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See the ", StyleBox["Mathematica ", FontSlant->"Italic"], "Help browser or wolfram.com for more information on using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ This notebook and those for the other seven chapters use a common stylesheet \ that has been imported into the notebook. This style sheet supports \ chapters, section, subsections, numbered equations, numbered figures and \ citations. A separate copy of the stylesheet can be download from the same \ page where this notebook was downloaded. \ \>", "Text"], Cell[TextData[{ StyleBox["COPYRIGHT ISSUES:", FontWeight->"Bold"], " The authors reserve all copyrights associated with this work. Any of \ the material appearing in these notebooks (such as the polyhedral meshes in \ chapter 7) can be used and modified without restriction as long as the use is \ non\[Hyphen]commercial. We simply ask that you acknowledge the authors when \ using material from these notebooks. For those readers interested in \ commercial use of the material in these notebooks, please contact \ jwarren@cs.rice.edu." }], "Text"], Cell[CellGroupData[{ Cell["Helper functions associated with previous chapters", "Subsubsection"], Cell[BoxData[ \(makeGenFun[coeff_, a_, x_] := Sum[coeff\[LeftDoubleBracket]i\[RightDoubleBracket] x\^\(i - 1 + a\), {i, 1, Length[coeff]}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {min_, max_}, k_, x_] := With[{expFun = Together[genFun]}, \n\t Table[Coefficient[expFun, x, i], {i, \(2\^k\) min, \(2\^k\) max}]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(subdivide[s_, p_, k_] := If[k \[Equal] 0, p, subdivide[s, s*\((p /. {x \[Rule] x\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(makeGenFun[coeff_, {a_, b_}, x_, y_] := Sum[coeff\[LeftDoubleBracket]i, j\[RightDoubleBracket] \(x\^\(i - 1 + a\)\) y\^\(j - 1 + b\), {i, \(Dimensions[ coeff]\)\[LeftDoubleBracket]1\[RightDoubleBracket]}, {j, \ \(Dimensions[coeff]\)\[LeftDoubleBracket]2\[RightDoubleBracket]}]\)], "Input",\ InitializationCell->True], Cell[BoxData[ \(makeCoeff[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_] := With[{expFun = Expand[genFun]}, \n\t Table[With[{rowY = Table[Coefficient[expFun, y, j], {j, \(2\^k\) minY, \(2\^k\) maxY}]}, \n\t\t\t\tTable[ Coefficient[rowY, x, i], {i, \(2\^k\) minX, \(2\^k\) maxX}]]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(subdivide2D[s_, p_, k_] := If[k \[Equal] 0, p, subdivide2D[s, s*\((p /. {x \[Rule] x\^2, y \[Rule] y\^2})\), k - 1]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(norm[t_, k_] := With[{coeff = Transpose[ Partition[ CoefficientList[t], {2\^k, 2\^k}, {2\^k, 2\^k}, {1, 1}, 0], {3, 4, 1, 2}]}, \[IndentingNewLine]Max[ Apply[Plus, Abs[coeff], {2, 4}]]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotGen2D[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_, opts___] := Module[{expFun = Expand[genFun], rowY}, \[IndentingNewLine]plotCoeff3D[ Table[rowY = Coefficient[expFun, y, j]; \[IndentingNewLine]Table[{i\/2\^k, j\/2\^k, Coefficient[rowY, x, i]}, {i, \(2\^k\) minX, \(2\^k\) maxX}], {j, \(2\^k\) minY, \(2\^k\) maxY}], opts]]\)], "Input", InitializationCell->True], Cell[BoxData[ \(plotCoeff3D[p_, plotArgs___] := Show[Graphics3D[\[IndentingNewLine]Table[ Polygon[{p\[LeftDoubleBracket]i, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], p\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], p\[LeftDoubleBracket]i, j + 1\[RightDoubleBracket]}], {i, \(Dimensions[ p]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - 1}, {j, \(Dimensions[ p]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - 1}]], Join[{plotArgs}, {Axes \[Rule] True}]]\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Subdivision for polyharmonic splines", "Section"], Cell[CellGroupData[{ Cell["The radial basis for polyharmonic splines", "Subsection"], Cell["\<\ General definition of radial basis functions for polyharmonic splines\ \>", "Text"], Cell[BoxData[ \(\[ScriptC][ m_, \[ScriptX]_, \[ScriptY]_] := \(\(\((\[ScriptX]\^2 + \ \[ScriptY]\^2)\)\^\(m - 1\)\) Log[\[ScriptX]\^2 + \ \[ScriptY]\^2]\)\/\(\(2\^\(2 m\)\) \(\((\(\((m - 1)\)!\))\)\^2\) \[Pi]\)\)], \ "Input", InitializationCell->True, CellTags->"def polyharm radial"], Cell[TextData[{ "Observe that for ", Cell[BoxData[ \(\[ScriptC][1, 0, 0]\)]], " is unbounded. 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.30091 L s .02767 .31501 m .03135 .31362 L s .0236 .34073 m .02728 .33935 L s .02153 .35374 m .02522 .35237 L s .01945 .36687 m .02315 .3655 L s .01736 .3801 m .02105 .37874 L s .01311 .4069 m .01681 .40555 L s .01096 .42047 m .01466 .41913 L s .00879 .43415 m .0125 .43282 L s .00661 .44794 m .01032 .44663 L s .00218 .47589 m .00589 .47459 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L 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 .745 .61 .708 r .37632 .74671 .39806 .80578 .43793 .75562 .4163 .69866 Metetra .732 .607 .718 r .4163 .69866 .43793 .75562 .47739 .71337 .45587 .65832 Metetra .717 .604 .729 r .45587 .65832 .47739 .71337 .5167 .6781 .49529 .62475 Metetra .702 .6 .74 r .49529 .62475 .5167 .6781 .55608 .649 .53478 .59704 Metetra .686 .597 .75 r .53478 .59704 .55608 .649 .59573 .62538 .57454 .57442 Metetra .669 .592 .76 r .57454 .57442 .59573 .62538 .63584 .6067 .61475 .55623 Metetra .651 .585 .767 r .61475 .55623 .63584 .6067 .67657 .59263 .65557 .54205 Metetra .632 .576 .773 r .65557 .54205 .67657 .59263 .71811 .58308 .69714 .53174 Metetra .61 .563 .777 r .69714 .53174 .71811 .58308 .76067 .57823 .73965 .52553 Metetra .584 .547 .778 r .73965 .52553 .76067 .57823 .80447 .57855 .78333 .52393 Metetra .554 .526 .778 r .78333 .52393 .80447 .57855 .84982 .58471 .82846 .52772 Metetra .52 .501 .775 r .82846 .52772 .84982 .58471 .89708 .59757 .87539 .53786 Metetra .481 .472 .77 r .87539 .53786 .89708 .59757 .9467 .61819 .92456 .55545 Metetra .439 .441 .764 r .92456 .55545 .9467 .61819 .99922 .6478 .97651 .58172 Metetra .755 .629 .721 r .35412 .69495 .37632 .74671 .4163 .69866 .39427 .6488 Metetra .74 .628 .734 r .39427 .6488 .4163 .69866 .45587 .65832 .43401 .61028 Metetra .725 .626 .747 r .43401 .61028 .45587 .65832 .49529 .62475 .4736 .57837 Metetra .708 .624 .761 r .4736 .57837 .49529 .62475 .53478 .59704 .51327 .55211 Metetra .691 .622 .774 r .51327 .55211 .53478 .59704 .57454 .57442 .55322 .5306 Metetra .675 .62 .786 r .55322 .5306 .57454 .57442 .61475 .55623 .59362 .51302 Metetra .658 .615 .795 r .59362 .51302 .61475 .55623 .65557 .54205 .6346 .49878 Metetra .639 .608 .802 r .6346 .49878 .65557 .54205 .69714 .53174 .67631 .4877 Metetra .618 .596 .806 r .67631 .4877 .69714 .53174 .73965 .52553 .71889 .48004 Metetra .591 .578 .807 r .71889 .48004 .73965 .52553 .78333 .52393 .76255 .47646 Metetra .559 .555 .806 r .76255 .47646 .78333 .52393 .82846 .52772 .80758 .47788 Metetra .521 .526 .802 r .80758 .47788 .82846 .52772 .87539 .53786 .85431 .48537 Metetra .476 .493 .796 r .85431 .48537 .87539 .53786 .92456 .55545 .90318 .5001 Metetra .427 .457 .788 r .90318 .5001 .92456 .55545 .97651 .58172 .95472 .52333 Metetra .764 .651 .736 r .33134 .64964 .35412 .69495 .39427 .6488 .37173 .60516 Metetra .749 .65 .751 r .37173 .60516 .39427 .6488 .43401 .61028 .41169 .56829 Metetra .731 .65 .768 r .41169 .56829 .43401 .61028 .4736 .57837 .45151 .53801 Metetra .713 .65 .785 r .45151 .53801 .4736 .57837 .51327 .55211 .49141 .51327 Metetra .695 .65 .801 r .49141 .51327 .51327 .55211 .55322 .5306 .53162 .49301 Metetra .678 .651 .816 r .53162 .49301 .55322 .5306 .59362 .51302 .57228 .47619 Metetra .663 .65 .827 r .57228 .47619 .59362 .51302 .6346 .49878 .61352 .46198 Metetra .647 .645 .834 r .61352 .46198 .6346 .49878 .67631 .4877 .65544 .45008 Metetra .627 .634 .838 r .65544 .45008 .67631 .4877 .71889 .48004 .69819 .44084 Metetra .601 .615 .838 r .69819 .44084 .71889 .48004 .76255 .47646 .74194 .43515 Metetra .566 .589 .836 r .74194 .43515 .76255 .47646 .80758 .47788 .78697 .43416 Metetra .523 .556 .831 r .78697 .43416 .80758 .47788 .85431 .48537 .83362 .43909 Metetra .472 .518 .824 r .83362 .43909 .85431 .48537 .90318 .5001 .88233 .45116 Metetra .415 .476 .814 r .88233 .45116 .90318 .5001 .95472 .52333 .93362 .47166 Metetra .775 .673 .752 r .30787 .61002 .33134 .64964 .37173 .60516 .34856 .56689 Metetra .757 .674 .77 r .34856 .56689 .37173 .60516 .41169 .56829 .38881 .53145 Metetra .737 .675 .79 r .38881 .53145 .41169 .56829 .45151 .53801 .42889 .50267 Metetra .716 .676 .81 r .42889 .50267 .45151 .53801 .49141 .51327 .46908 .47947 Metetra .695 .679 .83 r .46908 .47947 .49141 .51327 .53162 .49301 .50958 .46062 Metetra .678 .683 .848 r .50958 .46062 .53162 .49301 .57228 .47619 .55057 .44479 Metetra .665 .687 .861 r .55057 .44479 .57228 .47619 .61352 .46198 .59214 .43073 Metetra .655 .687 .868 r .59214 .43073 .61352 .46198 .65544 .45008 .63438 .4179 Metetra .639 .677 .869 r .63438 .4179 .65544 .45008 .69819 .44084 .67738 .40687 Metetra .613 .656 .868 r .67738 .40687 .69819 .44084 .74194 .43515 .72131 .39893 Metetra .575 .626 .865 r .72131 .39893 .74194 .43515 .78697 .43416 .76645 .39554 Metetra .526 .589 .861 r .76645 .39554 .78697 .43416 .83362 .43909 .81314 .39805 Metetra .468 .546 .853 r .81314 .39805 .83362 .43909 .88233 .45116 .86182 .40773 Metetra .4 .497 .842 r .86182 .40773 .88233 .45116 .93362 .47166 .91301 .42586 Metetra .785 .698 .767 r .28361 .57543 .30787 .61002 .34856 .56689 .32467 .53324 Metetra .765 .699 .789 r .32467 .53324 .34856 .56689 .38881 .53145 .36525 .49886 Metetra .742 .7 .812 r .36525 .49886 .38881 .53145 .42889 .50267 .40566 .47132 Metetra .716 .702 .836 r .40566 .47132 .42889 .50267 .46908 .47947 .44616 .44952 Metetra .69 .705 .859 r .44616 .44952 .46908 .47947 .50958 .46062 .487 .43218 Metetra .67 .712 .879 r .487 .43218 .50958 .46062 .55057 .44479 .52836 .41762 Metetra .663 .722 .893 r .52836 .41762 .55057 .44479 .59214 .43073 .57033 .4039 Metetra .663 .728 .897 r .57033 .4039 .59214 .43073 .63438 .4179 .61296 .38995 Metetra .654 .719 .896 r .61296 .38995 .63438 .4179 .67738 .40687 .65628 .37684 Metetra .628 .695 .893 r .65628 .37684 .67738 .40687 .72131 .39893 .70048 .36657 Metetra .586 .663 .891 r .70048 .36657 .72131 .39893 .76645 .39554 .74583 .36091 Metetra .53 .622 .888 r .74583 .36091 .76645 .39554 .81314 .39805 .79267 .36129 Metetra .462 .574 .881 r .79267 .36129 .81314 .39805 .86182 .40773 .84146 .36898 Metetra .382 .518 .869 r .84146 .36898 .86182 .40773 .91301 .42586 .8927 .3852 Metetra .797 .723 .783 r .25844 .54535 .28361 .57543 .32467 .53324 .29996 .50356 Metetra .774 .725 .807 r .29996 .50356 .32467 .53324 .36525 .49886 .34095 .46974 Metetra .746 .725 .832 r .34095 .46974 .36525 .49886 .40566 .47132 .38173 .44292 Metetra .715 .725 .858 r .38173 .44292 .40566 .47132 .44616 .44952 .42258 .42209 Metetra .682 .726 .883 r .42258 .42209 .44616 .44952 .487 .43218 .46378 .40602 Metetra .656 .73 .904 r .46378 .40602 .487 .43218 .52836 .41762 .50553 .39294 Metetra .653 .745 .917 r .50553 .39294 .52836 .41762 .57033 .4039 .54795 .37991 Metetra .673 .761 .916 r .54795 .37991 .57033 .4039 .61296 .38995 .59101 .36448 Metetra .671 .751 .91 r .59101 .36448 .61296 .38995 .65628 .37684 .63473 .34907 Metetra .642 .725 .908 r .63473 .34907 .65628 .37684 .70048 .36657 .67928 .33669 Metetra .595 .691 .91 r .67928 .33669 .70048 .36657 .74583 .36091 .72494 .3292 Metetra .532 .65 .91 r .72494 .3292 .74583 .36091 .79267 .36129 .77206 .32798 Metetra .452 .599 .907 r .77206 .32798 .79267 .36129 .84146 .36898 .82108 .33422 Metetra .358 .536 .894 r .82108 .33422 .84146 .36898 .8927 .3852 .87254 .34911 Metetra .809 .75 .797 r .23225 .51946 .25844 .54535 .29996 .50356 .27433 .47745 Metetra .784 .751 .823 r .27433 .47745 .29996 .50356 .34095 .46974 .31582 .44349 Metetra .753 .75 .85 r .31582 .44349 .34095 .46974 .38173 .44292 .35704 .41664 Metetra .716 .746 .875 r .35704 .41664 .38173 .44292 .42258 .42209 .3983 .39592 Metetra .678 .74 .897 r .3983 .39592 .42258 .42209 .46378 .40602 .43989 .38018 Metetra .645 .736 .915 r .43989 .38018 .46378 .40602 .50553 .39294 .48203 .3678 Metetra .64 .743 .923 r .48203 .3678 .50553 .39294 .54795 .37991 .52489 .35573 Metetra .686 .769 .915 r .52489 .35573 .54795 .37991 .59101 .36448 .56842 .33851 Metetra .681 .756 .908 r .56842 .33851 .59101 .36448 .63473 .34907 .6126 .32157 Metetra .648 .736 .913 r .6126 .32157 .63473 .34907 .67928 .33669 .65758 .30801 Metetra .597 .709 .922 r .65758 .30801 .67928 .33669 .72494 .3292 .70365 .29956 Metetra .526 .671 .929 r .70365 .29956 .72494 .3292 .77206 .32798 .75116 .2975 Metetra .435 .619 .929 r .75116 .2975 .77206 .32798 .82108 .33422 .80056 .303 Metetra .323 .55 .915 r .80056 .303 .82108 .33422 .87254 .34911 .85238 .31721 Metetra .824 .779 .81 r .20488 .49771 .23225 .51946 .27433 .47745 .24765 .45479 Metetra .797 .779 .837 r .24765 .45479 .27433 .47745 .31582 .44349 .28974 .41994 Metetra .763 .775 .863 r .28974 .41994 .31582 .44349 .35704 .41664 .33151 .39218 Metetra .724 .767 .887 r .33151 .39218 .35704 .41664 .3983 .39592 .37326 .37048 Metetra .685 .753 .903 r .37326 .37048 .3983 .39592 .43989 .38018 .4153 .35358 Metetra .656 .739 .911 r .4153 .35358 .43989 .38018 .48203 .3678 .45788 .33969 Metetra .653 .729 .905 r .45788 .33969 .48203 .3678 .52489 .35573 .50115 .32584 Metetra .671 .725 .89 r .50115 .32584 .52489 .35573 .56842 .33851 .54513 .30953 Metetra .67 .735 .899 r .54513 .30953 .56842 .33851 .6126 .32157 .5898 .29323 Metetra .642 .733 .915 r .5898 .29323 .6126 .32157 .65758 .30801 .63529 .27998 Metetra .589 .717 .933 r .63529 .27998 .65758 .30801 .70365 .29956 .68186 .27165 Metetra .51 .685 .946 r .68186 .27165 .70365 .29956 .75116 .2975 .72988 .26965 Metetra .403 .632 .948 r .72988 .26965 .75116 .2975 .80056 .