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Notebook[{
Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*Examples\ of\ matrix\ conditioning*) \)\)\
\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(A = {{1, 3, \(-2\), a}, {2, b, \(-1\), 1}, {0, 4, 
          0, \(-2\)}, \[IndentingNewLine]{1, 1, \(-2\), 4}}\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"1", "3", \(-2\), "a"},
            {"2", "b", \(-1\), "1"},
            {"0", "4", "0", \(-2\)},
            {"1", "1", \(-2\), "4"}
            }], "\[NoBreak]", ")"}],
      (MatrixForm[ #]&)]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Det[A]\)], "Input"],

Cell[BoxData[
    \(\(-36\) + 12\ a\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*\ A\ is\ invertible\ except\ when\ a\  = \ 3*) \)], "Input"],

Cell[BoxData[
    \(\(b = 0;\)\)], "Input"],

Cell[BoxData[
    \(\(a = 0;\)\)], "Input"],

Cell[BoxData[
    \(\(J = Inverse[A];\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(MatrixForm[J]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {\(-\(5\/18\)\), \(2\/3\), \(2\/9\), \(-\(1\/18\)\)},
            {\(-\(1\/6\)\), "0", \(1\/3\), \(1\/6\)},
            {\(-\(8\/9\)\), \(1\/3\), \(11\/18\), \(2\/9\)},
            {\(-\(1\/3\)\), "0", \(1\/6\), \(1\/3\)}
            }], "\[NoBreak]", ")"}],
      (MatrixForm[ #]&)]], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*Estimate\ the\ Euclidean\ norm\ of\ A*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(N[Eigenvalues[A . Transpose[A]]]\)], "Input"],

Cell[BoxData[
    \({\(\(0.5266308506696022`\)\(\[InvisibleSpace]\)\) - 
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\[InvisibleSpace]\)\) + 7.880591113537341`*^-17\ \[ImaginaryI]}\)], "Output"]
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Cell[BoxData[
    \( (*The\ norm\ of\ A\ is\ approximately\ Sqrt[31.13]*) \)], "Input"],

Cell[BoxData[
    \( (*Estimate\ the\ norm\ of\ Inverse[A]*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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\[InvisibleSpace]\)\) + 1.4449134976920402`*^-20\ \[ImaginaryI]}\)], "Output"]
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Cell[BoxData[
    \( (*The\ norm\ of\ Inverse[A]\ is\ approximately\ Sqrt[
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Cell[BoxData[
    \( (*Estimate\ the\ condition\ number\ of\ A*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(K = Sqrt[31.2]\ Sqrt[1.9]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
    \( (*We\ see\ that\ A\ is\ fairly\ well\ conditioned*) \)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
    \( (*Suppose\ Inverse[
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Cell[CellGroupData[{

Cell[BoxData[
    \(JJ\  = \ N[J]\)], "Input"],

Cell[BoxData[
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        0.3333333333333333`, 0.6111111111111112`, 
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Cell[BoxData[
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Cell[BoxData[
    \( (*The\ exact\ solution\ to\ A . xx\  = \ bb, \ 
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Cell[CellGroupData[{

Cell[BoxData[
    \(xx\  = \ J . bb\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
    \( (*The\ approximate\ \(\(solution\)\(:\)\)\ *) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(maxRelError\  = K\ relB\)], "Input"],

Cell[BoxData[
    \(0.3202111979110126`\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*Compute\ the\ actual\ relative\ error\ in\ xx*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(delX = \ xx - xxnew\)], "Input"],

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
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existing symbol \"\!\(delB\)\"."\)], "Message"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(relX\  = \ Sqrt[delX . delX]/Sqrt[xx . xx]\)], "Input"],

Cell[BoxData[
    \(General::"spell" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(relX\)\" is similar to \
existing symbols \!\({delX, relB}\)."\)], "Message"],

Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
    \( (*We\ see\ that\ relX\ is\ smaller\ than\ maxRelError, \
\[IndentingNewLine]\ \ \ as\ the\ theory\ predicts*) \)], "Input"],

Cell[BoxData[
    \( (*Another\ example, \ same\ procedure*) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(bb = {1015.7, 342.9, \(-44.8\), \(-661.5\)}\)], "Input"],

Cell[BoxData[
    \({1015.7`, 342.9`, \(-44.8`\), \(-661.5`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(bbnew = {1000, 340, \ \(-45\), \(-660\)}\)], "Input"],

Cell[BoxData[
    \({1000, 340, \(-45\), \(-660\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(xx = J . bb\)], "Input"],

Cell[BoxData[
    \({\(-26.744444444444504`\), \(-294.4666666666667`\), \
\(-962.9222222222222`\), \(-566.5333333333333`\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(xxnew = JJ . bbnew\)], "Input"],

Cell[BoxData[
    \({\(-24.44444444444445`\), \(-291.66666666666663`\), \
\(-949.7222222222221`\), \(-560.8333333333333`\)}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*large\ absolute\ error\ in\ \(\(bb\)\(:\)\)\ *) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(delB\  = \ bb\  - \ bbnew\)], "Input"],

Cell[BoxData[
    \({15.700000000000045`, 2.8999999999999773`, 
      0.20000000000000284`, \(-1.5`\)}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*small\ relative\ \(\(error\)\(:\)\)\ *) \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(relB\  = \ Sqrt[delB . delB]/Sqrt[bb . bb]\)], "Input"],

Cell[BoxData[
    \(0.012723020030472317`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

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

Cell[BoxData[
    \(7.699350621968063`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(maxRelError\  = \ K\ relB\)], "Input"],

Cell[BoxData[
    \(0.09795899218492915`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(delX = \ xx\  - \ xxnew\)], "Input"],

Cell[BoxData[
    \({\(-2.300000000000054`\), \(-2.800000000000068`\), \
\(-13.20000000000016`\), \(-5.7000000000000455`\)}\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(relX\  = Sqrt[delX . delX]/Sqrt[xx . xx]\)], "Input"],

Cell[BoxData[
    \(0.012830219203260851`\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \( (*The\ relative\ error\ is\ well\ within\ the\ prediction*) \)], \
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Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ Example\ 3; \ same\ method, \ 
      this\ time\ with\ a\ large\[IndentingNewLine]\ \ \ \ relative\ error\ \
in\ bb*) \)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(bb = \ {0.123456, \
\ 2.41415, \ \(-4.44444\), \ 3.14159}\)\)\)], "Input"],

Cell[BoxData[
    \({0.123456`, 2.41415`, \(-4.44444`\), 3.14159`}\)], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(relB\  = \ Sqrt[delB . delB]/Sqrt[bb . bb]\)], "Input"],

Cell[BoxData[
    \(0.10677393702837486`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(delX = \ xx - xxnew\)], "Input"],

Cell[BoxData[
    \({\(-2.300000000000054`\), \(-2.800000000000068`\), \
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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            {\(-0.8888888888888888`\), "0.3333333333333333`", 
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Cell[BoxData[
    \(r =. \)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[""], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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      the\ relative\ error\ in\ xx\ is\ well\ within\ the\ predicted\ value, \
\[IndentingNewLine]but\ it\ is\ very\ large\ \((150  %)\), \ 
      and\ coincides\ with\ the\ 150  %\ relative\[IndentingNewLine]\ 
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[""], "Input"]
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*)




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