(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). 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In this paper we propose to calculate the enclosure of \ range of multivariate polynomial using Bernstein basis function. \ \>", \ "Abstract"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "The problem of reliability of computer computations is one of great \ concern to specialists in many areas of science and engineering. The notion \ of computing estimates of numerical error in computer simulations is not new. \ In recent years, considerable progress has been made in determining \ theoretical and computational techniques that aid to improve the reliability \ of results of simulations. An important advance in this area has been the \ recent discovery of methods to determine upper and lower bounds of local \ approximation error in any given simulation. It is predicted that computable \ bounds giving upper and lower limits to computed quantities of interest could \ be a natural by-product in every simulation [Ode2003].\n\tThe idea of \ validated computation (verified computation) is based on the rigorous \ estimation of the influences of uncertainties on the calculation. Such \ uncertainties arise mainly from two reasons. First is associated with \ computational inaccuracies based on the finite accuracy of the computational \ environments. Second reason is the fact that there are uncertainties in the \ variables of the model to be analyzed.\n\tA fundamental problem in range \ methods is computing the ranges of values of real functions or polynomials. \ Interval methods provide rigorous enclosures of range of expressions, however \ the limitation of the methods is the overestimation. The one of tool to \ control overestimation was proposed by Moore [Moo1966, Moo1979]. Author \ proposed the tool of subdivision to compute the range of function to a \ desired accuracy. Moore actually suggested two different strategies for \ subdivision called uniform subdivision and adaptive subdivision. Both these \ strategies are well known and are extensively used in various applications of \ interval analysis. \nNataraj and Sheela in the paper [Nat2002] presented a \ new subdivision strategy in interval analysis for computing the ranges of \ functions. The authors showed through several real-world examples that the \ proposed in the paper parallel-adaptive strategy is more efficient than the \ widely used uniform and adaptive subdivision strategies. In this work, \ parallel means simultaneous processing of all boxes present in a subdivision \ partition. Unfortunately, in case of multivariate functions, the \ computational expense by simple interval methods increases astronomically, \ especially with subdivision strategy.\n\tIn paper [Usc2005] a ", StyleBox["Mathematica", FontSlant->"Italic"], " package for affine arithmetic was presented and applied for validated \ computations of initial value problems of ordinary differential equations. \ Affine arithmetic was proposed by Stolfi and Comba [Sto2003] in order to \ tackle the conservativeness problem caused by standard interval arithmetic. \ Similarly to interval arithmetic, affine arithmetic can be used for \ manipulating imprecise values and for evaluating functions over intervals. \ Also, likewise interval arithmetic, it provides guaranteed bounds for \ computed results, although affine arithmetic also takes into consideration \ the correlation or dependencies between the error sources. \n\tAnother well \ known method for range analysis is the Bernstein coefficient method based on \ the Bernstein convex hull property. This method relies on the simple idea \ that if a polynomial is written in the Bernstein basis, the range of the \ polynomial is bounded by the values of the minimum and maximum Bernstein \ coefficients. A modification of this method elevates the degree of the \ Bernstein polynomials before using the coefficients to find the range. It is \ known that as the degree is elevated, the bounds become tighter, but an \ additional computational cost is involved.\n\tIn this paper the Bernstein \ form for multivariate polynomials [Ber2000] is used to construct for \ approximating the range of the multivariate polynomial ([Lin1995], \ [Lin1996]). This method produce much tighter and more accurate intervals than \ interval arithmetic. The paper [Mar2002] compares the performance and \ efficiency of different function range interval methods for plotting f(x, \ y)=0 on a rectangular region based on a subdivision scheme, where f (x, y) is \ a polynomial.\n\tThe bounds obtained by the Bernstein coefficients method can \ be tightened if the unit box is bisected into subboxes and Bernstein \ expansion is applied to the polynomial on these subboxes [Gar2001]. In paper \ [Gar2001] authors applied the expansion of a multivariate polynomial into \ Bernstein polynomial to compute solution of systems of polynomial equations. \ Systems of polynomial equations appear in a great variety applications, e.g. \ in geometric intersection computations, chemical equilibrium problems, \ combustions, and kinematics, to name only a few." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Interval arithmetic", "Section"], Cell["\<\ Interval arithmetic is the arithmetic defined on sets of intervals, \ rather than sets of real numbers. A form of interval arithmetic perhaps first \ appeared in 1924 and 1931 in [Bur1924], [You1931], than later in [Sun1958]. \ Modern development of interval arithmetic began with Moore\[CloseCurlyQuote]s \ dissertation [Moo1962], [Moo1966]. Since then, thousands of research articles \ and books have been published on the subject [Ale1974], [Ale1998], [Ale2000]. \ There is an interesting amount of software support for interval computations. \tThe primary motivation for using interval analysis in almost all \ applications is that the interval extension of a function provides bounds for \ the variation of the function. It means that the interval arithmetic \ evaluation of real function always contains the range of the function. This \ comes from the fundamental property of interval arithmetic. \tUnfortunately the conventional interval methods do not carry the \ information on functional dependency. Let\[CloseCurlyQuote]s consider simple \ computation of powers. Note that the computation of powers can be done in \ several ways. To see this consider the following examples. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(xint1 = Interval[{\(-1\), 2}]\), "\[IndentingNewLine]", \(xint2 = xint1*xint1\), "\[IndentingNewLine]", \(xint3 = xint1^2\)}], "Input"], Cell[BoxData[ \(Interval[{\(-1\), 2}]\)], "Output"], Cell[BoxData[ \(Interval[{\(-2\), 4}]\)], "Output"], Cell[BoxData[ \(Interval[{0, 4}]\)], "Output"] }, Open ]], Cell["\<\ The dependency problem in interval arithmetic is typically caused \ by cancellation effects. For example, if x=[a,b], then y=x-x, if not \ recognised to represent the same number, is computed as \ [a-b]-[a,b]=[a,b]+[-b,-a]=[a-b,b-a], resulting in a width that is not zero, \ but twice as large as before. Such a situation can not be avoided in \ conventional interval methods in practice if the cancelling terms appear as a \ result of preceding complicated science computations.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(f1[x_] := x*\((10 - x)\); x0 = Interval[{4, 6}]; r1 = f1[x0]\)], "Input"], Cell[BoxData[ \(Interval[{16, 36}]\)], "Output"] }, Open ]], Cell["\<\ The overestimation of enclosures of range of given polynomial using \ interval arithmetic is really large. 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\>"], ImageRangeCache->{{{180, 479}, {231.188, 143.938}} -> {-1.33399, 1.00486, \ 0.00708345, 0.00708345}, {{188.312, 322.75}, {229.062, 146}} -> {-1.85575, \ 0.77591, 0.00893121, 0.00591909}, {{336.188, 470.625}, {229.062, 146}} -> \ {-3.17645, 0.838229, 0.00893121, 0.00642266}}] }, Open ]], Cell[BoxData[ \(Two\ examples\ of\ Bernstein\ basis\ function\ B\_\(4, 5\)\ \((left\ \ plot)\) and\ B\_\(1, 3\)\ \((right\ plot)\)\)], "NumberedFigureCaption"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[Sum[Bernstein[x, k, j], {j, 0, k}]]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k0 = 30;\)\), "\[IndentingNewLine]", \(\(i0 = 10;\)\), "\[IndentingNewLine]", \(Simplify[ Sum[Binomial[j, i0]/Binomial[k0, i0]*Bernstein[x, k0, j], {j, 0, k0}]]\)}], "Input"], Cell[BoxData[ \(x\^10\)], "Output"] }, Open ]], Cell["We can write polynomial (1) using (2)", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(p(x\_1, ... , x\_s)\), "=", RowBox[{ RowBox[{ RowBox[{\(\[Sum]\+\(i\_1 = 0\)\%\(n\_1\) ... \), RowBox[{\(\[Sum]\+\(i\_s = 0\)\%\(n\_s\)\), RowBox[{\(a\_\(\(i\_1 ... \) i\_s\)\), "\[Cross]", RowBox[{\(\[Sum]\+\(j\_1 = 0\)\%\(k\_1\)\), RowBox[{ FractionBox[ RowBox[{"(", GridBox[{ {\(j\_1\)}, {\(i\_1\)} }], ")"}], RowBox[{"(", GridBox[{ {\(k\_1\)}, {\(i\_1\)} }], ")"}]], \(\(B\_\(j\_1, k\_1\)\)( x\_1)\)}]}]}]}]}], "..."}], RowBox[{\(\[Sum]\+\(j\_s = 0\)\%\(k\_s\)\), RowBox[{ FractionBox[ RowBox[{"(", GridBox[{ {\(j\_s\)}, {\(i\_s\)} }], ")"}], RowBox[{"(", GridBox[{ {\(k\_s\)}, {\(i\_s\)} }], ")"}]], \(\(B\_\(j\_s, k\_s\)\)(x\_s)\)}]}]}]}], TraditionalForm]], "NumberedEquation"], Cell["\<\ After some mathematical manipulations we can write polynomial in \ the Bernstein form\ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`p(x\_1, ... , x\_s) = \(\(\[Sum]\+\(j\_1 = 1\)\%\(k\_1\) ... \) \(\[Sum]\+\(j\_s = \ 1\)\%\(k\_s\)b\_\(\(i\_1 .. \) i\_s\)*\(\(B\_\(j\_1, k\_1\)\)( x\_1)\)\) ... \) \(\(B\_\(j\_s, k\_s\)\)( x\_s)\)\)], "NumberedEquation"], Cell["where ", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(b\_\(\(j\_1 ... \) j\_s\)\), "=", RowBox[{ RowBox[{ RowBox[{\(\[Sum]\+\(i\_1 = 0\)\%\(min(j\_1, n\_1)\) ... \), RowBox[{\(\[Sum]\+\(i\_s = 0\)\%\(min(j\_s, n\_s)\)\), RowBox[{\(a\_\(\(i\_1 .. \) i\_s\)\), FormBox[ FractionBox[ RowBox[{"(", GridBox[{ {\(j\_1\)}, {\(i\_1\)} }], ")"}], RowBox[{"(", GridBox[{ {\(k\_1\)}, {\(i\_1\)} }], ")"}]], "TraditionalForm"]}]}]}], "..."}], FormBox[ FractionBox[ RowBox[{"(", GridBox[{ {\(j\_s\)}, {\(i\_s\)} }], ")"}], RowBox[{"(", GridBox[{ {\(k\_s\)}, {\(i\_s\)} }], ")"}]], "TraditionalForm"]}]}], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "and ", Cell[BoxData[ \(TraditionalForm\`k\_i \[GreaterEqual] n\_i\ for\ i = \(1. .. \) \(\(s\)\(\ \ \)\(.\)\)\)]] }], "Text"], Cell["\<\ The quantities b of Bernstein form for multivariate polynomials (2) \ can be used to construct for approximating the range p([0,1],...,[0,1]) of \ the polynomial (1). It can be proved that ([Lin1995], [Lin1996])\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{\(p(\([0, 1]\), ... , \([0, 1]\))\), "\[SubsetEqual]", RowBox[{ RowBox[{"[", RowBox[{ RowBox[{\(min\+\(\(j\_1 ... \) j\_s\)\), StyleBox[\(b\_\(\(j\_1 ... \) j\_s\)\), FontSlant->"Italic"]}], StyleBox[",", FontSlant->"Italic"], RowBox[{\(max\+\(\(j\_1 ... \) j\_s\)\), StyleBox[\(b\_\(\(j\_1 ... \) j\_s\)\), FontSlant->"Italic"]}]}], "]"}], "."}]}], TraditionalForm]], "NumberedEquation"], Cell[BoxData[ \(\(powxi[x_, k_, i_] := Module[{j}, \[IndentingNewLine]Sum[\((Binomial[j, i]/ Binomial[k, i])\)*\(Subscript[B, j, k]\)[x], {j, i, k}]];\)\)], "Input"], Cell[BoxData[ \(\(pow2ber[pp_, vars_, k_, prnout_] := Module[{pb, pj, pj1, pj2, epj, blj, lofp, n}, \[IndentingNewLine]pb = 0; \[IndentingNewLine]lofp = Length[pp]; \[IndentingNewLine]n = Length[k]; \[IndentingNewLine]coef = poly2coeffast[pp, vars]; \[IndentingNewLine]Do[ pj = pp[\([j]\)]; \[IndentingNewLine]epj = Exponent[pj, vars]; \[IndentingNewLine]blj = Table[powxi[Subscript[x, i], k[\([i]\)], epj[\([i]\)]], {i, 1, n}]; \[IndentingNewLine]pj1 = coef[\([j]\)]*Apply[Times, blj]; \[IndentingNewLine]pj2 = Expand[pj1]; \[IndentingNewLine]If[ prnout \[Equal] 1, \[IndentingNewLine]Print["\", j]; \[IndentingNewLine]Print[{pj, epj}]; \[IndentingNewLine]Print[{blj}]; \ \[IndentingNewLine]Print[{pj1, pj2}]\[IndentingNewLine]]; \[IndentingNewLine]pb += pj2\[IndentingNewLine], {j, 1, lofp}\[IndentingNewLine]]; \[IndentingNewLine]Return[ pb]\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(poly2coeffast[pp_, vars_] := Module[{pos1, pos2, pos, coef}, \[IndentingNewLine]pos1 = Position[pp, _Rational]; \[IndentingNewLine]pos2 = Position[pp, _Integer, 2]; \[IndentingNewLine]pos3 = Position[pp, _Real, 2]; \[IndentingNewLine]pos = Join[pos1, pos2, pos3]; \[IndentingNewLine]coef = Extract[pp, pos]; \[IndentingNewLine]Return[ coef]\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[ \(\(tosubform[poly_] := Module[{vars, varsnew}, \[IndentingNewLine]vars = Variables[poly]; \[IndentingNewLine]varsnew = Table[Subscript[x, i], {i, Length[vars]}]; \[IndentingNewLine]polynew = poly /. Thread[vars -> varsnew]; \[IndentingNewLine]Return[ polynew]\[IndentingNewLine]];\)\)], "Input"], Cell[BoxData[{ \(\(to01form[{xmin_, xmax_}, x_] := \((xmax - xmin)\)*x + xmin;\)\), "\[IndentingNewLine]", \(widthint[int_] := Max[int] - Min[int]\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Numerical results", "Section"], Cell[CellGroupData[{ Cell["Example 1", "Subsection"], Cell["Let us consider polynomial ", "Text"], Cell[BoxData[ RowBox[{ StyleBox[\(p \((x\_1, x\_2)\)\), FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox["=", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox["\[InvisibleSpace]", FontSlant->"Italic"], StyleBox[\(2\ x\_1 + 3\ x\_2 - x\_1\ x\_2\), FontSlant->"Italic"]}]], "NumberedEquation"], Cell[TextData[{ "over domain ", Cell[BoxData[ StyleBox[\((x\_1, x\_2)\), FontSlant->"Italic"]], FontFamily->"Times New Roman"], StyleBox["\[Element] [", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`4, 6\)], FontFamily->"Times New Roman"], StyleBox["]\[Times][4.6] with ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`k\_1 = \(k\_2 = 2\)\)]], StyleBox[".", FontFamily->"Times New Roman"] }], "Text"], Cell["\<\ First we choose the power form polynomial which will be transformed \ to Bernstein form.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly = 2*x + 3*y - x*y;\)\), "\[IndentingNewLine]", \(\(poly0 = tosubform[poly];\)\), "\[IndentingNewLine]", \(\(vars0 = Variables[poly0];\)\), "\[IndentingNewLine]", \(\(ndim = Length[vars0];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly, poly0}];\)\), "\[IndentingNewLine]", \(\(Print["\", vars0];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial: "\[InvisibleSpace]{2\ x + 3\ y - x\ y, 2\ x\_1 + 3\ x\_2 - x\_1\ x\_2}\), SequenceForm[ "Polynomial: ", { Plus[ Times[ 2, x], Times[ 3, y], Times[ -1, x, y]], Plus[ Times[ 2, Subscript[ x, 1]], Times[ 3, Subscript[ x, 2]], Times[ -1, Subscript[ x, 1], Subscript[ x, 2]]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Variables: "\[InvisibleSpace]{x\_1, x\_2}\), SequenceForm[ "Variables: ", { Subscript[ x, 1], Subscript[ x, 2]}], Editable->False]], "Print"] }, Open ]], Cell["\<\ We will calculate enclosure of real range of polynomial for \ following intervals values of variables of polynomial \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(xint = Table[Interval[{Subscript[x, i, 1], Subscript[x, i, 2]}], {i, 1, ndim}]\)], "Input"], Cell[BoxData[ \({Interval[{x\_\(1, 1\), x\_\(1, 2\)}], Interval[{x\_\(2, 1\), x\_\(2, 2\)}]}\)], "Output"] }, Open ]], Cell["Numerical values of intervals are given below", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vxint = Table[Interval[{4, 6}], {i, 1, ndim}]\)], "Input"], Cell[BoxData[ \({Interval[{4, 6}], Interval[{4, 6}]}\)], "Output"] }, Open ]], Cell[TextData[{ "To apply Bernstein form of polynomial we have to first transform a", StyleBox["ny finite interval [a,b] to [0,1]. ", FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(varsnew0 = Table[to01form[{Subscript[x, i, 1], Subscript[x, i, 2]}, Subscript[x, i]], {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(\(vxran = Table[{Min[vxint[\([i]\)]], Max[vxint[\([i]\)]]}, {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(varsnew1 = Table[to01form[{vxran[\([i, 1]\)], vxran[\([i, 2]\)]}, Subscript[x, i]], {i, 1, ndim}]\)}], "Input"], Cell[BoxData[ \({4 + 2\ x\_1, 4 + 2\ x\_2}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Next we have to transform polynomial variables to form with ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`x\_i = \([0, 1]\)\)]], StyleBox[". ", FontFamily->"Times New Roman"], "Considered polynomial (10) defined over [0,1] x [0,1] we can write in th \ following form ", StyleBox[" ", FontFamily->"Times New Roman"], " " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly1 = poly0 /. Thread[vars0 -> varsnew1];\)\), "\[IndentingNewLine]", \(\(poly2 = Expand[poly1];\)\), "\[IndentingNewLine]", \(\(Print["\", poly2];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial: "\[InvisibleSpace]\(4 - 4\ x\_1 - 2\ x\_2 - 4\ x\_1\ x\_2\)\), SequenceForm[ "Polynomial: ", Plus[ 4, Times[ -4, Subscript[ x, 1]], Times[ -2, Subscript[ x, 2]], Times[ -4, Subscript[ x, 1], Subscript[ x, 2]]]], Editable->False]], "Print"] }, Open ]], Cell["Now we can derive Bernstein form of polynomial, pb0.", "Text"], Cell[BoxData[{ \(\(klist = Exponent[poly2, vars0];\)\), "\[IndentingNewLine]", \(\(klist = Max[klist]*Table[1, {ndim}] + 1;\)\), "\[IndentingNewLine]", \(\(prnout = 0;\)\), "\[IndentingNewLine]", \(\(pb0 = pow2ber[poly2, vars0, klist, prnout];\)\), "\[IndentingNewLine]", \(\(vars1 = Variables[pb0];\)\), "\[IndentingNewLine]", \(\(Length[vars1];\)\)}], "Input"], Cell["\<\ The quantities (coefficient) of Bernstein form for multivariate \ polynomials are used to construct for approximating the range of the \ polynomial. The width of interval which represent enclosure of range of \ considered polynomial is calculated too. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(coeflist = poly2coeffast[pb0, vars1];\)\), "\[IndentingNewLine]", \(\(nofc = Length[coeflist];\)\), "\[IndentingNewLine]", \(\(If[nofc < 5, Print[coeflist]; Table[{pb0[\([i]\)], coeflist[\([i]\)]}, {i, 1, nofc}]\[IndentingNewLine]];\)\), "\[IndentingNewLine]", \(\(coefrange = Interval[{Min[coeflist], Max[coeflist]}];\)\), "\[IndentingNewLine]", \(\(Print["\", coefrange];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[coefrange] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: "\[InvisibleSpace]Interval[{\(-6\ \), 4}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-6, 4}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]10.`\), SequenceForm[ "Width of enclosure: ", .1*^2], Editable->False]], "Print"] }, Open ]], Cell["\<\ We can calculate the enclosure of real range of polynomial using \ interval arithmetic.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(polyintval = poly0 /. Thread[vars0 -> vxint];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly0, vxint}];\)\), "\[IndentingNewLine]", \(\(Print["\", polyintval];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[polyintval] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial and variables range: "\[InvisibleSpace]{2\ \ x\_1 + 3\ x\_2 - x\_1\ x\_2, {Interval[{4, 6}], Interval[{4, 6}]}}\), SequenceForm[ "Polynomial and variables range: ", { Plus[ Times[ 2, Subscript[ x, 1]], Times[ 3, Subscript[ x, 2]], Times[ -1, Subscript[ x, 1], Subscript[ x, 2]]], { Interval[ {4, 6}], Interval[ {4, 6}]}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-16\), 14}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-16, 14}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]30.`\), SequenceForm[ "Width of enclosure: ", .3*^2], Editable->False]], "Print"] }, Open ]], Cell["\<\ As we can see in considered case the width of resulting interval is \ three times less when we calculate the enclosure of range of polynomial using \ Bernstein form method. 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s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid .753 .802 .892 r .38606 .75402 .39954 .76348 .4221 .75046 .40878 .74102 Metetra .753 .804 .894 r .40878 .74102 .4221 .75046 .44477 .73738 .4316 .72795 Metetra .754 .807 .895 r .4316 .72795 .44477 .73738 .46753 .72424 .45452 .71483 Metetra .754 .809 .897 r .45452 .71483 .46753 .72424 .4904 .71104 .47755 .70165 Metetra .754 .812 .898 r .47755 .70165 .4904 .71104 .51336 .69779 .50068 .6884 Metetra .755 .814 .899 r .50068 .6884 .51336 .69779 .53643 .68448 .52392 .6751 Metetra .755 .817 .901 r .52392 .6751 .53643 .68448 .55961 .6711 .54726 .66173 Metetra .755 .819 .902 r .54726 .66173 .55961 .6711 .58289 .65767 .57071 .64831 Metetra .756 .822 .904 r .57071 .64831 .58289 .65767 .60627 .64417 .59427 .63482 Metetra .756 .824 .905 r .59427 .63482 .60627 .64417 .62975 .63062 .61794 .62127 Metetra .756 .827 .907 r .61794 .62127 .62975 .63062 .65335 .617 .64172 .60765 Metetra .757 .829 .908 r .64172 .60765 .65335 .617 .67705 .60333 .66561 .59398 Metetra .757 .832 .909 r .66561 .59398 .67705 .60333 .70085 .58959 .68961 .58024 Metetra .757 .834 .911 r .68961 .58024 .70085 .58959 .72477 .57579 .71373 .56643 Metetra .758 .837 .912 r .71373 .56643 .72477 .57579 .74879 .56192 .73795 .55256 Metetra .758 .84 .914 r .73795 .55256 .74879 .56192 .77293 .54799 .76229 .53863 Metetra .758 .842 .915 r .76229 .53863 .77293 .54799 .79717 .534 .78674 .52463 Metetra .758 .845 .917 r .78674 .52463 .79717 .534 .82152 .51995 .81131 .51056 Metetra .758 .848 .918 r .81131 .51056 .82152 .51995 .84599 .50583 .836 .49643 Metetra .759 .85 .919 r .836 .49643 .84599 .50583 .87057 .49164 .8608 .48223 Metetra .759 .853 .921 r .8608 .48223 .87057 .49164 .89526 .47739 .88572 .46796 Metetra .759 .856 .922 r .88572 .46796 .89526 .47739 .92007 .46307 .91076 .45363 Metetra .759 .858 .923 r .91076 .45363 .92007 .46307 .94499 .44869 .93592 .43923 Metetra .759 .861 .925 r .93592 .43923 .94499 .44869 .97003 .43424 .96119 .42476 Metetra .751 .802 .893 r .37237 .7444 .38606 .75402 .40878 .74102 .39523 .73142 Metetra .751 .804 .895 r .39523 .73142 .40878 .74102 .4316 .72795 .41821 .71837 Metetra .752 .807 .896 r .41821 .71837 .4316 .72795 .45452 .71483 .44129 .70526 Metetra .752 .809 .898 r .44129 .70526 .45452 .71483 .47755 .70165 .46448 .69209 Metetra .753 .812 .899 r .46448 .69209 .47755 .70165 .50068 .6884 .48777 .67885 Metetra .753 .814 .901 r .48777 .67885 .50068 .6884 .52392 .6751 .51118 .66556 Metetra .753 .816 .902 r .51118 .66556 .52392 .6751 .54726 .66173 .5347 .6522 Metetra .754 .819 .903 r .5347 .6522 .54726 .66173 .57071 .64831 .55833 .63878 Metetra .754 .821 .905 r .55833 .63878 .57071 .64831 .59427 .63482 .58207 .6253 Metetra .754 .824 .906 r .58207 .6253 .59427 .63482 .61794 .62127 .60592 .61175 Metetra .755 .827 .908 r .60592 .61175 .61794 .62127 .64172 .60765 .62989 .59814 Metetra .755 .829 .909 r .62989 .59814 .64172 .60765 .66561 .59398 .65397 .58446 Metetra .755 .832 .911 r .65397 .58446 .66561 .59398 .68961 .58024 .67817 .57071 Metetra .756 .834 .912 r .67817 .57071 .68961 .58024 .71373 .56643 .70248 .55691 Metetra .756 .837 .913 r .70248 .55691 .71373 .56643 .73795 .55256 .72691 .54303 Metetra .756 .839 .915 r .72691 .54303 .73795 .55256 .76229 .53863 .75146 .52909 Metetra .756 .842 .916 r .75146 .52909 .76229 .53863 .78674 .52463 .77613 .51508 Metetra .756 .845 .918 r .77613 .51508 .78674 .52463 .81131 .51056 .80091 .501 Metetra .757 .847 .919 r .80091 .501 .81131 .51056 .836 .49643 .82582 .48685 Metetra .757 .85 .92 r .82582 .48685 .836 .49643 .8608 .48223 .85084 .47264 Metetra .757 .853 .922 r .85084 .47264 .8608 .48223 .88572 .46796 .87599 .45836 Metetra .757 .855 .923 r .87599 .45836 .88572 .46796 .91076 .45363 .90127 .444 Metetra .757 .858 .924 r .90127 .444 .91076 .45363 .93592 .43923 .92666 .42958 Metetra .757 .86 .926 r .92666 .42958 .93592 .43923 .96119 .42476 .95218 .41508 Metetra .749 .802 .895 r .35844 .73463 .37237 .7444 .39523 .73142 .38146 .72165 Metetra .75 .804 .896 r .38146 .72165 .39523 .73142 .41821 .71837 .40459 .70862 Metetra .75 .807 .898 r .40459 .70862 .41821 .71837 .44129 .70526 .42783 .69552 Metetra .751 .809 .899 r .42783 .69552 .44129 .70526 .46448 .69209 .45118 .68236 Metetra .751 .811 .9 r .45118 .68236 .46448 .69209 .48777 .67885 .47465 .66914 Metetra .751 .814 .902 r .47465 .66914 .48777 .67885 .51118 .66556 .49822 .65585 Metetra .752 .816 .903 r .49822 .65585 .51118 .66556 .5347 .6522 .52192 .6425 Metetra .752 .819 .905 r .52192 .6425 .5347 .6522 .55833 .63878 .54572 .62909 Metetra .752 .821 .906 r .54572 .62909 .55833 .63878 .58207 .6253 .56965 .61561 Metetra .753 .824 .908 r .56965 .61561 .58207 .6253 .60592 .61175 .59369 .60206 Metetra .753 .826 .909 r .59369 .60206 .60592 .61175 .62989 .59814 .61785 .58845 Metetra .753 .829 .91 r .61785 .58845 .62989 .59814 .65397 .58446 .64212 .57477 Metetra .753 .831 .912 r .64212 .57477 .65397 .58446 .67817 .57071 .66652 .56102 Metetra .754 .834 .913 r .66652 .56102 .67817 .57071 .70248 .55691 .69103 .5472 Metetra .754 .836 .915 r .69103 .5472 .70248 .55691 .72691 .54303 .71567 .53332 Metetra .754 .839 .916 r .71567 .53332 .72691 .54303 .75146 .52909 .74043 .51937 Metetra .754 .842 .917 r .74043 .51937 .75146 .52909 .77613 .51508 .76531 .50535 Metetra .754 .844 .919 r .76531 .50535 .77613 .51508 .80091 .501 .79031 .49126 Metetra .755 .847 .92 r .79031 .49126 .80091 .501 .82582 .48685 .81544 .4771 Metetra .755 .849 .921 r .81544 .4771 .82582 .48685 .85084 .47264 .8407 .46287 Metetra .755 .852 .923 r .8407 .46287 .85084 .47264 .87599 .45836 .86608 .44856 Metetra .755 .855 .924 r .86608 .44856 .87599 .45836 .90127 .444 .89159 .43419 Metetra .755 .857 .925 r .89159 .43419 .90127 .444 .92666 .42958 .91722 .41974 Metetra .755 .86 .927 r .91722 .41974 .92666 .42958 .95218 .41508 .94299 .40522 Metetra .748 .802 .896 r .34428 .72468 .35844 .73463 .38146 .72165 .36745 .71173 Metetra .748 .804 .897 r .36745 .71173 .38146 .72165 .40459 .70862 .39074 .69871 Metetra .748 .806 .899 r .39074 .69871 .40459 .70862 .42783 .69552 .41414 .68562 Metetra .749 .809 .9 r .41414 .68562 .42783 .69552 .45118 .68236 .43766 .67247 Metetra .749 .811 .902 r .43766 .67247 .45118 .68236 .47465 .66914 .46129 .65926 Metetra .749 .814 .903 r .46129 .65926 .47465 .66914 .49822 .65585 .48504 .64598 Metetra .75 .816 .904 r .48504 .64598 .49822 .65585 .52192 .6425 .50891 .63263 Metetra .75 .819 .906 r .50891 .63263 .52192 .6425 .54572 .62909 .5329 .61922 Metetra .75 .821 .907 r .5329 .61922 .54572 .62909 .56965 .61561 .557 .60574 Metetra .751 .824 .909 r .557 .60574 .56965 .61561 .59369 .60206 .58123 .59219 Metetra .751 .826 .91 r .58123 .59219 .59369 .60206 .61785 .58845 .60558 .57858 Metetra .751 .829 .912 r .60558 .57858 .61785 .58845 .64212 .57477 .63005 .5649 Metetra .752 .831 .913 r .63005 .5649 .64212 .57477 .66652 .56102 .65465 .55114 Metetra .752 .834 .914 r .65465 .55114 .66652 .56102 .69103 .5472 .67937 .53732 Metetra .752 .836 .916 r .67937 .53732 .69103 .5472 .71567 .53332 .70421 .52343 Metetra .752 .839 .917 r .70421 .52343 .71567 .53332 .74043 .51937 .72918 .50947 Metetra .752 .841 .918 r .72918 .50947 .74043 .51937 .76531 .50535 .75428 .49543 Metetra .753 .844 .92 r .75428 .49543 .76531 .50535 .79031 .49126 .77951 .48133 Metetra .753 .846 .921 r .77951 .48133 .79031 .49126 .81544 .4771 .80486 .46715 Metetra .753 .849 .923 r .80486 .46715 .81544 .4771 .8407 .46287 .83035 .4529 Metetra .753 .852 .924 r .83035 .4529 .8407 .46287 .86608 .44856 .85597 .43858 Metetra .753 .854 .925 r .85597 .43858 .86608 .44856 .89159 .43419 .88172 .42418 Metetra .753 .857 .927 r .88172 .42418 .89159 .43419 .91722 .41974 .9076 .40971 Metetra .753 .86 .928 r .9076 .40971 .91722 .41974 .94299 .40522 .93362 .39516 Metetra .746 .802 .897 r .32988 .71457 .34428 .72468 .36745 .71173 .35321 .70163 Metetra .746 .804 .899 r .35321 .70163 .36745 .71173 .39074 .69871 .37665 .68862 Metetra .747 .806 .9 r .37665 .68862 .39074 .69871 .41414 .68562 .40021 .67555 Metetra .747 .809 .901 r .40021 .67555 .41414 .68562 .43766 .67247 .4239 .66241 Metetra .747 .811 .903 r .4239 .66241 .43766 .67247 .46129 .65926 .4477 .6492 Metetra .748 .814 .904 r .4477 .6492 .46129 .65926 .48504 .64598 .47162 .63593 Metetra .748 .816 .906 r .47162 .63593 .48504 .64598 .50891 .63263 .49567 .62259 Metetra .748 .818 .907 r .49567 .62259 .50891 .63263 .5329 .61922 .51984 .60918 Metetra .749 .821 .908 r .51984 .60918 .5329 .61922 .557 .60574 .54413 .5957 Metetra .749 .823 .91 r .54413 .5957 .557 .60574 .58123 .59219 .56855 .58215 Metetra .749 .826 .911 r .56855 .58215 .58123 .59219 .60558 .57858 .59309 .56853 Metetra .749 .828 .913 r .59309 .56853 .60558 .57858 .63005 .5649 .61776 .55484 Metetra .75 .831 .914 r .61776 .55484 .63005 .5649 .65465 .55114 .64256 .54108 Metetra .75 .833 .915 r .64256 .54108 .65465 .55114 .67937 .53732 .66748 .52725 Metetra .75 .836 .917 r .66748 .52725 .67937 .53732 .70421 .52343 .69254 .51335 Metetra .75 .838 .918 r .69254 .51335 .70421 .52343 .72918 .50947 .71773 .49938 Metetra .75 .841 .92 r .71773 .49938 .72918 .50947 .75428 .49543 .74304 .48533 Metetra .751 .844 .921 r .74304 .48533 .75428 .49543 .77951 .48133 .7685 .47121 Metetra .751 .846 .922 r .7685 .47121 .77951 .48133 .80486 .46715 .79408 .45701 Metetra .751 .849 .924 r .79408 .45701 .80486 .46715 .83035 .4529 .8198 .44274 Metetra .751 .851 .925 r .8198 .44274 .83035 .4529 .85597 .43858 .84566 .42839 Metetra .751 .854 .926 r .84566 .42839 .85597 .43858 .88172 .42418 .87165 .41397 Metetra .751 .856 .928 r .87165 .41397 .88172 .42418 .9076 .40971 .89778 .39947 Metetra .751 .859 .929 r .89778 .39947 .9076 .40971 .93362 .39516 .92405 .3849 Metetra .744 .802 .898 r .31523 .70429 .32988 .71457 .35321 .70163 .33871 .69136 Metetra .745 .804 .9 r .33871 .69136 .35321 .70163 .37665 .68862 .36232 .67836 Metetra .745 .806 .901 r .36232 .67836 .37665 .68862 .40021 .67555 .38604 .6653 Metetra .745 .809 .903 r .38604 .6653 .40021 .67555 .4239 .66241 .40989 .65217 Metetra .746 .811 .904 r .40989 .65217 .4239 .66241 .4477 .6492 .43387 .63897 Metetra .746 .813 .905 r .43387 .63897 .4477 .6492 .47162 .63593 .45796 .6257 Metetra .746 .816 .907 r .45796 .6257 .47162 .63593 .49567 .62259 .48219 .61236 Metetra .747 .818 .908 r .48219 .61236 .49567 .62259 .51984 .60918 .50654 .59895 Metetra .747 .821 .91 r .50654 .59895 .51984 .60918 .54413 .5957 .53102 .58547 Metetra .747 .823 .911 r .53102 .58547 .54413 .5957 .56855 .58215 .55563 .57192 Metetra .747 .826 .912 r .55563 .57192 .56855 .58215 .59309 .56853 .58037 .5583 Metetra .748 .828 .914 r .58037 .5583 .59309 .56853 .61776 .55484 .60524 .5446 Metetra .748 .831 .915 r .60524 .5446 .61776 .55484 .64256 .54108 .63024 .53084 Metetra .748 .833 .917 r .63024 .53084 .64256 .54108 .66748 .52725 .65537 .517 Metetra .748 .836 .918 r .65537 .517 .66748 .52725 .69254 .51335 .68064 .50308 Metetra .748 .838 .919 r .68064 .50308 .69254 .51335 .71773 .49938 .70605 .48909 Metetra .749 .841 .921 r .70605 .48909 .71773 .49938 .74304 .48533 .73159 .47503 Metetra .749 .843 .922 r .73159 .47503 .74304 .48533 .7685 .47121 .75727 .46089 Metetra .749 .846 .923 r .75727 .46089 .7685 .47121 .79408 .45701 .78309 .44667 Metetra .749 .848 .925 r .78309 .44667 .79408 .45701 .8198 .44274 .80904 .43238 Metetra .749 .851 .926 r .80904 .43238 .8198 .44274 .84566 .42839 .83514 .41801 Metetra .749 .853 .927 r .83514 .41801 .84566 .42839 .87165 .41397 .86138 .40356 Metetra .749 .856 .929 r .86138 .40356 .87165 .41397 .89778 .39947 .88777 .38903 Metetra .749 .858 .93 r .88777 .38903 .89778 .39947 .92405 .3849 .9143 .37442 Metetra .742 .802 .9 r .30033 .69383 .31523 .70429 .33871 .69136 .32397 .68091 Metetra .743 .804 .901 r .32397 .68091 .33871 .69136 .36232 .67836 .34773 .66792 Metetra .743 .806 .902 r .34773 .66792 .36232 .67836 .38604 .6653 .37162 .65487 Metetra .743 .809 .904 r .37162 .65487 .38604 .6653 .40989 .65217 .39564 .64174 Metetra .744 .811 .905 r .39564 .64174 .40989 .65217 .43387 .63897 .41978 .62855 Metetra .744 .813 .907 r .41978 .62855 .43387 .63897 .45796 .6257 .44406 .61528 Metetra .744 .816 .908 r .44406 .61528 .45796 .6257 .48219 .61236 .46846 .60194 Metetra .745 .818 .909 r .46846 .60194 .48219 .61236 .50654 .59895 .493 .58853 Metetra .745 .821 .911 r .493 .58853 .50654 .59895 .53102 .58547 .51767 .57505 Metetra .745 .823 .912 r .51767 .57505 .53102 .58547 .55563 .57192 .54247 .5615 Metetra .745 .825 .914 r .54247 .5615 .55563 .57192 .58037 .5583 .5674 .54787 Metetra .746 .828 .915 r .5674 .54787 .58037 .5583 .60524 .5446 .59248 .53417 Metetra .746 .83 .916 r .59248 .53417 .60524 .5446 .63024 .53084 .61768 .52039 Metetra .746 .833 .918 r .61768 .52039 .63024 .53084 .65537 .517 .64303 .50654 Metetra .746 .835 .919 r .64303 .50654 .65537 .517 .68064 .50308 .66852 .49261 Metetra .746 .838 .92 r .66852 .49261 .68064 .50308 .70605 .48909 .69414 .47861 Metetra .747 .84 .922 r .69414 .47861 .70605 .48909 .73159 .47503 .71991 .46452 Metetra .747 .843 .923 r .71991 .46452 .73159 .47503 .75727 .46089 .74582 .45036 Metetra .747 .845 .924 r .74582 .45036 .75727 .46089 .78309 .44667 .77187 .43613 Metetra .747 .848 .926 r .77187 .43613 .78309 .44667 .80904 .43238 .79807 .42181 Metetra .747 .85 .927 r .79807 .42181 .80904 .43238 .83514 .41801 .82441 .40741 Metetra .747 .853 .928 r .82441 .40741 .83514 .41801 .86138 .40356 .85091 .39293 Metetra .747 .855 .93 r .85091 .39293 .86138 .40356 .88777 .38903 .87755 .37837 Metetra .747 .858 .931 r .87755 .37837 .88777 .38903 .9143 .37442 .90434 .36373 Metetra .741 .801 .901 r .28517 .68319 .30033 .69383 .32397 .68091 .30897 .67028 Metetra .741 .804 .902 r .30897 .67028 .32397 .68091 .34773 .66792 .33289 .6573 Metetra .741 .806 .904 r .33289 .6573 .34773 .66792 .37162 .65487 .35695 .64425 Metetra .742 .808 .905 r .35695 .64425 .37162 .65487 .39564 .64174 .38113 .63113 Metetra .742 .811 .906 r .38113 .63113 .39564 .64174 .41978 .62855 .40545 .61794 Metetra .742 .813 .908 r .40545 .61794 .41978 .62855 .44406 .61528 .4299 .60468 Metetra .742 .816 .909 r .4299 .60468 .44406 .61528 .46846 .60194 .45449 .59134 Metetra .743 .818 .911 r .45449 .59134 .46846 .60194 .493 .58853 .47921 .57793 Metetra .743 .82 .912 r .47921 .57793 .493 .58853 .51767 .57505 .50407 .56444 Metetra .743 .823 .913 r .50407 .56444 .51767 .57505 .54247 .5615 .52906 .55088 Metetra .744 .825 .915 r .52906 .55088 .54247 .5615 .5674 .54787 .5542 .53725 Metetra .744 .828 .916 r .5542 .53725 .5674 .54787 .59248 .53417 .57947 .52354 Metetra .744 .83 .917 r .57947 .52354 .59248 .53417 .61768 .52039 .60489 .50975 Metetra .744 .832 .919 r .60489 .50975 .61768 .52039 .64303 .50654 .63045 .49588 Metetra .744 .835 .92 r .63045 .49588 .64303 .50654 .66852 .49261 .65615 .48194 Metetra .744 .837 .922 r .65615 .48194 .66852 .49261 .69414 .47861 .682 .46792 Metetra .745 .84 .923 r .682 .46792 .69414 .47861 .71991 .46452 .708 .45381 Metetra .745 .842 .924 r .708 .45381 .71991 .46452 .74582 .45036 .73414 .43963 Metetra .745 .845 .926 r .73414 .43963 .74582 .45036 .77187 .43613 .76043 .42537 Metetra .745 .847 .927 r .76043 .42537 .77187 .43613 .79807 .42181 .78688 .41103 Metetra .745 .85 .928 r .78688 .41103 .79807 .42181 .82441 .40741 .81347 .3966 Metetra .745 .852 .929 r .81347 .3966 .82441 .40741 .85091 .39293 .84021 .38209 Metetra .745 .855 .931 r .84021 .38209 .85091 .39293 .87755 .37837 .86711 .3675 Metetra .745 .857 .932 r .86711 .3675 .87755 .37837 .90434 .36373 .89417 .35282 Metetra .739 .801 .902 r .26975 .67236 .28517 .68319 .30897 .67028 .2937 .65946 Metetra .739 .804 .903 r .2937 .65946 .30897 .67028 .33289 .6573 .31779 .64649 Metetra .739 .806 .905 r .31779 .64649 .33289 .6573 .35695 .64425 .34201 .63344 Metetra .74 .808 .906 r .34201 .63344 .35695 .64425 .38113 .63113 .36636 .62033 Metetra .74 .811 .908 r .36636 .62033 .38113 .63113 .40545 .61794 .39085 .60714 Metetra .74 .813 .909 r .39085 .60714 .40545 .61794 .4299 .60468 .41548 .59388 Metetra .741 .815 .91 r .41548 .59388 .4299 .60468 .45449 .59134 .44025 .58054 Metetra .741 .818 .912 r .44025 .58054 .45449 .59134 .47921 .57793 .46516 .56712 Metetra .741 .82 .913 r .46516 .56712 .47921 .57793 .50407 .56444 .49021 .55363 Metetra .741 .822 .915 r .49021 .55363 .50407 .56444 .52906 .55088 .5154 .54007 Metetra .742 .825 .916 r .5154 .54007 .52906 .55088 .5542 .53725 .54074 .52642 Metetra .742 .827 .917 r .54074 .52642 .5542 .53725 .57947 .52354 .56622 .5127 Metetra .742 .83 .919 r .56622 .5127 .57947 .52354 .60489 .50975 .59185 .4989 Metetra .742 .832 .92 r .59185 .4989 .60489 .50975 .63045 .49588 .61762 .48502 Metetra .742 .835 .921 r .61762 .48502 .63045 .49588 .65615 .48194 .64355 .47105 Metetra .742 .837 .923 r .64355 .47105 .65615 .48194 .682 .46792 .66962 .45701 Metetra .743 .839 .924 r .66962 .45701 .682 .46792 .708 .45381 .69585 .44289 Metetra .743 .842 .925 r .69585 .44289 .708 .45381 .73414 .43963 .72223 .42868 Metetra .743 .844 .927 r .72223 .42868 .73414 .43963 .76043 .42537 .74876 .41439 Metetra .743 .847 .928 r .74876 .41439 .76043 .42537 .78688 .41103 .77545 .40002 Metetra .743 .849 .929 r .77545 .40002 .78688 .41103 .81347 .3966 .8023 .38556 Metetra .743 .852 .931 r .8023 .38556 .81347 .3966 .84021 .38209 .8293 .37102 Metetra .743 .854 .932 r .8293 .37102 .84021 .38209 .86711 .3675 .85646 .35639 Metetra .743 .857 .933 r .85646 .35639 .86711 .3675 .89417 .35282 .88379 .34168 Metetra .737 .801 .903 r .25405 .66133 .26975 .67236 .2937 .65946 .27816 .64844 Metetra .737 .803 .905 r .27816 .64844 .2937 .65946 .31779 .64649 .30241 .63548 Metetra .738 .806 .906 r .30241 .63548 .31779 .64649 .34201 .63344 .32679 .62244 Metetra .738 .808 .907 r .32679 .62244 .34201 .63344 .36636 .62033 .35132 .60933 Metetra .738 .81 .909 r .35132 .60933 .36636 .62033 .39085 .60714 .37599 .59614 Metetra .738 .813 .91 r .37599 .59614 .39085 .60714 .41548 .59388 .4008 .58287 Metetra .739 .815 .912 r .4008 .58287 .41548 .59388 .44025 .58054 .42575 .56953 Metetra .739 .817 .913 r .42575 .56953 .44025 .58054 .46516 .56712 .45085 .55611 Metetra .739 .82 .914 r .45085 .55611 .46516 .56712 .49021 .55363 .47609 .54262 Metetra .739 .822 .916 r .47609 .54262 .49021 .55363 .5154 .54007 .50148 .52904 Metetra .74 .825 .917 r .50148 .52904 .5154 .54007 .54074 .52642 .52702 .51539 Metetra .74 .827 .918 r .52702 .51539 .54074 .52642 .56622 .5127 .55271 .50165 Metetra .74 .829 .92 r .55271 .50165 .56622 .5127 .59185 .4989 .57855 .48783 Metetra .74 .832 .921 r .57855 .48783 .59185 .4989 .61762 .48502 .60455 .47394 Metetra .74 .834 .922 r .60455 .47394 .61762 .48502 .64355 .47105 .63069 .45995 Metetra .74 .837 .924 r .63069 .45995 .64355 .47105 .66962 .45701 .657 .44589 Metetra .741 .839 .925 r .657 .44589 .66962 .45701 .69585 .44289 .68346 .43174 Metetra .741 .842 .926 r .68346 .43174 .69585 .44289 .72223 .42868 .71007 .41751 Metetra .741 .844 .928 r .71007 .41751 .72223 .42868 .74876 .41439 .73685 .40319 Metetra .741 .846 .929 r .73685 .40319 .74876 .41439 .77545 .40002 .76379 .38879 Metetra .741 .849 .93 r .76379 .38879 .77545 .40002 .8023 .38556 .79089 .3743 Metetra .741 .851 .932 r .79089 .3743 .8023 .38556 .8293 .37102 .81816 .35972 Metetra .741 .854 .933 r .81816 .35972 .8293 .37102 .85646 .35639 .84559 .34506 Metetra .741 .856 .934 r .84559 .34506 .85646 .35639 .88379 .34168 .87319 .3303 Metetra .735 .801 .905 r .23807 .65012 .25405 .66133 .27816 .64844 .26234 .63723 Metetra .735 .803 .906 r .26234 .63723 .27816 .64844 .30241 .63548 .28675 .62427 Metetra .736 .806 .907 r .28675 .62427 .30241 .63548 .32679 .62244 .3113 .61124 Metetra .736 .808 .909 r .3113 .61124 .32679 .62244 .35132 .60933 .336 .59812 Metetra .736 .81 .91 r .336 .59812 .35132 .60933 .37599 .59614 .36084 .58494 Metetra .737 .813 .911 r .36084 .58494 .37599 .59614 .4008 .58287 .38583 .57167 Metetra .737 .815 .913 r .38583 .57167 .4008 .58287 .42575 .56953 .41097 .55832 Metetra .737 .817 .914 r .41097 .55832 .42575 .56953 .45085 .55611 .43626 .5449 Metetra .737 .82 .916 r .43626 .5449 .45085 .55611 .47609 .54262 .4617 .53139 Metetra .737 .822 .917 r .4617 .53139 .47609 .54262 .50148 .52904 .48729 .5178 Metetra .738 .824 .918 r .48729 .5178 .50148 .52904 .52702 .51539 .51304 .50414 Metetra .738 .827 .92 r .51304 .50414 .52702 .51539 .55271 .50165 .53894 .49038 Metetra .738 .829 .921 r .53894 .49038 .55271 .50165 .57855 .48783 .56499 .47655 Metetra .738 .831 .922 r .56499 .47655 .57855 .48783 .60455 .47394 .59121 .46263 Metetra .738 .834 .924 r .59121 .46263 .60455 .47394 .63069 .45995 .61758 .44863 Metetra .738 .836 .925 r .61758 .44863 .63069 .45995 .657 .44589 .64411 .43454 Metetra .738 .839 .926 r .64411 .43454 .657 .44589 .68346 .43174 .67081 .42037 Metetra .739 .841 .928 r .67081 .42037 .68346 .43174 .71007 .41751 .69767 .40611 Metetra .739 .844 .929 r .69767 .40611 .71007 .41751 .73685 .40319 .7247 .39176 Metetra .739 .846 .93 r .7247 .39176 .73685 .40319 .76379 .38879 .75189 .37733 Metetra .739 .848 .931 r .75189 .37733 .76379 .38879 .79089 .3743 .77925 .3628 Metetra .739 .851 .933 r .77925 .3628 .79089 .3743 .81816 .35972 .80678 .34818 Metetra .739 .853 .934 r .80678 .34818 .81816 .35972 .84559 .34506 .83448 .33348 Metetra .739 .856 .935 r .83448 .33348 .84559 .34506 .87319 .3303 .86236 .31868 Metetra .733 .801 .906 r .22181 .6387 .23807 .65012 .26234 .63723 .24623 .62582 Metetra .733 .803 .907 r .24623 .62582 .26234 .63723 .28675 .62427 .27081 .61286 Metetra .734 .805 .909 r .27081 .61286 .28675 .62427 .3113 .61124 .29553 .59983 Metetra .734 .808 .91 r .29553 .59983 .3113 .61124 .336 .59812 .3204 .58671 Metetra .734 .81 .911 r .3204 .58671 .336 .59812 .36084 .58494 .34542 .57352 Metetra .735 .812 .913 r .34542 .57352 .36084 .58494 .38583 .57167 .37059 .56025 Metetra .735 .815 .914 r .37059 .56025 .38583 .57167 .41097 .55832 .39592 .5469 Metetra .735 .817 .915 r .39592 .5469 .41097 .55832 .43626 .5449 .42139 .53346 Metetra .735 .819 .917 r .42139 .53346 .43626 .5449 .4617 .53139 .44703 .51995 Metetra .735 .822 .918 r .44703 .51995 .4617 .53139 .48729 .5178 .47282 .50635 Metetra .736 .824 .919 r .47282 .50635 .48729 .5178 .51304 .50414 .49877 .49266 Metetra .736 .826 .921 r .49877 .49266 .51304 .50414 .53894 .49038 .52489 .4789 Metetra .736 .829 .922 r .52489 .4789 .53894 .49038 .56499 .47655 .55116 .46504 Metetra .736 .831 .923 r .55116 .46504 .56499 .47655 .59121 .46263 .5776 .45111 Metetra .736 .833 .925 r .5776 .45111 .59121 .46263 .61758 .44863 .6042 .43708 Metetra .736 .836 .926 r .6042 .43708 .61758 .44863 .64411 .43454 .63097 .42297 Metetra .736 .838 .927 r .63097 .42297 .64411 .43454 .67081 .42037 .6579 .40876 Metetra .737 .841 .929 r .6579 .40876 .67081 .42037 .69767 .40611 .68501 .39447 Metetra .737 .843 .93 r .68501 .39447 .69767 .40611 .7247 .39176 .71228 .38009 Metetra .737 .845 .931 r .71228 .38009 .7247 .39176 .75189 .37733 .73973 .36562 Metetra .737 .848 .932 r .73973 .36562 .75189 .37733 .77925 .3628 .76736 .35105 Metetra .737 .85 .934 r .76736 .35105 .77925 .3628 .80678 .34818 .79516 .3364 Metetra .737 .853 .935 r .79516 .3364 .80678 .34818 .83448 .33348 .82313 .32165 Metetra .737 .855 .936 r .82313 .32165 .83448 .33348 .86236 .31868 .85129 .3068 Metetra .731 .801 .907 r .20524 .62707 .22181 .6387 .24623 .62582 .22983 .61419 Metetra .732 .803 .908 r .22983 .61419 .24623 .62582 .27081 .61286 .25457 .60124 Metetra .732 .805 .91 r .25457 .60124 .27081 .61286 .29553 .59983 .27946 .5882 Metetra .732 .808 .911 r .27946 .5882 .29553 .59983 .3204 .58671 .3045 .57509 Metetra .732 .81 .912 r .3045 .57509 .3204 .58671 .34542 .57352 .3297 .56189 Metetra .733 .812 .914 r .3297 .56189 .34542 .57352 .37059 .56025 .35506 .54861 Metetra .733 .814 .915 r .35506 .54861 .37059 .56025 .39592 .5469 .38057 .53525 Metetra .733 .817 .917 r .38057 .53525 .39592 .5469 .42139 .53346 .40624 .52181 Metetra .733 .819 .918 r .40624 .52181 .42139 .53346 .44703 .51995 .43207 .50828 Metetra .733 .821 .919 r .43207 .50828 .44703 .51995 .47282 .50635 .45807 .49466 Metetra .734 .824 .921 r .45807 .49466 .47282 .50635 .49877 .49266 .48423 .48096 Metetra .734 .826 .922 r .48423 .48096 .49877 .49266 .52489 .4789 .51056 .46718 Metetra .734 .828 .923 r .51056 .46718 .52489 .4789 .55116 .46504 .53705 .4533 Metetra .734 .831 .925 r .53705 .4533 .55116 .46504 .5776 .45111 .56371 .43934 Metetra .734 .833 .926 r .56371 .43934 .5776 .45111 .6042 .43708 .59054 .42529 Metetra .734 .835 .927 r .59054 .42529 .6042 .43708 .63097 .42297 .61755 .41115 Metetra .734 .838 .928 r .61755 .41115 .63097 .42297 .6579 .40876 .64473 .39692 Metetra .734 .84 .93 r .64473 .39692 .6579 .40876 .68501 .39447 .67208 .38259 Metetra .734 .843 .931 r .67208 .38259 .68501 .39447 .71228 .38009 .69961 .36817 Metetra .734 .845 .932 r .69961 .36817 .71228 .38009 .73973 .36562 .72732 .35366 Metetra .734 .847 .933 r .72732 .35366 .73973 .36562 .76736 .35105 .75521 .33906 Metetra .734 .85 .935 r .75521 .33906 .76736 .35105 .79516 .3364 .78328 .32436 Metetra .734 .852 .936 r .78328 .32436 .79516 .3364 .82313 .32165 .81154 .30956 Metetra .734 .855 .937 r .81154 .30956 .82313 .32165 .85129 .3068 .83998 .29466 Metetra .729 .801 .908 r .18838 .61522 .20524 .62707 .22983 .61419 .21312 .60235 Metetra .73 .803 .91 r .21312 .60235 .22983 .61419 .25457 .60124 .23803 .5894 Metetra .73 .805 .911 r .23803 .5894 .25457 .60124 .27946 .5882 .26309 .57636 Metetra .73 .807 .912 r .26309 .57636 .27946 .5882 .3045 .57509 .2883 .56324 Metetra .73 .81 .914 r .2883 .56324 .3045 .57509 .3297 .56189 .31368 .55004 Metetra .731 .812 .915 r .31368 .55004 .3297 .56189 .35506 .54861 .33922 .53675 Metetra .731 .814 .916 r .33922 .53675 .35506 .54861 .38057 .53525 .36492 .52338 Metetra .731 .816 .918 r .36492 .52338 .38057 .53525 .40624 .52181 .39079 .50992 Metetra .731 .819 .919 r .39079 .50992 .40624 .52181 .43207 .50828 .41682 .49638 Metetra .731 .821 .92 r .41682 .49638 .43207 .50828 .45807 .49466 .44303 .48275 Metetra .732 .823 .922 r .44303 .48275 .45807 .49466 .48423 .48096 .4694 .46903 Metetra .732 .826 .923 r .4694 .46903 .48423 .48096 .51056 .46718 .49594 .45522 Metetra .732 .828 .924 r .49594 .45522 .51056 .46718 .53705 .4533 .52265 .44133 Metetra .732 .83 .926 r .52265 .44133 .53705 .4533 .56371 .43934 .54954 .42734 Metetra .732 .833 .927 r .54954 .42734 .56371 .43934 .59054 .42529 .57661 .41326 Metetra .732 .835 .928 r .57661 .41326 .59054 .42529 .61755 .41115 .60385 .39908 Metetra .732 .837 .93 r .60385 .39908 .61755 .41115 .64473 .39692 .63127 .38482 Metetra .732 .84 .931 r .63127 .38482 .64473 .39692 .67208 .38259 .65888 .37046 Metetra .732 .842 .932 r .65888 .37046 .67208 .38259 .69961 .36817 .68667 .356 Metetra .732 .844 .933 r .68667 .356 .69961 .36817 .72732 .35366 .71464 .34145 Metetra .732 .847 .935 r .71464 .34145 .72732 .35366 .75521 .33906 .7428 .3268 Metetra .732 .849 .936 r .7428 .3268 .75521 .33906 .78328 .32436 .77115 .31205 Metetra .732 .852 .937 r .77115 .31205 .78328 .32436 .81154 .30956 .79969 .2972 Metetra .732 .854 .938 r .79969 .2972 .81154 .30956 .83998 .29466 .82842 .28226 Metetra .727 .801 .91 r .1712 .60316 .18838 .61522 .21312 .60235 .1961 .59029 Metetra .728 .803 .911 r .1961 .59029 .21312 .60235 .23803 .5894 .22117 .57733 Metetra .728 .805 .912 r .22117 .57733 .23803 .5894 .26309 .57636 .2464 .56429 Metetra .728 .807 .914 r .2464 .56429 .26309 .57636 .2883 .56324 .27179 .55117 Metetra .728 .809 .915 r .27179 .55117 .2883 .56324 .31368 .55004 .29735 .53796 Metetra .729 .812 .916 r .29735 .53796 .31368 .55004 .33922 .53675 .32308 .52466 Metetra .729 .814 .918 r .32308 .52466 .33922 .53675 .36492 .52338 .34897 .51128 Metetra .729 .816 .919 r .34897 .51128 .36492 .52338 .39079 .50992 .37503 .49781 Metetra .729 .818 .92 r .37503 .49781 .39079 .50992 .41682 .49638 .40127 .48425 Metetra .729 .821 .922 r .40127 .48425 .41682 .49638 .44303 .48275 .42768 .4706 Metetra .729 .823 .923 r .42768 .4706 .44303 .48275 .4694 .46903 .45426 .45686 Metetra .73 .825 .924 r .45426 .45686 .4694 .46903 .49594 .45522 .48102 .44302 Metetra .73 .828 .925 r .48102 .44302 .49594 .45522 .52265 .44133 .50796 .4291 Metetra .73 .83 .927 r .50796 .4291 .52265 .44133 .54954 .42734 .53508 .41508 Metetra .73 .832 .928 r .53508 .41508 .54954 .42734 .57661 .41326 .56238 .40097 Metetra .73 .835 .929 r .56238 .40097 .57661 .41326 .60385 .39908 .58986 .38677 Metetra .73 .837 .931 r .58986 .38677 .60385 .39908 .63127 .38482 .61754 .37246 Metetra .73 .839 .932 r .61754 .37246 .63127 .38482 .65888 .37046 .64539 .35806 Metetra .73 .842 .933 r .64539 .35806 .65888 .37046 .68667 .356 .67344 .34357 Metetra .73 .844 .934 r .67344 .34357 .68667 .356 .71464 .34145 .70168 .32897 Metetra .73 .846 .936 r .70168 .32897 .71464 .34145 .7428 .3268 .73012 .31427 Metetra .73 .849 .937 r .73012 .31427 .7428 .3268 .77115 .31205 .75875 .29947 Metetra .73 .851 .938 r .75875 .29947 .77115 .31205 .79969 .2972 .78758 .28457 Metetra .73 .853 .939 r .78758 .28457 .79969 .2972 .82842 .28226 .8166 .26957 Metetra .725 .8 .911 r .1537 .59087 .1712 .60316 .1961 .59029 .17876 .578 Metetra .726 .803 .912 r .17876 .578 .1961 .59029 .22117 .57733 .204 .56504 Metetra .726 .805 .913 r .204 .56504 .22117 .57733 .2464 .56429 .2294 .55199 Metetra .726 .807 .915 r .2294 .55199 .2464 .56429 .27179 .55117 .25497 .53886 Metetra .726 .809 .916 r .25497 .53886 .27179 .55117 .29735 .53796 .28071 .52564 Metetra .727 .811 .917 r .28071 .52564 .29735 .53796 .32308 .52466 .30662 .51233 Metetra .727 .814 .919 r .30662 .51233 .32308 .52466 .34897 .51128 .3327 .49893 Metetra .727 .816 .92 r .3327 .49893 .34897 .51128 .37503 .49781 .35896 .48544 Metetra .727 .818 .921 r .35896 .48544 .37503 .49781 .40127 .48425 .3854 .47187 Metetra .727 .82 .923 r .3854 .47187 .40127 .48425 .42768 .4706 .41202 .45819 Metetra .727 .823 .924 r .41202 .45819 .42768 .4706 .45426 .45686 .43881 .44443 Metetra .728 .825 .925 r .43881 .44443 .45426 .45686 .48102 .44302 .46579 .43057 Metetra .728 .827 .927 r .46579 .43057 .48102 .44302 .50796 .4291 .49296 .41662 Metetra .728 .83 .928 r .49296 .41662 .50796 .4291 .53508 .41508 .52031 .40257 Metetra .728 .832 .929 r .52031 .40257 .53508 .41508 .56238 .40097 .54785 .38843 Metetra .728 .834 .93 r .54785 .38843 .56238 .40097 .58986 .38677 .57558 .37418 Metetra .728 .836 .932 r .57558 .37418 .58986 .38677 .61754 .37246 .6035 .35984 Metetra .728 .839 .933 r .6035 .35984 .61754 .37246 .64539 .35806 .63162 .3454 Metetra .728 .841 .934 r .63162 .3454 .64539 .35806 .67344 .34357 .65993 .33086 Metetra .728 .843 .935 r .65993 .33086 .67344 .34357 .70168 .32897 .68844 .31621 Metetra .728 .846 .937 r .68844 .31621 .70168 .32897 .73012 .31427 .71715 .30147 Metetra .728 .848 .938 r .71715 .30147 .73012 .31427 .75875 .29947 .74607 .28662 Metetra .728 .85 .939 r .74607 .28662 .75875 .29947 .78758 .28457 .77519 .27166 Metetra .728 .853 .94 r .77519 .27166 .78758 .28457 .8166 .26957 .80451 .2566 Metetra .723 .8 .912 r .13587 .57836 .1537 .59087 .17876 .578 .16109 .56548 Metetra .724 .802 .913 r .16109 .56548 .17876 .578 .204 .56504 .18649 .55251 Metetra .724 .805 .915 r .18649 .55251 .204 .56504 .2294 .55199 .21207 .53946 Metetra .724 .807 .916 r .21207 .53946 .2294 .55199 .25497 .53886 .23781 .52631 Metetra .724 .809 .917 r .23781 .52631 .25497 .53886 .28071 .52564 .26373 .51308 Metetra .724 .811 .919 r .26373 .51308 .28071 .52564 .30662 .51233 .28983 .49976 Metetra .725 .813 .92 r .28983 .49976 .30662 .51233 .3327 .49893 .31611 .48634 Metetra .725 .816 .921 r .31611 .48634 .3327 .49893 .35896 .48544 .34257 .47283 Metetra .725 .818 .923 r .34257 .47283 .35896 .48544 .3854 .47187 .36921 .45923 Metetra .725 .82 .924 r .36921 .45923 .3854 .47187 .41202 .45819 .39604 .44554 Metetra .725 .822 .925 r .39604 .44554 .41202 .45819 .43881 .44443 .42305 .43175 Metetra .725 .825 .926 r .42305 .43175 .43881 .44443 .46579 .43057 .45025 .41786 Metetra .726 .827 .928 r .45025 .41786 .46579 .43057 .49296 .41662 .47764 .40388 Metetra .726 .829 .929 r .47764 .40388 .49296 .41662 .52031 .40257 .50523 .3898 Metetra .726 .831 .93 r .50523 .3898 .52031 .40257 .54785 .38843 .53301 .37561 Metetra .726 .834 .932 r .53301 .37561 .54785 .38843 .57558 .37418 .56099 .36133 Metetra .726 .836 .933 r .56099 .36133 .57558 .37418 .6035 .35984 .58916 .34695 Metetra .726 .838 .934 r .58916 .34695 .6035 .35984 .63162 .3454 .61754 .33246 Metetra .726 .841 .935 r .61754 .33246 .63162 .3454 .65993 .33086 .64612 .31787 Metetra .726 .843 .936 r .64612 .31787 .65993 .33086 .68844 .31621 .67491 .30318 Metetra .726 .845 .938 r .67491 .30318 .68844 .31621 .71715 .30147 .7039 .28838 Metetra .726 .847 .939 r .7039 .28838 .71715 .30147 .74607 .28662 .7331 .27347 Metetra .726 .85 .94 r .7331 .27347 .74607 .28662 .77519 .27166 .76252 .25845 Metetra .726 .852 .941 r .76252 .25845 .77519 .27166 .80451 .2566 .79215 .24332 Metetra .721 .8 .913 r .11769 .5656 .13587 .57836 .16109 .56548 .14308 .55271 Metetra .721 .802 .915 r .14308 .55271 .16109 .56548 .18649 .55251 .16865 .53974 Metetra .722 .804 .916 r .16865 .53974 .18649 .55251 .21207 .53946 .19439 .52667 Metetra .722 .807 .917 r .19439 .52667 .21207 .53946 .23781 .52631 .22032 .51352 Metetra .722 .809 .919 r .22032 .51352 .23781 .52631 .26373 .51308 .24642 .50027 Metetra .722 .811 .92 r .24642 .50027 .26373 .51308 .28983 .49976 .27271 .48693 Metetra .723 .813 .921 r .27271 .48693 .28983 .49976 .31611 .48634 .29918 .4735 Metetra .723 .815 .922 r .29918 .4735 .31611 .48634 .34257 .47283 .32584 .45997 Metetra .723 .818 .924 r .32584 .45997 .34257 .47283 .36921 .45923 .35269 .44634 Metetra .723 .82 .925 r .35269 .44634 .36921 .45923 .39604 .44554 .37972 .43262 Metetra .723 .822 .926 r .37972 .43262 .39604 .44554 .42305 .43175 .40695 .4188 Metetra .723 .824 .928 r .40695 .4188 .42305 .43175 .45025 .41786 .43438 .40488 Metetra .723 .826 .929 r .43438 .40488 .45025 .41786 .47764 .40388 .462 .39087 Metetra .723 .829 .93 r .462 .39087 .47764 .40388 .50523 .3898 .48983 .37675 Metetra .723 .831 .931 r .48983 .37675 .50523 .3898 .53301 .37561 .51785 .36252 Metetra .724 .833 .933 r .51785 .36252 .53301 .37561 .56099 .36133 .54608 .3482 Metetra .724 .835 .934 r .54608 .3482 .56099 .36133 .58916 .34695 .57451 .33377 Metetra .724 .838 .935 r .57451 .33377 .58916 .34695 .61754 .33246 .60315 .31924 Metetra .724 .84 .936 r .60315 .31924 .61754 .33246 .64612 .31787 .632 .30459 Metetra .724 .842 .938 r .632 .30459 .64612 .31787 .67491 .30318 .66107 .28984 Metetra .724 .845 .939 r .66107 .28984 .67491 .30318 .7039 .28838 .69034 .27499 Metetra .723 .847 .94 r .69034 .27499 .7039 .28838 .7331 .27347 .71984 .26002 Metetra .723 .849 .941 r .71984 .26002 .7331 .27347 .76252 .25845 .74956 .24494 Metetra .723 .851 .942 r .74956 .24494 .76252 .25845 .79215 .24332 .77949 .22974 Metetra .719 .8 .915 r .09917 .55259 .11769 .5656 .14308 .55271 .12473 .5397 Metetra .719 .802 .916 r .12473 .5397 .14308 .55271 .16865 .53974 .15046 .52672 Metetra .72 .804 .917 r .15046 .52672 .16865 .53974 .19439 .52667 .17637 .51364 Metetra .72 .806 .918 r .17637 .51364 .19439 .52667 .22032 .51352 .20247 .50047 Metetra .72 .808 .92 r .20247 .50047 .22032 .51352 .24642 .50027 .22876 .4872 Metetra .72 .811 .921 r .22876 .4872 .24642 .50027 .27271 .48693 .25524 .47384 Metetra .72 .813 .922 r .25524 .47384 .27271 .48693 .29918 .4735 .2819 .46039 Metetra .721 .815 .924 r .2819 .46039 .29918 .4735 .32584 .45997 .30876 .44684 Metetra .721 .817 .925 r .30876 .44684 .32584 .45997 .35269 .44634 .33582 .43318 Metetra .721 .819 .926 r .33582 .43318 .35269 .44634 .37972 .43262 .36307 .41943 Metetra .721 .822 .927 r .36307 .41943 .37972 .43262 .40695 .4188 .39052 .40558 Metetra .721 .824 .929 r .39052 .40558 .40695 .4188 .43438 .40488 .41817 .39163 Metetra .721 .826 .93 r .41817 .39163 .43438 .40488 .462 .39087 .44603 .37757 Metetra .721 .828 .931 r .44603 .37757 .462 .39087 .48983 .37675 .47409 .36341 Metetra .721 .83 .932 r .47409 .36341 .48983 .37675 .51785 .36252 .50236 .34915 Metetra .721 .833 .934 r .50236 .34915 .51785 .36252 .54608 .3482 .53084 .33478 Metetra .721 .835 .935 r .53084 .33478 .54608 .3482 .57451 .33377 .55953 .3203 Metetra .721 .837 .936 r .55953 .3203 .57451 .33377 .60315 .31924 .58844 .30571 Metetra .721 .839 .937 r .58844 .30571 .60315 .31924 .632 .30459 .61756 .29102 Metetra .721 .842 .939 r .61756 .29102 .632 .30459 .66107 .28984 .64691 .27621 Metetra .721 .844 .94 r .64691 .27621 .66107 .28984 .69034 .27499 .67648 .26129 Metetra .721 .846 .941 r .67648 .26129 .69034 .27499 .71984 .26002 .70627 .24626 Metetra .721 .848 .942 r .70627 .24626 .71984 .26002 .74956 .24494 .73629 .23111 Metetra .721 .851 .943 r .73629 .23111 .74956 .24494 .77949 .22974 .76654 .21584 Metetra .717 .8 .916 r .08029 .53934 .09917 .55259 .12473 .5397 .10601 .52643 Metetra .717 .802 .917 r .10601 .52643 .12473 .5397 .15046 .52672 .13191 .51344 Metetra .717 .804 .918 r .13191 .51344 .15046 .52672 .17637 .51364 .15799 .50034 Metetra .718 .806 .92 r .15799 .50034 .17637 .51364 .20247 .50047 .18427 .48716 Metetra .718 .808 .921 r .18427 .48716 .20247 .50047 .22876 .4872 .21074 .47387 Metetra .718 .81 .922 r .21074 .47387 .22876 .4872 .25524 .47384 .23741 .46049 Metetra .718 .812 .924 r .23741 .46049 .25524 .47384 .2819 .46039 .26427 .44701 Metetra .718 .815 .925 r .26427 .44701 .2819 .46039 .30876 .44684 .29133 .43343 Metetra .719 .817 .926 r .29133 .43343 .30876 .44684 .33582 .43318 .3186 .41975 Metetra .719 .819 .927 r .3186 .41975 .33582 .43318 .36307 .41943 .34606 .40597 Metetra .719 .821 .929 r .34606 .40597 .36307 .41943 .39052 .40558 .37373 .39208 Metetra .719 .823 .93 r .37373 .39208 .39052 .40558 .41817 .39163 .40161 .37809 Metetra .719 .826 .931 r .40161 .37809 .41817 .39163 .44603 .37757 .4297 .36399 Metetra .719 .828 .932 r .4297 .36399 .44603 .37757 .47409 .36341 .45801 .34979 Metetra .719 .83 .934 r .45801 .34979 .47409 .36341 .50236 .34915 .48652 .33548 Metetra .719 .832 .935 r .48652 .33548 .50236 .34915 .53084 .33478 .51526 .32106 Metetra .719 .834 .936 r .51526 .32106 .53084 .33478 .55953 .3203 .54422 .30653 Metetra .719 .837 .937 r .54422 .30653 .55953 .3203 .58844 .30571 .57339 .29189 Metetra .719 .839 .938 r .57339 .29189 .58844 .30571 .61756 .29102 .6028 .27713 Metetra .719 .841 .94 r .6028 .27713 .61756 .29102 .64691 .27621 .63243 .26226 Metetra .719 .843 .941 r .63243 .26226 .64691 .27621 .67648 .26129 .66229 .24728 Metetra .719 .846 .942 r .66229 .24728 .67648 .26129 .70627 .24626 .69238 .23217 Metetra .719 .848 .943 r .69238 .23217 .70627 .24626 .73629 .23111 .72271 .21695 Metetra .719 .85 .944 r .72271 .21695 .73629 .23111 .76654 .21584 .75328 .20161 Metetra .715 .799 .917 r .06104 .52582 .08029 .53934 .10601 .52643 .08692 .5129 Metetra .715 .801 .918 r .08692 .5129 .10601 .52643 .13191 .51344 .11298 .49989 Metetra .715 .804 .92 r .11298 .49989 .13191 .51344 .15799 .50034 .13924 .48678 Metetra .716 .806 .921 r .13924 .48678 .15799 .50034 .18427 .48716 .1657 .47357 Metetra .716 .808 .922 r .1657 .47357 .18427 .48716 .21074 .47387 .19236 .46027 Metetra .716 .81 .923 r .19236 .46027 .21074 .47387 .23741 .46049 .21922 .44686 Metetra .716 .812 .925 r .21922 .44686 .23741 .46049 .26427 .44701 .24627 .43335 Metetra .716 .814 .926 r .24627 .43335 .26427 .44701 .29133 .43343 .27354 .41974 Metetra .716 .816 .927 r .27354 .41974 .29133 .43343 .3186 .41975 .30101 .40603 Metetra .716 .819 .929 r .30101 .40603 .3186 .41975 .34606 .40597 .32869 .39221 Metetra .717 .821 .93 r .32869 .39221 .34606 .40597 .37373 .39208 .35659 .37829 Metetra .717 .823 .931 r .35659 .37829 .37373 .39208 .40161 .37809 .3847 .36425 Metetra .717 .825 .932 r .3847 .36425 .40161 .37809 .4297 .36399 .41302 .35011 Metetra .717 .827 .933 r .41302 .35011 .4297 .36399 .45801 .34979 .44157 .33586 Metetra .717 .83 .935 r .44157 .33586 .45801 .34979 .48652 .33548 .47034 .3215 Metetra .717 .832 .936 r .47034 .3215 .48652 .33548 .51526 .32106 .49933 .30703 Metetra .717 .834 .937 r .49933 .30703 .51526 .32106 .54422 .30653 .52855 .29244 Metetra .717 .836 .938 r .52855 .29244 .54422 .30653 .57339 .29189 .558 .27774 Metetra .717 .838 .939 r .558 .27774 .57339 .29189 .6028 .27713 .58769 .26292 Metetra .717 .841 .941 r .58769 .26292 .6028 .27713 .63243 .26226 .61761 .24799 Metetra .717 .843 .942 r .61761 .24799 .63243 .26226 .66229 .24728 .64777 .23293 Metetra .717 .845 .943 r .64777 .23293 .66229 .24728 .69238 .23217 .67817 .21776 Metetra .717 .847 .944 r .67817 .21776 .69238 .23217 .72271 .21695 .70881 .20246 Metetra .716 .849 .945 r .70881 .20246 .72271 .21695 .75328 .20161 .7397 .18704 Metetra .713 .799 .918 r .04141 .51204 .06104 .52582 .08692 .5129 .06744 .4991 Metetra .713 .801 .92 r .06744 .4991 .08692 .5129 .11298 .49989 .09368 .48607 Metetra .713 .803 .921 r .09368 .48607 .11298 .49989 .13924 .48678 .12011 .47294 Metetra .713 .805 .922 r .12011 .47294 .13924 .48678 .1657 .47357 .14675 .45971 Metetra .713 .808 .923 r .14675 .45971 .1657 .47357 .19236 .46027 .17359 .44638 Metetra .714 .81 .925 r .17359 .44638 .19236 .46027 .21922 .44686 .20064 .43295 Metetra .714 .812 .926 r .20064 .43295 .21922 .44686 .24627 .43335 .2279 .41941 Metetra .714 .814 .927 r .2279 .41941 .24627 .43335 .27354 .41974 .25537 .40577 Metetra .714 .816 .928 r .25537 .40577 .27354 .41974 .30101 .40603 .28305 .39202 Metetra .714 .818 .93 r .28305 .39202 .30101 .40603 .32869 .39221 .31095 .37816 Metetra .714 .82 .931 r .31095 .37816 .32869 .39221 .35659 .37829 .33907 .36419 Metetra .714 .823 .932 r .33907 .36419 .35659 .37829 .3847 .36425 .36741 .35012 Metetra .714 .825 .933 r .36741 .35012 .3847 .36425 .41302 .35011 .39597 .33593 Metetra .714 .827 .935 r .39597 .33593 .41302 .35011 .44157 .33586 .42476 .32163 Metetra .715 .829 .936 r .42476 .32163 .44157 .33586 .47034 .3215 .45379 .30721 Metetra .715 .831 .937 r .45379 .30721 .47034 .3215 .49933 .30703 .48304 .29268 Metetra .715 .833 .938 r .48304 .29268 .49933 .30703 .52855 .29244 .51253 .27803 Metetra .715 .836 .939 r .51253 .27803 .52855 .29244 .558 .27774 .54226 .26327 Metetra .715 .838 .94 r .54226 .26327 .558 .27774 .58769 .26292 .57223 .24838 Metetra .715 .84 .942 r .57223 .24838 .58769 .26292 .61761 .24799 .60244 .23338 Metetra .714 .842 .943 r .60244 .23338 .61761 .24799 .64777 .23293 .6329 .21825 Metetra .714 .844 .944 r .6329 .21825 .64777 .23293 .67817 .21776 .66361 .20299 Metetra .714 .846 .945 r .66361 .20299 .67817 .21776 .70881 .20246 .69457 .18761 Metetra .714 .849 .946 r .69457 .18761 .70881 .20246 .7397 .18704 .72579 .17211 Metetra .711 .799 .92 r .02138 .49798 .04141 .51204 .06744 .4991 .04758 .48503 Metetra .711 .801 .921 r .04758 .48503 .06744 .4991 .09368 .48607 .07398 .47197 Metetra .711 .803 .922 r .07398 .47197 .09368 .48607 .12011 .47294 .10059 .45882 Metetra .711 .805 .923 r .10059 .45882 .12011 .47294 .14675 .45971 .12741 .44556 Metetra .711 .807 .925 r .12741 .44556 .14675 .45971 .17359 .44638 .15443 .4322 Metetra .711 .809 .926 r .15443 .4322 .17359 .44638 .20064 .43295 .18167 .41874 Metetra .712 .811 .927 r .18167 .41874 .20064 .43295 .2279 .41941 .20913 .40517 Metetra .712 .814 .928 r .20913 .40517 .2279 .41941 .25537 .40577 .2368 .39149 Metetra .712 .816 .93 r .2368 .39149 .25537 .40577 .28305 .39202 .2647 .3777 Metetra .712 .818 .931 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FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox["=", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox["\[InvisibleSpace]", FontSlant->"Italic"], StyleBox[\(\(-4\)\ x\_1\ x\_2\ x\_3 + 2\ x\_2\%2\ x\_3\%3\), FontSlant->"Italic"]}]], "NumberedEquation"], Cell[TextData[{ "over domain ", Cell[BoxData[ StyleBox[\((x\_1, x\_2, x\_3)\), FontSlant->"Italic"]], FontFamily->"Times New Roman"], StyleBox["\[Element] [", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`\(-1\), 2\)], FontFamily->"Times New Roman"], StyleBox["]\[Times][-1,2]\[Times][-1,2] with ", FontFamily->"Times New Roman"], Cell[BoxData[ \(TraditionalForm\`k\_1 = \(k\_2 = \(k\_3 = 4\)\)\)]], StyleBox[".", FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly = \(-4\)*x*y*z + 2*y^2*z^3;\)\), "\[IndentingNewLine]", \(\(poly0 = tosubform[poly];\)\), "\[IndentingNewLine]", \(\(vars0 = Variables[poly0];\)\), "\[IndentingNewLine]", \(\(ndim = Length[vars0];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly, poly0}];\)\), "\[IndentingNewLine]", \(\(Print["\", vars0];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial: "\[InvisibleSpace]{\(-4\)\ x\ y\ z + 2\ y\^2\ z\^3, \(-4\)\ x\_1\ x\_2\ x\_3 + 2\ x\_2\%2\ x\_3\%3}\), SequenceForm[ "Polynomial: ", { Plus[ Times[ -4, x, y, z], Times[ 2, Power[ y, 2], Power[ z, 3]]], Plus[ Times[ -4, Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3]], Times[ 2, Power[ Subscript[ x, 2], 2], Power[ Subscript[ x, 3], 3]]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Variables: "\[InvisibleSpace]{x\_1, x\_2, x\_3}\), SequenceForm[ "Variables: ", { Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3]}], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(xint = Table[Interval[{Subscript[x, i, 1], Subscript[x, i, 2]}], {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(vxint = Table[Interval[{\(-1\), 2}], {i, 1, ndim}]\), "\[IndentingNewLine]", \(\(\(varsnew0 = Table[to01form[{Subscript[x, i, 1], Subscript[x, i, 2]}, Subscript[x, i]], {i, 1, ndim}];\)\(\[IndentingNewLine]\) \) (**) \), "\[IndentingNewLine]", \(\(vxran = Table[{Min[vxint[\([i]\)]], Max[vxint[\([i]\)]]}, {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(varsnew1 = Table[to01form[{vxran[\([i, 1]\)], vxran[\([i, 2]\)]}, Subscript[x, i]], {i, 1, ndim}]\)}], "Input"], Cell[BoxData[ \({Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}]}\)], "Output"], Cell[BoxData[ \({\(-1\) + 3\ x\_1, \(-1\) + 3\ x\_2, \(-1\) + 3\ x\_3}\)], "Output"] }, Open ]], Cell["\<\ Considered polynomial (11) defined over [0,1] x [0,1] x [0,1] we \ can write in th following form \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly1 = poly0 /. Thread[vars0 -> varsnew1];\)\), "\[IndentingNewLine]", \(poly2 = Expand[poly1]\)}], "Input"], Cell[BoxData[ \(2 - 12\ x\_1 + 36\ x\_1\ x\_2 - 18\ x\_2\%2 + 6\ x\_3 + 36\ x\_1\ x\_3 - 72\ x\_2\ x\_3 - 108\ x\_1\ x\_2\ x\_3 + 162\ x\_2\%2\ x\_3 - 54\ x\_3\%2 + 324\ x\_2\ x\_3\%2 - 486\ x\_2\%2\ x\_3\%2 + 54\ x\_3\%3 - 324\ x\_2\ x\_3\%3 + 486\ x\_2\%2\ x\_3\%3\)], "Output"] }, Open ]], Cell["Next we can derive Bernstein form of polynomial, pb1.", "Text"], Cell[BoxData[{ \(klist = Exponent[poly2, vars0]; klist1 = Max[klist]*Table[1, {ndim}] + 1;\), "\[IndentingNewLine]", \(\(prnout = 0;\)\), "\[IndentingNewLine]", \(\(pb1 = pow2ber[poly2, vars0, klist1, prnout];\)\), "\[IndentingNewLine]", \(\(vars1 = Variables[pb1];\)\), "\[IndentingNewLine]", \(\(Length[vars1];\)\)}], "Input"], Cell["\<\ Now we can calculate the enclosure of range of considered \ polynomial.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(coeflist = poly2coeffast[pb1, vars1];\)\), "\[IndentingNewLine]", \(\(nofc = Length[coeflist];\)\), "\[IndentingNewLine]", \(\(coefrange = Interval[{Min[coeflist], Max[coeflist]}];\)\), "\[IndentingNewLine]", \(\(Print["\", coefrange];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[coefrange] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-28\), 80}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-28, 80}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]108.`\), SequenceForm[ "Width of enclosure: ", 108.0], Editable->False]], "Print"] }, Open ]], Cell["\<\ The bounds obtained by the Bernstein coefficients method can be \ tightened if the degree of basis Bernstein function is elevated, but an \ additional computational cost is involved.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(klist2 = Max[klist]*Table[1, {ndim}] + 4;\)\), "\[IndentingNewLine]", \(\(pb1 = pow2ber[poly2, vars0, klist2, prnout];\)\), "\[IndentingNewLine]", \(\(vars1 = Variables[pb1];\)\), "\[IndentingNewLine]", \(\(coeflist = poly2coeffast[pb1, vars1];\)\), "\[IndentingNewLine]", \(\(coefrange = Interval[{Min[coeflist], Max[coeflist]}];\)\), "\[IndentingNewLine]", \(\(Print["\", coefrange];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[coefrange] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-16\), 80}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-16, 80}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]96.`\), SequenceForm[ "Width of enclosure: ", 96.0], Editable->False]], "Print"] }, Open ]], Cell["\<\ The width of enclosure of range of considered polynomial calculated \ using interval arithmetic is greater. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(polyintval = poly0 /. Thread[vars0 -> vxint];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly0, vxint}];\)\), "\[IndentingNewLine]", \(\(Print["\", polyintval];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[polyintval] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial and variables range: \ "\[InvisibleSpace]{\(-4\)\ x\_1\ x\_2\ x\_3 + 2\ x\_2\%2\ x\_3\%3, {Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}]}}\), SequenceForm[ "Polynomial and variables range: ", { Plus[ Times[ -4, Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3]], Times[ 2, Power[ Subscript[ x, 2], 2], Power[ Subscript[ x, 3], 3]]], { Interval[ {-1, 2}], Interval[ {-1, 2}], Interval[ {-1, 2}]}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-40\), 80}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-40, 80}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]120.`\), SequenceForm[ "Width of enclosure: ", .12*^3], Editable->False]], "Print"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Example 3", "Subsection"], Cell["\<\ Now we can use presented method for calculating enclosure of \ fifth-degree polynomial.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly = 2*y^2*z^3 - 4*v - 4*x*y*z*w*v;\)\), "\[IndentingNewLine]", \(\(poly0 = tosubform[poly];\)\), "\[IndentingNewLine]", \(\(vars0 = Variables[poly0];\)\), "\[IndentingNewLine]", \(\(ndim = Length[vars0];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly, poly0}];\)\), "\[IndentingNewLine]", \(\(Print["\", vars0];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial: "\[InvisibleSpace]{\(-4\)\ v - 4\ v\ w\ x\ y\ z + 2\ y\^2\ z\^3, \(-4\)\ x\_1 - 4\ x\_1\ x\_2\ x\_3\ x\_4\ x\_5 + 2\ x\_4\%2\ x\_5\%3}\), SequenceForm[ "Polynomial: ", { Plus[ Times[ -4, v], Times[ -4, v, w, x, y, z], Times[ 2, Power[ y, 2], Power[ z, 3]]], Plus[ Times[ -4, Subscript[ x, 1]], Times[ -4, Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3], Subscript[ x, 4], Subscript[ x, 5]], Times[ 2, Power[ Subscript[ x, 4], 2], Power[ Subscript[ x, 5], 3]]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Variables: "\[InvisibleSpace]{x\_1, x\_2, x\_3, x\_4, x\_5}\), SequenceForm[ "Variables: ", { Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3], Subscript[ x, 4], Subscript[ x, 5]}], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(xint = Table[Interval[{Subscript[x, i, 1], Subscript[x, i, 2]}], {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(vxint = Table[Interval[{\(-1\), 2}], {i, 1, ndim}]\), "\[IndentingNewLine]", \(\(\(varsnew0 = Table[to01form[{Subscript[x, i, 1], Subscript[x, i, 2]}, Subscript[x, i]], {i, 1, ndim}];\)\(\[IndentingNewLine]\) \) (**) \), "\[IndentingNewLine]", \(\(vxran = Table[{Min[vxint[\([i]\)]], Max[vxint[\([i]\)]]}, {i, 1, ndim}];\)\), "\[IndentingNewLine]", \(varsnew1 = Table[to01form[{vxran[\([i, 1]\)], vxran[\([i, 2]\)]}, Subscript[x, i]], {i, 1, ndim}]\)}], "Input"], Cell[BoxData[ \({Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}]}\)], "Output"], Cell[BoxData[ \({\(-1\) + 3\ x\_1, \(-1\) + 3\ x\_2, \(-1\) + 3\ x\_3, \(-1\) + 3\ x\_4, \(-1\) + 3\ x\_5}\)], "Output"] }, Open ]], Cell["\<\ Considered polynomial defined over [0,1] x [0,1] x [0,1] x [0,1] x \ [0,1] we can write in th following form \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(poly1 = poly0 /. Thread[vars0 -> varsnew1];\)\), "\[IndentingNewLine]", \(poly2 = Expand[poly1]\)}], "Input"], Cell[BoxData[ \(6 - 24\ x\_1 - 12\ x\_2 + 36\ x\_1\ x\_2 - 12\ x\_3 + 36\ x\_1\ x\_3 + 36\ x\_2\ x\_3 - 108\ x\_1\ x\_2\ x\_3 + 36\ x\_1\ x\_4 + 36\ x\_2\ x\_4 - 108\ x\_1\ x\_2\ x\_4 + 36\ x\_3\ x\_4 - 108\ x\_1\ x\_3\ x\_4 - 108\ x\_2\ x\_3\ x\_4 + 324\ x\_1\ x\_2\ x\_3\ x\_4 - 18\ x\_4\%2 + 6\ x\_5 + 36\ x\_1\ x\_5 + 36\ x\_2\ x\_5 - 108\ x\_1\ x\_2\ x\_5 + 36\ x\_3\ x\_5 - 108\ x\_1\ x\_3\ x\_5 - 108\ x\_2\ x\_3\ x\_5 + 324\ x\_1\ x\_2\ x\_3\ x\_5 - 72\ x\_4\ x\_5 - 108\ x\_1\ x\_4\ x\_5 - 108\ x\_2\ x\_4\ x\_5 + 324\ x\_1\ x\_2\ x\_4\ x\_5 - 108\ x\_3\ x\_4\ x\_5 + 324\ x\_1\ x\_3\ x\_4\ x\_5 + 324\ x\_2\ x\_3\ x\_4\ x\_5 - 972\ x\_1\ x\_2\ x\_3\ x\_4\ x\_5 + 162\ x\_4\%2\ x\_5 - 54\ x\_5\%2 + 324\ x\_4\ x\_5\%2 - 486\ x\_4\%2\ x\_5\%2 + 54\ x\_5\%3 - 324\ x\_4\ x\_5\%3 + 486\ x\_4\%2\ x\_5\%3\)], "Output"] }, Open ]], Cell["Next we can derive Bernstein form of polynomial, pb1.", "Text"], Cell[BoxData[{ \(klist = Exponent[poly2, vars0]; klist1 = Max[klist]*Table[1, {ndim}] + 3;\), "\[IndentingNewLine]", \(\(prnout = 0;\)\), "\[IndentingNewLine]", \(\(pb1 = pow2ber[poly2, vars0, klist1, prnout];\)\), "\[IndentingNewLine]", \(\(vars1 = Variables[pb1];\)\), "\[IndentingNewLine]", \(\(Length[vars1];\)\)}], "Input"], Cell["\<\ Now we can calculate the enclosure of range of considered \ polynomial.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(coeflist = poly2coeffast[pb1, vars1];\)\), "\[IndentingNewLine]", \(\(nofc = Length[coeflist];\)\), "\[IndentingNewLine]", \(\(coefrange = Interval[{Min[coeflist], Max[coeflist]}];\)\), "\[IndentingNewLine]", \(\(Print["\", coefrange];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[coefrange] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-88\), 132}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-88, 132}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]220.`\), SequenceForm[ "Width of enclosure: ", .22*^3], Editable->False]], "Print"] }, Open ]], Cell["\<\ The width of enclosure of range of considered polynomial calculated \ using interval arithmetic is greater again. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(polyintval = poly0 /. Thread[vars0 -> vxint];\)\), "\[IndentingNewLine]", \(\(Print["\", {poly0, vxint}];\)\), "\[IndentingNewLine]", \(\(Print["\", polyintval];\)\), "\[IndentingNewLine]", \(\(Print["\", widthint[polyintval] // N];\)\)}], "Input"], Cell[BoxData[ InterpretationBox[\("Polynomial and variables range: \ "\[InvisibleSpace]{\(-4\)\ x\_1 - 4\ x\_1\ x\_2\ x\_3\ x\_4\ x\_5 + 2\ x\_4\%2\ x\_5\%3, {Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}], Interval[{\(-1\), 2}]}}\), SequenceForm[ "Polynomial and variables range: ", { Plus[ Times[ -4, Subscript[ x, 1]], Times[ -4, Subscript[ x, 1], Subscript[ x, 2], Subscript[ x, 3], Subscript[ x, 4], Subscript[ x, 5]], Times[ 2, Power[ Subscript[ x, 4], 2], Power[ Subscript[ x, 5], 3]]], { Interval[ {-1, 2}], Interval[ {-1, 2}], Interval[ {-1, 2}], Interval[ {-1, 2}], Interval[ {-1, 2}]}}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Enclosure of range: \ "\[InvisibleSpace]Interval[{\(-144\), 132}]\), SequenceForm[ "Enclosure of range: ", Interval[ {-144, 132}]], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("Width of enclosure: "\[InvisibleSpace]276.`\), SequenceForm[ "Width of enclosure: ", 276.0], Editable->False]], "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["About the Author", "Section"], Cell[TextData[{ StyleBox["Dr. Tomasz Strek works in the Institute of Applied Mechanics at \ the Poznan University of Technology, Poland. Strek is The Annual Stipends for \ Young Scientist of The Foundation for Polish Science (2001,2002) and First \ Prize of Mechanics Committee of Polish Academy of Science for Young Scientist \ (2005). \nHe is researcher in three primary areas: computational fluid \ dynamics, dynamical systems and verified computations. He uses ", "SmallText", FontSize->12], StyleBox["Mathematica", "SmallText", FontSize->12, FontSlant->"Italic"], StyleBox[" extensively in research and teaching. \n\nAddress\nInstitute of \ Applied Mechanics\nPoznan University of Technology\nPiotrowo 3, 60-965 \ Poznan, Poland\ntomasz.strek@put.poznan.pl\nwww.tomasz.strek.prv.pl", "SmallText", FontSize->12] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ StyleBox["\n[Ale1974] G. Alfeld and J. Herzberger, ", FontSize->10], StyleBox["Introduction to Interval Computations,", FontSize->10, FontSlant->"Italic"], StyleBox[" Wissenschaftsverlag, Mannheim, Wien, Z\[UDoubleDot]rich (1974).\ \n[Ale1998] G. Alefeld and D. Claudio, The basic properties of interval \ arithmetic, its software realization and some applications. ", FontSize->10], StyleBox["Computer and Structures", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["67,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.3-8 (1998).\n[Ale2000] G. Alfeld and G. Mayer, Interval \ analysis: theory and applications. ", FontSize->10], StyleBox["Journal of Computational and Applied Mathematics", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["121,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.421-464 (2000).\n[Ber2000] ", FontSize->10], "J. Berchtold, I. Voiculescu, A. Bowyer, Interval arithmetic applied to \ multivariate Bernstein-form polynomials, Technical report Number 31/98, \ School of Mechanical Engineering, University of Bath, 2000.", StyleBox["\n[Bur1924] J.C. Burkill, Function of intervals. ", FontSize->10], StyleBox["Proccedings of the London Mathematical Society ", FontSize->10, FontSlant->"Italic"], StyleBox["22,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.375-446 (1924).\n[Gar2001] J.Garloff, A.p. Smith, Solution of \ Systems of Polynomial Equations by Using Bernstein Expansion, Symbolic \ Algebraic Methods and Verification Methods, Editors: G. Alefeld, J. Rohn, S. \ Rump, T. Yamamoto, Springer Mathematics, Wien, pp.87-97, (2001). \n[Mar2002] \ ", FontSize->10], "Ralph Martin, Huahao Shou, Irina Voiculescu, Adrian Bowyer, Guojin Wang. \ Comparison of interval methods for plotting algebraic curves, Computer Aided \ Geometric Design 19, pp.553\[Dash]587 (2002).", StyleBox["\n[Moo1962] R.E. Moore, ", FontSize->10], StyleBox["Interval Arithmetic and Automatic Error Analysis in Digital \ Computing.", FontSize->10, FontSlant->"Italic"], StyleBox[" PhD thesis, Stanford University (1962). \n[Moo1966] R.E. Moore, \ ", FontSize->10], StyleBox["Interval Analysis, ", FontSize->10, FontSlant->"Italic"], StyleBox["Prentice Hall, Englewood Cliffs (1966).\n[Moo1979] R.E. Moore, ", FontSize->10], StyleBox["Methods and applications of interval analysis,", FontSize->10, FontSlant->"Italic"], StyleBox[" SIAM, Philadelphia (1979).\n[Nat2002] P.S.V. Nataraj and S.M. \ Sheela, A New Subdivision Strategy for Range Computations. ", FontSize->10], StyleBox["Reliable Computing", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["8,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.83-92 (2002).\n[Lin1995] Q. Lin and J.G. Rokne, Methods for \ bounding the range of a polynomial. ", FontSize->10], StyleBox["Journal of Computational and Applied Mathematics", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["58,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.193-199 (1995).\n[Lin1996] Q. Lin and J.G. Rokne, Interval \ Approximation of Higher Order to the Ranges of Functions, ", FontSize->10], StyleBox["Computers and Math. Applic.", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["31", FontSize->10, FontWeight->"Bold"], StyleBox[" (7), pp.101-109 (1996).\n[Ode2003] J.T. Oden, T. Belytschko, I. \ Babuska and T.J.R. Hughes, Research directions in computational mechanics, ", FontSize->10], StyleBox["Comput. Methods Appl. Mech. Engrg.", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["192,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp.913-922 (2003).\n[Sto2003] ", FontSize->10], "J. Stolfi and L. H. de Figueiredo, \[OpenCurlyDoubleQuote]An Introduction \ to Affine Arithmetic,\[CloseCurlyDoubleQuote] ", StyleBox["TEMA Tend. Mat. Appl. Comput", FontSlant->"Italic"], ", ", StyleBox["4", "SB"], "(3), pp. 297\[Dash]312 (2003).", StyleBox["\n[Sun1958] T. Sunaga, Theory of an interval algebra and its \ application to numerical analysis. ", FontSize->10], StyleBox["RAAG Memories", FontSize->10, FontSlant->"Italic"], StyleBox[" ", FontSize->10], StyleBox["2, ", FontSize->10, FontWeight->"Bold"], StyleBox["pp.2", FontSize->10, FontVariations->{"CompatibilityType"->0}], StyleBox["9-46 (1958).\n[You1931] Y.C. Young, The algebra of many-valued \ quantities. ", FontSize->10], StyleBox["Math. Ann.", FontSize->10, FontSlant->"Italic"], StyleBox[", ", FontSize->10], StyleBox["104,", FontSize->10, FontWeight->"Bold"], StyleBox[" pp. 260-290 (1931).\n", FontSize->10], StyleBox["[Usc2005] A.Uscilowska, ", FontFamily->"Arial"], "A Package for Affine Arithmetic", StyleBox[", ", FontFamily->"Arial"], StyleBox["Applied Mathematica: Proceedings of the 7", FontFamily->"Arial", FontSlant->"Italic"], StyleBox["th", FontFamily->"Arial", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox[" International Mathematica Symposium", FontFamily->"Arial", FontSlant->"Italic"], StyleBox[", eProceedings of IMS 2005, The University of Western Australia, \ Perth, Australia, 5-8.08.2005, Editors: Paul Abbott and Shane McCarthy, \ Wolfram Media Inc., Champaign USA, 2005.", FontFamily->"Arial"] }], "Reference"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for X", ScreenRectangle->{{0, 1920}, {0, 1200}}, WindowToolbars->"EditBar", WindowSize->{1016, 651}, WindowMargins->{{0, Automatic}, {Automatic, 25}}, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{ Inherited, Inherited, Cell[ "Tomasz Strek", "Header"]}, { Cell[ "Enclosure of range of multivariate polynomial", "Header"], Inherited, Inherited}}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "papegay", \ "ims06", "eproc", "articles"}, "Strek.nb.ps", CharacterEncoding -> \ "iso8859-1"], "Magnification"->1}, Magnification->1, StyleDefinitions -> "IMS2006styles.nb" ] (******************************************************************* Cached data follows. 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