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" }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/pro cdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 72 "A procedure for constructing an interpolating polynomia l approximation: " }{TEXT 0 9 "interpoly" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "interpoly: \+ usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 263 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 264 2 " \+ " }{TEXT -1 26 " interpoly( f, rng deg ) " }{TEXT 265 1 "\n" }{TEXT -1 34 " interpoly( f, rng deg, 'pts' )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 10 "Parameters" }{TEXT -1 4 ": " }} {PARA 0 "" 0 "" {TEXT 23 9 " f - " }{TEXT -1 61 " an expressi on f(x) involving a single variable, say x, " }}{PARA 0 "" 0 "" {TEXT -1 88 " which evaluates to a real floating poin t number, or a procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 282 68 "the ra nge x=a . . b or a . . b for the function to be approximated." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " deg - " }{TEXT 267 53 "the degree of the resulting polyno mial approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 23 12 " 'pts' - " }{TEXT -1 57 "(optional) for returning th e list of interpolation points" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT 283 57 ": deg is an optional argument with a default value of 10." }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 10 " " }}{PARA 0 "" 0 "" {TEXT 285 11 "Description " }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 9 "interpoly" }{TEXT -1 112 " attempts to find an interpolatin g polynomial of specified degree which approximates f(x) on the interv al [a,b]." }}{PARA 0 "" 0 "" {TEXT -1 138 "The sample points can be ev enly spaced or they can be spaced according to the distribution of the roots of various orthogonal polynomials." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 8 "Options:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 38 "spacing=even/chebyshev/legendre/jacobi" } }{PARA 0 "" 0 "" {TEXT -1 194 "This option provides a choice for the s pacing of the sample points.\nThey can be evenly spaced or they can be spaced according to the distribution of the roots of various orthogon al polynomials. " }}{PARA 0 "" 0 "" {TEXT -1 23 "The Jacobi polynomial s " }{XPPEDIT 18 0 "P(n,alpha,beta,x)" "6#-%\"PG6&%\"nG%&alphaG%%betaG %\"xG" }{TEXT -1 36 " satisfy the orthogonality relation:" }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(w(x)*P(n,alpha,beta,x)*P(n,a lpha,beta,x),x = -1 .. 1) = PIECEWISE([nonzero, `if`*alpha = beta],[0, `if`*alpha <> beta]);" "6#/-%$IntG6$*(-%\"wG6#%\"xG\"\"\"-%\"PG6&%\"n G%&alphaG%%betaGF+F,-F.6&F0F1F2F+F,/F+;,$F,!\"\"F,-%*PIECEWISEG6$7$%(n onzeroG/*&%#ifGF,F1F,F27$\"\"!0*&F@F,F1F,F2" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 36 "with respect to the weight function " } {XPPEDIT 18 0 "w(x) = (1-x)^alpha*(1+x)^beta;" "6#/-%\"wG6#%\"xG*&),& \"\"\"F+F'!\"\"%&alphaGF+),&F+F+F'F+%%betaGF+" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 15 "The parameters " }{XPPEDIT 18 0 "alpha" " 6#%&alphaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" } {TEXT -1 45 " should be specified via the option \"params\"." }}{PARA 0 "" 0 "" {TEXT -1 178 "This option can be given in the alternative fo rms: spacing=Even/Chebyshev/Legendre/Jacobi or spacing=EVEN/CHEBYSHE V/LEGENDRE/JACOBI or spacing=even/chebyshev/legendre/jacobi." }} {PARA 0 "" 0 "" {TEXT -1 30 "The default is \"spacing=even\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "params=[a lpha,beta]" }}{PARA 0 "" 0 "" {TEXT -1 48 "The parameter values for th e Jacobi polynomials." }}{PARA 0 "" 0 "" {TEXT -1 109 "If this option \+ is included, the option \"spacing=Jacobi\" can be omitted, although th ere are default values of " }{XPPEDIT 18 0 "alpha=-1/4" "6#/%&alphaG,$ *&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "beta=-1/4 " "6#/%%betaG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 35 ", that is, \"par ams=[-0.25,-0.25]\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 126 "With t he option \"info=true\" information concerning the interpolation point s is displayed while the computation is in progress." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }{TEXT -1 156 "\nTo make the procedure s active open the subsection, place the cursor anywhere after the prom pt [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "interpoly: implementation" }} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13089 "interpoly := proc(f,rng,arg3,arg4)\n local n,i,j,k,b,s,c,phi, p,spcng,xvals,yvals,fn,alf,bet,\n x0,xn,m1,m2,x,h,Options,saveDig its,rs,yi,s2,d2,zrs,prntflg,\n xpow,u,chp,startopts,rg,t,vars,mak eproc,proctype,ff,\n assignpts,deg,pts,par;\n\n if nargs<2 then \n error \"at least 2 arguments are required; the basic syntax is : 'interpoly(f(x),x=a..b)'.\"\n end if;\n \n if type(f,procedure) then\n if nops([op(1,eval(f))])<>1 then\n error \"the 1s t argument, %1, is invalid .. it should be a procedure with a single a rgument\",f;\n end if;\n proctype := true;\n if type(rn g,realcons..realcons) then\n rg := rng\n else\n e rror \"the 2nd argument, %1, is invalid .. when the 1st argument is a \+ procedure, the 2nd argument should have the form 'a..b', where a and b are real constants, to provide the interval over which to construct t he interpolating polynomial\",rng;\n end if;\n elif type(f,alge braic) then\n vars := indets(f,name) minus indets(f,realcons);\n \+ if nops(vars)<>1 then \n if not has(indets(f),\{Int,Sum\} ) then\n error \"the 1st argument, %1, is invalid .. it sho uld be an expression which depends only on a single variable\",f;\n \+ end if;\n end if;\n if type(rng,name=realcons..realcon s) then\n proctype := false;\n x := op(1,rng);\n \+ if not member(x,vars) then\n error \"the 1st argument, % 1, is invalid .. it should be an expression which depends only on the \+ variable %2\",f,x;\n end if;\n rg := op(2,rng);\n \+ else\n error \"the 2nd argument, %1, is invalid .. it should \+ have the form 'x=a..b', where a and b are real constants, to provide t he interval over which to construct the interpolating polynomial\",rng ;\n end if;\n else\n error \"the 1st argument, %1, is inva lid .. it should be an algebraic expression in a single variable, or a procedure with a single real argument\",f;\n end if;\n\n startopt s := 3;\n deg := 10;\n assignpts := 0;\n if nargs>2 then\n \+ if type(arg3,integer) then\n deg := arg3;\n startopts \+ := startopts + 1;\n elif type(arg3,name) then\n startopts := startopts + 1;\n assignpts := 3;\n elif not type(arg3 ,`=`) then\n error \"3rd argument must be an integer or a vari able to which the list of interpolation points is assigned\"\n en d if;\n if nargs>3 then\n if type(arg4,name) then\n \+ if type(arg3,integer) then\n startopts := startopt s + 1;\n assignpts := 4;\n else\n \+ error \"when the 4th argument is a variable to which the list of in terpolation points is assigned, the 3rd argument must be an integer\" \n end if;\n elif not type(arg4,`=`) then\n \+ if not type(arg3,integer) then\n error \"3rd argumen t must be an integer\"\n else\n error \"4th a rgument must be a variable to which the list of interpolation points i s assigned\"\n end if;\n end if;\n end if;\n \+ end if;\n\n # Get the options.\n # Set the default values to star t with.\n prntflg := false;\n n := Digits*2;\n spcng := 1;\n a lf := -0.25;\n bet := -0.25;\n if nargs>3 and not type(args[4],`=` ) then startopts := 5 end if;\n if startopts <= nargs then\n Op tions:=[args[startopts..nargs]];\n if not type(Options,list(equat ion)) then\n error \"each optional argument after the %-1 argu ment must be an equation\",startopts-1;\n end if;\n if hasop tion(Options,'numpoints','n','Options') then \n if not type(n, posint) then\n error \"\\\"numpoints\\\" must be a positive integer\"\n end if;\n end if;\n if hasoption(Option s,'spacing','spcng','Options') then\n if not member(spcng,\{ 'Even','even','EVEN','Chebyshev','chebyshev','CHEBYSHEV', \+ 'Jacobi','jacobi','JACOBI','Legend re','legendre','LEGENDRE'\}) then\n error \"\\\"spacing\\\" must be 'Even' <-> 'even' <-> 'EVEN', 'Chebyshev' <-> 'chebyshev' <-> 'CHEBYSHEV', 'Jacobi' <-> 'jacobi' <-> 'JACOBI' or 'Legendre' <-> 'le gendre' <-> 'LEGENDRE'\"; \n end if;\n if member(spcng ,\{'Even','even','EVEN'\}) then spcng := 1 end if;\n if member (spcng,\{'Chebyshev','chebyshev','CHEBYSHEV'\}) then spcng := 2 end if ;\n if member(spcng,\{'Jacobi','jacobi','JACOBI'\}) then spcng := 3 end if;\n if member(spcng,\{'Legendre','legendre','LEGEN DRE'\}) then spcng := 4 end if;\n end if;\n if hasoption(Opt ions,'params','prms','Options') then \n if not (type(prms,list (realcons)) and nops(prms)=2) then\n error \"\\\"params\\\" must be a list of two real constants\"\n end if;\n al f := op(1,prms);\n bet := op(2,prms);\n if signum(0,al f+1,0)<=0 or signum(0,bet+1,0)<=0 then\n error \"Jacobi pol ynomial is not defined for current parameter values\"\n end if ;\n spcng := 3;\n end if;\n if hasoption(Options,'in fo','prntflg','Options') then\n if prntflg<>true then prntflg \+ := false end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2 .. the recognised options a re \\\"numpoints\\\", \\\"spacing\\\",\\\"params\\\" and \\\"info\\\" \",op(1,Options),procname;\n end if\n end if;\n\n # Increase \+ precision for the computation considerably\n saveDigits := Digits;\n Digits := Digits + min(Digits,20);\n\n x0 := evalf(op(1,rg));\n \+ xn := evalf(op(2,rg));\n if not type([x0,xn],[numeric, numeric]) th en\n error \"expecting a numeric range in the second argument\"\n end if;\n if x0>=xn then\n t := x0; x0 := xn; xn := t;\n e nd if;\n\n makeproc := proc(fx,x,a,b)\n proc(_x)\n loca l y;\n y := traperror(evalf(eval(subs(x=_x,fx))));\n i f y=lasterror or not type(y,numeric) then\n if evalf(_x-a)= 0 then\n y := evalf(limit(fx,x=_x,'right'));\n \+ elif evalf(_x-b)=0 then\n y := evalf(limit(fx,x=_x,'le ft'));\n else\n y := evalf(limit(fx,x=_x,'rea l'));\n end if;\n end if;\n y;\n end p roc;\n end proc;\n\n if proctype then\n if type(f,procedure) \+ then fn := eval(f)\n else\n try fn := subs(_body=f(_t),pr oc(_t) evalf(_body) end proc)\n catch:\n error \"expec ting the 1st argument to be an operator, but received %1\",f\n \+ end try;\n fn := subs(_t='t', eval(fn))\n end if\n els e\n fn := makeproc(f,x,a,b)\n end if;\n\n if not type([fn(x0) ,fn(.7101449275*x0+.2898550725*xn),\n fn(.381966011*x0+.618033989 *xn),fn(xn)],list(numeric)) then\n error \"function does not eval uate to a numeric\"\n end if;\n\n n := deg+1;\n\n if spcng=1 the n\n if prntflg then\n print(`interpolating polynomial has evenly spaced nodes`);\n print(``);\n end if; \n h := (xn-x0)/deg;\n m1 := iquo(deg,2);\n m2 := deg-m1;\n \+ xvals := [seq(x0+h*i,i=0..m1),seq(xn-h*(m2-i),i=1..m2)];\n elif spc ng=2 then\n if prntflg then\n print(`nodes are distribute d according to the zeros of a Chebyshev polynomial`);\n print( `weight function: `,1/sqrt(1-x^2));\n print(``);\n end if ; \n d2 := (xn-x0)*0.5;\n s2 := (xn+x0)*0.5;\n xvals \+ := [seq(evalf(cos((2*(n-i)+1)*Pi/(2*n)))*d2+s2,i=1..n)];\n elif spcn g=3 then\n if prntflg then\n print(`nodes are distributed according to the zeros of a Jacobi polynomial`);\n print(`wei ght function: `,(1-x)^alf*(1+x)^bet);\n print(``);\n end \+ if;\n d2 := (xn-x0)*0.5;\n s2 := (xn+x0)*0.5;\n zrs := \+ jacobizeros(n,alf,bet);\n xvals := map(u->u*d2+s2,zrs);\n else \n if prntflg then\n print(`nodes are distributed accordi ng to the zeros of a Legendre polynomial`);\n print(`weight fu nction: `,1);\n print(``);\n end if;\n d2 := (xn-x0) *0.5;\n s2 := (xn+x0)*0.5;\n zrs := legendrezeros(n);\n \+ xvals := map(u->u*d2+s2,zrs);\n end if;\n \n if prntflg then\n \+ print(`nodes:`);\n print(xvals);\n print(``);\n end i f;\n yvals := map(fn,xvals);\n if prntflg then\n print(`value s:`);\n print(yvals);\n print(``);\n end if;\n\n for i f rom 1 to n do\n s[i] := 0;\n c[i] := 0;\n end do;\n\n s[ n] := -xvals[1];\n\n for i from 2 to n do\n for j from n-i to n- 1 do\n s[j] := s[j] - xvals[i]*s[j+1]\n end do;\n s[ n] := s[n] - xvals[i];\n end do;\n\n for j from 1 to n do\n ph i := n;\n for k from n-1 to 1 by -1 do\n phi := k*s[k+1] \+ + xvals[j]*phi;\n end do;\n if phi=0 then\n error \" division by zero error\"\n end if;\n ff := yvals[j]/phi;\n \+ b := 1;\n for k from n to 1 by -1 do\n c[k] := c[k] + b*ff; \n b := s[k] + xvals[j]*b;\n end do;\n end do;\n\n xpow \+ := 1:\n p := c[1]:\n for i from 2 to n do\n xpow := x*xpow;\n p := p + c[i]*xpow;\n end do;\n\n if not proctype and x0+xn= 0 and member(spcng,\{1,2,4\}) then\n par := false;\n par := \+ traperror(type(f,oddfunc(x)));\n if par<>true then par := false e nd if;\n if par then\n chp := proc(u) if irem(degree(u),2 )=0 then 0 else u end if end proc;\n p := map(chp,p);\n e nd if;\n if not par then\n par := traperror(type(f,evenfu nc(x)));\n if par<>true then par := false end if;\n if par then\n chp := proc(u) if irem(degree(u),2)=1 then 0 el se u end if end proc;\n p := map(chp,p);\n end if; \n end if;\n end if; \n\n Digits := saveDigits;\n if assign pts>0 then\n pts := evalf(zip((x,y)->[x,y],xvals,yvals));\n \+ if assignpts=3 then\n arg3 := pts;\n elif assignpts=4 the n\n arg4 := pts;\n end if;\n end if;\n\n if proctype \+ then\n p := unapply(evalf(p),x);\n eval(p);\n else\n \+ evalf(p);\n end if; \nend proc: # of interpoly\n\n# calculate zero s of a Legendre polynomial\nlegendrezeros := proc(n::posint)\n local eps,z,h,j,k,p1,p2,p3,m,p,x;\n\n x := [];\n eps := Float(5,-Digits ); \n m := trunc((n+1)/2);\n p := ceil((n-1)/2);\n for k from \+ 1 to m do\n # starting approximation\n z := evalf(cos(Pi*(k- 0.25)/(n+0.5)));\n for j from 1 to Digits*3 do\n # evalua te Legendre poly at z .. p1\n p1 := 1;\n p2 := 0;\n \+ for j from 1 to n do\n p3 := p2;\n p2 := p 1;\n p1 := ((2*j-1)*z*p2-(j-1)*p3)/j;\n end do;\n\n # get the derivative and apply the Newton formula\n h := p1*(z*z-1)/(n*(z*p1-p2));\n z := z - h;\n if abs(h ) <= eps then break end if;\n end do;\n x := [op(x),z];\n \+ end do;\n [seq(-x[i],i=1..p),seq(x[m-i],i=0..m-1)];\nend proc: # of \+ legendrezeros\n\n#calculate zeros of a Jacobi polynomial\njacobizeros \+ := proc(n::posint,alf::realcons,bet::realcons)\n local r1,r2,r3,ns,z ,x,h,eps,i,j,k,a,b,c,p1,p2,p3,\n temp,ab,ds,an,bn;\n\n if signum(0 ,alf+1,0)<0 or signum(0,bet+1,0)<0 then\n error \"Jacobi polynomi al is not defined for current parameter values\";\n end if;\n\n x \+ := [];\n # obtain initial estimate for root \n ns := n*n;\n \+ for i from 1 to n do\n if irem(n,2)=1 and i=iquo(n,2)+1 and sig num(0,alf-bet,0)=0 then\n z := 0;\n goto(1111);\n \+ elif i=1 then\n an := alf/n;\n bn := bet/n;\n \+ r1 := (1+alf)*(2.78/(4+ns)+0.768*an/n);\n r2 := 1+(1.48+0.452 *an)*an+(0.96+0.83*an)*bn;\n z := 1-r1/r2;\n elif i=2 the n\n r1 := (4.1+alf)/((1+alf)*(1+0.156*alf));\n r2 := 1 +0.06*(n-8)*(1+0.12*alf)/n;\n r3 := 1+0.012*bet*(1+0.25*abs(al f))/n;\n z := z - (1-z)*r1*r2*r3; \n elif i=3 then\n \+ r1 := (1.67+0.28*alf)/(1+0.37*alf);\n r2 := 1+0.22*(n-8)/n ;\n r3 := 1+8*bet/((6.28+bet)*ns);\n z := z-(x[1]-z)*r 1*r2*r3;\n elif i=n-1 then\n r1 := (1+0.235*bet)/(0.766+0 .119*bet);\n r2 := 1/(1+0.639*(n-4)/(1+0.71*(n-4)));\n \+ r3 := 1/(1+20*alf/((7.5*alf)*ns));\n z := z+(z-x[n-3])*r1*r2* r3;\n elif i=n then\n r1 := (1+0.37*bet)/(1.67+0.28*bet); \n r2 := 1/(1+0.22*(n-8)/n);\n r3 := 1/(1+8*alf/((6.28 +alf)*ns));\n z := z+(z-x[n-2])*r1*r2*r3;\n else\n \+ z := 3*x[i-1]-3*x[i-2]+x[i-3];\n end if;\n \n eps := \+ Float(5,-Digits);\n ab := alf+bet;\n ds := alf*alf-bet*bet; \n for k from 1 to Digits*5 do\n # evaluate Jacobi poly a t z .. p1\n temp := 2+ab;\n p1 := (alf-bet+temp*z)/2; \n p2 := 1;\n for j from 2 to n do\n p3 := \+ p2;\n p2 := p1;\n temp := 2*j+ab;\n a := 2*j*(j+ab)*(temp-2);\n b := (temp-1)*(ds+temp*(temp-2)* z);\n c := 2*(j-1+alf)*(j-1+bet)*temp;\n p1 := ( b*p2-c*p3)/a;\n end do;\n\n # get the derivative and a pply the Newton formula\n h := p1*(temp*(1-z*z))/(n*(alf-bet-t emp*z)*p1+2*(n+alf)*(n+bet)*p2);\n z := z-h;\n if abs( h)<=eps then break end if;\n end do;\n 1111:\n x := [op (x),z];\n end do;\n [seq(x[n-i],i=0..n-1)];\nend proc: # of jacobi zeros" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "Examples are given in the next section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "interpoly" }{TEXT -1 35 ": examples with evenly spaced node s" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 272 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) Find a degree 10 polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = x/(x+exp(x));" "6#/-%\"fG6#% \"xG*&F'\"\"\",&F'F)-%$expG6#F'F)!\"\"" }{TEXT -1 18 " on the interva l " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 34 ", which ag rees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " at 11 " }{TEXT 296 14 "equally spaced" }{TEXT -1 1 " " } {TEXT 286 1 "x" }{TEXT -1 38 " values between and including 0 and 2." }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Draw a picture which shows the inte rpolation points on the graph of " }{XPPEDIT 18 0 "y = x/(x+exp(x)); " "6#/%\"yG*&%\"xG\"\"\",&F&F'-%$expG6#F&F'!\"\"" }{TEXT -1 57 " toget her with the graph of the interpolating polynomial." }}{PARA 0 "" 0 " " {TEXT -1 17 "(c) Estimate the " }{TEXT 296 22 "maximum absolute erro r" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6 #-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 2 ];" "6#7$\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "(d ) Compare the value of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 48 " with the value of the interpolating polynomial \+ " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "x = sqrt(3);" "6#/%\"xG-%%sqrtG6#\"\"$" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 98 " Estimate the absolute and relati ve errors in using the second value to approximate the first?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "Solution " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) and (b) " }} {PARA 0 "" 0 "" {TEXT -1 43 "The degree 10 polynomial approximation fo r " }{XPPEDIT 18 0 "f(x) = x/(x+exp(x));" "6#/-%\"fG6#%\"xG*&F'\"\"\", &F'F)-%$expG6#F'F)!\"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 92 ", based on equally space d nodes between 0 and 2 inclusive can be found using the procedure " }{TEXT 0 9 "interpoly" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 " Note" }{TEXT -1 121 ": Including the optional 4th argument 'pts' in qu otes causes this variable to be given the list of interpolation points . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "f := x -> x/(x+exp(x));\ninterpoly(f(x),x=0..2,10,'p ts'):\np := unapply(%,x);\nplot([p(x),pts],x=0..2,style=[line,point],s ymbol=circle,color=[coral,black]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F-F.-%$expG6#F-F. !\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(,6*&$\"+\\fx]**!#5\"\"\"9$F1F1*&$\"+nGB>>!\"*F1)F2\" \"#F1!\"\"*&$\"+M%*3DHF6F1)F2\"\"$F1F1*&$\"+)z`qs$F6F1)F2\"\"%F1F9*&$ \"+u1awOF6F1)F2\"\"&F1F1*&$\"+2[peEF6F1)F2\"\"'F1F9*&$\"+f:zV8F6F1)F2 \"\"(F1F1*&$\"+P+nfWF0F1)F2\"\")F1F9*&$\"+uhO!p)!#6F1)F2\"\"*F1F1*&$\" +hr-8v!#7F1)F2\"#5F1F9F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3u8z _&3Z23#F-7$$\"39LLLL3VfVF-$\"3qSA\"e(G<'*RF-7$$\"31++]i&*)fD'F-$\"36F9 z@RISbF-7$$\"3'pmm;H[D:)F-$\"3g0QD;(*4!)pF-7$$\"3-++v$pU&G5!#=$\"3))\\ vQ;1%\\[)F-7$$\"3LLLLe0$=C\"FB$\"3S)R>irG*y)*F-7$$\"3KLLLLA`c9FB$\"37i AGc`)z6\"FB7$$\"3ILLL3RBr;FB$\"3D(f.'HSoQ7FB7$$\"3Ymm;zjf)4#FB$\"3aY:) zp$*RX\"FB7$$\"3=LL$e4;[\\#FB$\"3![w_U?vwi\"FB7$$\"3p****\\i'y]!HFB$\" 3))p.\"4Zl\\y\"FB7$$\"3,LL$ezs$HLFB$\"358mtIFyE>FB7$$\"3_****\\7iI_PFB $\"3OogG0X$)\\?FB7$$\"3#pmmm@Xt=%FB$\"3Y_j1,Hyf@FB7$$\"3QLLL3y_qXFB$\" 37FS+i@KWAFB7$$\"3i******\\1!>+&FB$\"3@!\\&*3C$GFBFB7$$\"3()******\\Z/ NaFB$\"3uxa`Vz**)R#FB7$$\"3'*******\\$fC&eFB$\"3G#*pKdEJeCFB7$$\"3ELL$ ez6:B'FB$\"32&[m\\2&o/DFB7$$\"3Smmm;=C#o'FB$\"3y#*\\3p$3lh#FB7$$\"3sl mmm(y8!zFB$\"3IcU3RZ@REFB7$$\"3V++]i.tK$)FB$\"3K^0w!**>(eEFB7$$\"39++] (3zMu)FB$\"3%y%p@od[sEFB7$$\"3#pmm;H_?<*FB$\"31_5RA*zAo#FB7$$\"3emm;zi hl&*FB$\"3i'p!eO7](o#FB7$$\"39LLL3#G,***FB$\"3M7hMzJT*o#FB7$$\"3EFB7$$\"3_mmmwanL8Fet$\"3Qz%) p%\\S/g#FB7$$\"3'******\\2goP\"Fet$\"3E*p#=\"*\\syDFB7$$\"3CLLeR<*fT\" Fet$\"3+g9.J0`dDFB7$$\"3'******\\)Hxe9Fet$\"3Q;\\9G-%G`#FB7$$\"3Ymm\"H !o-*\\\"Fet$\"32'Q!*p,t#3DFB7$$\"3))***\\7k.6a\"Fet$\"3/>4>o9K\"[#FB7$ $\"3emmmT9C#e\"Fet$\"3M^&GC0=QX#FB7$$\"3\"****\\i!*3`i\"Fet$\"3b/,*yk4 RU#FB7$$\"3QLLL$*zym;Fet$\"3-&yVL'f7%R#FB7$$\"3GLL$3N1#4Fet$\"3G^\\)>G#)**>#FB7$$\"3/++v.Uac> Fet$\"3oLeby&Gm;#FB7$$\"\"#F)$\"3L'>gh[R,8#FB-%'COLOURG6&%$RGBG$\"*+++ +\"!\")$\")AR!)\\Fa\\lF(-%&STYLEG6#%%LINEG-F$6%7-F'7$$\"35+++++++?FB$ \"3)******zsgqS\"FB7$$\"3A+++++++SFB$\"3)*******43O9@FB7$$\"3w******** ******fFB$\"3/+++))*prZ#FB7$$\"3U+++++++!)FB$\"33+++t_:WEFB7$$\"\"\"F) $\"3#******R@9%*o#FB7$$\"3%**************>\"Fet$\"3#)*****He)zaEFB7$$ \"3!**************R\"Fet$\"3y******=(ejc#FB7$$\"33+++++++;Fet$\"3.+++2 xhTCFB7$$\"3/+++++++=Fet$\"31+++6m4$H#FB7$Fg[l$\"3()*****zdR,8#FB-F\\ \\l6&F^\\lF)F)F)-Fe\\l6#%&POINTG-%+AXESLABELSG6$Q\"x6\"Q!Fd`l-%'SYMBOL G6#%'CIRCLEG-%%VIEWG6$;F(Fg[l%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "We can view the coordi nates of the interpolation points by accessing the variable \"pts\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "pts;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7-7$$\"\"!F&F%7$$\"+++++? !#5$\"+G2129F*7$$\"+++++SF*$\"+53O9@F*7$$\"+++++gF*$\"+))*prZ#F*7$$\"+ ++++!)F*$\"+t_:WEF*7$$\"+++++5!\"*$\"+9UT*o#F*7$$\"+++++7F?$\"+$e)zaEF *7$$\"+++++9F?$\"+>(ejc#F*7$$\"+++++;F?$\"+2xhTCF*7$$\"+++++=F?$\"+6m4 $H#F*7$$\"\"#F&$\"+y&R,8#F*" }}}{PARA 0 "" 0 "" {TEXT -1 87 "(c) We ca n see from the following error curve that the maximum absolute error i n using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to ap proximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[0,2]" "6#7$\"\"!\"\"#" }{TEXT -1 10 " \+ is about " }{XPPEDIT 18 0 "10^(-4);" "6#)\"#5,$\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(f(x)-p(x),x=0..2,color=blue);" }}{PARA 13 "" 1 " " {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7cq7$$\"\"!F)F(7$$ \"3immTN@Ki8!#?$\"3U:+nt\"\\ub'!#B7$$\"3ALL$3FWYs#F-$\"3]+%)HF3P#G\"!# A7$$\"3%)***\\iSmp3%F-$\"3SFN?H,s!)=F67$$\"3WmmmT&)G\\aF-$\"3WCmY\\eh^ CF67$$\"3m****\\7G$R<)F-$\"3I$\"3QLBiY oYwWF67$$\"3gmmTN@Ki8FI$\"3R3'\\MT4UM&F67$$\"3$*****\\ilyM;FI$\"3'>b:, bAJ7'F67$$\"3DLLe*)4D2>FI$\"3@81<&R$f=oF67$$\"3emmm;arz@FI$\"3#>:*=Igv NuF67$$F3FI$\"3U]!>7L)[a%)F67$$\"3')*****\\7t&pKFI$\"3f8!f!)*zq:#*F67$ $\"3]mm;z>]9QFI$\"3q;7&QWnBv*F67$$\"39LLLL3VfVFI$\"3rWd[!yD%45!#@7$$\" 3;m\"zWn+lf%FI$\"3QYmPt2\"*=5F]p7$$\"3()**\\i:0dL[FI$\"3+&zYA\"pUD5F]p 7$$\"3eL3xc.kq]FI$\"3%4#))e1-5xI&FI$\"3OU#)*[zF..\" F]p7$$\"3h*\\i!R+yWbFI$\"3\"\\su2Fy!H5F]p7$$\"3KL$3-))\\=y&FI$\"3Bl#)Q )*>fD5F]p7$$\"3.nTN@(>*=gFI$\"36$oXwTJ+-\"F]p7$$\"31++]i&*)fD'FI$\"3&[ )fPR8b75F]p7$$\"3]LL3F*oU?(FI$\"3eLA\"py4Gm*F67$$\"3'pmm;H[D:)FI$\"3#o VRx-i))**)F67$$\"3bLLe9w)*=#*FI$\"3I$=AXhPg4)F67$$\"3-++v$pU&G5!#=$\"3 tiZ5(o4;5(F67$$\"3nm;/Em=N6Fhs$\"3eKr>U%*)\\2'F67$$\"3LLLLe0$=C\"Fhs$ \"3#f@K_]_41&F67$$\"3=LL$eR\"=\\8Fhs$\"3u(fs(ff-'3%F67$$\"3KLLLLA`c9Fh s$\"3I,y0CAP!=$F67$$\"3WLL$32$)Qc\"Fhs$\"3e8,WbT@fBF67$$\"3ILLL3RBr;Fh s$\"3(o=7I^)GJ;F67$$\"3-++vV^\"\\)=Fhs$\"3dO#[*yt\"Rq%F07$$\"3Ymm;zjf) 4#Fhs$!3!yF(pt.0FKF07$$\"3q****\\Piq'H#Fhs$!3gQn`>c*Hq(F07$$\"3=LL$e4; [\\#Fhs$!3Og@9*)pFC**F07$$\"3Wmmm;*)4YDFhs$!3ze4c)*\\u>5F67$$\"3m**** \\P^rH\\`WO5F67$$\"3#HLL$eXm[EFhs$!3+7)))e%>UV5F67$$\"3;mm ;zt%**p#Fhs$!3#=ma?FY:/\"F67$$\"3lKL$3-8D!GFhs$!3*[,rX%ef95F67$$\"3p** **\\i'y]!HFhs$!3]:Xz&=m,i*F07$$\"32mm;HdA!#C7$$\"3#pmmm@Xt=%Fhs$\"3$\\L7XG1!R5F07$$\"3; ++]7l$*yVFhs$\"3E,J))zCDl+&Fhs$\"3XjB=VUCQ@F07$$ \"3()******\\Z/NaFhs$\"3&Q%p_ODP.8F07$$\"3'*******\\$fC&eFhs$\"38Q*3,I /h7$F_z7$$\"3ELL$ez6:B'Fhs$!3S>quB!GR1%F_z7$$\"3Smmm;=C#o'Fhs$!3()QE@N 5!*=$)F_z7$$\"3-mmmm#pS1(Fhs$!3e@uXDG.\"=)F_z7$$\"3]****\\i`A3vFhs$!3' 3E,#)*=-f\\F_z7$$\"3slmmm(y8!zFhs$!3eio4@8)3y*!#D7$$\"3V++]i.tK$)Fhs$ \"3koJ,>lVbGF_z7$$\"39++](3zMu)Fhs$\"3[bzru\\gQ[F_z7$$\"3#pmm;H_?<*Fhs $\"3y3I*)Hm[FZF_z7$$\"3emm;zihl&*Fhs$\"3YneRVDq[HF_z7$$\"39LLL3#G,***F hs$\"3;#[=iE-4d(!#E7$$\"3F_z7$$\"3' ******\\)Hxe9Fj_l$!3%pZ'f0!**4R(F_z7$$\"3Ymm\"H!o-*\\\"Fj_l$!3G6$3KzYm 4\"F07$$\"3))***\\7k.6a\"Fj_l$!3w\\D&pa3U1\"F07$$\"3emmmT9C#e\"Fj_l$!3 wZ9m7Lb0XF_z7$$\"3\"****\\i!*3`i\"Fj_l$\"38euX>%3'>#)F_z7$$\"3QLLL$*zy m;Fj_l$\"3AR,(fn_jP#F07$$\"3KLL3sr*zo\"Fj_l$\"3P;1]wBw,JF07$$\"3GLL$3N 1#4AZ#>c'[Q\"F67$$\"3am;zW?)\\*=Fj_l$!3I5mD^0L\")=F67$$\"30++DOl5;>Fj_l $!3**yM2\"f%))3BF67$$\"30+]7`f@E>Fj_l$!3)HvM[B!*\\X#F67$$\"3/+++q`KO>F j_l$!3Dxii=W7SDF67$$\"3:+vVy+QT>Fj_l$!3)f'eY@SS`DF67$$\"3/+](oyMk%>Fj_ l$!3RSwCD5HVDF67$$\"3%**\\7`\\*[^>Fj_l$!3%37i<(fe1DF67$$\"3/++v.Uac>Fj _l$!3Oav[WS$)RCF67$$\"3/+D\"G:3u'>Fj_l$!3SfW&[vgo<#F67$$\"3-+](=5s#y>F j_l$!3W_#*3cI#fr\"F67$$\"3-+v$40O\"*)>Fj_l$!3K%***p*Q]'35F67$$\"\"#F)$ \"3bxTn/$zBA*Ff_l-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q \"x6\"Q!Ff\\m-%%VIEWG6$;F(Fg[m%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "numapprox[infnorm] (f(x)-p(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+e*G..\"!#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "(d) \+ First compute the function value and the approximate value using the i nterpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "xx := sqrt(3);\nval := evalf(f(xx)) ;\napproxval := evalf(p(xx));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xx G*$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valG$\"+u0f XB!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*approxvalG$\"(`bM#!\"(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The abso lute error in using the first value to approximate the second is about . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "abserr := abs(val-approxval);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&uv$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The relative error is about . . . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(abserr/val,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&>g\"! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 268 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 47 "(a) Find a degree 10 polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = arctan(x^2);" "6#/-%\"fG6#%\"xG-%'arctanG6#*$F'\"\"#" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\" \"#" }{TEXT -1 34 ", which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 11 " }{TEXT 296 14 "equally \+ spaced" }{TEXT -1 11 " values of " }{TEXT 302 1 "x" }{TEXT -1 28 " bet ween 0 and 2 inclusive." }}{PARA 0 "" 0 "" {TEXT -1 72 "(b) Draw a pi cture which shows the interpolation points on the graph of " } {XPPEDIT 18 0 "y=arctan(x^2)" "6#/%\"yG-%'arctanG6#*$%\"xG\"\"#" } {TEXT -1 57 " together with the graph of the interpolating polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) Estimate the " }{TEXT 296 22 "max imum absolute error" }{TEXT -1 25 " in using the polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interv al " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 48 "(d) Repeat part (a), but change the inter val to " }{XPPEDIT 18 0 "[-5,5]" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 53 ", \+ that is, find a degree 10 polynomial approximation " }{XPPEDIT 18 0 "q (x)" "6#-%\"qG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = arct an(x^2);" "6#/-%\"fG6#%\"xG-%'arctanG6#*$F'\"\"#" }{TEXT -1 17 " on th e interval " }{XPPEDIT 18 0 "[-5, 5];" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 34 ", which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 32 " at 11 equally spaced values of " }{TEXT 287 1 "x" }{TEXT -1 29 " between -5 and 5 inclusive." }}{PARA 0 "" 0 "" {TEXT -1 37 "(e) Plot the graphs of the polyomial " }{XPPEDIT 18 0 "q( x)" "6#-%\"qG6#%\"xG" }{TEXT -1 41 " from part(d) together with the gr aph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 42 " in the same picture. Does the polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6# %\"xG" }{TEXT -1 47 " provide a good approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-5,5]" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 17 "? Why or why not?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 12 "(a) a nd (b) " }}{PARA 0 "" 0 "" {TEXT -1 43 "The degree 10 polynomial appro ximation for " }{XPPEDIT 18 0 "f(x) = arctan(x^2);" "6#/-%\"fG6#%\"xG- %'arctanG6#*$F'\"\"#" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 92 ", based on equally spaced nod es between 0 and 2 inclusive can be found using the procedure " } {TEXT 0 9 "interpoly" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 4 "N ote" }{TEXT -1 121 ": Including the optional 4th argument 'pts' in quo tes causes this variable to be given the list of interpolation points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "f := x -> arctan(x^2);\ninterpoly(f(x),x=0..2,10,'pt s'):\np := unapply(%,x);\nplot([p(x),pts],x=0..2,style=[line,point],sy mbol=circle,color=[red,black]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%'arctanG6#*$)9$\"\"#\"\"\"F(F (F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,6*&$\"+C?RHK!#6\"\"\"9$F1!\"\"*&$\"+z&>SS\"!\"*F1)F2\"\"#F1 F1*&$\"+fF\"\\)=F7F1)F2\"\"$F1F3*&$\"+N<@?TF7F1)F2\"\"%F1F1*&$\"+o(Hy$ QF7F1)F2\"\"&F1F3*&$\"+!yJJo(!#5F1)F2\"\"'F1F3*&$\"+*R$\\(o$F7F1)F2\" \"(F1F1*&$\"+(Hy[g#F7F1)F2\"\")F1F3*&$\"+Gc\"*[zFLF1)F2\"\"*F1F1*&$\"+ ?*=!*H*F0F1)F2\"#5F1F3F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"39LLLL3VfV!#>$\"3]%4 fv+m&=6!#?7$$\"3'pmm;H[D:)F-$\"3m!zexK2c%eF07$$\"3LLLLe0$=C\"!#=$\"3QE N)[87(*[\"F-7$$\"3ILLL3RBr;F9$\"3#yQ;7H8Gx#F-7$$\"3Ymm;zjf)4#F9$\"3a)p s>(oo0WF-7$$\"3=LL$e4;[\\#F9$\"3NiV#)R#p9B'F-7$$\"3p****\\i'y]!HF9$\"3 S0c'yInjV)F-7$$\"3,LL$ezs$HLF9$\"33!=***=<806F97$$\"3_****\\7iI_PF9$\" 3#G1d[c&>*R\"F97$$\"3#pmmm@Xt=%F9$\"3k5Z.Ta[N+&F9$\"37>+FJ!R4X#F97$$\"3()******\\Z/NaF9 $\"3%p\\$3(*='=(GF97$$\"3'*******\\$fC&eF9$\"3gtHLs=y*H$F97$$\"3ELL$ez 6:B'F9$\"3G$GkAr+Tq$F97$$\"3Smmm;=C#o'F9$\"3gEv[;v#**>%F97$$\"3-mmmm#p S1(F9$\"3Rr[z5t!*GYF97$$\"3]****\\i`A3vF9$\"33@R)pE!\\L^F97$$\"3slmmm( y8!zF9$\"395`!3JK6e&F97$$\"3V++]i.tK$)F9$\"3e!)fK3)f!pgF97$$\"39++](3z Mu)F9$\"30(GqUBDo_'F97$$\"3#pmm;H_?<*F9$\"3qDvje*pP*pF97$$\"3emm;zihl& *F9$\"3^^V[R'4.T(F97$$\"39LLL3#G,***F9$\"3^?'fOF,T%yF97$$\"3gT#3@`.!*F97$$\"3#*******pvxl6Fbs$\"3!H\"G#e`$[k$*F 97$$\"3z****\\_qn27Fbs$\"3YRd#e#fn(p*F97$$\"3%)***\\i&p@[7Fbs$\"3&*ze4 3t;+5Fbs7$$\"3#)****\\2'HKH\"Fbs$\"3\"*fJs>C(=.\"Fbs7$$\"3_mmmwanL8Fbs $\"3kb[v&=)fe5Fbs7$$\"3'******\\2goP\"Fbs$\"3Q4o:q1Q&3\"Fbs7$$\"3CLLeR <*fT\"Fbs$\"3XJ$)*\\Fj\"36Fbs7$$\"3'******\\)Hxe9Fbs$\"3`L*y'Qw`J6Fbs7 $$\"3Ymm\"H!o-*\\\"Fbs$\"3-'>K\"eW9_6Fbs7$$\"3))***\\7k.6a\"Fbs$\"3/Wx tYfKs6Fbs7$$\"3emmmT9C#e\"Fbs$\"3?owWuAz!>\"Fbs7$$\"3\"****\\i!*3`i\"F bs$\"3%3pBusu)37Fbs7$$\"3QLLL$*zym;Fbs$\"3\"=@>/Gx^A\"Fbs7$$\"3GLL$3N1 #4Fbs$\"3[NJ^(eWeI \"Fbs7$$\"3/++v.Uac>Fbs$\"3,-frs_G;8Fbs7$$\"\"#F)$\"3G$)*>$[w\"eK\"Fbs -%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-F$6%7-F'7$$\"35 +++++++?F9$\"3?+++7(oy*RF-7$$\"3A+++++++SF9$\"33+++AEb'e\"F97$$\"3w*** ***********fF9$\"3A+++1ebbMF97$$\"3U+++++++!)F9$\"3/+++6>8$p&F97$$\"\" \"F)$\"3;+++M;)R&yF97$$\"3%**************>\"Fbs$\"3&)*****pi'3Q'*F97$$ \"3!**************R\"Fbs$\"31+++;)=!*4\"Fbs7$$\"33+++++++;Fbs$\"3!**** ***zyR)>\"Fbs7$$\"3/+++++++=Fbs$\"3)******H4I9F\"Fbs7$Fdz$\"3!******Rm " 0 "" {MPLTEXT 1 0 4 "pts;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7-7$$\"\"!F&F%7$$\"+++++? !#5$\"+7(oy*R!#67$$\"+++++SF*$\"+AEb'e\"F*7$$\"+++++gF*$\"+1ebbMF*7$$ \"+++++!)F*$\"+6>8$p&F*7$$\"+++++5!\"*$\"+M;)R&yF*7$$\"+++++7F@$\"+Fm3 Q'*F*7$$\"+++++9F@$\"+;)=!*4\"F@7$$\"+++++;F@$\"+!)yR)>\"F@7$$\"+++++= F@$\"+$4I9F\"F@7$$\"\"#F&$\"+kw\"eK\"F@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "(c) We can see from the following \+ error curve that the maximum absolute error in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,2]" "6#7$\"\"!\"\"#" }{TEXT -1 10 " is about " }{XPPEDIT 18 0 "10^(-3)" "6#)\"#5,$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "plot(f(x)-p( x),x=0..2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7er7$$\"\"!F)F(7$$\"3WmmmT&)G\\a!#?$\"3=&4V &Q-$Gk\"!#@7$$\"3ILLL3x&)*3\"!#>$\"3cu=/kE^jIF07$$\"3$*****\\ilyM;F4$ \"3a)R,i_n!zUF07$$\"3emmm;arz@F4$\"3q%*>k?vo0`F07$$\"3')*****\\7t&pKF4 $\"3mDB$[T*)G&oF07$$\"39LLLL3VfVF4$\"3S7$)*\\Ba*=yF07$$\"3()**\\i:0dL[ F4$\"3a%yj%4i3%3)F07$$\"3fmm\"z>5xI&F4$\"3')\\f$p+6kE)F07$$\"3h*\\i!R+ yWbF4$\"39\"*ej*yI)G$)F07$$\"3KL$3-))\\=y&F4$\"39u'\\.WtKP)F07$$\"3.nT N@(>*=gF4$\"3TA\"y(p$)f+%)F07$$\"31++]i&*)fD'F4$\"3m72*e:U;T)F07$$\"33 m;zW#H,t'F4$\"3;CEgn/7)Q)F07$$\"3]LL3F*oU?(F4$\"3%**G%\\Cr&*3$)F07$$\" 3#4+v$4'3%ywF4$\"39#))zK+d+=)F07$$\"3'pmm;H[D:)F4$\"3F<:6?<*p+)F07$$\" 3-++v$pU&G5!#=$\"3**o>$y)=p:oF07$$\"3LLLLe0$=C\"Fjp$\"33Ta71n(3B&F07$$ \"3KLLLLA`c9Fjp$\"3;Rm,z=IONF07$$\"3ILLL3RBr;Fjp$\"3CWiz3[N[>F07$$\"3- ++vV^\"\\)=Fjp$\"3(Q]e()=IP-'!#A7$$\"3Ymm;zjf)4#Fjp$!36OW%=%\\ACWFar7$ $\"3q****\\Piq'H#Fjp$!35qeO>a`B6F07$$\"3=LL$e4;[\\#Fjp$!3()Rz]z`0Q:F07 $$\"3Wmmm;*)4YDFjp$!3#o#>k\\M20;F07$$\"3m****\\PI7v#Fjp$!3a!y8d$e?Em6F07$$\"3_****\\7iI_PFjp$!3=b1UO$HA5%Far7$$\"3#pmmm@Xt=%Fjp$\"3/E$> jNgTc#Far7$$\"3;++]7l$*yVFjp$\"3!4%[,djNjXFar7$$\"3QLLL3y_qXFjp$\"3%GY lD(f%=(eFar7$$\"3'****\\(=5PyYFjp$\"3'QOV0Ol>I'Far7$$\"3]mm;HU@'y%Fjp$ \"3S#fB-H5$>lFar7$$\"31LLeRu0%*[Fjp$\"3+Wa#**o[d`'Far7$$\"3i******\\1! 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The situation is espec ially bad towards the edges of the interval where the derivative of " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 39 " differs widely \+ from the derivative of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 30 " at the interpolation points. " }}{PARA 0 "" 0 "" {TEXT -1 301 "Since the construction of the interpolating polynomial only ut ilises the function values, there is nothing to prevent this sort of b ehaviour occuring, especially when the graph of the function involved \+ has characteristics which are different from polynomial characteristic s. In this case the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 30 " has the horizontal asymptote " }{XPPEDIT 18 0 "y=Pi/2 " "6#/%\"yG*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "f(x)->Pi/2" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrow G6\"*&%#PiG\"\"\"\"\"#!\"\"F-F-F-" }{TEXT -1 4 " as " }{XPPEDIT 18 0 " abs(x)->infinity" "6#f*6#-%$absG6#%\"xG7\"6$%)operatorG%&arrowG6\"%)in finityGF-F-F-" }{TEXT -1 10 ", whereas " }{XPPEDIT 18 0 "abs(q(x))->in finity" "6#f*6#-%$absG6#-%\"qG6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infi nityGF0F0F0" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "abs(x)->infinity" "6#f *6#-%$absG6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "f := x -> arctan(x^2);\ninterpoly(f(x),x=-5..5,10): \nq := unapply(%,x);\nplot([f(x),q(x)],x=-5..5,y=-2..2.5,color=[green, red],linestyle=[1,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%'arctanG6#*$)9$\"\"#\"\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,,*$)9$\"\"#\"\"\"$\"+P?FW5!\"**&$\"+27RBH!#5F1)F/\"\"%F1!\"\"*&$\" +6WpBN!#6F1)F/\"\"'F1F1*&$\"+#edO!=!#7F1)F/\"\")F1F;*&$\"+v:.'>$!#9F1) F/\"#5F1F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6%7co7$$!\"&\"\"!$\"3n1;nRw\"3`\"!#<7$$!3YLLLe%G?y%F-$ \"3$RW$*3b%4F:F-7$$!3OmmT&esBf%F-$\"3$R!)Q1s:M_\"F-7$$!3ALL$3s%3zVF-$ \"37%)efTgp=:F-7$$!3_LL$e/$QkTF-$\"3$3Qq4#p>8:F-7$$!3ommT5=q]RF-$\"3.h %eRD9o]\"F-7$$!3ILL3_>f_PF-$\"3sMf#\\U-**\\\"F-7$$!3K++vo1YZNF-$\"3Am# fM(**\\\"\\\"F-7$$!3;LL3-OJNLF-$\"3)*yoYMT9\"[\"F-7$$!3p***\\P*o%Q7$F- $\"3Az+/zrno9F-7$$!3Kmmm\"RFj!HF-$\"3KnrhYf&HX\"F-7$$!33LL$e4OZr#F-$\" 3cJ$4_&4$fV\"F-7$$!3u*****\\n\\!*\\#F-$\"3jwc*QRA?T\"F-7$$!3%)*****\\i xCG#F-$\"3i[O2)\\`6Q\"F-7$$!3#******\\KqP2#F-$\"3q:9>PlKU8F-7$$!39LL3- TC%)=F-$\"3W>Pes'[iH\"F-7$$!3[mmm\"4z)e;F-$\"30bn=MZCA7F-7$$!3Mmmmm`'z Y\"F-$\"3QilYuoJO6F-7$$!37L$3FMEpN\"F-$\"3'\\&ePj'eK2\"F-7$$!3#****\\( =t)eC\"F-$\"3geGVvW'[)**!#=7$$!3OL$3x'*)fZ6F-$\"3tNZ_)\\FO@*Fgq7$$!3!o mmmh5$\\5F-$\"3YZ=!y$QdM$)Fgq7$$!3tIL3xrs9%*Fgq$\"3)49x\"y'RBD(Fgq7$$! 3S$***\\(=[jL)Fgq$\"3kw!\\2SmK2'Fgq7$$!3q%****\\Pw%4tFgq$\"3ncu*e@vp! \\Fgq7$$!3)f***\\iXg#G'Fgq$\"3A/A%=p(QfPFgq7$$!3$oK$3_:<6_Fgq$\"35\\Z% GV\"o^EFgq7$$!3ndmmT&Q(RTFgq$\"3yrv(*=gD(p\"Fgq7$$!3Ihm\"HdGe:$Fgq$\"3 &eDw7.@l#**!#>7$$!3%\\mmTg=><#Fgq$\"3&yqJ*=ht8ZFet7$$!3FK$3Fpy7k\"Fgq$ \"3+EF?VW9$p#Fet7$$!3g***\\7yQ16\"Fgq$\"3zl!*QXfXL7Fet7$$!3iK$3_D)=`%) Fet$\"3?xtO_v^Xr!#?7$$!3Epm\"zp))**z&Fet$\"3i]kE+U(RO$Fju7$$!3#f+D19*y YJFet$\"3#\\O!**e'yA!**!#@7$$!3vDMLLe*e$\\Fju$\"3!ozpsw1jV#!#A7$$\"3+l ;a)3RBE#Fet$\"3<\"fGYqx\"=^Fev7$$\"3bsmTgxE=]Fet$\"3;;#*)z!eH=DFju7$$ \"37!o\"HKk>uxFet$\"3MAVeH%RP/'Fju7$$\"3womT5D,`5Fgq$\"3\")oD41**y36Fe t7$$\"3Gq;zW#)>/;Fgq$\"3)R?z88%)Gd#Fet7$$\"3!=nm\"zRQb@Fgq$\"3G\\WTjAM UYFet7$$\"3mOLL$e,]6$Fgq$\"3MaHnoG&Hn*Fet7$$\"3_,+](=>Y2%Fgq$\"3y;2fw \\CX;Fgq7$$\"36QLe*[K56&Fgq$\"3u9k?Ee;bDFgq7$$\"3summ\"zXu9'Fgq$\"3*) \\*fgW-Kh$Fgq7$$\"3#yLLe9i\"=sFgq$\"35Y)3%)*y?.[Fgq7$$\"3#4+++]y))G)Fg q$\"3)Hg/Sa^)>gFgq7$$\"3%>++DcljL*Fgq$\"3#HUc#)o\\%prFgq7$$\"3H++]i_QQ 5F-$\"3Vz3Y0YHI#)Fgq7$$\"3U+](=-N(R6F-$\"3kdS^aDFZ\"*Fgq7$$\"3b++D\"y% 3T7F-$\"3!=g*QO:k\\**Fgq7$$\"3G+]P4kh`8F-$\"3%)*o=!pt?r5F-7$$\"3+++]P! 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" }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Draw a pict ure which shows the interpolation points on the graph of " }{XPPEDIT 18 0 "y = sin(x)/sqrt(1+x^2);" "6#/%\"yG*&-%$sinG6#%\"xG\"\"\"-%%sqrtG 6#,&F*F**$F)\"\"#F*!\"\"" }{TEXT -1 57 " together with the graph of th e interpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "(c) Verify by means of a graph that the maximu m absolute error meets the requirement given in (a), and determine the maximum absolute error exhibited by your approximating polynomial on \+ the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "` `<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "By trial and error the required minimal degree for the interpolating polynomial is 11. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "f := x -> sin(x)/sqrt(x^2+1);\ninterpoly(f(x),x=0..1,11,'pts' ):\np := unapply(%,x);\nplot([p(x),pts],x=0..1,style=[line,point],symb ol=circle,color=[coral,black]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$sinG6#9$\"\"\"-%%sqrtG6#,& *$)F0\"\"#F1F1F1F1!\"\"F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"p Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,:$\"+%)ph=Q!#B\"\"\"*&$\"+K-#**** *!#5F09$F0F0*&$\"+')*HVn#!#8F0)F5\"\"#F0F0*&$\"++B!Rq'F4F0)F5\"\"$F0! \"\"*&$\"+*3%z()G!#6F0)F5\"\"%F0F0*&$\"+6^\\oKF4F0)F5\"\"&F0F0*&$\"+q8 >XWF4F0)F5\"\"'F0F0*&$\"+8a-=8!\"*F0)F5\"\"(F0FA*&$\"+\"RVOF\"FUF0)F5 \"\")F0F0*&$\"+7l.&R'F4F0)F5\"\"*F0FA*&$\"+8!Rpl\"F4F0)F5\"#5F0F0*&$\" +)Q?9p\"FEF0)F5\"#6F0FAF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)$\"35+++%)ph=Q!#J7$$\"3emmm ;arz@!#>$\"3GYR)\\G'F07$$\"3[mmmT&phN)F0$\"3*4j%)3rfuJ)F07$$\"3CLLe* =)H\\5!#=$\"3nFy**e]lT5FE7$$\"3gmm\"z/3uC\"FE$\"3`X;tCtgM7FE7$$\"3%)** *\\7LRDX\"FE$\"3!*[E'>$\\SK9FE7$$\"3]mm\"zR'ok;FE$\"3(HUz]c:Xj\"FE7$$ \"3w***\\i5`h(=FE$\"3TQ7IM?FE7$$\" 3qmm;/RE&G#FE$\"3tG@IZ,\\3AFE7$$\"3\")*****\\K]4]#FE$\"3o'3xd_55S#FE7$ $\"3$******\\PAvr#FE$\"3)o0P'))pD!f#FE7$$\"3)******\\nHi#HFE$\"3Oh;A/# [&oFFE7$$\"3jmm\"z*ev:JFE$\"3@9=Tu5\"o#HFE7$$\"3?LLL347TLFE$\"3W3r[yVH 5JFE7$$\"3,LLLLY.KNFE$\"3#>8$yk[ehKFE7$$\"3w***\\7o7Tv$FE$\"3M>,#G8NEV $FE7$$\"3'GLLLQ*o]RFE$\"3=*oY2aFE$\"3mk.Z/s7GX FE7$$\"39mm;zXu9cFE$\"3Sm(>600Ek%FE7$$\"3l******\\y))GeFE$\"3S^%Q$3s[b ZFE7$$\"3'*)***\\i_QQgFE$\"3!HL`J%)[1'[FE7$$\"3@***\\7y%3TiFE$\"37a._# Qnu&\\FE7$$\"35****\\P![hY'FE$\"3'zo8S;M$f]FE7$$\"3kKLL$Qx$omFE$\"3-O$ z`**fe9&FE7$$\"3!)*****\\P+V)oFE$\"38`.K9f2L_FE7$$\"3?mm\"zpe*zqFE$\"3 l>K\"*oZb2`FE7$$\"3%)*****\\#\\'QH(FE$\"3a)3fjQ)4%Q&FE7$$\"3GKLe9S8&\\ (FE$\"3AKum`T`^aFE7$$\"3R***\\i?=bq(FE$\"3J(RP#o(\\t^&FE7$$\"3\"HLL$3s ?6zFE$\"35CC%R!y:xbFE7$$\"3a***\\7`Wl7)FE$\"3wziR5J0NcFE7$$\"3#pmmm'*R RL)FE$\"3cE240)Rjo&FE7$$\"3Qmm;a<.Y&)FE$\"3&[%>))H=MMdFE7$$\"3=LLe9tOc ()FE$\"339`[\"\\+wx&FE7$$\"3u******\\Qk\\*)FE$\"3dbDG3qh8eFE7$$\"3CLL$ 3dg6<*FE$\"3-)Rb?(of]eFE7$$\"3ImmmmxGp$*FE$\"3Q_y(Q')p)zeFE7$$\"3A++D \"oK0e*FE$\"3_gjY.0A2fFE7$$\"3A++v=5s#y*FE$\"3rrQj,_vHfFE7$$\"\"\"F)$ \"3'QQ'3%R)4]fFE-%'COLOURG6&%$RGBG$\"*++++\"!\")$\")AR!)\\F_[lF(-%&STY LEG6#%%LINEG-F$6%7.7$F(F(7$$\"3Q+++\"4444*F0$\"3$******zE46/*F07$$\"31 +++=====FE$\"3-+++8Z,z+++V " 0 "" {MPLTEXT 1 0 34 "plot(f(x)-p(x),x=0..1,color= blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CUR VESG6#7`r7$$\"\"!F)$!35+++%)ph=Q!#J7$$\"3/LL3x1h6o!#@$\"3*4M%ynV`5`!#E 7$$\"3immTN@Ki8!#?$\"32?W'\\S\"*z.\"!#D7$$\"3#***\\7.K[V?F7$\"3\"H_0(* 4%\\@:F:7$$\"3ALL$3FWYs#F7$\"3gkno*4JA)>F:7$$\"3%)***\\iSmp3%F7$\"3H]1 D!e:!QGF:7$$\"3WmmmT&)G\\aF7$\"37cB\"ou@/h$F:7$$F/F7$\"3)y9>%f/J/VF:7$ $\"3m****\\7G$R<)F7$\"3sCV3@PLC\\F:7$$\"3Emm\"z%\\DO&*F7$\"3YYabr;%\\Z &F:7$$\"3ILLL3x&)*3\"!#>$\"31WzA'R*QgfF:7$$\"3gmmTN@Ki8Fjn$\"3GqZGJ))) =v'F:7$$\"3$*****\\ilyM;Fjn$\"3$**z7v4>#HtF:7$$\"3DLLe*)4D2>Fjn$\"3V>D W#\\_+s(F:7$$\"3emmm;arz@Fjn$\"3kYXH0T`\\zF:7$$\"33$eRsL]#)H#Fjn$\"39B ne4!e[+)F:7$$\"3%**\\7yD&y;CFjn$\"3e@+`n5tN!)F:7$$\"3z;aQy,KNDFjn$\"3c WY9))H#Q/)F:7$$\"3IL$e*)4bQl#Fjn$\"3#))fKLII2.)F:7$$\"3mmT5S\\#4*GFjn$ \"3-D%>6eDq%zF:7$$\"3.++D\"y%*z7$Fjn$\"35TCbBY,'z(F:7$$\"3vm;ajW8-OFjn $\"3uUr8u-hKtF:7$$\"3[LL$e9ui2%Fjn$\"3)fHzWP2Cr'F:7$$\"3xm;H2Q\\4YFjn$ \"3%3h--D?+!fF:7$$\"33++voMrU^Fjn$\"32=T'H&f8H]F:7$$\"3QL$3-8Lfn&Fjn$ \"3'4&)p&49F]TF:7$$\"3nmmm\"z_\"4iFjn$\"3xX'p79X4I$F:7$$\"3%fmm\"zp!fu 'Fjn$\"3>![*GZ2)G]#F:7$$\"3emmmm6m#G(Fjn$\"3/Ilh3v4!y\"F:7$$\"3Anm;a`T >yFjn$\"3brsvf%GN9\"F:7$$\"3[mmmT&phN)Fjn$\"3y5;L=n%G)fF37$$\"36++v=dd C%*Fjn$!3!R@3q3^J;#F37$$\"3CLLe*=)H\\5!#=$!3-A3#4%\\@5qF37$$\"3gmm;ac# ))4\"Fiu$!3o9Qy#y_$)H)F37$$\"3')***\\(=JN[6Fiu$!3AbihQ_lt!*F37$$\"3Ym; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "numapprox[infnorm](f(x)-p(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.<+W!)!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 271 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 45 "This question is concer ned with the function " }{XPPEDIT 18 0 "f(x) = exp(x^2)*arctan(x);" "6 #/-%\"fG6#%\"xG*&-%$expG6#*$F'\"\"#\"\"\"-%'arctanG6#F'F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 46 "(a) Find a degree 5 polynomial a pproximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " f or " }{XPPEDIT 18 0 "f(x) = exp(x^2)*arctan(x);" "6#/-%\"fG6#%\"xG*&-% $expG6#*$F'\"\"#\"\"\"-%'arctanG6#F'F." }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 34 ", which ag rees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 6 " at 6 " }{TEXT 299 14 "equally spaced" }{TEXT -1 11 " valu es of " }{TEXT 288 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) Estimate the " }{TEXT 296 22 "maximum absolute error" } {TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"p G6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0, 1]; " "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 "(c) Find the value of " }{XPPEDIT 18 0 "Int(p(x),x=0..1)" "6#-%$IntG6$-% \"pG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 29 " correct to about 10 digits ." }}{PARA 0 "" 0 "" {TEXT -1 34 "(d) Use numerical integration via " }{TEXT 261 9 "evalf/Int" }{TEXT -1 13 " to evaluate " }{XPPEDIT 18 0 " Int(f(x),x = 0 .. 1)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"\"" } {TEXT -1 79 " correct to about 10 digits. Hence estimate the absolute \+ error in the value of " }{XPPEDIT 18 0 "Int(p(x),x = 0 .. 1)" "6#-%$In tG6$-%\"pG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 46 " computed in part (c) \+ as an approximation for " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 1)" "6#-%$ IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 42 "T he degree 5 polynomial approximation for " }{XPPEDIT 18 0 "f(x) = exp( x^2)*arctan(x);" "6#/-%\"fG6#%\"xG*&-%$expG6#*$F'\"\"#\"\"\"-%'arctanG 6#F'F." }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7 $\"\"!\"\"\"" }{TEXT -1 92 ", based on equally spaced nodes between 0 \+ and 1 inclusive can be found using the procedure " }{TEXT 0 9 "interp oly" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> exp(x^2)*arctan(x);\ninterpoly(f( x),x=0..1,5,'pts'):\np := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&-%$expG6#*$)9$\"\"#\" \"\"F4-%'arctanG6#F2F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,,9$$\"+XzG?5!\"**&$\"+cPS/A!#5\"\" \")F-\"\"#F5!\"\"*&$\"+Tyu%\\\"F0F5)F-\"\"$F5F5*&$\"+))>cC8F0F5)F-\"\" %F5F8*&$\"+M,!\\;\"F0F5)F-\"\"&F5F5F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "(b) We can see from the followi ng error curve that the maximum absolute error in using " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval \+ " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 10 " is about " } {XPPEDIT 18 0 "10^(-3)" "6#)\"#5,$\"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pl ot(f(x)-p(x),x=0..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7^q7$$\"\"!F)F(7$$\"3WmmmT&)G\\a!#? $!3_zHktxTT5!#@7$$\"3ILLL3x&)*3\"!#>$!3[#y(yLXzf>F07$$\"3$*****\\ilyM; F4$!3M7XpL`viFF07$$\"3emmm;arz@F4$!3-MKg=TldMF07$$\"3.++D\"y%*z7$F4$!3 i(37Xf)=IWF07$$\"3[LL$e9ui2%F4$!3*oitj:#GK^F07$$\"33++voMrU^F4$!3I*\\4 #eB#*RcF07$$\"3nmmm\"z_\"4iF4$!3ymql\"G<6*eF07$$\"3ImmT&))HvZ'F4$!3BkZ \"=&zx=fF07$$\"3%fmm\"zp!fu'F4$!3zOI?t=PLfF07$$\"3clm\"H2%G9qF4$!3&*)) o2>O\\NfF07$$\"3emmmm6m#G(F4$!3,9iW:YsDfF07$$\"3Anm;a`T>yF4$!3-i$Q<7vF (eF07$$\"3[mmmT&phN)F4$!3-o&3S)o\"*ydF07$$\"36++v=ddC%*F4$!3%)QKW(>Pn[ &F07$$\"3CLLe*=)H\\5!#=$!3i=Po6!R'z]F07$$\"3gmm\"z/3uC\"Fep$!3=Y\")GR# \\I6%F07$$\"3%)***\\7LRDX\"Fep$!3SYU**ej\"f&HF07$$\"3]mm\"zR'ok;Fep$!3 \\Hi;2&H@t\"F07$$\"3w***\\i5`h(=Fep$!373(yKYJ*[f!#A7$$\"3WLLL3En$4#Fep $\"3Iainl,ceSF\\r7$$\"3qmm;/RE&G#Fep$\"3BlQ+K'>=5\"F07$$\"3\")*****\\K ]4]#Fep$\"3Kc&G8y2>l\"F07$$\"3()******\\jB4EFep$\"3Y%[,A_iB$=F07$$\"3$ ******\\PAvr#Fep$\"3uX!R6mg)\\>F07$$\"3!)*******>*ppFFep$\"3o/%=9B(f%) >F07$$\"3n*****\\-w=#GFep$\"3PEN!o@Cb+#F07$$\"3c******\\G0uGFep$\"3:]g \">H7I,#F07$$\"3)******\\nHi#HFep$\"3g&\\#z>uZ2?F07$$\"3-L$eky#*4-$Fep $\"3Wkc?Zrjl>F07$$\"3jmm\"z*ev:JFep$\"3UTtP(GTa)=F07$$\"3>+]7.%Q%GKFep $\"3*30c%H+$\\u\"F07$$\"3?LLL347TLFep$\"3M\"*eEHcQh:F07$$\"3,LLLLY.KNF ep$\"3E5%)p*zF:<\"F07$$\"3w***\\7o7Tv$Fep$\"3y-sq0syTjF\\r7$$\"3'GLLLQ *o]RFep$\"3_L\"=&3z!)p7F\\r7$$\"3A++D\"=lj;%Fep$!3wv/Ly?OaTF\\r7$$\"31 ++vV&RW,B$=$\\\"F07 $$\"3%***\\(=7O*))[Fep$!3?I&Rtnzlc\"F07$$\"3cmm;/T1&*\\Fep$!3;OCo_&R** f\"F07$$\"3@m;/^7I0^Fep$!3_[#**3_q3f\"F07$$\"3&em;zRQb@&Fep$!3))=fa(=7 n`\"F07$$\"3ALLLe,]6`Fep$!3%)y]z*G%4`9F07$$\"3\\***\\(=>Y2aFep$!3w'GCg $HKO8F07$$\"39mm;zXu9cFep$!3Shx\\6w8y(*F\\r7$$\"3l******\\y))GeFep$!3U y,c-*y)\\ZF\\r7$$\"3'*)***\\i_QQgFep$\"37i9r&*R8H6F\\r7$$\"3@***\\7y%3 TiFep$\"3GQ6=V(\\$*H(F\\r7$$\"35****\\P![hY'Fep$\"311t>i[e59F07$$\"3kK LL$Qx$omFep$\"3W_&zMwAx&>F07$$\"3!)*****\\P+V)oFep$\"3WL()*yM[^S#F07$$ \"3WK$ek`H@)pFep$\"3wUby3*R?a#F07$$\"3?mm\"zpe*zqFep$\"3-%G&*f:O%GEF07 $$\"31*\\(oa_VLrFep$\"37-E&zn$*>l#F07$$\"3-L$e9\"=\"p=(Fep$\"3S&=JkOCv l#F07$$\"3)p;H#o$)QSsFep$\"32%e&Qj%fSk#F07$$\"3%)*****\\#\\'QH(Fep$\"3 :L9r>gn5EF07$$\"3im;zp%*\\%R(Fep$\"3o%G)o#fM3\\#F07$$\"3GKLe9S8&\\(Fep $\"3GHZW7%>AH#F07$$\"3R***\\i?=bq(Fep$\"3YX#4x*[.0;F07$$\"3:m;H2FO3yFe p$\"30')4%\\_?*H6F07$$\"3\"HLL$3s?6zFep$\"3p?C$>Y@hi&F\\r7$$\"3Am;zpe( )=!)Fep$!3/78Hg4\"3G\"F\\r7$$\"3a***\\7`Wl7)Fep$!3F#oA0x?28*F\\r7$$\"3 nK$e*[ACI#)Fep$!3W%)*Q9?!*4v\"F07$$\"3#pmmm'*RRL)Fep$!3OvblPv6fEF07$$ \"3lmmTge)*R%)Fep$!3u6/gh*Rkk$F07$$\"3Qmm;a<.Y&)Fep$!3?(3z!)puZn%F07$$ \"3M+]PM&*>^')Fep$!3L^F.@b]7dF07$$\"3=LLe9tOc()Fep$!3i=X)fe;2u'F07$$\" 3Ym;H#e0I&))Fep$!39SH^*R5#\\wF07$$\"3u******\\Qk\\*)Fep$!3Ivu!HS(Q#\\) F07$$\"31nmT5ASg!*Fep$!3H(zdyK5RL*F07$$\"3CLL$3dg6<*Fep$!3#HP;aq]X)**F 07$$\"3^m;zpBp?#*Fep$!3.,*\\hI%e>5F-7$$\"3y***\\(oTAq#*Fep$!3!RGdW4T[. \"F-7$$\"3Sm\"H#o+*\\H*Fep$!3PA'GL'=/S5F-7$$\"3.L$3x'fv>$*Fep$!3ldyPw% )\\V5F-7$$\"3m*\\(=n=_W$*Fep$!3['>q>58^/\"F-7$$\"3ImmmmxGp$*Fep$!3j*zi :-&yW5F-7$$\"3sK$eRA5\\Z*Fep$!3XZ.n]s3>5F-7$$\"3A++D\"oK0e*Fep$!3Ycu12 +=u%*F07$$\"3B+]il(z5j*Fep$!3h>e`/e\\S*)F07$$\"3C+++]oi\"o*Fep$!37??#= lrxE)F07$$\"3B+]PMR&F07$$\"3e\\(=nj+U')*Fep$!3!Q*\\'3#))[#[%F07$$\"35+]P40O\"*)*Fep$! 3]Tm9;7v6PF07$$\"3k]7.#Q?&=**Fep$!3Si8w\"z*f!)GF07$$\"31+voa-oX**Fep$! 3--)o!*H,m)>F07$$\"3[\\PMF,%G(**Fep$!3#3lJ1 " 0 "" {MPLTEXT 1 0 37 "numapprox[infnorm](f(x)-p(x) ,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+oYAX5!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(p(x),x=0..1);\nint1 := value(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,,%\"xG$\"+XzG?5!\"**&$ \"+cPS/A!#5\"\"\")F'\"\"#F/!\"\"*&$\"+Tyu%\\\"F*F/)F'\"\"$F/F/*&$\"+)) >cC8F*F/)F'\"\"%F/F2*&$\"+M,!\\;\"F*F/)F'\"\"&F/F//F';\"\"!F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%int1G$\"+BV)eR(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(d)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(f(x),x=0..1);\nint2 := evalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$expG6#*$)%\"xG\"\"#\"\" \"F.-%'arctanG6#F,F./F,;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% int2G$\"+Z4H%R(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 28 "The absolute error in using " }{XPPEDIT 18 0 "Int(p(x), x = 0 .. 1)" "6#-%$IntG6$-%\"pG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 16 " \+ to approximate " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 1)" "6#-%$IntG6$-% \"fG6#%\"xG/F);\"\"!\"\"\"" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "abs(int2-i nt1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(wLf\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 71 "Using an interpolating polynomial to emulate a finite C hebyshev series " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 9 "The n th " }{TEXT 260 20 "Chebyshev polyno mial" }{TEXT -1 33 " of the first kind is defined by " }{XPPEDIT 18 0 "T(n,x) = cos(n*arccos(x));" "6#/-%\"TG6$%\"nG%\"xG-%$cosG6#*&F'\"\"\" -%'arccosG6#F(F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "Cheby shev polynomials are available in Maple via the procedure " }{TEXT 0 10 "ChebyshevT" }{TEXT -1 52 ", and also in the package of orthogonal \+ polynomials " }{TEXT 0 9 "orthopoly" }{TEXT -1 4 " as " }{TEXT 0 12 "o rthopoly[T]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 46 "The first few Chebyshev polynomials are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "alias(T=ChebyshevT):\nfo r n from 0 to 7 do\n print(`T(`||n||`,x)`=expand(T(n,x)));\nend do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T(0,x)G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T(1,x)G%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %'T(2,x)G,&*$)%\"xG\"\"#\"\"\"F)F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T(3,x)G,&*$)%\"xG\"\"$\"\"\"\"\"%*&F)F*F(F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T(4,x)G,(*$)%\"xG\"\"%\"\"\"\"\")*&F+F*)F(\" \"#F*!\"\"F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T(5,x)G,(*$)%\"xG \"\"&\"\"\"\"#;*&\"#?F*)F(\"\"$F*!\"\"*&F)F*F(F*F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%'T(6,x)G,**$)%\"xG\"\"'\"\"\"\"#K*&\"#[F*)F(\"\"%F* !\"\"*&\"#=F*)F(\"\"#F*F*F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'T( 7,x)G,**$)%\"xG\"\"(\"\"\"\"#k*&\"$7\"F*)F(\"\"&F*!\"\"*&\"#cF*)F(\"\" $F*F**&F)F*F(F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "For an ar bitrary function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " defined on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"! \"\"F%" }{TEXT -1 46 ", we can define the infinite Chebyshev series " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(``^`*`*c[k]*T( k,x),k = 0 .. infinity)" "6#-%$SumG6$*()%!G%\"*G\"\"\"&%\"cG6#%\"kGF*- %\"TG6$F.%\"xGF*/F.;\"\"!%)infinityG" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "c[0]/2 + c[1]*T(1,x) + c[2]*T(2,x) + ` . . . `+ c[k]*T(k,x) + ` . . . `" "6#,.*&&%\"cG6#\"\"!\"\"\"\"\"#!\"\"F)*&&F&6#F)F)-%\"TG6$F)%\"xG F)F)*&&F&6#F*F)-F06$F*F2F)F)%(~.~.~.~GF)*&&F&6#%\"kGF)-F06$F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Calculation of the first few coefficients" }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }{XPPEDIT 18 0 "c[0] = 2/Pi;" "6#/&%\"cG6#\"\"!*&\"\"#\"\"\"%#PiG!\"\"" } {XPPEDIT 18 0 "Int(sin(Pi*x/2)/sqrt(1-x^2),x = -1 .. 1);" "6#-%$IntG6$ *&-%$sinG6#*(%#PiG\"\"\"%\"xGF,\"\"#!\"\"F,-%%sqrtG6#,&F,F,*$F-F.F/F// F-;,$F,F/F," }{TEXT -1 14 " = 0, because " }{XPPEDIT 18 0 "sin(Pi*x/2) /sqrt(1-x^2)" "6#*&-%$sinG6#*(%#PiG\"\"\"%\"xGF)\"\"#!\"\"F)-%%sqrtG6# ,&F)F)*$F*F+F,F," }{TEXT -1 20 " is an odd function." }}{PARA 0 "" 0 " " {TEXT -1 43 "Similarly, all the even coefficients are 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int( sin(Pi/2*x)/sqrt(1-x^2),x=-1..1);\nvalue(%);\n\n\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#,$*&%#PiG\"\"\"%\"xGF-#F-\"\"#F-* $-%%sqrtG6#,&F-F-*$)F.F0F-!\"\"F-F8/F.;F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "c[1]=2/Pi" "6#/&%\"cG6#\"\"\"*&\"\"#F'%#PiG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(x*sin(Pi/2*x)/sqrt(1-x^2),x=-1..1)" "6#-%$IntG6$ *(%\"xG\"\"\"-%$sinG6#*(%#PiGF(\"\"#!\"\"F'F(F(-%%sqrtG6#,&F(F(*$F'F.F /F//F';,$F(F/F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*J[1](Pi/2)" "6#*& \"\"#\"\"\"-&%\"JG6#F%6#*&%#PiGF%F$!\"\"F%" }{TEXT -1 8 ", where " } {TEXT 289 1 "J" }{TEXT -1 24 " is the Bessel function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "alias(J=BesselJ):\n2/Pi*Int(x*sin(P i/2*x)/sqrt(1-x^2),x=-1..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$IntG6$*&*&-%$sinG6#,$*&%#PiG\"\"\"%\"xGF0#F0\"\"#F0F1F0F0 *$-%%sqrtG6#,&F0F0*$)F1F3F0!\"\"F0F;/F1;F;F0F0F/F;F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%\"JG6$\"\"\",$%#PiG#F'\"\"#F+" }}}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3] = 2/Pi;" "6#/&%\"cG6#\"\"$*&\"\" #\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(T(3,x)*sin(Pi/2 *x)/sqrt(1-x^2),x = -1 .. 1);" "6#-%$IntG6$*(-%\"TG6$\"\"$%\"xG\"\"\"- %$sinG6#*(%#PiGF,\"\"#!\"\"F+F,F,-%%sqrtG6#,&F,F,*$F+F2F3F3/F+;,$F,F3F ," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-2*J[3](Pi/2);" "6#,$*&\"\"#\"\" \"-&%\"JG6#\"\"$6#*&%#PiGF&F%!\"\"F&F/" }{TEXT -1 3 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "alias(J=BesselJ):alias(T=ChebyshevT):\n2/P i*Int(T(3,x)*sin(Pi/2*x)/sqrt(1-x^2),x=-1..1);\nvalue(expand(%));\nnor mal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$IntG6$*&*&-%$sinG6#, $*&%#PiG\"\"\"%\"xGF0#F0\"\"#F0-%\"TG6$\"\"$F1F0F0*$-%%sqrtG6#,&F0F0*$ )F1F3F0!\"\"F0F?/F1;F?F0F0F/F?F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&,&*&,&*&)%#PiG\"\"#\"\"\"-%\"JG6$\"\"!,$F*#F,F+F,#F,\"#K**#F,\"#;F,F *F,,&*$F)F,#F,\"\"%F+!\"\"F,-F.6$F,F1F,F,F,*$F)F,F<\"$c#*(\"\"$F,F*F,F =F,F " 0 "" {MPLTEXT 1 0 42 "-2*J(3,Pi/2)=normal(expand(-2*J(3,Pi/2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$-%\"JG6$\"\"$,$%#PiG#\"\"\"\"\"#!\"#,$*&,(*&F*F,-F&6 $\"\"!F)F,\"\")*&-F&6$F,F)F,)F*F-F,F,*&\"#KF,F8F,!\"\"F,*$F:F,F=F-" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The following function " }{TEXT 0 1 "d" }{TEXT -1 52 " gives the c oefficients in the Chebyshev series for " }{XPPEDIT 18 0 "f(x)=sin(Pi/ 2*x)" "6#/-%\"fG6#%\"xG-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F'F-" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "d[k]" "6#&%\"dG6#%\"kG" }{TEXT -1 3 " \+ = " }{XPPEDIT 18 0 "c[2*k+1]=2*(-1)^k*BesselJ(2*k+1,Pi/2)" "6#/&%\"cG6 #,&*&\"\"#\"\"\"%\"kGF*F*F*F**(F)F*),$F*!\"\"F+F*-%(BesselJG6$,&*&F)F* F+F*F*F*F**&%#PiGF*F)F/F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "alias(J=BesselJ):\nd \+ := k -> 2*(-1)^k*BesselJ(2*k+1,Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dGf*6#%\"kG6\"6$%)operatorG%&arrowGF(,$*&)!\"\"9$\"\"\"-%\"JG6$ ,&F0\"\"#F1F1,$%#PiG#F1F6F1F6F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The first few coefficients are . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "seq(d(k),k=0..6);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) ,$-%\"JG6$\"\"\",$%#PiG#F'\"\"#F+,$-F%6$\"\"$F(!\"#,$-F%6$\"\"&F(F+,$- F%6$\"\"(F(F0,$-F%6$\"\"*F(F+,$-F%6$\"#6F(F0,$-F%6$\"#8F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+y\"[O8\"!\"*$!+mxr!Q\"!#5$\"+]Ur!\\%!#7$ !+!fF,x'!#9$\"+Q`H\"*e!#;$!+9%f!QL!#=$\"+TGqH8!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We can construct finite C hebyshev series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " using the following function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "alias(T=ChebyshevT):\nCS := (x,n)->Sum(d(k)*T(2*k+1,x),k=0..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CSGf*6$%\"xG%\"nG6\"6$%)operatorG%&arrowGF)-%$SumG6$*&-%\"dG6 #%\"kG\"\"\"-%\"TG6$,&F4\"\"#F5F59$F5/F4;\"\"!9%F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Taking " }{XPPEDIT 18 0 "n=4" "6#/%\"nG\"\"%" }{TEXT -1 47 " gives a degree 9 polynomial \+ approximation for " }{XPPEDIT 18 0 "f(x)=sin(Pi/2*x)" "6#/-%\"fG6#%\"x G-%$sinG6#*(%#PiG\"\"\"\"\"#!\"\"F'F-" }{TEXT -1 17 " on the interval \+ " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(evalf(CS(x,4),15));\np := unapply(expand(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%\"TG6$\"\"\"%\"xG$\"+y\"[O8\"!\"**&$\"+mxr! Q\"!#5F'-F%6$\"\"$F(F'!\"\"*&$\"+ZUr!\\%!#7F'-F%6$\"\"&F(F'F'*&$\"+%eF ,x'!#9F'-F%6$\"\"(F(F'F3*&$\"+I`H\"*e!#;F'-F%6$\"\"*F(F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,,9 $$\"+!H'zq:!\"**&$\"+#eL'fk!#5\"\"\")F-\"\"$F5!\"\"*&$\"+![Z)oz!#6F5)F -\"\"&F5F5*&$\"+l-AsY!#7F5)F-\"\"(F5F8*&$\"+/;<3:!#8F5)F-\"\"*F5F5F(F( F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Th e first omitted term of the Chebyshev series is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "d(5)*T( 11,x);\nevalf(evalf(expand(%),30)):\nt := unapply(%,x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$*&-%\"JG6$\"#6,$%#PiG#\"\"\"\"\"#F,-%\"TG6$F(% \"xGF,!\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"tGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(,.9$$\"+]`'=n$!#<*&$\"+NG<=M!#:\"\"\")F-\"#6F5!\"\"*& $\"+'Hv**R*F4F5)F-\"\"*F5F5*&$\"+'Hv**R*F4F5)F-\"\"(F5F8*&$\"+#>*[7TF4 F5)F-\"\"&F5F5*&$\"++2tVt!#;F5)F-\"\"$F5F8F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 277 37 "____________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 1 " " } {TEXT 275 1 "*" }{TEXT 262 18 " The graph of the " }{TEXT 260 18 "firs t omitted term" }{TEXT 262 41 " is virtually indistinguishable from th e " }{TEXT 260 11 "total error" }{TEXT -1 2 ". 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7$F-7$Fgin$!3]P^!\\!=s/LF-7$$\"37Le9Tq3n$)F1$!3w3I>x7)RK$F-7$F\\jn$!3V O*)p(y@^L$F-7$$\"3RmT5S)=mT)F1$!3)[ZG`2WzL$F-7$Fajn$!3;ig,'HiAL$F-7$Ff jn$!3G!pul0F-7$Fd\\o$!3KoKV@ 8]3;F-7$Fi\\o$!3`#fIvt](>7F-7$F^]o$!3_YkG(RJz+)FM7$Fc]o$!3/9L$>_]#oNFM 7$Fh]o$\"3%G]^6\\G!e5FM7$F]^o$\"3wdfGJiu%z&FM7$Fb^o$\"3@Q4Y+#*)[.\"F-7 $Fg^o$\"3Sjd**3W+$[\"F-7$F\\_o$\"3T$e;2CmK\">F-7$Fa_o$\"3`&zFy!oj8BF-7 $Ff_o$\"3i'y)yhvToHF-7$F[`o$\"3`)>ER)Gr;LF-7$F``o$\"3F&**zp'=#zL$F-7$F e`o$\"3x@s'>Q$*=K$F-7$Fj`o$\"3+k!4xIY[E$F-7$F_ao$\"3-gK$30%ziJF-7$Fdao $\"3D0F9R_l1GF-7$Fiao$\"3K[2-\"*f= " 0 "" {MPLTEXT 1 0 50 "n := 11:\nseq(evalf(cos((2*k-1)*Pi/ (2*n))),k=1..n);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$\"+>W@)*)*!#5$\"+ a*>j4*F%$\"+Wd\\dvF%$\"+v\"3kS&F%$\"+obKj4*F%$!+>W@)*)*F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Constructing an interpolating polyn omial with these zeros as nodes will give a polynomial approximation f or " }{XPPEDIT 18 0 "f(x)=sin(Pi/2*x)" "6#/-%\"fG6#%\"xG-%$sinG6#*(%#P iG\"\"\"\"\"#!\"\"F'F-" }{TEXT -1 127 " which is not very different fr om the Chebyshev series, and therefore also not very different from a \+ minimax approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 72 "Such an interpolating po lynomial can be constructed using the procedure " }{TEXT 0 9 "interpol y" }{TEXT -1 37 " with the option \"spacing=Chebyshev\"." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "f := x -> sin(Pi/2*x);\ninterpoly(f(x),x=-1..1,10,'pts',spacing=Chebyshev) :\nq := unapply(%,x);\nplot([pts,q(x)],x=-1..1,color=[navy,coral],styl e=[point,line],symbol=circle);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$sinG6#,$*&%#PiG\"\"\"9$F2#F2\" \"#F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,,9$$\"+!H'zq:!\"**&$\"+nNjfk!#5\"\"\")F-\"\"$F5!\"\"* &$\"+0p%)oz!#6F5)F-\"\"&F5F5*&$\"+.E@sY!#7F5)F-\"\"(F5F8*&$\"+kv83:!#8 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" }}{PARA 0 "" 0 " " {TEXT -1 14 "The procedure " }{TEXT 0 9 "interpoly" }{TEXT -1 18 " w ith the option \"" }{TEXT 296 17 "spacing=Chebyshev" }{TEXT -1 23 "\" \+ can be used for this." }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot an (absolut e) error curve and estimate the maximum absolute error in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interv al " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "(b) Construct a polynomial approximation " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 17 " of degree 9 for \+ " }{XPPEDIT 18 0 "f(x) = sin(x^2)/(x^2);" "6#/-%\"fG6#%\"xG*&-%$sinG6# *$F'\"\"#\"\"\"*$F'F-!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 43 ", as a truncated (finite) Chebyshev series." }}{PARA 0 "" 0 "" {TEXT -1 14 "The proced ure " }{TEXT 0 10 "chebseries" }{TEXT -1 22 " can be used for this." } }{PARA 0 "" 0 "" {TEXT -1 37 "Compare the absolute error curve for " } {XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 25 " as an approximat ion for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 55 " with \+ error curve drawn in part(a). What do you notice?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "f := x -> sin(x^2)/x^2;\ninterpoly(f(x),x=0..1,9,spa cing=Chebyshev):\np := unapply(%,x);\nplot1 := plot(f(x)-p(x),x=0..1,c olor=blue):\nplot1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG 6\"6$%)operatorG%&arrowGF(*&-%$sinG6#*$)9$\"\"#\"\"\"F4F2!\"#F(F(F(" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,6$\"+0+++5!\"*\"\"\"*&$\"+_Ah]%*!#;F09$F0!\"\"*&$\"+mq*\\4$!#9F0) F5\"\"#F0F0*&$\"+g()R/R!#8F0)F5\"\"$F0F6*&$\"+GM(=k\"!#5F0)F5\"\"%F0F6 *&$\"+hTaI*)!#7F0)F5\"\"&F0F6*&$\"+nu.?>!#6F0)F5\"\"'F0F0*&$\"+5oFmCFR F0)F5\"\"(F0F6*&$\"+?&)=4EFRF0)F5\"\")F0F0*&$\"+'z$=!o&FLF0)F5\"\"*F0F 6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CU RVESG6#7jw7$$\"3`*****\\n5;\"o!#@$!3)GZ/18)\\qV!#E7$$\"3#******\\8ABO 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L[/s&F-7$$\"3a******HESw(*Fcv$\"3o!QN*\\EL&p&F-7$Fjfn$\"3sy6/_=QfcF-7$ F_gn$\"3g!REtpn " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 278 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 "(a) Construct a polynom ial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " of degree 10 for " }{XPPEDIT 18 0 "f(x) = exp(x^2);" "6#/-%\"fG6# %\"xG-%$expG6#*$F'\"\"#" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 90 " as an interpolating polynomial with nodes given by the zeros of the Chebyshev polynomial \+ " }{XPPEDIT 18 0 "T(12,x)" "6#-%\"TG6$\"#7%\"xG" }{TEXT -1 27 ". This \+ is possible because " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 22 " is an even function. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The proc edure " }{TEXT 0 9 "interpoly" }{TEXT -1 18 " with the option \"" } {TEXT 296 17 "spacing=Chebyshev" }{TEXT -1 32 "\" can be used for this . Because " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 149 " is an even function, setting the degree argument to 11 rather than 10 wi ll still give a degree 10 polynomial. Check that the nodes are the zer os of " }{XPPEDIT 18 0 "T(12,x)" "6#-%\"TG6$\"#7%\"xG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "Plot an (absolute) error curve and es timate the maximum absolute error in using " }{XPPEDIT 18 0 "p(x)" "6# -%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1, 1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "(b) Construct a polynomial approximation " }{XPPEDIT 18 0 "q(x)" " 6#-%\"qG6#%\"xG" }{TEXT -1 18 " of degree 10 for " }{XPPEDIT 18 0 "f(x ) = exp(x^2);" "6#/-%\"fG6#%\"xG-%$expG6#*$F'\"\"#" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 43 ", as a truncated (finite) Chebyshev series." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 10 "chebseries" }{TEXT -1 22 " c an be used for this." }}{PARA 0 "" 0 "" {TEXT -1 37 "Compare the absol ute error curve for " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" }{TEXT -1 25 " as an approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\" xG" }{TEXT -1 55 " with error curve drawn in part(a). What do you noti ce?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 8 "So lution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "f := x -> exp(x^2);\ninterpoly(f(x ),x=-1..1,11,'pts',spacing=chebyshev):\np := unapply(%,x);\nplot1 := p lot(f(x)-p(x),x=-1..1,color=blue):\nplot1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#*$)9$ \"\"#\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,.$\"+*\\*)*****!#5\"\"\"*&$\"+ka2+5!\"*F0)9$ \"\"#F0F0*&$\"+(R\"G\"*\\F/F0)F6\"\"%F0F0*&$\"+`07.9@\"!#B7$$!3]LekynP')**!#=$\"31rvEtYD5)*!#C7$ $!3-n;HdNvs**F1$\"3w)Q27O*ecwF47$$!3_+v$fLI\"f**F1$\"3[lU[HjNZcF47$$!3 /MLe9r]X**F1$\"3-he&H*e(ox$F47$$!3an\"HK*Q)=$**F1$\"36E4u6OiR?F47$$!3/ ,](=ng#=**F1$\"3X'[$*R)3F-V!#D7$$!3bM3_]uj/**F1$!3!*3+@,r`c5F47$$!3%pm m\"HU,\"*)*F1$!3k?-=rluDCF47$$!3M*\\7y+\"Rx)*F1$!3CoA)H3OBo$F47$$!3'GL ekynP')*F1$!39u74y05J[F47$$!3OmT5lX9])*F1$!39?W5#3)pweF47$$!3()***\\PM @l$)*F1$!3G%o+yg_O#oF47$$!3)omT5!\\F4)*F1$!3$*z*[$Hc2R%)F47$$!3!RLL$e% G?y*F1$!3h=t/L!))4r*F47$$!3#om;HdNvs*F1$!3z@oSq,!\\8\"F-7$$!3u****\\(o UIn*F1$!3;p\"[v,zo>\"F-7$$!3xLL3-)\\&='*F1$!3K(R'z&Q'Qx6F-7$$!3ommm;p0 k&*F1$!3#[(eA,WD%4\"F-7$$!3D*\\P%[Hk;&*F1$!3UM\"o&z.$H#)*F47$$!3#HL3-) *G#p%*F1$!3ijb\"y!=))G%)F47$$!3gm\"z>,:=U*F1$!35y'4SAE)QoF47$$!3E++vV5 Su$*F1$!3JCdz]-7@^F47$$!3&R$3_vq)pK*F1$!3p05ZQd!\\L$F47$$!3_m;H2Jdz#*F 1$!34Qt.3E\"4`\"F47$$!33*\\i!R\"f@B*F1$\"3;0NZ'y&*yZ#FN7$$!3wKL$3B**e)3l>F47$$!3!))\\(o/KUJ\"*F1$\"3_QCA0X9'y$F47$$!3'fmT&Q75y !*F1$\"3i\"z(*\\5M\"eaF47$$!36LeRs#zZ-*F1$\"3WBDm(ps_&pF47$$!39***\\iI d9(*)F1$\"3#)y2zo%4*e#)F47$$!3YL$eRP8['))F1$\"33^)Hc0gT-\"F-7$$!3mmmmT %p\"e()F1$\"3VRYBK&Qo8\"F-7$$!3En;HK:\"F-7$$!3umm\" H-%\\/()F1$\"3M]Xx?6%=;\"F-7$$!3Am;a8jlx')F1$\"3MX@Eo#*pm6F-7$$!3qlm;/ '=3l)F1$\"3#y?\"3&**Gm;\"F-7$$!3wlmT&=Vrf)F1$\"3u'o(*>\"oG_6F-7$$!3&em mmwnMa)F1$\"3bS$zm#o4?6F-7$$!3+mm;Hp6O%)F1$\"3+n&*>$yQ!35F-7$$!3:mmm\" 4m(G$)F1$\"3C#=hN#[\\J%)F47$$!3iL$eRZD>A)F1$\"3sG@+/Z@+kF47$$!3)****\\ i&[3:\")F1$\"3)4/n\"*[LQ6%F47$$!3gKeRZXmh!)F1$\"3*))f;[l8a\"HF47$$!3Mm ;aQUC3!)F1$\"38S8cpff+@#)F47$$!3tKL$3i_+I(F1$!3vK.w(=`O/\"F-7$$!3u****\\P8#\\4(F1 $!3i()fs`J8D6F-7$$!3#RLL3FuF)oF1$!3'G%e\"Q\"y)f1\"F-7$$!3+nm;/siqmF1$! 3`/)[,oB1\"))F47$$!3#)******\\Q*[c'F1$!3-`6L0A'R^(F47$$!3uLL$e\\g\"fkF 1$!3.`!)Qkl\\FgF47$$!3mnmmTrU`jF1$!3)=Q![tZ,*R%F47$$!3[++](y$pZiF1$!3Y >PuKZ5xEF47$$!3YM$ek.M*QhF1$!37HX*y\\E6f)FN7$$!3MnmT&Gu,.'F1$\"31XOYt \"Q?c*FN7$$!3?+]PMXT@fF1$\"3)G7)4fV7@FF47$$!33LLL$yaE\"eF1$\"39(=c6#=x \"R%F47$$!3YmmTN\"for&F1$\"3=F5C\"*G%Qv&F47$$!3%)****\\([j5i&F1$\"3'z \"z#=HY())pF47$$!3BLLeRyEDbF1$\"3WNRwjeOw!)F47$$!3hmmm\">s%HaF1$\"3'oO wAmC,+*F47$$!3]***\\7)*G;K&F1$\"3-u]S*Q#*z#)*F47$$!3]LL$3x&y8_F1$\"3\\ .usub*=/\"F-7$$!31,]ilT')f^F1$\"3A5Q#Gk9B1\"F-7$$!3]nmTgD%f5&F1$\"3Wkc *e)zew5F-7$$!3%RL3_&4-_]F1$\"3!G16N.9Z3\"F-7$$!3Q+++]$*4)*\\F1$\"3-,&* G]Ys'3\"F-7$$!3E++++t_\"y%F1$\"3t2yEDr&\\.\"F-7$$!39+++]_&\\c%F1$\"3M? 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&=**F1$\"3SX2uQGX(f%FN7$$\"3!fPf$=.5K**F1$\"3'QGn!f*3i1#F47$$\"31+voa- oX**F1$\"3K%p(y$z+)*z$F47$$\"3ACc,\">g#f**F1$\"3%f\"R(pBWem&F47$$\"3[ \\PMF,%G(**F1$\"3/x+hVH\")pwF47$$\"3uu=nj+U')**F1$\"3V1G>jQL<)*F47$$\" \"\"F*F+-%'COLOURG6&%$RGBG$F*F*Fhdo$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6 \"Q!F`eo-%%VIEWG6$;F(Fbdo%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "The maximum absolute error in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)=exp(x^2)" "6#/-%\"fG6#%\"xG-%$expG6#*$F'\"\"#" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\" !\"\"F%" }{TEXT -1 10 " is about " }{XPPEDIT 18 0 "10^(-6)" "6#)\"#5,$ \"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "numapprox[infnorm](f(x)-p(x),x=0..1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+X!>9@\"!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 293 1 "x" }{TEXT -1 35 " coordinates of the nodes are . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "map(u->u[1], pts);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7.$!+9'[W\"**!#5$!+D`zQ#*F&$! +.M`LzF&$!+!H9w3'F&$!+CV$o#QF&$!+A>E08F&$\"+A>E08F&$\"+CV$o#QF&$\"+!H9 w3'F&$\"+.M`LzF&$\"+D`zQ#*F&$\"+9'[W\"**F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "These are the zeros of the Cheb yshev polynomial " }{XPPEDIT 18 0 "T(12,x)" "6#-%\"TG6$\"#7%\"xG" } {TEXT -1 30 " calculated directly as . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "fsolve(orthopoly[T ](12,x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6.$!+9'[W\"**!#5$!+D`zQ#*F% $!+.M`LzF%$!+!H9w3'F%$!+CV$o#QF%$!+A>E08F%$\"+A>E08F%$\"+CV$o#QF%$\"+! H9w3'F%$\"+.M`LzF%$\"+D`zQ#*F%$\"+9'[W\"**F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(b)" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 134 "f := x -> exp(x^2);\nchebseries(f(x),x=-1..1, 10,output=poly):\nq := unapply(%,x);\nplot2 := plot(f(x)-q(x),x=-1..1, color=magenta):\nplot2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(-%$expG6#*$)9$\"\"#\"\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,.*$)9$\"#5\"\"\"$\"+o-J)Q\"!#6*&$\"+qd@&[$F4F1)F/\"\")F1F1*&$\"+)4 oNq\"!#5F1)F/\"\"'F1F1*&$\"+4C7\"*\\F=F1)F/\"\"%F1F1*&$\"+Pu2+5!\"*F1) F/\"\"#F1F1$\"+4\"*)*****F=F1F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7iy7$$!\"\"\"\"!$\"3c:z$\\/z%p6!#B7 $$!3]LekynP')**!#=$\"3kG#*3*\\qbW*!#C7$$!3-n;HdNvs**F1$\"3AdPJ>))=WtF4 7$$!3_+v$fLI\"f**F1$\"3!plDG^q[Q&F47$$!3/MLe9r]X**F1$\"31oLRl!)*>c$F47 $$!3an\"HK*Q)=$**F1$\"3#[s'\\*33,(=F47$$!3/,](=ng#=**F1$\"3wmAKY;))QI! #D7$$!3bM3_]uj/**F1$!3HbH7Hi#=9\"F47$$!3%pmm\"HU,\"*)*F1$!31)\\z'*)[0s CF47$$!3M*\\7y+\"Rx)*F1$!3yGr<'yv;p$F47$$!3'GLekynP')*F1$!3d)Q=V6Ga![F 47$$!3OmT5lX9])*F1$!3=u`Sn^\"z\"eF47$$!3()***\\PM@l$)*F1$!3Hb?8NsgLnF4 7$$!3)omT5!\\F4)*F1$!3'R%G,:=U*F1$!3K2TIx>OVkF47$$!3E++vV5Su$*F1$!3?#=v(*f\")Ru%F47$$!3&R$3 _vq)pK*F1$!3--!Qms$G#)HF47$$!3_m;H2Jdz#*F1$!3*4iuL4hz?\"F47$$!33*\\i!R \"f@B*F1$\"3y*R=DA%po`FN7$$!3wKL$3*\\P6F-7$$!3umm\"H-%\\/()F1$\"3]!fEd]Z`9\"F-7$$!3A m;a8jlx')F1$\"3g<))owo?[6F-7$$!3qlm;/'=3l)F1$\"3/NEd[,@Y6F-7$$!3wlmT&= Vrf)F1$\"3QzI+^_CG6F-7$$!3&emmmwnMa)F1$\"3i\"GgS.eF4\"F-7$$!3+mm;Hp6O% )F1$\"3%zn*\\$)*z9v*F47$$!3:mmm\"4m(G$)F1$\"39'>xG,5;1)F47$$!3))*\\7Gy X`F)F1$\"3WW(ytEy72(F47$$!3iL$eRZD>A)F1$\"33>\"Q>c1U+'F47$$!3OnT5l^]o \")F1$\"3g@ViF(Rq([F47$$!3)****\\i&[3:\")F1$\"3EW@*\\(f-1PF47$$!3gKeRZ Xmh!)F1$\"3iScTn#oo]#F47$$!3Mm;aQUC3!)F1$\"3K,2cR%)f%H\"F47$$!33+voHR# [&zF1$\"3K\"QB!)p8sM)!#E7$$!3\"QLL3i.9!zF1$!3`*zG'pq686F47$$!3Mnmm\"p[ B!yF1$!3%=Me$4@x^KF47$$!3'3++Dw$H.xF1$!3?3+b\\(4zA&F47$$!3RMLLL)QUg(F1 $!3kIO--UV$)pF47$$!3\"ommT!R=0vF1$!3%ytCoueFZ)F47$$!3tKL$3i_+I(F1$!3Y8 3Cj'y!e5F-7$$!3u****\\P8#\\4(F1$!3$zkWk0Tz7\"F-7$$!3#RLL3FuF)oF1$!3se. b`j%p0\"F-7$$!3+nm;/siqmF1$!32RyrQy(Hh)F47$$!3#)******\\Q*[c'F1$!3KSJa ?7_psF47$$!3uLL$e\\g\"fkF1$!3\"o=OR]+C>n#['[FN7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 55 "(a) Obtain a interpolating polynomial approximation to " }{XPPEDIT 18 0 "f(x) = sin(x)/(1+arctan(x));" "6#/-%\"fG6#%\"xG*&-%$si nG6#F'\"\"\",&F,F,-%'arctanG6#F'F,!\"\"" }{TEXT -1 17 " on the interva l " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1% !G\"\"\"" }{TEXT -1 1 " " }{TEXT 299 29 "based on equally spaced nodes " }{TEXT -1 12 ", and having" }{TEXT 260 15 " minimal degree" }{TEXT -1 16 ", such that the " }{TEXT 296 22 "maximum absolute error" } {TEXT -1 14 " is less than " }{XPPEDIT 18 0 "10^(-7);" "6#)\"#5,$\"\"( !\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 73 "(b) Draw a pictu re which shows the interpolation points on the graph of " }{XPPEDIT 18 0 "y = sin(x)/(1+arctan(x));" "6#/%\"yG*&-%$sinG6#%\"xG\"\"\",&F*F* -%'arctanG6#F)F*!\"\"" }{TEXT -1 57 " together with the graph of the i nterpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 117 "(c) Verify by means of a graph that the maximum a bsolute error meets the requirement given in (a), and determine the " }{TEXT 299 22 "maximum absolute error" }{TEXT -1 60 " exhibited by you r approximating polynomial on the interval " }{XPPEDIT 18 0 "0<=x" "6# 1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 13 "(a) Obtain a " }{TEXT 299 24 "interpolati ng polynomial" }{TEXT -1 18 " approximation to " }{XPPEDIT 18 0 "f(x) \+ = ln(sin(x)+cos(x));" "6#/-%\"fG6#%\"xG-%#lnG6#,&-%$sinG6#F'\"\"\"-%$c osG6#F'F/" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1 \"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 " " } {TEXT 299 29 "based on equally spaced nodes" }{TEXT -1 12 ", and havin g" }{TEXT 260 15 " minimal degree" }{TEXT -1 16 ", such that the " } {TEXT 299 22 "maximum absolute error" }{TEXT -1 14 " is less than " } {XPPEDIT 18 0 "10^(-7);" "6#)\"#5,$\"\"(!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 73 "(b) Draw a picture which shows the interp olation points on the graph of " }{XPPEDIT 18 0 "y = ln(sin(x)+cos(x) );" "6#/%\"yG-%#lnG6#,&-%$sinG6#%\"xG\"\"\"-%$cosG6#F,F-" }{TEXT -1 57 " together with the graph of the interpolating polynomial." }} {PARA 0 "" 0 "" {TEXT -1 117 "(c) Verify by means of a graph that the \+ maximum absolute error meets the requirement given in (a), and determi ne the " }{TEXT 299 22 "maximum absolute error" }{TEXT -1 60 " exhibit ed by your approximating polynomial on the interval " }{XPPEDIT 18 0 " 0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 39 "________________________________ _______" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 47 "(a) Find a degree 10 p olynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = exp(-x^2);" "6#/-%\"fG6#%\"x G-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 34 ", which agrees \+ with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 11 " }{TEXT 296 14 "equally spaced" }{TEXT -1 11 " values of " }{TEXT 298 1 "x" }{TEXT -1 28 " between 0 and 2 inclusive." }} {PARA 0 "" 0 "" {TEXT -1 72 "(b) Draw a picture which shows the interp olation points on the graph of " }{XPPEDIT 18 0 "y = exp(-x^2);" "6#/% \"yG-%$expG6#,$*$%\"xG\"\"#!\"\"" }{TEXT -1 57 " together with the gra ph of the interpolating polynomial." }}{PARA 0 "" 0 "" {TEXT -1 66 "(c ) Estimate the absolute error, and the relative error, in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x = sqrt(2)" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "(d) Estimate the " }{TEXT 296 22 "maxim um absolute error" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval \+ " }{XPPEDIT 18 0 "[0, 2];" "6#7$\"\"!\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "(e) Repeat part (a), but change the interval to \+ " }{XPPEDIT 18 0 "[-5,5]" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 53 ", that is , find a degree 10 polynomial approximation " }{XPPEDIT 18 0 "q(x)" "6 #-%\"qG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = exp(-x^2); " "6#/-%\"fG6#%\"xG-%$expG6#,$*$F'\"\"#!\"\"" }{TEXT -1 17 " on the in terval " }{XPPEDIT 18 0 "[-5, 5];" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 34 " , which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 32 " at 11 equally spaced values of " }{TEXT 297 1 "x " }{TEXT -1 29 " between -5 and 5 inclusive." }}{PARA 0 "" 0 "" {TEXT -1 37 "(f) Plot the graphs of the polyomial " }{XPPEDIT 18 0 "q( x)" "6#-%\"qG6#%\"xG" }{TEXT -1 41 " from part(e) together with the gr aph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 42 " in the same picture. Does the polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6# %\"xG" }{TEXT -1 47 " provide a good approximation for the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-5,5]" "6#7$,$\"\"&!\"\"F%" }{TEXT -1 17 "? Why or why not?" }}{PARA 0 "" 0 "" {TEXT -1 39 "____________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 45 "(a) Find a degree10 pol ynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = ln(x+1)/(x^2-3*x+4);" "6#/-% \"fG6#%\"xG*&-%#lnG6#,&F'\"\"\"F-F-F-,(*$F'\"\"#F-*&\"\"$F-F'F-!\"\"\" \"%F-F3" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\" \"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 33 " which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 7 " at 11 " }{TEXT 299 14 "equally spaced" }{TEXT -1 11 " val ues of " }{TEXT 300 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 51 "(b) Plot the graph of the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 83 " found in part (a) togethe r with the interpolation points used in its construction." }}{PARA 0 " " 0 "" {TEXT -1 17 "(c) Estimate the " }{TEXT 299 22 "maximum absolute error" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x) " "6#-%\"pG6#%\"xG" }{TEXT -1 42 " found in (a) to approximate the fun ction " }{XPPEDIT 18 0 "f(x) = ln(x+1)/(x^2-3*x+4);" "6#/-%\"fG6#%\"xG *&-%#lnG6#,&F'\"\"\"F-F-F-,(*$F'\"\"#F-*&\"\"$F-F'F-!\"\"\"\"%F-F3" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "(d) For " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 49 " as in part (a), obtain an approximate value for " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2);" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"#" } {TEXT -1 16 " by calculating " }{XPPEDIT 18 0 "Int(p(x),x = 0 .. 2);" "6#-%$IntG6$-%\"pG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 41 " using \"standa rd\" analytical integration." }}{PARA 0 "" 0 "" {TEXT -1 35 " Compa re the value obtained for " }{XPPEDIT 18 0 "Int(p(x),x = 0 .. 2);" "6# -%$IntG6$-%\"pG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 28 " with an accurate \+ value for " }{XPPEDIT 18 0 "Int(f(x),x = 0 .. 2);" "6#-%$IntG6$-%\"fG6 #%\"xG/F);\"\"!\"\"#" }{TEXT -1 76 " ( a value which is correct to 10 \+ digits) obtained by numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 29 "(a) Approximate the function " }{XPPEDIT 18 0 "f(x) = PIE CEWISE([0, x <= 0],[exp(-1/(x^2)), 0 < x]);" "6#/-%\"fG6#%\"xG-%*PIECE WISEG6$7$\"\"!1F'F,7$-%$expG6#,$*&\"\"\"F4*$F'\"\"#!\"\"F72F,F'" } {TEXT -1 36 " using an interpolating polynomial " }{XPPEDIT 18 0 "p(x )" "6#-%\"pG6#%\"xG" }{TEXT -1 66 " of degree 10 based on equally spac ed grid points in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\" !\"\"F%" }{TEXT -1 47 ". Estimate the maximum absolute error in using \+ " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximat e " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on the inte rval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 45 "(b) What happens as the degree is increa sed? " }}{PARA 0 "" 0 "" {TEXT -1 29 "(c) Approximate the function " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 43 " in part(a) by an interpolating polynomial " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6#%\"xG" } {TEXT -1 73 " of degree 10 based on grid points which are distributed \+ in the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" } {TEXT -1 96 " according to the zeros of a Chebyshev polynomial. Estima te the maximum absolute error in using " }{XPPEDIT 18 0 "q(x)" "6#-%\" qG6#%\"xG" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "_ ______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________________ ___________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }