1}},
              ColumnWidths->0.999,
              ColumnAlignments->{Left}]}
          },
        GridBaseline->{Baseline, {1, 1}},
        ColumnAlignments->{Left}],
      Definition[ "prumer"],
      Editable->False]], "Print",
  CellTags->"Info3274077878-2704486"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Funkce typu Nest a Fold", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Nest[f, x0, 6]\)], "Input"],

Cell[BoxData[
    \(f[f[f[f[f[f[x0]]]]]]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(NestList[f, x0, 4]\)], "Input"],

Cell[BoxData[
    \({x0, f[x0], f[f[x0]], f[f[f[x0]]], f[f[f[f[x0]]]]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(NestList[f, x0, 0]\)], "Input"],

Cell[BoxData[
    \({x0}\)], "Output"]
}, Open  ]],

Cell["\<\
Vypocet druhe odmocniny ze tri Newtonovou metodou s pocatecni hodnotou 1:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Nest[Function[x, \((x + 3/x)\)/2], 1, 5]\)], "Input"],

Cell[BoxData[
    \(708158977\/408855776\)], "Output"]
}, Open  ]],

Cell["\<\
My ale vetsinou pozadujeme numerickou hodnotu. Staci za kterymkoli cislem \
napsat tecku:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Nest[Function[x, \((x + 3/x)\)/2], 1. , 5]\)], "Input"],

Cell[BoxData[
    \(1.7320508075688774`\)], "Output"]
}, Open  ]],

Cell["\<\
V predchozim pripade bylo nutne urcit pocet iteraci. Nasledujici funkce \
umoznuje najit pevny bod zobrazeni. Presnost druheho argumentu udava, kdy se \
proces ukonci (dve po sobe jdouci iterace se lisi o mene nez je presnost \
druheho argumentu). Zde na 40 mist.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(FixedPoint[Function[x, \((x + 3/x)\)/2], N[1, 40]]\)], "Input"],

Cell[BoxData[
    \(1.732050807568877293527446341505872366942805253810380628056`39.\
999999999999986\)], "Output"]
}, Open  ]],

Cell["Odvozena funkce NestList vraci vsechny iterace.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(anu = NestList[Function[x, x/\((1 + s)\)], c, 4]\)], "Input"],

Cell[BoxData[
    \({c, c\/\(1 + s\), c\/\((1 + s)\)\^2, c\/\((1 + s)\)\^3, 
      c\/\((1 + s)\)\^4}\)], "Output"]
}, Open  ]],

Cell["\<\
To je vlastne diskontovana anuita pri urokove mire s. Soucasna hodnota je \
tedy:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(presentvalue = Apply[Plus, anu]\), "\[IndentingNewLine]", 
    \(presentvalue = 
      Apply[Plus, anu] // FullSimplify\), "\[IndentingNewLine]", 
    \(FullSimplify[presentvalue = Apply[Plus, anu], 
      c > 0 && s > 0]\)}], "Input"],

Cell[BoxData[
    \(c + c\/\((1 + s)\)\^4 + c\/\((1 + s)\)\^3 + c\/\((1 + s)\)\^2 + 
      c\/\(1 + s\)\)], "Output"],

Cell[BoxData[
    \(\(c\ \((5 + s\ \((10 + s\ \((10 + s\ \((5 + s)\))\))\))\)\)\/\((1 + \
s)\)\^4\)], "Output"],

Cell[BoxData[
    \(\(c\ \((5 + s\ \((10 + s\ \((10 + s\ \((5 + s)\))\))\))\)\)\/\((1 + \
s)\)\^4\)], "Output"]
}, Open  ]],

Cell["\<\
Prekvapeni, ze vysledek neni ve tvaru castecneho soouctu geometricke rady, a \
to ani pri pouziti volby predpokladu v poslednim pripade. Zkusime vyjadreni \
pomoci diskontniho faktoru a pouziti funkce Sum a dalsi finty:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(rule1 = 
      Solve[v \[Equal] 1/\((1 + s)\), s] // 
        Flatten\), "\[IndentingNewLine]", 
    \(Sum[c/\((1 + s)\)\^i //. rule1, {i, 0, 4}] // 
      FullSimplify\), "\[IndentingNewLine]", 
    \({Sum[v\^i, {i, 0, 4}], Sum[v\^i, {i, 0, n}]}\)}], "Input"],

Cell[BoxData[
    \({s \[Rule] \(1 - v\)\/v}\)], "Output"],

Cell[BoxData[
    \(c\ \((1 + v + v\^2 + v\^3 + v\^4)\)\)], "Output"],

Cell[BoxData[
    \({1 + v + v\^2 + v\^3 + 
        v\^4, \(\(-1\) + v\^\(1 + n\)\)\/\(\(-1\) + v\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Sum[v^i, {i, 0, n}] /. n \[Rule] 4\)], "Input"],

Cell[BoxData[
    \(\(\(-1\) + v\^5\)\/\(\(-1\) + v\)\)], "Output"]
}, Open  ]],

Cell[TextData[StyleBox["Zahada, ze? Kdo znate pricinu, napiste mi.",
  FontColor->RGBColor[0, 0, 1]]], "Text",
  FontSize->24],

Cell[CellGroupData[{

Cell[BoxData[
    \(Fold[g, x0, {a, b, c}]\)], "Input"],

Cell[BoxData[
    \(g[g[g[x0, a], b], c]\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(FoldList[Plus, 0, {a, b, c}]\)], "Input"],

Cell[BoxData[
    \({0, a, a + b, a + b + c}\)], "Output"]
}, Open  ]],

Cell["Priklad nahodne prochazky startujici z bodu 100", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(SeedRandom[13];\)\), "\[IndentingNewLine]", 
    \(\(ceny1 = 
        FoldList[Plus, 100, 
          Table[Random[] - 0.5, {5000}]];\)\), "\[IndentingNewLine]", 
    \(\(ListPlot[ceny1, PlotJoined \[Rule] True];\)\)}], "Input"],

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*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)