{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 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 "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 Output" -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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Newton's interpolating pol ynomial" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "x:=[0,1,3,4, 7]: y:=[2,-3,0,1,-2]: n:=nops(y)-1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for j to n do y:=[seq((y[i+1]-y[i])/(x[i+j]-x[i]),i=1 ..n+1-j)] end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7&!\"&#\"\" $\"\"#\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7%#\"#8\"\" '#!\"\"F(#F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7$#!\"(\"#7 #!\"\"\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG7##\"#>\"$_#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Lagrange interpolating polynomial " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "lagrange:=proc(xn,y n)local n,pol,i; n:=nops(xn); pol:=0; for i to n do\npol:=pol+mu l(x-xn[k],k=1..i-1)*mul(x-xn[k],k=i+1..n)*yn[i]\n/mul(xn[i]-xn[k],k=1. .i-1)/mul(xn[i]-xn[k],k=i+1..n) end do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "lagrange([0,1,3,4,7],[2,-3,0,1,-2]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,**,\"#U!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"$F&F), &F(F)\"\"%F&F),&F(F)\"\"(F&F)F)*,\"#7F&F(F)F*F)F,F)F.F)F)*,\"#OF&F(F)F 'F)F*F)F.F)F&*,\"$_#F&F(F)F'F)F*F)F,F)F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Least-square fit to discrete data" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 79 "x:=[seq(i,i=1..12)]: y:=[26,31,26,16,9,12,23 ,31,29,19,10,10]: n:=nops(y): k:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A0:=matrix(k+1,k+1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "for i to k+1 do for j to k+1 do A0[i,j]:=add(x[L]^(i+ j-2),L=1..n) end do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "B:=[seq(add(y[L]*x[L]^i,L=1..n),i=0..k)]: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace hav e been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=augment(A0,B):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for i to k+1 do for j from i+1 to k+1 do\nA:=addrow(A ,i,j,-A[j,i]/A[i,i]) end do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "for i to k+1 do a[k+1-i]:=(A[k+2-i,k+2]-add(a[k+1-L] *A[k+2-i,k+2-L],L=1..i-1))/A[k+2-i,k+2-i]\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "pol:=add(a[L]*z^(L),L=0..k);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$polG,,#\"&RL\"\"$'R\"\"\"*&#\"'R&4$\"'_N7F)% \"zGF)!\"\"*&#\"'h,7\"&oB)F)*$)F.\"\"#F)F)F/*&#\"&&oVF-F)*$)F.\"\"$F)F )F)*&#\"$H&\"&cu#F)*$)F.\"\"%F)F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Same with weights" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x:=[seq(i,i=1..12)]: y:=[26,31,26,16,9,12,23,31,29,19,10,10]: n:=n ops(y): k:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "w:=[12,7,4 ,9,3,8,6,7,10,11,7,5]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A 0:=matrix(k+1,k+1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for \+ i to k+1 do for j to k+1 do A0[i,j]:=add(w[L]*x[L]^(i+j-2),L=1..n) end do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "B:=[seq(add( w[L]*y[L]*x[L]^i,L=1..n),i=0..k)]: with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been red efined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=augment(A0,B):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for i to k+1 do for j from i+1 to k+1 do\nA:=addrow(A,i,j,-A[j,i]/A[i,i]) e nd do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "for i to \+ k+1 do a[k+1-i]:=(A[k+2-i,k+2]-add(a[k+1-L]*A[k+2-i,k+2-L],L=1..i-1))/ A[k+2-i,k+2-i]\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " pol:=add(a[L]*z^(L),L=0..k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pol G,*#\".@Lo&*[%R\",*49`%\\*\"\"\"*&#\".Xwyp$p8F(F)%\"zGF)!\"\"*&#\"-8OS 6h]\"-)>G1*)*=F)*$)F-\"\"#F)F)F)*&#\"*B!pLW\"+=6'H6$F)*$)F-\"\"$F)F)F. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Linear model" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x:=[seq(i,i=1..12)]: y:=[26,31,26,1 6,9,12,23,31,29,19,10,10]: n:=nops(y): k:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f[1]:=x->exp(-1.*x):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "f[2]:=x->ln(1.*x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f[3]:=x->sin(1.*x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A0:=matrix(k,k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for i to k do for j to k do A0[i,j]:=add(f[i](x[L])*f [j](x[L]),L=1..n)\nend do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "B:=[seq(add(y[L]*f[i](x[L]),L=1..n),i=1..k)]: with(l inalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n orm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A:=stackmatrix(augment(A0,B),[seq(a[L],L= 1..k),_]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&$ \"+FkF,$\"+x*yc>%F,$\"+P-UZ:!\")7&F-$\"+h1\\dRF3$!+liS ]I!\"*$\"+m^LPP!\"(7&F/F7$\"+W:K*G'F9$\"+zUOnhF37&&%\"aG6#\"\"\"&FD6# \"\"#&FD6#\"\"$%\"_G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:= swaprow(A,1,2): A:=swapcol(A,1,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for j from 2 to k do A:=addrow(A,1,j,-A[j,1]/A[1,1]) \+ end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&$\"+h1\\dR!\")$\"+a[$3 #>!#5$!+liS]I!\"*$\"+m^LPP!\"(7&$\"\"!F6$\"+JL&eb\"F-$\"+\"fNPM%F-$\"+ qB-m8F*7&F5F9$\"+_$)>agF0$\"+?i2[!*F*7&&%\"aG6#\"\"#&FD6#\"\"\"&FD6#\" \"$%\"_G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=swaprow(A,2, 3): A:=swapcol(A,2,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " A:=addrow(A,2,3,-A[3,2]/A[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"AG-%'matrixG6#7&7&$\"+h1\\dR!\")$!+liS]I!\"*$\"+a[$3#>!#5$\"+m^LPP! \"(7&$\"\"!F8$\"+_$)>agF/$\"+\"fNPM%F2$\"+?i2[!*F,7&F7F7$\"+)=,UC\"F2$ \"+qdXorF/7&&%\"aG6#\"\"#&FF6#\"\"$&FF6#\"\"\"%\"_G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "a[1]:=A[3,4]/A[3,3]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "a[3]:=(A[2,4]-A[2,3]*a[1])/A[2,2]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a[2]:=(A[1,4]-A[1,3]*a[1]-A[ 1,2]*a[3])/A[1,1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "model :=add(a[L]*f[L](z),L=1..k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&mode lG,(*&$\"+lC\\hd!\")\"\"\"-%$expG6#,$*&$F*\"\"!F*%\"zGF*!\"\"F*F**&$\" +I2R(***!\"*F*-%#lnG6#,$*&$F*F1F*F2F*F*F*F**&$\"+P+9\"3\"F)F*-%$sinGF: F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Gram-Schmidt" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "w:=ln(x): A:=0: B:=3.:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "for i from 0 to 5 do psi[i]:=x^i-a dd(int(psi[i-k]*x^i*w,x=A..B)/alpha[i-k]*psi[i-k],k=1..i):\nalpha[i]:= int(w*psi[i]^2,x=A..B): print(psi[i],alpha[i]) end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+g'o$eH!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&%\"xG\"\"\"$\"+#)Ha0\"*!\"*!\"\"$!+1%fSw\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,(*$)%\"xG\"\"#\"\"\"F($\")JI)>#!\")F(*&$\"+Bf)4e# !\"*F(F&F(!\"\"$\"+\"y#\\R$)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,** $)%\"xG\"\"$\"\"\"F($\")ART**!\")!\"\"*&$\"+#HW^9&!\"*F()F&\"\"#F(F,*& $\"+/9;&)oF0F(F&F(F($!+Ik>xP!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,* $)%\"xG\"\"%\"\"\"F($\"()**\\e!\"(!\"\"*&$\"+U6acW!\"*F()F&\"\"$F(F,*& $\"*.G^z$!\")F()F&\"\"#F(F(*&$\"*w+!yDF6F(F&F(F($\"+%=dX'z!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$,.*$)%\"xG\"\"&\"\"\"F($\"(;q4$!\"(!\" \"*&$\"+vxW5v!\"*F()F&\"\"%F(F,*&$\"+K#e8&>!\")F()F&\"\"$F(F(*&$\"+\"y )4')>F6F()F&\"\"#F(F,*&$\"*>U+1'F6F(F&F(F($\"+c6YCX!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Least-square 'continuous' fit using Legen dre polynomial" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart: \+ k:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "for i from 0 to k do psi[i]:=X^i-add(int(psi[i-k]*X^i,X=-1..1)/alpha[i-k]*psi[i-k],k= 1..i):\nalpha[i]:=int(psi[i]^2,X=-1..1) end do:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "f:=exp(-x)/(1+x^2): A:=0.5: B:=3.5:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=eval(f,x=(A+B+(B-A)*X)/2) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "for i from 0 to k do a [i]:=int(psi[i]*F,X=-1..1)/alpha[i] end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "POL:=eval(add(a[i]*psi[i],i=0..k),I=0):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "LSpol:=expand(eval(POL,X=(2* x-A-B)/(B-A)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&LSpolG,*$\"+7<7+ ()!#5\"\"\"*&$\"+syYA5!\"*F)%\"xGF)!\"\"*&$\"+H)>j-%F(F))F.\"\"#F)F)*& $\"+OF^H_!#6F))F.\"\"$F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Sam e using Chebyshev" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart: k:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "for i from 0 to k do psi[i]:=X^i-add(int(psi[i-k]*X^ i/sqrt(1-X^2),X=-1..1)/alpha[i-k]*psi[i-k],k=1..i):\nalpha[i]:=int(psi [i]^2/sqrt(1-X^2),X=-1..1) end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=exp(-x)/(1+x^2): A:=0.5: B:=3.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F:=eval(f,x=(A+B+(B-A)*X)/2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "for i from 0 to k do a[i]:=e valf(int(psi[i]*F/sqrt(1-X^2),X=-1..1)/alpha[i]) end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "POL:=add(a[i]*psi[i],i=0..k):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "LSpol:=expand(eval(POL,X=(2* x-A-B)/(B-A)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&LSpolG,*$\"+e@Ei !*!#5\"\"\"*&$\"+@Jit5!\"*F)%\"xGF)!\"\"*&$\"+dmu>UF(F))F.\"\"#F)F)*&$ \"+$)y'zU&!#6F))F.\"\"$F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Tr apezoidal rule" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=x->1/sqrt(1+x^2): A:=2: B:=5: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "n:=8: m:=4:\nfor k to 3 do\nIn[k]:=evalf((f(A)+f(B )+2*add(f(A+(B-A)/n*i),i=1..n-1))/(2*n)*(B-A)):\nprint(n,In[k]); n:=n* m end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")$\"+MZa/()!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#K$\"+@D1*o)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$G\"$\"+vK4)o)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "for i to 2 do J[i]:=(m^2*In[i+1]-In[i])/(m^2-1) end d o;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"JG6#\"\"\"$\"+t..)o)!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"JG6#\"\"#$\"+g'G!)o)!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for i to 1 do K[i]:=(m^4*J[i +1]-J[i])/(m^4-1) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"KG6# \"\"\"$\"+b'G!)o)!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Simpson's rule" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "f:=x->sin(x)/x: A:=3: B:=6:\nn:=3: m:=4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "for k to 3 do In[k]:=evalf( (B-A)*(f(A)+f(B)+\n2*sum(f(A+2*( B-A)*i/n),i=1..n/2-1)+\n4*sum(f(A+(2*i-1)*(B-A)/n),i=1..n/2) )/3/n); \nprint(n,In[k]); n:=n*m end do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"\"$$!+2e5@D!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#7$!+5crRU!#5" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#[$!+A+lRU!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "for i from 1 to 2 do J[i]:=(m^4*In[i+1]-In[i] )/(m^4-1) end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"JG6#\"\"\"$! +b_XYU!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"JG6#\"\"#$!+l(\\'RU! #5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "(m^6*J[2]-J[1])/(m^6- 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+.\"['RU!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Gaussian rule" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart: n:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "for i from 0 to n do psi[i]:=X^i-add(int (psi[i-k]*X^i,X=-1..1)/alpha[i-k]*psi[i-k],k=1..i):\nalpha[i]:=int(psi [i]^2,X=-1..1.) end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "lagrange:=proc(xn)local n,pol,i; n:=nops(xn); pol:=0; for i to \+ n do\npol:=pol+mul(X-xn[k],k=1..i-1)*mul(X-xn[k],k=i+1..n)*y((A+B+xn[i ]*(B-A))/2)\n/mul(xn[i]-xn[k],k=1..i-1)/mul(xn[i]-xn[k],k=i+1..n) end \+ do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "rule:=(B-A)/2*in t(lagrange([solve(psi[n],X)]),X=-1..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "y:=x->sin(x)/x: A:=3.: B:=6.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "rule;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+y7 aSU!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "A:=3.: B:=4.5: re s1:=rule: A:=4.5: B:=6.: res2:=rule: res1+res2;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!+=9mRU!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Ow n rule" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:=3.: B:=6.: xn:=[3.,4.,5.,6.]: w:=exp(-x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "lagrange:=proc(xn)local n,pol,i; n:=nops(xn); po l:=0; for i to n do\npol:=pol+mul(x-xn[k],k=1..i-1)*mul(x-xn[k],k=i+ 1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1..i-1)/mul(xn[i]-xn[k],k=i+1..n) e nd do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rule:=int(w*l agrange(xn),x=A..B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,**&$ \"+([n<(G!#6\"\"\"-%\"yG6#\"\"%F*F**&$\"*xRFP#F)F*-F,6#\"\"&F*F**&$\"* b,j8#F)F*-F,6#\"\"'F*F**&$\"+/+;39F)F*-F,6#\"\"$F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "y:=x->ln(x): evalf(rule); # integrate s exp(-x)*ln(x) from 3 to 6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+X& )y#H'!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Gaussian version" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "restart : n:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "w:=exp(-x): A :=3.: B:=6.:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "for i from 0 to n do psi[i]:=x^i-add(int(psi[i-k]*x^i*w,x=A..B)/alpha[i-k]*psi[i-k],k=1 ..i):\nalpha[i]:=int(w*psi[i]^2,x=A..B) end do:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 200 "lagrange:=proc(xn)local n,pol,i; n:=nops(xn ); pol:=0; for i to n do\npol:=pol+mul(x-xn[k],k=1..i-1)*mul(x-xn[ k],k=i+1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1..i-1)/mul(xn[i]-xn[k],k=i+ 1..n) end do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "rule:= int(w*lagrange([solve(psi[n],x)]),x=A..B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,(*&$\"+%pVSO#!#6\"\"\"-%\"yG6##\"+6lC=;\"*++++ &F*F**&$\"+e:2&*=F)F*-F,6##\"+BFz(=%\"+++++5F*F**&$\"+sO; " 0 "" {MPLTEXT 1 0 26 "y :=x->ln(x): evalf(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M3R%H '!#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Rule for differentiation " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "lagrange:=proc(xn)l ocal n,pol,i; n:=nops(xn); pol:=0; for i to n do\npol:=pol+mul(x -xn[k],k=1..i-1)*mul(x-xn[k],k=i+1..n)*y(xn[i])\n/mul(xn[i]-xn[k],k=1. .i-1)/mul(xn[i]-xn[k],k=i+1..n) end do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pol:=lagrange([x0-2*h,x0-h,x0,x0+h,x0+2*h]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rule:=eval(diff(pol,x,x,x),x =x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,**&%\"hG!\"$-%\"yG6# ,&%#x0G\"\"\"F'!\"\"F.F.*&#F.\"\"#F.*&F'F(-F*6#,&F-F.*&F2F.F'F.F/F.F.F /*&F'F(-F*6#,&F-F.F'F.F.F/*&#F.F2F.*&F'F(-F*6#,&F-F.*&F2F.F'F.F.F.F.F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "mtaylor(rule,h,8);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(---%#@@G6$%\"DG\"\"$6#%\"yG6#%#x0G\" \"\"*&#F/\"\"%F/*&)%\"hG\"\"#F/---F'6$F)\"\"&F+F-F/F/F/*&#F/\"#SF/*&)F 5F2F/---F'6$F)\"\"(F+F-F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y:=x->sin(x)*exp(-x)/x: x0:=1: h:=0.01:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 51 "evalf(rule); # six digits lost to round off \+ error" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%HJ!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "LU decomposition" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 149 "LUdec:=proc(a,b,c) local L,U,k,n; n:=nops(b); L:=[ b[1]]; U:=[];\nfor k to n-1 do U:=[op(U),c[k]/L[k]];L:=[op(L),b[k+1]-a [k]*U[k]] end do;\n[a,L,U] end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 " M:=LUdec([3,-3,1],[2,-2,4,5],[-1,5,2]); #know how to interpret" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7%7%\"\"$!\"$\"\"\"7&\"\"##!\"\" F+!#E#\"#m\"#87%F,!#5#F-F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "tridset:=proc(M,r) local n,k,a,L,U,y,x; a:=M[1]; L:=M[2]; U:=M[3] ; n:=nops(L);\ny:=[r[1]/L[1]]; for k to n-1 do y:=[op(y),(r[k+1]-a[k]* y[k])/L[k+1]] end do;\nx:=[y[n]]; for k to n-1 do x:=[y[n-k]-U[n-k]*x[ 1],op(x)] end do; [y,x] end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "tri dset(M,[8,13,4,-5]); # the second vector is the solution" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7&\"\"%!\"##\"\"\"\"#8!\"\"7&\"\"$F&\"\"!F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Newton's technique - one unknow n" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:= exp(x^2/10)-sin(x)-x-x^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "df:=diff(f,x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x:=-2 .: # know how do we get the starting value(s)" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "for k to 4 do x:=x-f/df end do;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"xG$!+.8FU@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$!+!o$yM@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$!+ _OwM@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$!+_OwM@!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "More than one unknown" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "F:=[x[1]^3/(2+sin(x[2]))-x[2 ]^4/(3+cos(x[1])),exp(-x[1]^2)/(1+x[2]^2)-(x[1]-x[2])^3]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "J:=matrix(2,2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "for i to 2 do for j to 2 do J[i,j]:=diff(F [i],x[j]) end do end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected \+ names norm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x:=[0.5,-1.]: # know how to get th e intial values " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "for i t o 6 do x:=evalm(x-linsolve(J,F)) end do; # monitor convergence" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7$$\"+xj\\pL!#5$!+!Q S'[uF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6#7$$\"+)eLH 1$!#5$!+-!\\!RgF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG-%'vectorG6 #7$$\"+'Q;mD$!#5$!+u&=Ud&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG- %'vectorG6#7$$\"+DHTzK!#5$!+@Q'Ra&F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"xG-%'vectorG6#7$$\"+%\"xG-%'vectorG6#7$$\"+;dWzK!#5$!+^z!Ra&F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "linear 2nd-order ODE with boundary values " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "LUdec:=proc(a,b,c) \+ local L,U,k,n; n:=nops(b); L:=[b[1]]; U:=[];\nfor k to n-1 do U:=[op(U ),c[k]/L[k]];L:=[op(L),b[k+1]-a[k]*U[k]] end do;\n[a,L,U] end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "tridset:=proc(M,r) local n, k,a,L,U,y,x; a:=M[1]; L:=M[2]; U:=M[3]; n:=nops(L);\ny:=[r[1]/L[1]]; f or k to n-1 do y:=[op(y),(r[k+1]-a[k]*y[k])/L[k+1]] end do;\nx:=[y[n]] ; for k to n-1 do x:=[y[n-k]-U[n-k]*x[1],op(x)] end do end:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "n:=4: A:=0: B:=2: #specify these" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p:=x->-1./(1+x^2):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "q:=x->1./sqrt(3.+x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=x->x/2.:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "h:=(B-A )/n:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:=[seq(A+h*k,k=1..n-1)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "a:=[seq(1-h/2*p(x[k]),k=2..n-1)]: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "b:=[seq(-2+h^2*q(x[k]),k=1..n-1 )]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c:=[seq(1+h/2*p(x[k]),k=1..n -2)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R:=[seq(h^2*r(x[k]),k=1..n -1)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "R[1]:=R[1]-2*(1-h/2*p(x[1] )): # set the first boundary value" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "R[n-1]:=R[n-1]-3*(1+h/2*p(x[n-1])): # the seco nd boundary " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "y4:=tridset (LUdec(a,b,c),R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y4G7%$\"+`V-YC !\"*$\"+\">1Yy#F($\"+b>)\\'HF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "# know how to re-do with bigger n, followed by Richardson (th e error pattern similar to trapezoidal rule)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "non-linear same" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "LUdec:=proc(a,b,c) local L,U,k,n; n:=nops(b); L:=[b[ 1]]; U:=[];\nfor k to n-1 do U:=[op(U),c[k]/L[k]];L:=[op(L),b[k+1]-a[k ]*U[k]] end do;\n[a,L,U] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "tridset:=proc(M,r) local n,k,a,L,U,y,x; a:=M[1]; L:=M[2]; U:=M[ 3]; n:=nops(L);\ny:=[r[1]/L[1]]; for k to n-1 do y:=[op(y),(r[k+1]-a[k ]*y[k])/L[k+1]] end do;\nx:=[y[n]]; for k to n-1 do x:=[y[n-k]-U[n-k]* x[1],op(x)] end do end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "n:=10: A:=-2: B:=3: \+ y[0]:=1: y[n]:=-1: # set these" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "h:=(B-A )/evalf(n): x:=[seq(A+h*i,i=1..n-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "F:=[se q((y[i+1]-2*y[i]+y[i-1])/h^2-3*((y[i+1]-y[i-1])/2/h)^2*sin(x[i])/\n \+ (1+y[i]^4),i=1..n-1)]: # 'discretize' the expression for y''" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a:=[seq(diff(F[i],y[i-1]),i=2..n-1)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "b:=[seq(diff(F[i],y[i]),i=1. .n-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c:=[seq(diff(F[i -1],y[i]),i=2..n-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "y: =[seq(y[0]+(y[n]-y[0])*i/evalf(n),i=1..n-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "for k to 7 do y:=y-tridset(LUdec(a,b,c),F) end \+ do: # monitor convergence" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "y10:=y;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%$y10G7+$\"+4x$fv)!#5$\"+\\!f97(F($\"+gz8MYF($ \"+ikJEv!#6$!+qY()GJF($!+H\"f9e&F($!+'*QT=rF($!+Dc#fB)F($!+Kl&*e\"*F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "y:='y': # be able to \+ repeat with other values of n, and follow by Richardson" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Runge-Kutta" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "yd:=[y[2],-y[1]-y[2]*(y[1]^2-1)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "# be able to 'transcribe' yo ur (set of) equation(s) of any order into this form" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "t:=0: y:=[1.,2.]: h:=0.1: sol:=[y]: n:=20: # properly initia lize" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "for i to n do\nk1:=h*yd:\nt:=t+h/2: y:=y+ k1/2:\nk2:=h*yd:\ny:=y-k1/2+k2/2:\nk3:=h*yd:\nt:=t+h/2: y:=y-k2/2+k3: \nk4:=h*yd:\ny:=y-k3+(k1+2*k2+2*k3+k4)/6.:\nsol:=[op(sol),y] end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "y; # the final values re ached " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+`vqf5!\"*$!+n11l5F&" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "h:=-0.1: # all you have t o do for back-tracking" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Fitting a function by trigonometric polynomial" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "restart: A:=-2: B:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f:=x->abs(x): # often defined in a piecewise manner" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F:=eval(f(x),x=(B-A)*X/(2*Pi )+(A+B)/2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for k from \+ 0 to 7 do a[k]:=evalf(int(F*cos(k*X),X=-Pi..Pi)/Pi);\nb[k]:=evalf(int( F*sin(k*X), X=-Pi..Pi)/Pi) end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "trpol:=simplify(a[0]/2+add(a[k]*cos(k*X)+b[k]*sin(k*X ),k=1..7)): # set the number of terms" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "trpol:=eval(trpol,\{X=(2*x-A-B)*Pi/(B-A)\});" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%&trpolG,<$\"+NLLL$)!#5\"\"\"*&$\"+iK XfXF(F)-%$cosG6#,$*(\"\"$!\"\",&*&\"\"#F)%\"xGF)F)F)F)F)%#PiGF)F)F)F3* &$\"+I/]:eF(F)-%$sinGF/F)F3*&$\"+;L')R6F(F)-F.6#,$*($\"+nmmmmF(F)F4F)F 8F)F)F)F)*&$\"+*Q!\\M$*!#6F)-F=FBF)F)*&$\"+aH.h5F(F)-F=6#,$*($\"+++++5 !\"*F)F4F)F8F)F)F)F3*&$\"+*Ge'\\GFJF)-F.6#,$*($\"+LLLL8FUF)F4F)F8F)F)F )F)*&$\"+J\")*Hg*FJF)-F=FZF)F)*&$\"+08yB=FJF)-F.6#,$*($\"+nmmm;FUF)F4F )F8F)F)F)F3*&$\"+'4PKJ&FJF)-F=FaoF)F3*&$\"+pZ;0`FJF)-F=6#,$*($\"+++++? FUF)F4F)F8F)F)F)F)*&$\"+dm+0$*!#7F)-F.6#,$*($\"+LLLLBFUF)F4F)F8F)F)F)F 3*&$\"+(*)3X3&FJF)-F=FhpF)F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "L east-square fit to discrete data" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y:=[3,7,5,1,-2,3,6,11,5,2]: # must have even number o f values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "xfirst:=-4: xla st:=14: # need the fist and last value of x only (the rest must be eve nly spaced)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "m:=nops(y)/2: " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "xn:=[seq(evalf(-Pi+2*Pi*i/2/m),i=0..2*m-1)]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "n:=3: # set the number of terms (must be les s than m)!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for k from 0 to n do a[k]:=add(y[i ]*cos(k*xn[i]),i=1..2*m)/m;\n b[k]:=add(y[i]*sin(k*xn[i ]),i=1..2*m)/m end do: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 " trpol:=a[0]/2+add(a[k]*cos(k*X)+b[k]*sin(k*X),k=1..n-1)+a[n]*cos(n*X): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "trpol:=eval(trpol,X=((x -xfirst)/(xlast-xfirst)*(2-1/m)-1)*Pi);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&trpolG,.$\"+++++T!\"*\"\"\"*&$\"+k>5ao!#5F)-%$cosG6#*&,&*&\"# 5!\"\"%\"xGF)F)#\"\"$\"\"&F5F)%#PiGF)F)F5*&$\"+%=%yaAF(F)-%$sinGF0F)F) *&$\"+#fIib\"F(F)-F/6#,$*(\"\"#F)F2F)F:F)F)F)F5*&$\"+YuJ[OF(F)-F?FDF)F )*&$\"+?-)*e9!#6F)-F/6#,$*(F8F)F2F)F:F)F)F)F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Interpolating trigonometric polynomial" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "y:=[3,7,5,1,-2,3,6,11,5,2]: # must \+ have even number of values" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "xfirst:=-4: xlast:=14: # need the fist and last value of x only (t he rest must be evenly spaced)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "m:=nops(y)/2: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xn:=[seq(evalf(-Pi+2*Pi*i /2/m),i=0..2*m-1)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for k from 0 to m do a[k]:= add(y[i]*cos(k*xn[i]),i=1..2*m)/m;\n b[k]:=add(y[i]*sin (k*xn[i]),i=1..2*m)/m end do: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "trpol:=a[0]/2+add(a[k]*cos(k*X)+b[k]*sin(k*X),k=1..m-1)+a[m]/2 *cos(m*X):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "trpol:=eval(t rpol,X=((x-xfirst)/(xlast-xfirst)*(2-1/m))-1)*Pi;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&trpolG*&,6$\"+++++T!\"*\"\"\"*&$\"+k>5ao!#5F*-%$cosG 6#,&*&\"#5!\"\"%\"xGF*F*#\"\"$\"\"&F5F*F5*&$\"+%=%yaAF)F*-%$sinGF1F*F* *&$\"+#fIib\"F)F*-F06#,&*&F9F5F6F*F*#\"\"'F9F5F*F5*&$\"+YuJ[OF)F*-F>FC F*F**&$\"+?-)*e9!#6F*-F06#,&*(F8F*F4F5F6F*F*#\"\"*F9F5F*F5*&$\"+ofO\\g F.F*-F>FQF*F5*&$\"+[fIiXF.F*-F06#,&*(\"\"#F*F9F5F6F*F*#\"#7F9F5F*F**&$ \"+'z8(QPF.F*-F>FhnF*F5*&$\"+++++qF.F*-F06#,&*&F[oF5F6F*F*F8F5F*F*F*%# PiGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "198 0 \+ 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }