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8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Bold /Times-Bold-iso isovec ReEncode /Times-Italic /Times-Italic-iso isovec ReEncode /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit n -1000 10721 m -1000 -1000 l 11212 -1000 l 11212 10721 l cp clip 0.06000 0.06000 sc 7.500 slw % Ellipse n 5400 8400 75 75 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 6600 8400 75 75 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 7800 8400 75 75 0 360 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6567 l 6173 6528 l 6185 6490 l 6198 6454 l 6211 6420 l 6224 6387 l 6237 6355 l 6250 6326 l 6263 6298 l 6276 6272 l 6288 6248 l 6300 6225 l 6322 6187 l 6343 6154 l 6363 6126 l 6383 6102 l 6403 6083 l 6423 6066 l 6443 6052 l 6464 6040 l 6485 6030 l 6505 6022 l 6524 6016 l 6542 6011 l 6558 6007 l 6572 6005 l 6583 6003 l 6590 6001 l 6596 6001 l 6599 6000 l 6600 6000 l gs col0 s gr 7.500 slw % Ellipse n 4200 8400 75 75 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline 15.000 slw n 9000 8400 m 8999 8400 l 8996 8399 l 8991 8397 l 8983 8394 l 8972 8389 l 8957 8383 l 8938 8376 l 8915 8367 l 8887 8356 l 8854 8343 l 8818 8328 l 8777 8312 l 8732 8294 l 8683 8274 l 8632 8253 l 8578 8231 l 8522 8208 l 8464 8184 l 8405 8159 l 8346 8133 l 8286 8108 l 8227 8082 l 8169 8056 l 8111 8030 l 8056 8004 l 8002 7978 l 7950 7952 l 7900 7927 l 7853 7902 l 7807 7877 l 7765 7852 l 7724 7828 l 7686 7803 l 7650 7778 l 7617 7754 l 7585 7728 l 7555 7703 l 7527 7677 l 7500 7650 l 7473 7621 l 7447 7591 l 7423 7559 l 7399 7527 l 7377 7493 l 7355 7458 l 7335 7422 l 7315 7384 l 7297 7346 l 7279 7306 l 7262 7265 l 7245 7223 l 7229 7180 l 7214 7137 l 7199 7093 l 7185 7048 l 7171 7003 l 7157 6958 l 7144 6913 l 7131 6868 l 7117 6823 l 7105 6778 l 7092 6734 l 7079 6691 l 7066 6649 l 7053 6607 l 7040 6567 l 7027 6528 l 7015 6490 l 7002 6454 l 6989 6420 l 6976 6387 l 6963 6355 l 6950 6326 l 6937 6298 l 6924 6272 l 6912 6248 l 6900 6225 l 6878 6187 l 6857 6154 l 6837 6126 l 6817 6102 l 6797 6083 l 6777 6066 l 6757 6052 l 6736 6040 l 6715 6030 l 6695 6022 l 6676 6016 l 6658 6011 l 6642 6007 l 6628 6005 l 6617 6003 l 6610 6001 l 6604 6001 l 6601 6000 l 6600 6000 l gs col0 s gr /Times-Italic-iso ff 450.00 scf sf 8175 9600 m gs 1 -1 sc (B\(x\)) col0 sh gr /Times-Bold-iso ff 270.00 scf sf 3975 8775 m gs 1 -1 sc (-2h) col0 sh gr /Times-Bold-iso ff 270.00 scf sf 5250 8775 m gs 1 -1 sc (-h) col0 sh gr /Times-Bold-iso ff 270.00 scf sf 6525 8775 m gs 1 -1 sc (0) col0 sh gr /Times-Bold-iso ff 270.00 scf sf 7725 8775 m gs 1 -1 sc (h) col0 sh gr /Times-Bold-iso ff 270.00 scf sf 8850 8775 m gs 1 -1 sc (2h) col0 sh gr /Times-Bold-iso ff 450.00 scf sf 4425 9600 m gs 1 -1 sc (Figure of function ) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 1011 3233 a currentpoint currentpoint translate 1 .5 div 1 .5 div scale neg exch neg exch translate 1011 3233 a -118 3479 a Fu(No)m(w)33 b(de\014ne)808 3600 y Ft(B)882 3615 y Fs(i)910 3600 y Fu(\()p Ft(x)p Fu(\))28 b(=)g Ft(B)5 b Fu(\()p Ft(x)22 b Fm(\000)h Ft(x)1522 3615 y Fs(i)1551 3600 y Fu(\))p Ft(;)309 b(i)28 b Fu(=)f Fm(\000)p Fu(1)p Ft(;)17 b Fu(0)p Ft(;)g Fu(1)p Ft(;)g(:)g(:)g(:)e(;)i(n;)g(n)22 b Fu(+)g(1)-118 3772 y(Then)57 b Ft(B)234 3787 y Fs(i)263 3772 y Fu(\()p Ft(x)p Fu(\))40 b Fm(2)g Ft(\033)595 3787 y Fj(3)635 3772 y Fu([)p Ft(x)717 3787 y Fj(0)757 3772 y Ft(;)17 b(x)856 3787 y Fj(1)896 3772 y Ft(;)g(:)g(:)g(:)e(;)i(x)1169 3787 y Fs(n)1216 3772 y Fu(])57 b(and)40 b Ft(B)1571 3787 y Fl(\000)p Fj(1)1665 3772 y Fu(\()p Ft(x)p Fu(\))p Ft(;)17 b(B)1914 3787 y Fj(0)1954 3772 y Fu(\()p Ft(x)p Fu(\))p Ft(;)g(:)g(:)g(:)f(;)h(B)2378 3787 y Fs(n)2425 3772 y Fu(\()p Ft(x)p Fu(\))p Ft(;)g(B)2674 3787 y Fs(n)p Fj(+1)2811 3772 y Fu(\()p Ft(x)p Fu(\))40 b(are)f(linearly)f(indep)s (en-)-118 3892 y(den)m(t.)44 b(So)33 b(that)f(they)i(comp)s(ose)e(a)h (Basis)f(of)g Ft(\033)1601 3907 y Fj(3)1641 3892 y Fu([)p Ft(x)1723 3907 y Fj(0)1763 3892 y Ft(;)17 b(x)1862 3907 y Fj(1)1902 3892 y Ft(;)g(:)g(:)g(:)e(;)i(x)2175 3907 y Fs(n)2223 3892 y Fu(])32 b(and)h(are)f(called)g(B-spline.)42 b(The)34 b(B-spline)-118 4013 y Ft(B)-44 4028 y Fs(i)-16 4013 y Fu(\()p Ft(x)p Fu(\))f(cen)m(tered)h(at)f Ft(x)710 4028 y Fs(i)771 4013 y Fu(and)1072 4225 y Ft(B)1146 4240 y Fs(i)1175 4225 y Fu(\()p Ft(x)1268 4240 y Fs(i)1296 4225 y Fu(\))173 b(=)83 b Ft(B)5 b Fu(\(0\))217 b(=)83 b(1)1072 4431 y Ft(B)1146 4446 y Fs(i)1175 4431 y Fu(\()p Ft(x)1268 4446 y Fs(i)p Fl(\006)p Fj(1)1386 4431 y Fu(\))g(=)g Ft(B)5 b Fu(\()p Fm(\006)p Ft(h)p Fu(\))133 b(=)2256 4392 y Fj(1)p 2256 4409 36 4 v 2256 4466 a(4)1072 4638 y Ft(B)1146 4653 y Fs(i)1175 4638 y Fu(\()p Ft(x)1268 4653 y Fs(i)p Fl(\006)p Fj(2)1386 4638 y Fu(\))83 b(=)g Ft(B)5 b Fu(\()p Fm(\006)p Fu(2)p Ft(h)p Fu(\))84 b(=)f(0)1072 4844 y Ft(B)1146 4859 y Fs(i)1175 4844 y Fu(\()p Ft(x)1268 4859 y Fs(j)1305 4844 y Fu(\))164 b(=)83 b(0)p Ft(;)504 b Fm(j)p Ft(i)22 b Fm(\000)g Ft(j)6 b Fm(j)28 b(\025)g Fu(2)-118 5104 y(F)-8 b(or)32 b(an)m(y)h(cubic)g(spline)f Ft(S)833 5119 y Fs(h)877 5104 y Fu(\()p Ft(x)p Fu(\))c Fm(2)g Ft(\033)1185 5119 y Fj(3)1225 5104 y Fu([)p Ft(x)1307 5119 y Fj(0)1347 5104 y Ft(;)17 b(x)1446 5119 y Fj(1)1486 5104 y Ft(;)g(:)g(:)g(:)f(;)h(x)1760 5119 y Fs(n)1807 5104 y Fu(],)1427 5409 y Ft(S)1487 5424 y Fs(h)1532 5409 y Fu(\()p Ft(x)p Fu(\))28 b(=)1817 5285 y Fs(n)p Fj(+1)1811 5315 y Fi(X)1794 5525 y Fs(j)t Fj(=)p Fl(\000)p Fj(1)1988 5409 y Ft(\013)2050 5424 y Fs(j)2087 5409 y Ft(B)2161 5424 y Fs(j)2198 5409 y Fu(\()p Ft(x)p Fu(\))p eop %%Page: 25 4 25 16 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(25)-118 214 y Fk(2.4.6)136 b(In)l(terp)t(olation)46 b(using)f(B-splines)-118 422 y Fu(Giv)m(en)32 b(data)h Ft(y)435 437 y Fs(i)490 422 y Fu(=)28 b Ft(f)11 b Fu(\()p Ft(x)746 437 y Fs(i)774 422 y Fu(\))p Ft(;)44 b(i)28 b Fu(=)g(0)p Ft(;)17 b Fu(1)p Ft(;)g(:)g(:)g(:)e(;)i(n)60 b Fu(and)32 b(b)s(oundary)h(conditions)1548 547 y Fi(\()1670 613 y Ft(f)1729 577 y Fl(0)1752 613 y Fu(\()p Ft(x)1845 628 y Fj(0)1885 613 y Fu(\))28 b(=)f Ft(y)2106 577 y Fl(0)2102 638 y Fj(0)1670 819 y Ft(f)1729 783 y Fl(0)1752 819 y Fu(\()p Ft(x)1845 834 y Fs(n)1892 819 y Fu(\))h(=)g Ft(y)2114 783 y Fl(0)2110 844 y Fs(n)-118 1012 y Fu(\014nd)33 b Ft(S)137 1027 y Fs(h)182 1012 y Fu(\()p Ft(x)p Fu(\))28 b Fm(2)g Ft(\033)490 1027 y Fj(3)530 1012 y Fu([)p Ft(x)612 1027 y Fj(0)652 1012 y Ft(;)17 b(x)751 1027 y Fj(1)790 1012 y Ft(;)g(:)g(:)g(:)f(;)h(x)1064 1027 y Fs(n)1111 1012 y Fu(])33 b(suc)m(h)h(that)1242 1203 y Ft(S)1302 1218 y Fs(h)1347 1203 y Fu(\()p Ft(x)1440 1218 y Fs(i)1468 1203 y Fu(\))28 b(=)g Ft(y)1686 1218 y Fs(i)1713 1203 y Ft(;)191 b(i)28 b Fu(=)g(0)p Ft(;)17 b Fu(1)p Ft(;)g(:)g(:)g(:)e(;)i (n)1242 1409 y(S)1308 1373 y Fl(0)1302 1435 y Fs(h)1347 1409 y Fu(\()p Ft(x)1440 1424 y Fj(0)1480 1409 y Fu(\))27 b(=)h Ft(y)1701 1373 y Fl(0)1697 1434 y Fj(0)1242 1615 y Ft(S)1308 1579 y Fl(0)1302 1641 y Fs(h)1347 1615 y Fu(\()p Ft(x)1440 1630 y Fs(n)1487 1615 y Fu(\))g(=)f Ft(y)1708 1579 y Fl(0)1704 1640 y Fs(n)-118 1902 y Fk(2.4.7)136 b(F)-11 b(orm)l(ulation)-118 2176 y Fu(Let)66 b Ft(S)150 2191 y Fs(h)195 2176 y Fu(\()p Ft(x)p Fu(\))28 b(=)479 2051 y Fs(n)p Fj(+1)474 2081 y Fi(X)457 2291 y Fs(j)t Fj(=)p Fl(\000)p Fj(1)651 2176 y Ft(\013)713 2191 y Fs(j)750 2176 y Ft(B)824 2191 y Fs(j)860 2176 y Fu(\()p Ft(x)p Fu(\))q(.)49 b(Since)35 b Ft(B)1399 2191 y Fs(i)1427 2176 y Fu(\()p Ft(x)1520 2191 y Fs(j)1557 2176 y Fu(\))c(=)g(0)k (whenev)m(er)i Fm(j)p Ft(i)24 b Fm(\000)g Ft(j)6 b Fm(j)30 b(\025)i Fu(2,)j(the)g(conditions)f Ft(S)3456 2191 y Fs(h)3501 2176 y Fu(\()p Ft(x)3594 2191 y Fs(i)3622 2176 y Fu(\))d(=)g Ft(y)3846 2191 y Fs(i)-118 2391 y Fu(implies)544 2545 y Fs(n)p Fj(+1)539 2575 y Fi(X)522 2785 y Fs(j)t Fj(=)p Fl(\000)p Fj(1)716 2670 y Ft(\013)778 2685 y Fs(j)814 2670 y Ft(B)888 2685 y Fs(j)925 2670 y Fu(\()p Ft(x)1018 2685 y Fs(i)1047 2670 y Fu(\))83 b(=)f Ft(\013)1388 2685 y Fs(i)p Fl(\000)p Fj(1)1507 2670 y Ft(B)1581 2685 y Fs(i)p Fl(\000)p Fj(1)1699 2670 y Fu(\()p Ft(x)1792 2685 y Fs(i)1821 2670 y Fu(\))22 b(+)g Ft(\013)2041 2685 y Fs(i)2069 2670 y Ft(B)2143 2685 y Fs(i)2171 2670 y Fu(\()p Ft(x)2264 2685 y Fs(i)2293 2670 y Fu(\))g(+)g Ft(\013)2513 2685 y Fs(i)p Fj(+1)2631 2670 y Ft(B)2705 2685 y Fs(i)p Fj(+1)2824 2670 y Fu(\()p Ft(x)2917 2685 y Fs(i)2945 2670 y Fu(\))28 b(=)g Ft(y)3163 2685 y Fs(i)3190 2670 y Ft(;)1926 2906 y(i)g Fu(=)g(0)p Ft(;)17 b(:)g(:)g(:)f(;)h(n)-118 3139 y Fu(Because)34 b Ft(B)329 3154 y Fs(i)357 3139 y Fu(\()p Ft(x)450 3154 y Fs(i)479 3139 y Fu(\))27 b(=)h(1)k(and)h Ft(B)993 3154 y Fs(i)1021 3139 y Fu(\()p Ft(x)1114 3154 y Fs(i)p Fl(\006)p Fj(1)1233 3139 y Fu(\))28 b(=)1412 3072 y(1)p 1412 3116 49 4 v 1412 3208 a(4)1471 3139 y(,)k(w)m(e)i(ha)m (v)m(e)957 3354 y(1)p 957 3398 V 957 3489 a(4)1016 3421 y Ft(\013)1078 3436 y Fs(i)p Fl(\000)p Fj(1)1218 3421 y Fu(+)22 b Ft(\013)1378 3436 y Fs(i)1429 3421 y Fu(+)1537 3354 y(1)p 1537 3398 V 1537 3489 a(4)1595 3421 y Ft(\013)1657 3436 y Fs(i)p Fj(+1)1803 3421 y Fu(=)28 b Ft(y)1955 3436 y Fs(i)1983 3421 y Ft(;)309 b(i)28 b Fu(=)f(0)p Ft(;)17 b(:)g(:)g(:)f(;)h(n)-118 3646 y Fu(The)33 b(b)s(oundary)g(conditions)f Ft(S)1054 3610 y Fl(0)1048 3672 y Fs(h)1093 3646 y Fu(\()p Ft(x)1186 3661 y Fj(0)1226 3646 y Fu(\))27 b(=)h Ft(y)1447 3610 y Fl(0)1443 3670 y Fj(0)1514 3646 y Fu(and)33 b Ft(S)1770 3610 y Fl(0)1764 3672 y Fs(h)1809 3646 y Fu(\()p Ft(x)1902 3661 y Fs(n)1949 3646 y Fu(\))28 b(=)f Ft(y)2170 3610 y Fl(0)2166 3670 y Fs(n)2245 3646 y Fu(b)s(ecome)734 3755 y Fi(8)734 3844 y(<)734 4024 y(:)864 3838 y Ft(\013)926 3853 y Fl(\000)p Fj(1)1020 3838 y Ft(B)1099 3802 y Fl(0)1094 3862 y(\000)p Fj(1)1188 3838 y Fu(\()p Ft(x)1281 3853 y Fj(0)1321 3838 y Fu(\))22 b(+)g Ft(\013)1541 3853 y Fj(0)1581 3838 y Ft(B)1660 3802 y Fl(0)1655 3862 y Fj(0)1694 3838 y Fu(\()p Ft(x)1787 3853 y Fj(0)1827 3838 y Fu(\))g(+)g Ft(\013)2047 3853 y Fj(1)2087 3838 y Ft(B)2166 3802 y Fl(0)2161 3862 y Fj(1)2200 3838 y Fu(\()p Ft(x)2293 3853 y Fj(0)2333 3838 y Fu(\))402 b(=)27 b Ft(y)2928 3802 y Fl(0)2924 3862 y Fj(0)864 4078 y Ft(\013)926 4093 y Fs(n)p Fl(\000)p Fj(1)1063 4078 y Ft(B)1142 4042 y Fl(0)1137 4103 y Fs(n)p Fl(\000)p Fj(1)1274 4078 y Fu(\()p Ft(x)1367 4093 y Fs(n)1414 4078 y Fu(\))c(+)f Ft(\013)1635 4093 y Fs(n)1682 4078 y Ft(B)1761 4042 y Fl(0)1756 4103 y Fs(n)1803 4078 y Fu(\()p Ft(x)1896 4093 y Fs(n)1943 4078 y Fu(\))g(+)g Ft(\013)2163 4093 y Fs(n)p Fj(+1)2300 4078 y Ft(B)2379 4042 y Fl(0)2374 4103 y Fs(n)p Fj(+1)2511 4078 y Fu(\()p Ft(x)2604 4093 y Fs(n)2652 4078 y Fu(\))83 b(=)27 b Ft(y)2928 4042 y Fl(0)2924 4103 y Fs(n)-118 4265 y Fu(It)33 b(is)f(easy)h(to)f(c)m(hec)m(k)1051 4459 y Ft(B)1130 4418 y Fl(0)1125 4484 y Fj(0)1164 4459 y Fu(\()p Ft(x)1257 4474 y Fj(0)1297 4459 y Fu(\))83 b(=)g Ft(B)1656 4418 y Fl(0)1651 4484 y Fs(n)1698 4459 y Fu(\()p Ft(x)1791 4474 y Fs(n)1838 4459 y Fu(\))28 b(=)g Ft(B)2087 4418 y Fl(0)2110 4459 y Fu(\(0\))f(=)h(0)996 4649 y Ft(B)1075 4608 y Fl(0)1070 4673 y(\000)p Fj(1)1164 4649 y Fu(\()p Ft(x)1257 4664 y Fj(0)1297 4649 y Fu(\))83 b(=)g Ft(B)1656 4608 y Fl(0)1651 4673 y Fs(n)p Fl(\000)p Fj(1)1788 4649 y Fu(\()p Ft(x)1881 4664 y Fs(n)1929 4649 y Fu(\))27 b(=)h Ft(B)2177 4608 y Fl(0)2200 4649 y Fu(\()p Ft(h)p Fu(\))g(=)f Fm(\000)2550 4581 y Fu(3)p 2550 4626 V 2550 4717 a(4)2610 4649 y Ft(h)2666 4608 y Fl(\000)p Fj(1)1051 4882 y Ft(B)1130 4841 y Fl(0)1125 4907 y Fj(1)1164 4882 y Fu(\()p Ft(x)1257 4897 y Fj(0)1297 4882 y Fu(\))83 b(=)g Ft(B)1656 4841 y Fl(0)1651 4907 y Fs(n)p Fj(+1)1788 4882 y Fu(\()p Ft(x)1881 4897 y Fs(n)1929 4882 y Fu(\))27 b(=)h Ft(B)2177 4841 y Fl(0)2200 4882 y Fu(\()p Fm(\000)p Ft(h)p Fu(\))g(=)2551 4815 y(3)p 2551 4859 V 2551 4950 a(4)2610 4882 y Ft(h)2666 4841 y Fl(\000)p Fj(1)-118 5101 y Fu(Hence)34 b(w)m(e)f(ha)m(v)m(e)1252 5114 y Fi(8)1252 5204 y(>)1252 5234 y(>)1252 5264 y(<)1252 5443 y(>)1252 5473 y(>)1252 5503 y(:)1382 5241 y Fm(\000)1469 5174 y Fu(3)p 1469 5218 V 1469 5310 a(4)1528 5241 y Ft(\013)1590 5256 y Fl(\000)p Fj(1)1707 5241 y Fu(+)1815 5174 y(3)p 1815 5218 V 1815 5310 a(4)1873 5241 y Ft(\013)1935 5256 y Fj(1)2198 5241 y Fu(=)28 b Ft(hy)2410 5205 y Fl(0)2406 5266 y Fj(0)1382 5529 y Fm(\000)1469 5461 y Fu(3)p 1469 5506 V 1469 5597 a(4)1528 5529 y Ft(\013)1590 5544 y Fs(n)p Fl(\000)p Fj(1)1749 5529 y Fu(+)1857 5461 y(3)p 1857 5506 V 1857 5597 a(4)1916 5529 y Ft(\013)1978 5544 y Fs(n)p Fj(+1)2198 5529 y Fu(=)g Ft(hy)2410 5493 y Fl(0)2406 5554 y Fs(n)p eop %%Page: 26 5 26 17 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(26)-118 334 y(If)44 b(no)f(conditions)g(are)h(p)s(osed)h(on)e(the) i(b)s(oundary)-8 b(,)47 b(w)m(e)e(can)f(tak)m(e)g Ft(\013)2502 349 y Fl(\000)p Fj(1)2644 334 y Fu(=)i Ft(\013)2828 349 y Fs(n)p Fj(+1)3012 334 y Fu(=)h(0)d(to)f(simplify)f(the)-118 455 y(problem)31 b(and)i(get)g(the)g(follo)m(wing)c(linear)i(system)j (of)e(equations)h(to)f(determine)g Ft(\013)2972 470 y Fj(0)3012 455 y Ft(;)17 b(\013)3118 470 y Fj(1)3157 455 y Ft(;)g(:)g(:)g(:)f(;)h(\013)3438 470 y Fs(n)3517 455 y Fu(:)956 558 y Fi(2)956 733 y(6)956 793 y(6)956 853 y(6)956 913 y(6)956 973 y(6)956 1036 y(4)1067 625 y Fu(1)1242 585 y Fj(1)p 1242 602 36 4 v 1242 659 a(4)1074 706 y(1)p 1074 722 V 1074 779 a(4)1235 745 y Fu(1)1439 706 y Fj(1)p 1439 722 V 1439 779 a(4)1208 849 y Fu(.)1246 874 y(.)1284 899 y(.)1405 849 y(.)1443 874 y(.)1481 899 y(.)1603 849 y(.)1641 874 y(.)1679 899 y(.)1405 1011 y(.)1443 1036 y(.)1481 1061 y(.)1603 1011 y(.)1641 1036 y(.)1679 1061 y(.)1800 1011 y(.)1838 1036 y(.)1876 1061 y(.)1637 1150 y Fj(1)p 1637 1166 V 1637 1224 a(4)1827 1189 y Fu(1)1951 558 y Fi(3)1951 733 y(7)1951 793 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5433 y(.)1521 5561 y Ft(l)1550 5576 y Fs(nn)p Fl(\000)p Fj(1)1813 5561 y Ft(l)1842 5576 y Fs(nn)1973 4810 y Fi(3)1973 4985 y(7)1973 5045 y(7)1973 5105 y(7)1973 5165 y(7)1973 5224 y(7)1973 5284 y(7)1973 5344 y(7)1973 5408 y(5)2056 4810 y(2)2056 4985 y(6)2056 5045 y(6)2056 5105 y(6)2056 5165 y(6)2056 5224 y(6)2056 5284 y(6)2056 5344 y(6)2056 5408 y(4)2164 4876 y Fu(1)83 b Ft(u)2352 4891 y Fj(12)2337 4996 y Fu(1)123 b Ft(u)2565 5011 y Fj(23)2523 5100 y Fu(.)2561 5125 y(.)2599 5150 y(.)2728 5100 y(.)2766 5125 y(.)2804 5150 y(.)2755 5279 y(1)116 b Ft(u)2976 5294 y Fs(ii)p Fj(+1)2967 5383 y Fu(.)3005 5408 y(.)3043 5433 y(.)3206 5383 y(.)3245 5408 y(.)3283 5433 y(.)3234 5561 y(1)3357 4810 y Fi(3)3357 4985 y(7)3357 5045 y(7)3357 5105 y(7)3357 5165 y(7)3357 5224 y(7)3357 5284 y(7)3357 5344 y(7)3357 5408 y(5)p eop %%Page: 29 6 29 20 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(29)-118 214 y Fo(2.5)161 b(Least)53 b(Square)g(Appro)l(ximation) -118 457 y Fu(Supp)s(ose)34 b(that)e Ft(p)p Fu(\()p Ft(x)p Fu(\))h(is)g(a)f(p)s(olynomial)d(of)k(degree)g Ft(n)p Fu(,)g(and)g(that)g(w)m(e)h(wish)f(to)f(determine)g(its)h(co)s (e\016cien)m(ts.)-118 577 y(By)24 b(in)m(terp)s(olation,)f(it)g (su\016ces)i(to)f(kno)m(w)h Ft(p)p Fu(\()p Ft(x)p Fu(\))f(at)f(\()p Ft(n)t Fu(+)t(1\))g(distinct)g(p)s(oin)m(ts.)40 b(In)24 b(man)m(y)g(situations,)g(ho)m(w)m(ev)m(er,)-118 697 y(the)33 b(v)-5 b(alues)33 b(of)g Ft(p)p Fu(\()p Ft(x)p Fu(\))g(can)g(b)s(e)h(found)f(only)f(b)m(y)i(measuremen)m(ts)g(and)f (ma)m(y)g(b)s(e)g(in)g(error.)44 b(In)34 b(this)e(case,)i(it)e(is)-118 818 y(common)g(to)h(tak)m(e)i(more)e(than)g(\()p Ft(n)23 b Fu(+)g(1\))33 b(measuremen)m(ts)i(in)e(the)h(hop)s(e)f(that)h(these)h (measuremen)m(t)f(errors)-118 938 y(will)c Fh(aver)-5 b(age)34 b(out)p Fu(.)28 1082 y(The)f(w)m(a)m(y)h(in)d(whic)m(h)i (these)g(errors)f(a)m(v)m(erage)i(out)e(is)f(determined)h(b)m(y)h(the)g (metho)s(d)e(used)i(to)f(com)m(bined)-118 1203 y(the)40 b(measuremen)m(ts)g(to)e(obtain)h(the)g(co)s(e\016cien)m(ts)i(of)d Ft(p)p Fu(\()p Ft(x)p Fu(\).)64 b(F)-8 b(or)38 b(b)s(oth)h (computational)e(and)i(statistical)-118 1323 y(reasons,)49 b(the)c(metho)s(d)f(of)h(c)m(hoice)g(is)f(often)h(called)f Fh(le)-5 b(ast)46 b(squar)-5 b(es)44 b Fu(and)h(this)g(metho)s(d)f (also)g(enjo)m(ys)i(an)-118 1443 y(elegan)m(t)32 b(mathematical)e (simplicit)m(y)-8 b(.)-118 1696 y Fv(Example)36 b(:)49 b Fh(\(Spring)44 b(Constant\))h Fu(Ho)s(ok)m(e's)f(la)m(w)f(states)i (that)e(when)i(a)e(force)h(is)f(applied)f(to)h(a)g(spring)-118 1829 y(constructed)49 b(of)e(uniform)e(material,)j(the)g(length)f(of)g (the)g(spring)g(is)g(a)g(linear)f(function)g(of)h(the)h(force)-118 1961 y(applied:)1540 2093 y Ft(F)14 b Fu(\()p Ft(l)r Fu(\))27 b(=)h Ft(k)s Fu(\()p Ft(l)c Fm(\000)f Ft(E)6 b Fu(\))-118 2276 y(where)43 b Ft(F)56 b Fu(represen)m(ts)45 b(the)e(force,)i(constan)m(t)e Ft(E)49 b Fu(represen)m(ts)44 b(the)f(length)f(of)g(the)h(spring)f(with)g(no)g(force)-118 2408 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Fs(\013)2089 4816 y Fc(j)2122 4806 y Fs(x)2166 4847 y Ft(;)115 b(\013)2370 4862 y Fs(j)2439 4847 y Fu(real)31 b(n)m(um)m(b)s(ers.)-118 5052 y(W)-8 b(e)33 b(could)f(also)g(use)h (piecewise)g(p)s(olynomials)d(or)i(splines.)28 5196 y Fv(Problem)p Fu(:)299 b(T)-8 b(o)32 b(\014nd)h Ft(a)1161 5211 y Fj(0)1201 5196 y Ft(;)17 b(a)1296 5211 y Fj(1)1335 5196 y Ft(;)g(:)g(:)g(:)f(;)h(a)1605 5211 y Fs(n)1685 5196 y Fu(suc)m(h)34 b(that)1388 5347 y Fs(m)1348 5377 y Fi(X)1363 5587 y Fs(i)p Fj(=1)1508 5472 y Ft(w)1578 5487 y Fs(i)1606 5472 y Fu([)p Ft(\036)p Fu(\()p Ft(x)1784 5487 y Fs(i)1812 5472 y Fu(\))23 b Fm(\000)f Ft(y)2020 5487 y Fs(i)2048 5472 y Fu(])2075 5431 y Fj(2)2142 5472 y Fu(=)28 b(min)p eop %%Page: 32 9 32 23 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(32)-118 214 y(where)34 b(\()p Ft(x)257 229 y Fs(i)285 214 y Ft(;)17 b(y)377 229 y Fs(i)405 214 y Fu(\))32 b(are)h(giv)m(en)f (data.)28 434 y(Let)300 b Ft(g)t Fu(\()p Ft(a)610 449 y Fj(0)649 434 y Ft(;)17 b(a)744 449 y 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b(these)h(equal)g(to)f(zero,)k(collecting)43 b(co)s(e\016cien)m(ts)k(of)d(the)i Ft(a)2453 1150 y Fs(i)2482 1135 y Fu(,)i(w)m(e)f(obtain)d(the)i(follo)m(wing)d(linear)-118 1255 y(system)1008 1287 y Fi(2)1008 1462 y(6)1008 1522 y(6)1008 1582 y(6)1008 1645 y(4)1120 1375 y Ft(s)1166 1390 y Fj(00)1331 1375 y Ft(s)1377 1390 y Fj(01)1538 1375 y Fm(\001)17 b(\001)g(\001)85 b Ft(s)1787 1390 y Fj(0)p Fs(n)1120 1495 y Ft(s)1166 1510 y Fj(10)1331 1495 y Ft(s)1377 1510 y Fj(11)1538 1495 y Fm(\001)17 b(\001)g(\001)85 b Ft(s)1787 1510 y Fj(1)p Fs(n)1166 1591 y Fu(.)1166 1624 y(.)1166 1657 y(.)1378 1591 y(.)1378 1624 y(.)1378 1657 y(.)1545 1599 y(.)1583 1624 y(.)1621 1649 y(.)1792 1591 y(.)1792 1624 y(.)1792 1657 y(.)1116 1777 y Ft(s)1162 1792 y Fs(n)p Fj(0)1327 1777 y Ft(s)1373 1792 y Fs(n)p Fj(1)1538 1777 y Fm(\001)17 b(\001)g(\001)82 b Ft(s)1784 1792 y Fs(nn)1915 1287 y Fi(3)1915 1462 y(7)1915 1522 y(7)1915 1582 y(7)1915 1645 y(5)1998 1287 y(2)1998 1462 y(6)1998 1522 y(6)1998 1582 y(6)1998 1645 y(4)2110 1375 y Ft(a)2161 1390 y Fj(0)2110 1495 y Ft(a)2161 1510 y Fj(1)2141 1591 y Fu(.)2141 1624 y(.)2141 1657 y(.)2106 1777 y Ft(a)2157 1792 y Fs(n)2245 1287 y Fi(3)2245 1462 y(7)2245 1522 y(7)2245 1582 y(7)2245 1645 y(5)2340 1577 y Fu(=)2443 1287 y Fi(2)2443 1462 y(6)2443 1522 y(6)2443 1582 y(6)2443 1645 y(4)2555 1375 y Ft(c)2597 1390 y Fj(0)2555 1495 y Ft(c)2597 1510 y Fj(1)2582 1591 y Fu(.)2582 1624 y(.)2582 1657 y(.)2551 1777 y Ft(c)2593 1792 y Fs(n)2682 1287 y Fi(3)2682 1462 y(7)2682 1522 y(7)2682 1582 y(7)2682 1645 y(5)-118 1952 y Fu(where)770 2124 y Ft(s)816 2139 y Fs(j)t(i)904 2124 y Fu(=)1048 1999 y Fs(m)1008 2029 y Fi(X)1015 2241 y Fs(k)r Fj(=1)1168 2124 y Ft(w)1238 2139 y Fs(k)1280 2124 y Ft(\036)1338 2139 y Fs(j)1375 2124 y Fu(\()p Ft(x)1468 2139 y Fs(k)1511 2124 y Fu(\))p Ft(\036)1607 2139 y Fs(i)1635 2124 y Fu(\()p Ft(x)1728 2139 y Fs(k)1771 2124 y Fu(\))p Ft(;)309 b(c)2187 2139 y Fs(j)2251 2124 y Fu(=)2395 1999 y Fs(m)2355 2029 y Fi(X)2362 2241 y Fs(k)r Fj(=1)2515 2124 y Ft(w)2585 2139 y Fs(k)2628 2124 y Ft(\036)2686 2139 y Fs(j)2722 2124 y Fu(\()p Ft(x)2815 2139 y Fs(k)2858 2124 y Fu(\))p Ft(y)2944 2139 y Fs(k)-118 2383 y Fu(The)34 b(co)s(e\016cien)m(t)g(matrix)e(is)g (again)g(symmetric)g(but)i(in)e(order)i(to)e(conclude)i(that)f(it)f(is) h(p)s(ositiv)m(e)g(de\014nite,)-118 2503 y(suitable)f(conditions)f(m)m (ust)i(b)s(e)g(imp)s(osed)e(on)i Fm(f)p Ft(\036)1713 2518 y Fs(j)1749 2503 y Fm(g)1799 2467 y Fs(n)1799 2528 y(j)t Fj(=0)1958 2503 y Fu(and)f Fm(f)p Ft(x)2252 2518 y Fs(i)2281 2503 y Fm(g)2331 2467 y Fs(m)2331 2528 y(i)p Fj(=1)2449 2503 y Fu(.)-118 2951 y Fk(2.5.4)136 b(Orthogonal)46 b(P)l(olynomials)-118 3159 y Fu(The)33 b(normal)c(equtaions)j(are)g (useful)g(for)f(theoretical)f(purp)s(oses,)k(or)d(for)g(computation)f (when)j Ft(n)f Fu(is)f(small.)-118 3280 y(But)i(they)g(ha)m(v)m(e)h(a)e (tendency)j(to)d(b)s(ecome)h(v)m(ery)h(ill-co)s(ditioned)28 b(\(unstable\))k(as)h Ft(n)g Fu(increases.)-118 3543 y Fv(De\014nition)j(2.4)49 b Fh(L)-5 b(et)25 b Ft(q)798 3558 y Fj(0)838 3543 y Fu(\()p Ft(x)p Fu(\))p Ft(;)17 b(q)1056 3558 y Fj(1)1096 3543 y Fu(\()p Ft(x)p Fu(\))p Ft(;)g(:)g(:)g(:)f(;)h(q)1489 3558 y Fs(n)1536 3543 y Fu(\()p Ft(x)p Fu(\))25 b Fh(b)-5 b(e)24 b(p)-5 b(olynomials)24 b(of)h(de)-5 b(gr)g(e)g(e)24 b Fu(0)p Ft(;)17 b Fu(1)p Ft(;)g(:)g(:)g(:)e(;)i(n)25 b Fh(r)-5 b(esp)g(e)g(ctively.)41 b(Then)-118 3676 y(we)c(wil)5 b(l)37 b(say)g(that)h Ft(q)635 3691 y Fj(0)675 3676 y Ft(;)17 b(q)762 3691 y Fj(1)801 3676 y Ft(;)g(:)g(:)g(:)f(;)h(q)1063 3691 y Fs(n)1147 3676 y Fh(ar)-5 b(e)38 b(ortho)-5 b(gonal)36 b(with)i(r)-5 b(esp)g(e)g(ct)37 b(to)g(the)h(p)-5 b(oints)37 b Ft(x)2958 3691 y Fj(1)2998 3676 y Ft(;)17 b(:)g(:)g(:)f(;)h(x)3272 3691 y Fs(m)3376 3676 y Fh(and)37 b(weights)-118 3808 y Ft(w)-48 3823 y Fj(1)-9 3808 y Ft(;)17 b(:)g(:)g(:)f(;)h(w)280 3823 y Fs(m)381 3808 y Fh(if)1144 3875 y Fs(m)1104 3904 y Fi(X)1111 4117 y Fs(k)r Fj(=1)1264 3999 y Ft(w)1334 4014 y Fs(k)1376 3999 y Ft(q)1419 4014 y Fs(i)1448 3999 y Fu(\()p Ft(x)1541 4014 y Fs(k)1584 3999 y Fu(\))p Ft(q)1665 4014 y Fs(j)1702 3999 y Fu(\()p Ft(x)1795 4014 y Fs(k)1838 3999 y Fu(\))27 b(=)h(0)p Ft(;)315 b(i)28 b Fm(6)p Fu(=)g Ft(j)22 b(:)28 4347 y Fu(If)28 b(w)m(e)h(tak)m(e)56 b Fm(f)p Ft(q)587 4362 y Fs(i)616 4347 y Fu(\()p Ft(x)p Fu(\))p Fm(g)797 4310 y Fs(n)797 4371 y(i)p Fj(=0)970 4347 y Fu(as)29 b(basis)e(functions)h(instead)g(of)55 b Fm(f)p Ft(x)2307 4310 y Fs(i)2336 4347 y Fm(g)2386 4310 y Fs(n)2386 4371 y(i)p Fj(=0)2559 4347 y Fu(for)28 b(least)f(square)i(appro)m(ximation,)-118 4467 y(the)k(co)s(e\016cien)m (t)g(matrix)e Ft(S)38 b Fu(of)33 b(the)g(normal)d(equations)j(will)d(b) s(ecome)j(a)f(diagonal)e(matrix,)h(i.e.,)1016 4600 y Fi(2)1016 4775 y(6)1016 4835 y(6)1016 4894 y(6)1016 4958 y(4)1124 4687 y Ft(s)1170 4702 y Fj(00)1328 4808 y Ft(s)1374 4823 y Fj(11)1537 4912 y Fu(.)1575 4936 y(.)1614 4962 y(.)1729 5090 y Ft(s)1775 5105 y Fs(nn)1906 4600 y Fi(3)1906 4775 y(7)1906 4835 y(7)1906 4894 y(7)1906 4958 y(5)1989 4600 y(2)1989 4775 y(6)1989 4835 y(6)1989 4894 y(6)1989 4958 y(4)2101 4687 y Ft(a)2152 4702 y Fj(0)2101 4808 y Ft(a)2152 4823 y Fj(1)2133 4903 y Fu(.)2133 4936 y(.)2133 4970 y(.)2097 5090 y Ft(a)2148 5105 y Fs(n)2237 4600 y Fi(3)2237 4775 y(7)2237 4835 y(7)2237 4894 y(7)2237 4958 y(5)2331 4889 y Fu(=)2435 4600 y Fi(2)2435 4775 y(6)2435 4835 y(6)2435 4894 y(6)2435 4958 y(4)2546 4687 y Ft(c)2588 4702 y Fj(0)2546 4808 y Ft(c)2588 4823 y Fj(1)2574 4903 y Fu(.)2574 4936 y(.)2574 4970 y(.)2543 5090 y Ft(c)2585 5105 y Fs(n)2673 4600 y Fi(3)2673 4775 y(7)2673 4835 y(7)2673 4894 y(7)2673 4958 y(5)-118 5265 y Fu(where)924 5436 y Ft(s)970 5451 y Fs(ii)1049 5436 y Fu(=)1194 5312 y Fs(m)1153 5342 y Fi(X)1161 5554 y Fs(k)r Fj(=1)1313 5436 y Ft(w)1383 5451 y Fs(k)1426 5436 y Ft(q)1473 5395 y Fj(2)1469 5461 y Fs(i)1512 5436 y Fu(\()p Ft(x)1605 5451 y Fs(k)1648 5436 y Fu(\))p Ft(;)309 b(c)2064 5451 y Fs(i)2120 5436 y Fu(=)2265 5312 y Fs(m)2224 5342 y Fi(X)2231 5554 y Fs(k)r Fj(=1)2384 5436 y Ft(w)2454 5451 y Fs(k)2497 5436 y Ft(q)2540 5451 y Fs(i)2568 5436 y Fu(\()p Ft(x)2661 5451 y Fs(k)2704 5436 y Fu(\))p Ft(y)2790 5451 y Fs(k)p eop %%Page: 33 10 33 24 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(33)-118 214 y(Hence)34 b(w)m(e)f(obtain)f(the)h(least-squares)g(p) s(olynomial)1501 504 y Ft(p)p Fu(\()p Ft(x)p Fu(\))28 b(=)1863 380 y Fs(n)1812 410 y Fi(X)1827 620 y Fs(i)p Fj(=0)1973 504 y Ft(a)2024 519 y Fs(i)2052 504 y Ft(q)2095 519 y Fs(i)2124 504 y Fu(\()p Ft(x)p Fu(\))-118 807 y(with)k(co)s (e\016cien)m(ts)1458 1144 y Ft(a)1509 1159 y Fs(i)1565 1144 y Fu(=)1720 854 y Fs(m)1679 884 y Fi(X)1687 1096 y Fs(k)r Fj(=1)1839 978 y Ft(w)1909 993 y Fs(k)1952 978 y Ft(q)1995 993 y Fs(i)2023 978 y Fu(\()p Ft(x)2116 993 y Fs(k)2159 978 y Fu(\))p Ft(y)2245 993 y Fs(k)p 1679 1122 609 4 v 1757 1172 a(m)1717 1202 y Fi(X)1724 1414 y Fs(k)r Fj(=1)1877 1297 y Ft(w)1947 1312 y Fs(k)1990 1297 y Ft(q)2037 1255 y Fj(2)2033 1321 y Fs(i)2076 1297 y Fu(\()p Ft(x)2169 1312 y Fs(k)2212 1297 y Fu(\))-118 1144 y(\(2.3\))-118 1556 y(The)37 b(use)f(of)g(orthogonal)e(basis)h (reduces)j(the)e(normal)e(equations)i(to)g(a)f(diagonal)f(system)i(of)g (equations,)-118 1676 y(whic)m(h)d(is)f(trivial)e(to)i(solv)m(e.)-118 2003 y Fk(2.5.5)136 b(Computation)46 b(of)f(the)g Fb(q)1564 2021 y Fa(i)1602 2003 y Fk(\()p Fb(x)p Fk(\))-118 2212 y Fu(Assume)39 b(all)c(the)k(w)m(eigh)m(ts)f Ft(w)985 2227 y Fs(i)1050 2212 y Fu(=)e(1)i(\(the)g(general)f(case)i(left)e(as)h (an)g(assignmen)m(t\).)59 b(Let)38 b Ft(q)3272 2227 y Fj(0)3312 2212 y Fu(\()p Ft(x)p Fu(\))f Fm(\021)g Fu(1)74 b(and)-118 2332 y Ft(q)-75 2347 y Fj(1)-35 2332 y Fu(\()p Ft(x)p Fu(\))28 b(=)f Ft(x)c Fm(\000)f Ft(\013)466 2347 y Fj(1)533 2332 y Fu(,)33 b(where)h Ft(\013)937 2347 y Fj(1)1009 2332 y Fu(is)e(to)g(b)s(e)h(determined.)43 b(Since)849 2628 y(0)27 b(=)1069 2503 y Fs(m)1029 2533 y Fi(X)1036 2746 y Fs(k)r Fj(=1)1189 2628 y Ft(q)1232 2643 y Fj(0)1272 2628 y Fu(\()p Ft(x)1365 2643 y Fs(k)1408 2628 y Fu(\))p Ft(q)1489 2643 y Fj(1)1529 2628 y Fu(\()p Ft(x)1622 2643 y Fs(k)1665 2628 y Fu(\))g(=)1874 2503 y Fs(m)1834 2533 y Fi(X)1841 2746 y Fs(k)r Fj(=1)1978 2628 y Fu(\()p Ft(x)2071 2643 y Fs(k)2136 2628 y Fm(\000)22 b Ft(\013)2297 2643 y Fj(1)2337 2628 y Fu(\))28 b(=)2547 2503 y Fs(m)2506 2533 y Fi(X)2514 2746 y Fs(k)r Fj(=1)2667 2628 y Ft(x)2722 2643 y Fs(k)2787 2628 y Fm(\000)22 b Ft(m\013)3033 2643 y Fj(1)849 3118 y Fm(\))125 b Ft(\013)1136 3133 y Fj(1)1203 3118 y Fu(=)1357 2827 y Fs(m)1316 2857 y Fi(X)1324 3069 y Fs(k)r Fj(=1)1477 2952 y Ft(x)1532 2967 y Fs(k)p 1316 3095 259 4 v 1403 3186 a Ft(m)1618 3118 y(:)-1763 b Fu(\(2.4\))-118 3369 y(No)m(w)33 b(let)92 b Ft(q)348 3384 y Fj(2)388 3369 y Fu(\()p Ft(x)p Fu(\))28 b(=)f Ft(xq)748 3384 y Fj(1)788 3369 y Fu(\()p Ft(x)p Fu(\))c Fm(\000)f Ft(\013)1103 3384 y Fj(2)1143 3369 y Ft(q)1186 3384 y Fj(1)1226 3369 y Fu(\()p Ft(x)p Fu(\))g Fm(\000)h Ft(\014)1534 3384 y Fj(1)1590 3369 y Ft(;)32 b Fu(where)i Ft(\013)1993 3384 y Fj(2)2065 3369 y Fu(and)f Ft(\014)2310 3384 y Fj(1)2382 3369 y Fu(are)f(to)h(b)s(e)f(determined.) 44 b(Because)522 3665 y(0)83 b(=)853 3541 y Fs(m)813 3570 y Fi(X)820 3783 y Fs(k)r Fj(=1)973 3665 y Ft(q)1016 3680 y Fj(2)1056 3665 y Fu(\()p Ft(x)1149 3680 y Fs(k)1192 3665 y Fu(\))p Ft(q)1273 3680 y Fj(0)1312 3665 y Fu(\()p Ft(x)1405 3680 y Fs(k)1448 3665 y Fu(\))g(=)1769 3541 y Fs(m)1728 3570 y Fi(X)1736 3783 y Fs(k)r Fj(=1)1872 3665 y Fu([)p Ft(x)1954 3680 y Fs(k)1997 3665 y Ft(q)2040 3680 y Fj(1)2080 3665 y Fu(\()p Ft(x)2173 3680 y Fs(k)2216 3665 y Fu(\))22 b Fm(\000)h Ft(\013)2438 3680 y Fj(2)2477 3665 y Ft(q)2520 3680 y Fj(1)2560 3665 y Fu(\()p Ft(x)2653 3680 y Fs(k)2696 3665 y Fu(\))f Fm(\000)h Ft(\014)2911 3680 y Fj(1)2950 3665 y Fu(])522 3956 y(0)83 b(=)853 3831 y Fs(m)813 3861 y Fi(X)820 4073 y Fs(k)r Fj(=1)973 3956 y Ft(q)1016 3971 y Fj(2)1056 3956 y Fu(\()p Ft(x)1149 3971 y Fs(k)1192 3956 y Fu(\))p Ft(q)1273 3971 y Fj(1)1312 3956 y Fu(\()p Ft(x)1405 3971 y Fs(k)1448 3956 y Fu(\))g(=)1769 3831 y Fs(m)1728 3861 y Fi(X)1736 4073 y Fs(k)r Fj(=1)1872 3956 y Fu([)p Ft(x)1954 3971 y Fs(k)1997 3956 y Ft(q)2040 3971 y Fj(1)2080 3956 y Fu(\()p Ft(x)2173 3971 y Fs(k)2216 3956 y Fu(\))22 b Fm(\000)h Ft(\013)2438 3971 y Fj(2)2477 3956 y Ft(q)2520 3971 y Fj(1)2560 3956 y Fu(\()p Ft(x)2653 3971 y Fs(k)2696 3956 y Fu(\))f Fm(\000)h Ft(\014)2911 3971 y Fj(1)2950 3956 y Fu(])p Ft(q)3020 3971 y Fj(1)3060 3956 y Fu(\()p Ft(x)3153 3971 y Fs(k)3196 3956 y Fu(\))-118 4354 y(and)33 b(noting)e(that)627 4230 y Fs(m)586 4259 y Fi(X)594 4472 y Fs(k)r Fj(=1)730 4354 y Ft(q)773 4369 y Fj(1)813 4354 y Fu(\()p Ft(x)906 4369 y Fs(k)949 4354 y Fu(\))d(=)f(0)p Ft(;)33 b Fu(w)m(e)g(ha)m(v)m(e)714 4628 y Fs(m)674 4658 y Fi(X)681 4870 y Fs(k)r Fj(=1)834 4753 y Ft(x)889 4768 y Fs(k)932 4753 y Ft(q)975 4768 y Fj(1)1015 4753 y Fu(\()p Ft(x)1108 4768 y Fs(k)1151 4753 y Fu(\))22 b Fm(\000)h Ft(m\014)1451 4768 y Fj(1)1518 4753 y Fu(=)28 b(0)p Ft(;)2048 4628 y Fs(m)2007 4658 y Fi(X)2014 4870 y Fs(k)r Fj(=1)2151 4753 y Fu(\()p Ft(x)2244 4768 y Fs(k)2287 4753 y Ft(q)2334 4712 y Fj(2)2330 4777 y(1)2373 4753 y Fu(\()p Ft(x)2466 4768 y Fs(k)2509 4753 y Fu(\))22 b Fm(\000)h Ft(\013)2731 4768 y Fj(2)2770 4753 y Ft(q)2817 4712 y Fj(2)2813 4777 y(1)2856 4753 y Fu(\()p Ft(x)2949 4768 y Fs(k)2993 4753 y Fu(\)\))k(=)h(0)674 5243 y Fm(\))124 b Ft(\014)953 5258 y Fj(1)1021 5243 y Fu(=)1175 4952 y Fs(m)1134 4982 y Fi(X)1142 5194 y Fs(k)r Fj(=1)1278 5076 y Ft(x)1333 5091 y Fs(k)1376 5076 y Ft(q)1419 5091 y Fj(1)1459 5076 y Fu(\()p Ft(x)1552 5091 y Fs(k)1595 5076 y Fu(\))p 1134 5220 499 4 v 1341 5311 a Ft(m)1643 5243 y(;)309 b(\013)2041 5258 y Fj(2)2108 5243 y Fu(=)2262 4952 y Fs(m)2222 4982 y Fi(X)2229 5194 y Fs(k)r Fj(=1)2366 5076 y Ft(x)2421 5091 y Fs(k)2464 5076 y Ft(q)2511 5040 y Fj(2)2507 5101 y(1)2550 5076 y Fu(\()p Ft(x)2643 5091 y Fs(k)2686 5076 y Fu(\))p 2222 5220 503 4 v 2312 5270 a Fs(m)2271 5300 y Fi(X)2278 5512 y Fs(k)r Fj(=1)2415 5395 y Ft(q)2462 5361 y Fj(2)2458 5419 y(1)2501 5395 y Fu(\()p Ft(x)2594 5410 y Fs(k)2637 5395 y Fu(\))2734 5243 y Ft(:)p eop %%Page: 34 11 34 25 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(34)28 214 y(The)34 b(computation)d(for)h(the)h(remaining)d Ft(q)1613 229 y Fs(i)1641 214 y Fu(\()p Ft(x)p Fu(\))j(pro)s(ceeds)h (in)e(an)g(analogous)g(fashion.)28 358 y Fn(Three-T)-8 b(erm)42 b(Recurrence)h(Relation)p Fu(:)61 b(Assume)42 b(that)f(w)m(e)i(ha)m(v)m(e)g(determined)e Ft(q)2997 373 y Fj(0)3037 358 y Ft(;)17 b(q)3124 373 y Fj(1)3163 358 y Ft(;)g(:)g(:)g(:)f(;)h(q)3425 373 y Fs(j)3461 358 y Fu(,)44 b(then)e(w)m(e)-118 478 y(can)33 b(de\014ne)h Ft(q)386 493 y Fs(j)t Fj(+1)545 478 y Fu(in)e(terms)g(of)g Ft(q)1084 493 y Fs(j)1154 478 y Fu(and)g Ft(q)1386 493 y Fs(j)t Fl(\000)p Fj(1)1546 478 y Fu(as)h(follo)m(wing)c(so)k(called)f (three-term)g(recurrence)i(form)m(ula)955 698 y Ft(q)998 713 y Fs(j)t Fj(+1)1125 698 y Fu(\()p Ft(x)p Fu(\))83 b(=)g Ft(xq)1596 713 y Fs(j)1633 698 y Fu(\()p Ft(x)p Fu(\))23 b Fm(\000)f Ft(\013)1948 713 y Fs(j)t Fj(+1)2075 698 y Ft(q)2118 713 y Fs(j)2155 698 y Fu(\()p Ft(x)p Fu(\))g Fm(\000)h Ft(\014)2463 713 y Fs(j)2500 698 y Ft(q)2543 713 y Fs(j)t Fl(\000)p Fj(1)2670 698 y Fu(\()p Ft(x)p Fu(\))-2919 b(\(2.5\))-118 1039 y(where)34 b Ft(\013)226 1054 y Fs(j)t Fj(+1)385 1039 y Fu(and)f Ft(\014)630 1054 y Fs(j)699 1039 y Fu(are)f(to)h(b)s(e)f(determined)h(from)e(the)i (orthogonalit)m(y)e(requiremen)m(ts)1439 1213 y Fs(m)1399 1243 y Fi(X)1406 1455 y Fs(k)r Fj(=1)1559 1338 y Ft(q)1602 1353 y Fs(j)t Fj(+1)1729 1338 y Fu(\()p Ft(x)1822 1353 y Fs(k)1865 1338 y Fu(\))p Ft(q)1946 1353 y Fs(j)1983 1338 y Fu(\()p Ft(x)2076 1353 y Fs(k)2119 1338 y Fu(\))83 b(=)g(0)-2566 b(\(2.6\))1349 1623 y Fs(m)1308 1653 y Fi(X)1316 1865 y Fs(k)r Fj(=1)1469 1747 y Ft(q)1512 1762 y Fs(j)t Fj(+1)1639 1747 y Fu(\()p Ft(x)1732 1762 y Fs(k)1775 1747 y Fu(\))p Ft(q)1856 1762 y Fs(j)t Fl(\000)p Fj(1)1983 1747 y Fu(\()p Ft(x)2076 1762 y Fs(k)2119 1747 y Fu(\))83 b(=)g(0)-2566 b(\(2.7\))-118 2103 y(If)30 b(these)g(t)m(w)m(o)h (relations)d(are)i(satis\014ed,)h(then)f Ft(q)1620 2118 y Fs(j)t Fj(+1)1777 2103 y Fu(m)m(ust)f(b)s(e)h(orthogonal)e(to)i(all)d (the)j(previous)g Ft(q)3472 2118 y Fs(l)3499 2103 y Fu(,)g(0)d Fm(\024)i Ft(l)h Fm(\024)-118 2223 y Ft(j)e Fm(\000)23 b Fu(2.)43 b(Since)33 b(b)m(y)g(\(2.5\))465 2395 y Fs(m)424 2425 y Fi(X)432 2637 y Fs(k)r Fj(=1)585 2519 y Ft(q)628 2534 y Fs(j)t Fj(+1)755 2519 y Fu(\()p Ft(x)848 2534 y Fs(k)891 2519 y Fu(\))p Ft(q)972 2534 y Fs(l)998 2519 y Fu(\()p Ft(x)1091 2534 y Fs(k)1134 2519 y Fu(\))83 b(=)1455 2395 y Fs(m)1414 2425 y Fi(X)1421 2637 y Fs(k)r Fj(=1)1574 2519 y Ft(x)1629 2534 y Fs(k)1672 2519 y Ft(q)1715 2534 y Fs(j)1752 2519 y Fu(\()p Ft(x)1845 2534 y Fs(k)1888 2519 y Fu(\))p Ft(q)1969 2534 y Fs(l)1995 2519 y Fu(\()p Ft(x)2088 2534 y Fs(k)2131 2519 y Fu(\))1414 2843 y Fm(\000)p Ft(\013)1553 2858 y Fs(j)t Fj(+1)1737 2719 y Fs(m)1697 2748 y Fi(X)1704 2961 y Fs(k)r Fj(=1)1857 2843 y Ft(q)1900 2858 y Fs(j)1937 2843 y Fu(\()p Ft(x)2030 2858 y Fs(k)2073 2843 y Fu(\))p Ft(q)2154 2858 y Fs(l)2180 2843 y Fu(\()p Ft(x)2273 2858 y Fs(k)2316 2843 y Fu(\))22 b Fm(\000)h Ft(\014)2531 2858 y Fs(j)2625 2719 y(m)2584 2748 y Fi(X)2592 2961 y Fs(k)r Fj(=1)2745 2843 y Ft(q)2788 2858 y Fs(j)t Fl(\000)p Fj(1)2914 2843 y Fu(\()p Ft(x)3007 2858 y Fs(k)3051 2843 y Fu(\))p Ft(q)3132 2858 y Fs(l)3158 2843 y Fu(\()p Ft(x)3251 2858 y Fs(k)3294 2843 y Fu(\))-118 3153 y(The)38 b(last)e(t)m(w)m(o)i(terms)e(are)h(zero)h(b)m(y)g(assumption,)f (whereas)h Ft(xq)2277 3168 y Fs(l)2304 3153 y Fu(\()p Ft(x)p Fu(\))f(is)g(a)f(p)s(olynomial)e(of)i(degree)i(\()p Ft(l)27 b Fu(+)e(1\))-118 3274 y(and)33 b(can)f(b)s(e)h(expressed)j(as) c(a)h(linear)e(com)m(bination)f(of)i Ft(q)2002 3289 y Fj(0)2042 3274 y Ft(;)17 b(q)2129 3289 y Fj(1)2168 3274 y Ft(;)g(:)g(:)g(:)f(;)h(q)2430 3289 y Fs(l)q Fj(+1)2546 3274 y Fu(.)44 b(Hence,)34 b(the)f(\014rst)g(term)f(is)g(zero.)-118 3394 y(Returning)g(to)g(\(2.6\)\(2.7\))g(and)g(putting)g(in)g Ft(q)1590 3409 y Fs(j)t Fj(+1)1749 3394 y Fu(from)g(\(2.5\))g(leads)g (to)850 3693 y Ft(\013)912 3708 y Fs(j)t Fj(+1)1122 3693 y Fu(=)1310 3625 y(1)p 1291 3670 88 4 v 1291 3761 a Ft(\015)1342 3776 y Fs(j)1445 3568 y(m)1404 3598 y Fi(X)1412 3810 y Fs(k)r Fj(=1)1565 3693 y Ft(x)1620 3708 y Fs(k)1663 3693 y Ft(q)1710 3652 y Fj(2)1706 3718 y Fs(j)1749 3693 y Fu(\()p Ft(x)1842 3708 y Fs(k)1885 3693 y Fu(\))p Ft(;)317 b(\015)2318 3708 y Fs(j)2382 3693 y Fu(=)2526 3568 y Fs(m)2485 3598 y Fi(X)2493 3810 y Fs(k)r Fj(=1)2646 3693 y Ft(q)2693 3652 y Fj(2)2689 3718 y Fs(j)2732 3693 y Fu(\()p Ft(x)2825 3708 y Fs(k)2868 3693 y Fu(\))-3024 b(\(2.8\))947 4077 y Ft(\014)1002 4092 y Fs(j)1122 4077 y Fu(=)1355 4009 y(1)p 1291 4054 178 4 v 1291 4145 a Ft(\015)1342 4160 y Fs(j)t Fl(\000)p Fj(1)1535 3952 y Fs(m)1494 3982 y Fi(X)1502 4194 y Fs(k)r Fj(=1)1655 4077 y Ft(x)1710 4092 y Fs(k)1753 4077 y Ft(q)1796 4092 y Fs(j)1833 4077 y Fu(\()p Ft(x)1926 4092 y Fs(k)1969 4077 y Fu(\))p Ft(q)2050 4092 y Fs(j)t Fl(\000)p Fj(1)2177 4077 y Fu(\()p Ft(x)2270 4092 y Fs(k)2313 4077 y Fu(\))-118 4387 y(The)28 b Ft(\014)132 4402 y Fs(j)197 4387 y Fu(can)g(b)s(e)f (computed)h(in)f(a)h(b)s(etter)g(w)m(a)m(y)g(if)f(w)m(e)i(substitute)f (for)f Ft(xq)2523 4402 y Fs(j)t Fl(\000)p Fj(1)2650 4387 y Fu(\()p Ft(x)p Fu(\))h(using)g(\(2.5\))f(and)g(then)i(note)-118 4507 y(that)568 4554 y Fs(m)527 4584 y Fi(X)535 4796 y Fs(k)r Fj(=1)688 4679 y Ft(q)731 4694 y Fs(j)768 4679 y Fu(\()p Ft(x)861 4694 y Fs(k)904 4679 y Fu(\)[)p Ft(q)1012 4694 y Fs(j)1049 4679 y Fu(\()p Ft(x)1142 4694 y Fs(k)1185 4679 y Fu(\))22 b(+)g Ft(\013)1405 4694 y Fs(j)1441 4679 y Ft(q)1484 4694 y Fs(j)t Fl(\000)p Fj(1)1611 4679 y Fu(\()p Ft(x)1704 4694 y Fs(k)1747 4679 y Fu(\))g(+)g Ft(\014)1960 4694 y Fs(j)t Fl(\000)p Fj(1)2087 4679 y Ft(q)2130 4694 y Fs(j)t Fl(\000)p Fj(2)2257 4679 y Fu(\()p Ft(x)2350 4694 y Fs(k)2393 4679 y Fu(\)])28 b(=)2630 4554 y Fs(m)2589 4584 y Fi(X)2597 4796 y Fs(k)r Fj(=1)2750 4679 y Ft(q)2797 4637 y Fj(2)2793 4703 y Fs(j)2836 4679 y Fu(\()p Ft(x)2929 4694 y Fs(k)2972 4679 y Fu(\))g(=)f Ft(\015)3192 4694 y Fs(j)-118 4938 y Fu(b)m(y)33 b(the)g(orthogonalit)m (y)e(of)h(the)h Ft(q)t Fu('s.)44 b(Th)m(us)1646 5175 y Ft(\014)1701 5190 y Fs(j)1765 5175 y Fu(=)1924 5108 y Ft(\015)1975 5123 y Fs(j)p 1879 5152 V 1879 5243 a Ft(\015)1930 5258 y Fs(j)t Fl(\000)p Fj(1)2083 5175 y Ft(:)-2228 b Fu(\(2.9\))28 5472 y(Summarize)31 b(the)i(orthogonal)e(p)s (olynomial)e(algorithm)g(as)k(follo)m(ws:)p eop %%Page: 35 12 35 26 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(35)p 121 118 3515 4 v 121 2235 4 2118 v 327 414 a Fm(\017)48 b Fu(\(i\))32 b(Set)h Ft(q)772 429 y Fj(0)812 414 y Fu(\()p Ft(x)p Fu(\))28 b Fm(\021)g Fu(1)p Ft(;)211 b(q)1406 429 y Fj(1)1446 414 y Fu(\()p Ft(x)p Fu(\))28 b(=)g Ft(x)22 b Fm(\000)1909 374 y Fj(1)p 1896 391 63 4 v 1896 448 a Fs(m)2009 289 y(m)1968 319 y Fi(X)1976 531 y Fs(k)r Fj(=1)2112 414 y Ft(x)2167 429 y Fs(k)2210 414 y Ft(;)212 b(\015)2500 429 y Fj(0)2567 414 y Fu(=)27 b Ft(m;)327 707 y Fm(\017)48 b Fu(\(ii\))31 b(Compute)i(for)f Ft(j)h Fu(=)28 b(1)k(to)h Ft(n)22 b Fm(\000)h Fu(1)1307 1003 y Ft(\015)1358 1018 y Fs(j)1477 1003 y Fu(=)1677 879 y Fs(m)1636 908 y Fi(X)1644 1121 y Fs(k)r Fj(=1)1797 1003 y Ft(q)1844 962 y Fj(2)1840 1028 y Fs(j)1883 1003 y Fu(\()p Ft(x)1976 1018 y Fs(k)2019 1003 y Fu(\))1206 1327 y Ft(\013)1268 1342 y Fs(j)t Fj(+1)1477 1327 y Fu(=)1666 1259 y(1)p 1646 1304 88 4 v 1646 1395 a Ft(\015)1697 1410 y Fs(j)1801 1202 y(m)1760 1232 y Fi(X)1768 1444 y Fs(k)r Fj(=1)1921 1327 y Ft(x)1976 1342 y Fs(k)2019 1327 y Ft(q)2066 1286 y Fj(2)2062 1351 y Fs(j)2105 1327 y Fu(\()p Ft(x)2198 1342 y Fs(k)2241 1327 y Fu(\))p Ft(;)212 b(\014)2573 1342 y Fs(j)2637 1327 y Fu(=)2796 1259 y Ft(\015)2847 1274 y Fs(j)p 2750 1304 178 4 v 2750 1395 a Ft(\015)2801 1410 y Fs(j)t Fl(\000)p Fj(1)1093 1562 y Ft(q)1136 1577 y Fs(j)t Fj(+1)1263 1562 y Fu(\()p Ft(x)p Fu(\))83 b(=)g Ft(xq)1734 1577 y Fs(j)1772 1562 y Fu(\()p Ft(x)p Fu(\))22 b Fm(\000)h Ft(\013)2087 1577 y Fs(j)t Fj(+1)2213 1562 y Ft(q)2256 1577 y Fs(j)2293 1562 y Fu(\()p Ft(x)p Fu(\))g Fm(\000)f Ft(\014)2601 1577 y Fs(j)2638 1562 y Ft(q)2681 1577 y Fs(j)t Fl(\000)p Fj(1)2808 1562 y Fu(\()p Ft(x)p Fu(\))327 1824 y Fm(\017)48 b Fu(\(iii\))30 b(Compute)j(the)g(co)s(e\016cien)m(ts)h Ft(a)1751 1839 y Fj(0)1790 1824 y Ft(;)17 b(a)1885 1839 y Fj(1)1925 1824 y Ft(;)g(:)g(:)g(:)e(;)i(a)2194 1839 y Fs(n)2274 1824 y Fu(of)32 b(the)h(least-squares)g(p)s(olynomial)1144 2044 y Ft(p)p Fu(\()p Ft(x)p Fu(\))28 b(=)g Ft(a)1507 2059 y Fj(0)1546 2044 y Ft(q)1589 2059 y Fj(0)1629 2044 y Fu(\()p Ft(x)p Fu(\))22 b(+)g Ft(a)1931 2059 y Fj(1)1971 2044 y Ft(q)2014 2059 y Fj(1)2054 2044 y Fu(\()p Ft(x)p Fu(\))g(+)g Fm(\001)17 b(\001)g(\001)j Fu(+)i Ft(a)2592 2059 y Fs(n)2640 2044 y Ft(q)2683 2059 y Fs(n)2730 2044 y Fu(\()p Ft(x)p Fu(\))p Ft(:)p 3632 2235 4 2118 v 121 2238 3515 4 v 28 2447 a Fn(Advantages:)40 b Fu(Orthogonal)22 b(system)k(is)e(often)g(used)h(n)m(umerically)e(since)i(it)e(a)m(v)m (oids)i(solving)e(the)i(p)s(ossibly)-118 2567 y(ill-conditioned)35 b(system.)65 b(Anpther)41 b(adv)-5 b(an)m(tage)39 b(is)g(that)g(w)m(e)i (are)e(able)g(to)g(build)f(up)i(the)g(least-squares)-118 2688 y(p)s(olynomial)25 b(degree)k(b)m(y)h(degree.)43 b(F)-8 b(or)27 b(example,)i(if)e(w)m(e)j(do)e(not)h(kno)m(w)g(what)g (degree)g(p)s(olynomial)c(w)m(e)30 b(wish)-118 2808 y(to)36 b(use,)j(w)m(e)e(migh)m(t)e(start)i(with)f(a)g(\014rst-degree)i(p)s (olynomial,)c(then)j(a)f(second-degree,)j(and)e(so)g(on,)g(un)m(til) -118 2929 y(w)m(e)d(obtain)d(a)h(\014t)h(that)f(w)m(e)i(b)s(eliev)m(e)e (is)g(suitable.)-118 3303 y Fo(2.6)161 b(A)53 b(Numerical)i(Example) -118 3546 y Fu(W)-8 b(e)33 b(no)m(w)g(giv)m(e)g(a)f(simple)f(example,)h (using)g(the)h(data)1049 3805 y Ft(x)1104 3820 y Fj(1)1171 3805 y Fu(=)28 b(0)97 b Ft(x)1476 3820 y Fj(2)1544 3805 y Fu(=)1657 3737 y(1)p 1657 3782 49 4 v 1657 3873 a(4)1814 3805 y Ft(x)1869 3820 y Fj(3)1936 3805 y Fu(=)2050 3737 y(1)p 2050 3782 V 2050 3873 a(2)2206 3805 y Ft(x)2261 3820 y Fj(4)2328 3805 y Fu(=)2442 3737 y(3)p 2442 3782 V 2442 3873 a(4)2598 3805 y Ft(x)2653 3820 y Fj(5)2721 3805 y Fu(=)27 b(1)1049 3971 y Ft(y)t Fu(1)f(=)i(1)97 b Ft(y)1474 3986 y Fj(2)1541 3971 y Fu(=)28 b(2)97 b Ft(y)1839 3986 y Fj(3)1906 3971 y Fu(=)27 b(1)97 b Ft(y)2203 3986 y Fj(4)2270 3971 y Fu(=)28 b(0)97 b Ft(y)2568 3986 y Fj(5)2635 3971 y Fu(=)27 b(1)17 b Ft(:)-118 4191 y Fu(Since)31 b(there)g(are)g(\014v)m(e)h(p)s(oin)m(ts)e Ft(x)1066 4206 y Fs(i)1095 4191 y Fu(,)h(these)h(data)e(will)e (uniquely)j(determine)f(a)h(fourth-degree)g(in)m(terp)s(olating)-118 4311 y(p)s(olynomial.)52 b(W)-8 b(e)36 b(will)f(compute)h(the)h(linear) e(and)i(quadratic)f(least)g(square)h(p)s(olynomials)d(b)m(y)j(b)s(oth)g (the)-118 4431 y(normal)30 b(equations)j(and)g(the)g(orthorgonal)e(p)s (olynomial)e(approac)m(h.)-118 4759 y Fk(2.6.1)136 b(Linear)45 b(p)t(olynomial)-118 4967 y Fu(If)25 b(without)g(sp)s(ecial)f(men)m (tion,)i(w)m(e)g(tak)m(e)g Ft(w)1446 4982 y Fs(k)1516 4967 y Fm(\021)i Fu(1.)41 b(It)25 b(follo)m(ws)f(from)f(\(2.2\))i(that) g(w)m(e)h(will)d(need)j(the)g(follo)m(wing)-118 5088 y(quan)m(tities:)775 5168 y Fj(5)720 5198 y Fi(X)735 5408 y Fs(i)p Fj(=1)881 5292 y Ft(x)936 5307 y Fs(i)992 5292 y Fu(=)1106 5225 y(5)p 1106 5270 V 1106 5361 a(2)1164 5292 y Ft(;)1360 5168 y Fj(5)1306 5198 y Fi(X)1321 5408 y Fs(i)p Fj(=1)1466 5292 y Ft(x)1521 5251 y Fj(2)1521 5317 y Fs(i)1589 5292 y Fu(=)1702 5225 y(15)p 1702 5270 98 4 v 1727 5361 a(8)1810 5292 y Ft(;)2005 5168 y Fj(5)1951 5198 y Fi(X)1966 5408 y Fs(i)p Fj(=1)2112 5292 y Ft(y)2160 5307 y Fs(i)2215 5292 y Fu(=)i(5)p Ft(;)2563 5168 y Fj(5)2509 5198 y Fi(X)2523 5408 y Fs(i)p Fj(=1)2669 5292 y Ft(x)2724 5307 y Fs(i)2753 5292 y Ft(y)2801 5307 y Fs(i)2856 5292 y Fu(=)g(2)p Ft(:)-3154 b Fu(\(2.10\))p eop %%Page: 36 13 36 27 bop 1153 -34 a Fp(HKBU)37 b(MSc)g(Course)h(SCI4010)1206 b Fu(36)-118 214 y(Then)34 b(solving)d(the)i(normal)e(equation)h (\(2.1\))g(giv)m(es)1464 460 y Ft(a)1515 475 y Fj(0)1583 460 y Fu(=)1696 393 y(7)p 1696 437 49 4 v 1696 528 a(5)1755 460 y Ft(;)114 b(a)1947 475 y Fj(1)2015 460 y Fu(=)27 b Fm(\000)2205 393 y Fu(4)p 2205 437 V 2205 528 a(5)2264 460 y Ft(:)-118 687 y Fu(Th)m(us,)34 b(the)f(linear)e(least-squares)i (p)s(olynomial)c(is)1532 928 y Ft(p)1581 943 y Fj(1)1620 928 y Fu(\()p Ft(x)p Fu(\))f(=)1893 861 y(7)p 1893 905 V 1893 996 a(5)1973 928 y Fm(\000)2083 861 y Fu(4)p 2083 905 V 2083 996 a(5)2142 928 y Ft(x:)28 1179 y Fu(If)h(w)m(e)h(compute)g (the)f(same)g(p)s(olynomial)d(b)m(y)k(orthorgonal)d(p)s(olynomials,)g (the)i(p)s(olynomial)d(is)i(giv)m(en)i(in)-118 1299 y(the)j(form)1106 1420 y Ft(a)1157 1435 y Fj(0)1197 1420 y Ft(q)1240 1435 y Fj(0)1279 1420 y Fu(\()p Ft(x)p Fu(\))23 b(+)f Ft(a)1582 1435 y Fj(1)1622 1420 y Ft(q)1665 1435 y Fj(1)1704 1420 y Fu(\()p Ft(x)p Fu(\))28 b(=)g Ft(a)2018 1435 y Fj(0)2080 1420 y Fu(+)22 b Ft(a)2229 1435 y Fj(1)2268 1420 y Fu(\()p Ft(x)h Fm(\000)f Ft(\013)2545 1435 y Fj(1)2585 1420 y Fu(\))p Ft(;)-2768 b Fu(\(2.11\))-118 1586 y(where)34 b Ft(a)215 1601 y Fj(0)287 1586 y Fu(and)f Ft(a)528 1601 y Fj(1)600 1586 y Fu(are)f(giv)m(en)h(b)m(y)g(\(2.3\))g(and)f (\(2.10\),)g(and)h Ft(\013)2136 1601 y Fj(1)2208 1586 y Fu(b)m(y)g(\(2.4\):)1245 1832 y Ft(a)1296 1847 y Fj(0)1363 1832 y Fu(=)28 b(1)p Ft(;)114 b(a)1708 1847 y Fj(1)1775 1832 y Fu(=)28 b Fm(\000)1966 1765 y Fu(4)p 1966 1809 V 1966 1901 a(5)2025 1832 y Ft(;)114 b(\013)2228 1847 y Fj(1)2295 1832 y Fu(=)2409 1765 y(1)p 2409 1809 V 2409 1901 a(2)2484 1832 y Ft(:)-118 2072 y Fu(Therefore,)39 b(the)e(p)s(olynomial)c(\(2.11\))j(is)g(1)25 b Fm(\000)1611 2033 y Fj(4)p 1611 2049 36 4 v 1611 2107 a(5)1656 2072 y Fu(\()p Ft(x)g Fm(\000)1887 2033 y Fj(1)p 1887 2049 V 1887 2107 a(2)1932 2072 y Fu(\),)38 b(whic)m(h)f(is,)g(as)g(exp)s (ected,)j(the)d(same)g(as)g(the)g(one)-118 2193 y(obtained)32 b(with)g(the)h(normal)e(equation)h(approac)m(h.)-118 2517 y Fk(2.6.2)136 b(Quadratic)45 b(P)l(olynomial)-118 2726 y Fu(T)-8 b(o)31 b(compute)f(the)i(least-squares)f(quadratic)g(p)s (olynomial)c(b)m(y)k(the)g(normal)e(equations,)j(w)m(e)f(need)h(to)f (solv)m(e)-118 2846 y(the)i(system)g(\(2.1\))f(for)h Ft(n)27 b Fu(=)h(2,)k(whic)m(h)h(for)f(our)h(data)f(is)1037 2966 y Fi(2)1037 3145 y(4)1145 3045 y Fu(640)83 b(320)f(240)1145 3165 y(320)h(240)f(200)1145 3286 y(240)h(200)f(177)1792 2966 y Fi(3)1792 3145 y(5)1875 2966 y(2)1875 3145 y(4)1983 3045 y Ft(a)2034 3060 y Fj(0)1983 3165 y Ft(a)2034 3180 y Fj(1)1983 3286 y Ft(a)2034 3301 y Fj(2)2115 2966 y Fi(3)2115 3145 y(5)2209 3166 y Fu(=)2313 2966 y Fi(2)2313 3145 y(4)2421 3045 y Fu(640)2421 3165 y(256)2421 3286 y(176)2608 2966 y Fi(3)2608 3145 y(5)2691 3166 y Ft(:)-118 3481 y Fu(The)33 b(solution)e(of)i(this)f(system)h(is)1161 3721 y Ft(a)1212 3736 y Fj(0)1279 3721 y Fu(=)1393 3654 y(7)p 1393 3698 49 4 v 1393 3790 a(5)1452 3721 y Ft(;)211 b(a)1741 3736 y Fj(1)1809 3721 y Fu(=)1922 3654 y Fm(\000)p Fu(4)p 1922 3698 127 4 v 1961 3790 a(5)2058 3721 y Ft(;)h(a)2348 3736 y Fj(2)2416 3721 y Fu(=)27 b(0)p Ft(:)-2713 b Fu(\(2.12\))-118 3948 y(Th)m(us,)31 b(the)e(b)s(est)g(least-squares)g(p)s(olynomial)c (appro)m(ximation)i(turns)i(out)f(to)g(b)s(e)h(the)g(linear)e (least-squares)-118 4069 y(p)s(olynomial;)41 b(that)g(is,)h(no)e(impro) m(v)m(emen)m(t)h(to)f(the)h(linear)e(appro)m(ximation)g(can)i(b)s(e)g (made)f(b)m(y)h(adding)f(a)-118 4189 y(quadratic)32 b(term.)43 b(That)33 b(\(2.12\))e(is)h(correct)h(is)f(v)m(eri\014ed)h(b)m(y)h (computing)d(the)i(least-squares)g(quadratic)f(b)m(y)-118 4310 y(orthogonal)f(p)s(olynomials.)28 4454 y(The)j(orthogonal)c(p)s (olynomial)f(represen)m(tation)34 b(will)c(b)s(e)233 4707 y Ft(a)284 4722 y Fj(0)324 4707 y Ft(q)367 4722 y Fj(0)406 4707 y Fu(\()p Ft(x)p Fu(\))23 b(+)f Ft(a)709 4722 y Fj(1)749 4707 y Ft(q)792 4722 y Fj(1)831 4707 y Fu(\()p Ft(x)p Fu(\))h(+)f Ft(a)1134 4722 y Fj(2)1173 4707 y Ft(q)1216 4722 y Fj(2)1256 4707 y Fu(\()p Ft(x)p Fu(\))28 b(=)1529 4640 y(7)p 1529 4684 49 4 v 1529 4775 a(5)1609 4707 y Fm(\000)1719 4640 y Fu(4)p 1719 4684 V 1719 4775 a(5)1778 4707 y Ft(x)22 b Fu(+)g Ft(a)2004 4722 y Fj(2)2061 4567 y Fi(\024)2113 4707 y Ft(x)2185 4567 y Fi(\022)2259 4707 y Ft(x)g Fm(\000)2446 4640 y Fu(1)p 2446 4684 V 2446 4775 a(2)2504 4567 y Fi(\023)2600 4707 y Fm(\000)h Ft(\013)2762 4722 y Fj(2)2818 4567 y Fi(\022)2891 4707 y Ft(x)g Fm(\000)3078 4640 y Fu(1)p 3078 4684 V 3078 4775 a(2)3137 4567 y Fi(\023)3232 4707 y Fm(\000)g Ft(\014)3387 4722 y Fj(1)3426 4567 y Fi(\025)3496 4707 y Ft(;)-118 4966 y Fu(where)34 b Ft(\013)226 4981 y Fj(2)298 4966 y Fu(and)e Ft(\014)542 4981 y Fj(1)614 4966 y Fu(are)h(computed)g(from)e(\(2.8\))h(and)h(\(2.9\))f(as)676 5236 y Ft(\013)738 5251 y Fj(2)805 5236 y Fu(=)918 5088 y Fi(P)1040 5163 y Ft(x)1095 5178 y Fs(i)1124 5163 y Fu(\()p Ft(x)1217 5178 y Fs(i)1267 5163 y Fm(\000)1377 5124 y Fj(1)p 1377 5140 36 4 v 1377 5197 a(2)1422 5163 y Fu(\))1460 5127 y Fj(2)p 918 5213 582 4 v 968 5232 a Fi(P)1074 5307 y Fu(\()p Ft(x)1167 5322 y Fs(i)1217 5307 y Fm(\000)1327 5268 y Fj(1)p 1327 5284 36 4 v 1327 5341 a(2)1372 5307 y Fu(\))1410 5278 y Fj(2)1537 5236 y Fu(=)1651 5169 y(1)p 1651 5213 49 4 v 1651 5304 a(2)1709 5236 y Ft(;)212 b(\014)2003 5251 y Fj(1)2070 5236 y Fu(=)2184 5169 y(1)p 2184 5213 V 2184 5304 a(5)2259 5141 y Fi(X)2420 5236 y Ft(x)2475 5251 y Fs(i)2503 5236 y Fu(\()p Ft(x)2596 5251 y Fs(i)2647 5236 y Fm(\000)2757 5169 y Fu(1)p 2757 5213 V 2757 5304 a(2)2815 5236 y(\))28 b(=)2994 5169 y(1)p 2994 5213 V 2994 5304 a(8)3053 5236 y Ft(:)-118 5513 y Fu(Th)m(us,)34 b Ft(q)199 5528 y Fj(2)239 5513 y Fu(\()p Ft(x)p Fu(\))28 b(=)f Ft(x)556 5477 y Fj(2)618 5513 y Fm(\000)c Ft(x)g Fm(\000)905 5474 y Fj(1)p 905 5490 36 4 v 905 5547 a(8)950 5513 y Fu(,)33 b(and)g(from)e(\(2.3\),)h (w)m(e)i(\014nd)f(that)f Ft(a)2291 5528 y Fj(2)2359 5513 y Fu(=)27 b(0.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF