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Fialkow\[IndentingNewLine][ Houston\ J . \ Math . \ 17 \((1991)\), \ 603 - 635], \[IndentingNewLine]\ \ \ \ \ \ MR\ 93 \( a : 47016\)*) \)\(\ \ \ \ \ \ \ \ \)\(\[IndentingNewLine]\)\(Gaussian[ n_, a_, b_] := \((\[IndentingNewLine]For[i = 0, i \[LessEqual] 2 n + 1, \(i++\), \[IndentingNewLine]bb[ i]\ = \ \((b^\((i + 1)\) - a^\((i + 1)\))\)/\((i + 1)\)]; \[IndentingNewLine]For[i = 0, i \[LessEqual] n, i = i + 1, \[IndentingNewLine]For[j = 0, j \[LessEqual] n, j = j + 1, \[IndentingNewLine]Mtab[i, j] = bb[i + j]]]; \[IndentingNewLine]H = Table[Mtab[r, s], {r, 0, n}, {s, 0, n}]; \[IndentingNewLine]J\ = \ N[Inverse[H]]; \[IndentingNewLine]For[i = 0, \ i \[LessEqual] n, i = i + 1, \[IndentingNewLine]v[i]\ = \ bb[n + 1 + i]]; \[IndentingNewLine]w\ = Table[v[i], {i, 0, n}]; \[IndentingNewLine]z\ = \ J . w; \[IndentingNewLine]p\ = \ 0; \[IndentingNewLine]For[i = 0, i \[LessEqual] n, i = i + 1, \[IndentingNewLine]p\ = \ p\ + \ z[\([i + 1]\)]\ t^i]; \[IndentingNewLine]q\ = \ t^\((n + 1)\) - p; \[IndentingNewLine]roots\ = \ Solve[q \[Equal] 0, t]; \[IndentingNewLine]For[i = 1, i \[LessEqual] n + 1, i = i + 1, \[IndentingNewLine]x[ i - 1]\ = \ \(\(roots[\([i]\)]\)[\([1]\)]\)[\([2]\)]]; \ \[IndentingNewLine]For[\ i = 0, i \[LessEqual] n, i = i + 1, \[IndentingNewLine]For[j = 0, j \[LessEqual] n, j = j + 1, \[IndentingNewLine]TTab[i, j] = \ x[j]^i]]; \[IndentingNewLine]V = \ Table[TTab[rr, ss], {rr, 0, n}, {ss, 0, n}]; \[IndentingNewLine]For[ i = 0, \ i \[LessEqual] n, i = i + 1, \[IndentingNewLine]vvv[i]\ = \ bb[i]]; \[IndentingNewLine]w\ = Table[vvv[i], {i, 0, n}]; \[IndentingNewLine]weights = N[Inverse[V] . w];)\)\)\)\)], "Input"], Cell[BoxData[ \(\(\( (*Gaussian\ quadrature\ estimate\ of\[IndentingNewLine]degree\ 2 n + 1\ for\ integral\ of\ \[IndentingNewLine]predefined\ function\ \ f[x]\[IndentingNewLine]over\ [a, b]; \ uses\ results\ of\ \[IndentingNewLine]previous\ call\ to\ Gaussian*) \)\ \(\[IndentingNewLine]\)\(G[ j_] := \((\ \ \ \ \ \ (*j\ is\ ignored*) \[IndentingNewLine]ss = 0; \[IndentingNewLine]For[i = 0, \ i \[LessEqual] n, \(i++\), \[IndentingNewLine]ss\ = \ ss\ + \ weights[\([i + 1]\)]\ f[ x[i]]]; \[IndentingNewLine]ss = \ N[ss])\);\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(a = 1\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = 3\)], "Input"], Cell[BoxData[ \(3\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(n = 2\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[BoxData[ \(f[x_] := x^5\)], "Input"], Cell[BoxData[ \(Gaussian[n, a, b]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(G[1]\)], "Input"], Cell[BoxData[ \(121.33333333325524`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(NIntegrate[f[x], {x, 1, 3}]\)], "Input"], Cell[BoxData[ \(121.33333333333334`\)], "Output"] }, Open ]], Cell[BoxData[ \( (*Exact\ result\ up\ to\ degree\ 2 n + 1\ = \ 5\ \((ignoring\ possible\ roundoff)\)*) \)], "Input"], Cell[BoxData[ \(f[x_] 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Cell[CellGroupData[{ Cell[BoxData[ \(n = 5\)], "Input"], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[BoxData[ \(Gaussian[n, a, b]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(G[1]\)], "Input"], Cell[BoxData[ \(0.20875831331161662`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(n = 6\)], "Input"], Cell[BoxData[ \(6\)], "Output"] }, Open ]], Cell[BoxData[ \(Gaussian[n, a, b]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(G[1]\)], "Input"], Cell[BoxData[ \(0.2119811374795203`\)], "Output"] }, Open ]], Cell[BoxData[ \( (*Depending\ on\ n, a, b, \ the\ matices\ in\ Gaussian\ may\ be\ poorly\ conditioned, \ \[IndentingNewLine]leading\ to\ undependable\ results, \ as\ the\ last\ result\ shows*) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(n = 5\)], "Input"], Cell[BoxData[ \(5\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a = 0\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(b = 2\)], 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