{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier " 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Norm al" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT -1 9 "P\370\355klad:\n" }}{PARA 0 "" 0 "" {TEXT -1 208 "(Napi\232te proceduru Aproximace, kter\341 k z adan\375m bod\371m, hopdnot\341m v t\354chto bodech a funkc\355m, najd e line\341rn\355 kombinaci on\354ch funkc\355, kter\351 bude nejl\351p e aproximovat ony hodnoty v zadan\375ch bodech)\n\nYou have given:" }} {PARA 15 "" 0 "" {TEXT -1 9 "n points " }{XPPEDIT 18 0 "x[1] .. x[n]; " "6#;&%\"xG6#\"\"\"&F%6#%\"nG" }}{PARA 15 "" 0 "" {TEXT -1 9 "n value s " }{XPPEDIT 18 0 "y[1] .. y[n];" "6#;&%\"yG6#\"\"\"&F%6#%\"nG" }} {PARA 15 "" 0 "" {TEXT -1 12 "k functions " }{XPPEDIT 18 0 "F[1] .. F[ k];" "6#;&%\"FG6#\"\"\"&F%6#%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 59 "and \+ you are looking for linear combination of the functions" }}{PARA 256 " " 0 "" {XPPEDIT 18 0 "sum(c[i]*F[i]),i = 1 .. k;" "6$-%$sumG6#*&&%\"cG 6#%\"iG\"\"\"&%\"FG6#F*F+/F*;F+%\"kG" }}{PARA 0 "" 0 "" {TEXT -1 56 "w hich are the best aproximation of the expected funtion " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 16 ", which fulfield" }}{PARA 257 "" 0 " " {TEXT -1 16 "for any i=1..n: " }{XPPEDIT 18 0 "phi(x[i]) = y[i];" "6 #/-%$phiG6#&%\"xG6#%\"iG&%\"yG6#F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "We find the coefficients " }{XPPEDIT 18 0 "c[i];" "6#&%\"cG6#% \"iG" }{TEXT -1 50 " as coefficiens of ortogonal projection of vector \+ " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 36 " into the space generate d by vectors" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "[seq(F[i](x[j]),j = 1 .. n)];" "6#7#-%$seqG6$-&%\"FG6#%\"iG6#&%\"xG 6#%\"jG/F0;\"\"\"%\"nG" }{TEXT -1 8 ", i=1..k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "\n\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined an d unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x:=[1,2, 3,5,7];\ny:=[1,1.1,2,2.2,3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG 7'\"\"\"\"\"#\"\"$\"\"&\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG 7'\"\"\"$\"#6!\"\"\"\"#$\"#AF)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F:=[x->1,x->x^2,x->ln(x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG7%j+6#%\"xG6\"6$%)operatorG%&arrowGF)\"\"\"F)F)F) 6#\"+'yu.3#j+F'F)F*F)*$)9$\"\"#F-F)F)F)6#\"+'yu.3#j+F'F)F*F)-%#lnG6#F3 F)F)F)6#\"+'yu.3#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "for i \+ from 1 to nops(F) do\nw[i]:=[seq(F[i](x[j]),j=1..nops(x))];\nod;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"wG6#\"\"\"7'F'F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"wG6#\"\"#7'\"\"\"\"\"%\"\"*\"#D\"#\\" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"wG6#\"\"$7'\"\"!-%#lnG6#\"\"#-F+ F&-F+6#\"\"&-F+6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "S calMul:=(a,b)->sum(a[i]*b[i],i=1..nops(a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(ScalMulGj+6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$s umG6$*&&9$6#%\"iG\"\"\"&9%F3F5/F4;F5-%%nopsG6#F2F)F)F)6$\"\"!F>" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "i:='i';\nAprox:=unapply(sum( c['i']*F['i'](z),'i'=1..nops(F)),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AproxGj+6#%\"zG6\"6$% )operatorG%&arrowGF(,(&%\"cG6#\"\"\"F0*&&F.6#\"\"#F0)9$F4F0F0*&&F.6#\" \"$F0-%#lnG6#F6F0F0F(F(F(6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "i:='i';j:='j';\nfor p from 1 to nops(F) do\neq[p]:=e valf(ScalMul(y,w[p])=sum(c[j]*ScalMul(w[p],w[j]),j=1..nops(F)));\nod; " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"jGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#eqG6#\"\"\"/$\"#$*!\"\",(*&$\"\"&\"\"!F' 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