(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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[ 11297, 260]*) (*NotebookOutlinePosition[ 11944, 282]*) (* CellTagsIndexPosition[ 11900, 278]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Now suppose that I took a given matrix whose limit as n or m goes to infinity \ is zero. I know the B matrix, and its inverse, so I will use it to \ experiment. The concept is that if the matrix converges to zero on its \ borders, I can just invert the upper left portion to approximate.\ \>", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(B[n_, \ m_]\ := \(\((\(-1\))\)\^\(\(2 n - 1 + 2 \((m)\) + 1\)\/2\)\/\[Pi]\ \) \((1\/\(\((2 n - 1)\) + 2 m\) - 1\/\(\((2 n - 1)\) - 2 m\))\);\)\[IndentingNewLine] \(Binv[n_, \ m_]\ := 4\ \(\((\(-1\))\)\^\(\(2 m - 1 + 2 \((n)\) + 1\)\/2\)\/\[Pi]\) \ \((1\/\(\((2 m - 1)\) + 2 n\) - 1\/\(\((2 m - 1)\) - 2 n\))\);\)\[IndentingNewLine] \(ApproximateInverse[f_, \ approx_, n_, \ m_]\ := \ N[\(\(Inverse[ Array[f, \ {approx, \ approx}, {1, 1}]]\)[\([n]\)]\)[\([m]\)]];\)\[IndentingNewLine] \(ApproximateInverseArray[f_, \ approx_]\ := \ Inverse[Array[f, \ {approx, \ approx}, {1, 1}]];\)\[IndentingNewLine] ApproximateInverseTable[f_, \ approx_, \ n_, \ m_]\ := \ Table[N[\(\(ApproximateInverseArray[f, \ approx]\)[\([a]\)]\)[\([b]\)]], \ {a, 1, n}, \ {b, 1, m}]\)\)\)], "Input"], Cell["\<\ Here is an example of how well this procedure works: Ask for ApproximateInverseTable[Matrix, Approximation Level, Rows, Columns] You see that this inverse is fairly close to B-inverse. It would be better \ if Approximation Level was higher, and approaches the actual inverse as \ Approximation Level goes to infinity.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(ApproximateInverseTable[B, \ 10, 5, 5]\[IndentingNewLine] \ Table[N[Binv[a, b]], \ {a, 1, 5}, \ {b, 1, 5}]\)\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1.703748394926573`", "1.0124667495209587`", \(-0.2363367762653965`\), "0.1068969418800409`", \(-0.0597262062880491`\)}, {\(-0.6915214073525502`\), "1.4676513806081122`", "1.1191241464332011`", \(-0.2958244247215035`\), "0.14358612398570356`"}, {"0.45594818067201115`", \(-0.5853880150573667`\), "1.4086877367690644`", "1.1552906527585345`", \(-0.31908027552378565`\)}, {\(-0.35072936974770086`\), "0.39790010533969267`", \(-0.5501384060754926`\), "1.3863487833102413`", "1.1694797428699837`"}, {"0.29420686801150114`", \(-0.31700832568272214`\), "0.37709487107356493`", \(-0.5374881646523663`\), "1.3793146249638804`"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1.6976527263135504`", "1.0185916357881302`", \(-0.2425218180447929`\), "0.11317684842090336`", \(-0.06614231401221625`\)}, {\(-0.6790610905254202`\), "1.4551309082687574`", "1.1317684842090334`", \(-0.30866413205700916`\), "0.1567064055058662`"}, {"0.43653927248062724`", \(-0.5658842421045167`\), "1.3889885942565412`", "1.1752980412939964`", \(-0.3395305452627101`\)}, {\(-0.3233624240597239`\), "0.370396958468411`", \(-0.522354685019554`\), "1.3581221810508404`", "1.198343100927212`"}, {"0.2572201100475076`", \(-0.27983286697476106`\), "0.3395305452627101`", \(-0.49930962538633833`\), "1.3402521523528028`"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]], Cell["Here is the matrix I need to invert", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(M1[n_, \ m_] := \(1\/\[Pi]\) \(\@\(\(2\ n\)\/\(2\ m\ - \ 1\)\)\) \((\(-1\))\)\^\(n + m\)\/\(n - \((m - \ 1\/2)\)\);\)\[IndentingNewLine] \(M2[n_, \ m_] := \(1\/\[Pi]\) \(\@\(\(2\ n\)\/\(2\ m\ - \ 1\)\)\) \((\(-1\))\)\^\(n + m\)\/\(n + \((m - \ 1\/2)\)\);\)\[IndentingNewLine] S[m_, \ n_]\ := \ \(\((\@\(\(2\ m\)\/\(2\ n\ + \ 1\)\))\)\^\(-1\)\) \((M1[ m, \ n + 1] + \(1\/2\) M2[m, \ n + 1])\)\)\)\)], "Input"], Cell["\<\ Now let me look at the first 5 entries in the corner to see what I can learn. \ S is indexed from 1.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ ApproximateInverseTable[S, 12, 8, 8]\[IndentingNewLine] \ Table[N[S[a, b]], \ {a, 1, 8}, \ {b, 1, 8}]\)\)\)], "Input"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1.130978470644623`", "0.45834619780349894`", \(-0.10158377247045708`\), "0.04645855354523715`", \(-0.0268266443657221`\), "0.017348955255749657`", \(-0.011942871079936907`\), "0.00850434736655689`"}, {\(-0.676808433152089`\), "0.8113119663898539`", "0.5263878455811167`", \(-0.13084222833491616`\), "0.06234673827216367`", \(-0.03650942303743617`\), "0.023623887587369996`", \(-0.016118831912073327`\)}, {"0.5825678932291463`", \(-0.38848605802920627`\), "0.7493850906316041`", "0.5526932422626751`", \(-0.14478042847668382`\), "0.07055401990409463`", \(-0.041587780480838325`\), "0.026786998611417073`"}, {\(-0.5469308306141525`\), "0.30790773664396653`", \(-0.32909936696732583`\), "0.724293035761706`", "0.5657737769157546`", \(-0.1522663471636874`\), "0.07497589310630343`", \(-0.04412703015705181`\)}, {"0.5351942706896303`", \(-0.2782690491009053`\), "0.2498594865122596`", \(-0.30478968793407285`\), "0.711866146943264`", "0.5726393096131552`", \(-0.1560706341210073`\), "0.07688476876522361`"}, {\(-0.5385787551265703`\), "0.26763373355030856`", \(-0.22098423521229021`\), "0.22627570346613488`", \(-0.2930983630307123`\), "0.7057566365073963`", "0.5756610483183096`", \(-0.15716151941883627`\)}, {"0.5553719282358079`", \(-0.2681570224329569`\), "0.21086832238902617`", \(-0.19832751048255842`\), "0.21563270821036218`", \(-0.2880957647520416`\), "0.7038878011129988`", "0.5755459451872041`"}, {\(-0.5878986531021657`\), "0.2783473246369562`", \(-0.21207193864879523`\), "0.18973633079961574`", \(-0.18942532307136706`\), "0.21245445284607878`", \(-0.2881094541322091`\), "0.7059559977070299`"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.5729577951308232`", \(-0.16673374990579512`\), "0.09195618934198396`", \(-0.06200841938645273`\), "0.04625015440277301`", \(-0.036653865681769836`\), "0.030246640768595495`", \(-0.025688166253428724`\)}, {"0.68209261325098`", "0.6012520072360491`", \(-0.1832693284088492`\), "0.1028385786132247`", \(-0.06972502268787796`\), "0.05201141931107691`", \(-0.04112137285627918`\), "0.03381313809278363`"}, {\(-0.2475743559207261`\), "0.6655570347479259`", "0.6121343965072898`", \(-0.19098593171027442`\), "0.10859984352152859`", \(-0.0741925298623873`\), "0.05557791663526504`", \(-0.044034964491828754`\)}, {"0.15626121685386088`", \(-0.23669196664948539`\), "0.6578404314465008`", "0.6178956614155937`", \(-0.19545343888478375`\), "0.11216634084571672`", \(-0.07710612149793689`\), "0.058003134815712974`"}, {\(-0.11543105762708893`\), "0.14854461355243564`", \(-0.23093070174118147`\), "0.6533729242719915`", "0.6214621587397818`", \(-0.19836703052033333`\), "0.11459155902616465`", \(-0.07915642672295324`\)}, {"0.09195618934198396`", \(-0.10966979271878503`\), "0.1440771063779263`", \(-0.22736420441699334`\), "0.650459332636442`", "0.6238873769202298`", \(-0.2004173357453497`\), "0.11634775150166143`"}, {\(-0.0765986357126769`\), "0.08748868216747463`", \(-0.1061032953945969`\), "0.14116351474237673`", \(-0.22493898623654543`\), "0.6484090274114256`", "0.6256435693957265`", \(-0.20193852994455538`\)}, {"0.06572390362499322`", \(-0.07303213838848878`\), "0.08457509053192507`", \(-0.10367807721414897`\), "0.1391132095173604`", \(-0.22318279376104866`\), "0.6468878332122198`", "0.6269740182407999`"} }, ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], TraditionalForm]], "Output"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{1125, 822}, WindowMargins->{{-14, Automatic}, {Automatic, 42}} ] (******************************************************************* Cached data follows. 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