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{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Determining all roots in an inter val" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Can ada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 28 "load root-finding procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m" }{TEXT -1 37 " \+ contains the code for the procedure " }{TEXT 0 8 "allroots" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can \+ be read into a Maple session by a command similar to the one that foll ows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "A proced ure for determining all roots in an interval: " }{TEXT 0 8 "allroots" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 26 83 "allroots, leastroot, greatestroot, leas t_nonzero_root, greatest_nonzero_root: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Acknowledgement: The procedure \+ " }{TEXT 0 8 "allroots" }{TEXT -1 39 " is a modification of one writte n by: " }}{PARA 0 "" 0 "" {TEXT -1 104 "Robert Israel, Department of \+ Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 " }}{PARA 0 "" 0 "" {TEXT -1 27 "e-mail: israel@math.ubc.ca " }} {SECT 1 {PARA 4 "" 0 "" {TEXT 263 29 "Text file for original code " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22810 "# \+ USAGE: leastroot( expr, x = a .. b) \n# finds floating-point appro ximation to least root of the expression\n# or equation expr in the c losed finite interval [a,b]\n# greatestroot( expr, x = a .. b) \n# does the same for the greatest root.\n# allsolve(expr, x = a .. b)\n# finds all the roots.\n#\n# expr, and all of its subexpre ssions, should be real-valued and twice \n# continuously differentiab le on [a,b].\n# In particular, you must make sure that expr does not \+ go to infinity on\n# this interval, nor does it involve functions tha t do so. \n#\n# Note: This code is experimental, and is not guarante ed to work. \n# In particular, \"allsolve\" uses \"evalr\" to do inte rval arithmetic, and is\n# therefore subject to the weaknesses of \"ev alr\". For example,\n# it doesn't work with the two-variable version \+ of \"arctan\".\n#\n# It also suffers from the usual weaknesses of roun ding error in floating-point \n# computations.\n# To some extent this \+ can be helped by increasing Digits. Execution\n# can be quite slow, p articularly for functions that are complicated or\n# change direction \+ rapidly on the given interval. \n#\n# An idea for further investigatio n: use the interval arithmetic\n# package of Connell and Corless in th e share library instead of \"evalr\".\n#\n# I would appreciate your se nding comments and bug reports to\n# israel@math.ubc.ca.\n\nif kernelo pts(version)='kernelopts(version)' then \n ERROR(`This package requir es at least Release 4`)\nfi;\n\n`evalf/INTERVAL`:= proc() map(evalf,IN TERVAL(args)) end;\n# Maple forgot this\n\n\nleastroot:= proc(eq::\{eq uation,algebraic\},\n r::(name = realcons .. realcons))\nopt ions \n`Maple Advisor Database 1.01 for Maple 6`,\n`Copyright (c) 2000 by Robert B. Israel. All rights reserved.`;\n allsolve(eq, r, -1); \nend;\n\ngreatestroot:= proc(eq::\{equation,algebraic\},\n \+ r::(name = realcons .. realcons))\noptions \n`Maple Advisor Database 1 .01 for Maple 6`,\n`Copyright (c) 2000 by Robert B. Israel. All right s reserved.`;\n allsolve(eq, r, 1);\nend;\n\nallsolve := proc(eq::\{e quation,algebraic\}, r::(name = realcons .. realcons))\nlocal x, a, b, fn, fnp, fnpp, fa, fb, f, fl, i, n, ai, bi, c, res, flag;\nglobal `al lsolve/eps1`, `allsolve/eps2`, `allsolve/eps3`;\noptions \n`Maple Advi sor Database 1.01 for Maple 6`,\n`Copyright (c) 1997 by Robert B. Isra el. All rights reserved.`;\n# third argument -1 for least, 1 for grea test, 0 for all\n# -2 or 2 for least or greatest not including 0\n \+ if nargs = 2 then flag:= 0 else flag:= args[3] fi;\n if not member( flag,\{-2,-1,0,1,2\}) then \n ERROR(\"Third argument of allsolve \+ must be integer -2 to 2, not \",flag);\n fi;\n Digits:= Digits+5 ;\n x := op(1, r);\n a := evalf(op(1, op(2, r)));\n b := eval f(op(2, op(2, r)));\n if a > b then\n ERROR(`invalid interva l `, [a, b])\n fi;\n if type(eq,`=`) then f:= lhs(eq)-rhs(eq)\n \+ else f:= eq\n fi;\n if has(f,\{Heaviside,abs,signum,piecewise ,min,max\}) then\n fl:= convert(f,pwlist,x);\n if has(fl,\{H eaviside,abs,signum,piecewise,min,max\}) then\n ERROR(`could no t convert to piecewise list`)\n fi;\n n:= (nops(fl)+1)/2;\n \+ Digits:= Digits-5;\n res:= NULL;\n if flag = 0 then\n \+ for i from 1 to n do\n if i=1 then ai:= a else ai:= max(a ,evalf(fl[2*i-2],Digits+5)) fi;\n if i=n then bi:= b else bi:= \+ min(b,evalf(fl[2*i],Digits+5)) fi;\n if ai > bi then next fi;\n res:= res, op(allsolve(fl[2*i-1],x=ai .. bi,0));\n o d;\n RETURN(sort([res]));\n elif flag <= -1 then\n \+ for i from 1 to n do\n if i=1 then ai:= a else ai:= max(a,ev alf(fl[2*i-2],Digits+5)) fi;\n if i=n then bi:= b else bi:= \+ min(b,evalf(fl[2*i],Digits+5)) fi;\n if ai > bi then next fi ;\n res:= allsolve(fl[2*i-1],x=ai .. bi,flag);\n i f res <> NULL then RETURN(res) fi;\n od;\n RETURN(res)\n else\n for i from n to 1 by -1 do\n if i=1 then ai:= a else ai:= max(a,evalf(fl[2*i-2],Digits+5)) fi;\n if \+ i=n then bi:= b else bi:= min(b,evalf(fl[2*i],Digits+5)) fi;\n \+ if ai > bi then next fi;\n res:= allsolve(fl[2*i-1],x=ai \+ .. bi, flag);\n if res <> NULL then RETURN(res) fi;\n \+ od;\n RETURN(res)\n fi;\n fi;\n if has(r, infinity) then\n Digits:= Digits-5;\n fn:= proc(t)\n local a,b; \n if t = 0 then NULL \n elif type(t,specfunc(anything,IN TERVAL)) then \n a:= op([1,1],t);\n if a = 0 then a:= \+ infinity else a:= 1/a fi;\n b:= op([1,2],t);\n if b = \+ 0 then b:= -infinity else b:= 1/b fi;\n INTERVAL(b..a)\n \+ else 1/t \n fi \n end;\n if a = Float(-infinity) then \n if b = Float(infinity) then # two-sided infinite\n# on (-inf inity, -10] transform v = 1/x, v = -.1 .. 0\n# on [10, infinity) trans form v = 1/x, v = 0 .. .1\n if flag = 0 then\n re s:= map(fn,\n \{ op(allsolve(subs(x=1/v, f), v = -0.1 .. \+ 0, 0)),\n op(allsolve(subs(x=1/v, f), v = 0 .. 0.1, 0)) \});\n res:= \{ op(res), op(allsolve(f, x = -10 .. 10, 0) )\};\n elif flag <= -1 then\n res:= allsolve(subs (x=1/v, f), v = -0.1 .. 0, 2);\n if res <> NULL then RETUR N(fn(res)) fi;\n res:= allsolve(f, x = -10 .. 10, -1);\n \+ if res <> NULL then RETURN(res) fi;\n res:= all solve(subs(x=1/v, f), v = 0 .. 0.1, 2);\n if res <> NULL t hen RETURN(fn(res)) fi;\n RETURN(NULL);\n else\n \+ res:= allsolve(subs(x=1/v,f), v = 0 .. 0.1, -2);\n \+ if res <> NULL then RETURN(fn(res)) fi;\n res:= allso lve(f, x = -10 .. 10, 1);\n if res <> NULL then RETURN(res ) fi;\n res:= allsolve(subs(x=1/v,f), v = -0.1 .. 0, -2); \n if res <> NULL then RETURN(fn(res)) fi;\n R ETURN(NULL);\n fi;\n else # one-sided left\n c:= -10 - abs(b);\n if flag = 0 then\n res:= map(fn, \n \{ op(allsolve(subs(x=1/v, f), v = 1/c .. 0, 0)) \}); \n res:= \{ op(res), op(allsolve(f, x = c .. b,0))\};\n \+ elif flag <= -1 then\n res:= allsolve(subs(x=1/v, f), v = 1/c .. 0, 2);\n if res <> NULL then RETURN(fn(res)) fi ;\n RETURN(allsolve(f, x = c .. b, -1));\n else\n \+ res:= allsolve(f, x = c .. b, 1);\n if res <> NU LL then RETURN(res) fi;\n res:= allsolve(subs(x=1/v, f), v= 1/c .. 0, -2);\n if res <> NULL then RETURN(fn(res)) fi;\n \+ RETURN(NULL);\n fi\n fi\n else # one-s ided right\n c:= 10 + abs(a);\n if flag = 0 then\n \+ res:= map(fn, \n \{ op(allsolve(subs(x=1/v, f), v = 0 . . 1/c, 0 )) \});\n res:= \{ op(res), op(allsolve(f, x = a .. \+ c, 0))\};\n elif flag <= -1 then\n res:= allsolve(f, x = a .. c, -1);\n if res <> NULL then RETURN(res) fi;\n \+ res:= allsolve(subs(x=1/v, f), v = 0 .. 1/c, 2);\n if res <> NULL then RETURN(fn(res)) fi;\n RETURN(NULL)\n els e\n res:= allsolve(subs(x=1/v, f), v = 0 .. 1/c, -2);\n \+ if res <> NULL then RETURN(fn(res)) fi;\n RETURN(allsolve (f, x = a .. c, 1));\n fi;\n fi;\n RETURN(sort([op(re s)]));\n fi;\n fn := unapply(f, x);\n fa := traperror(evalf(f n(a)));\n if (fa = lasterror) or type(fa,undefined) then fa:= evalf (limit(f,x=a,right)) fi;\n if not type(fa,realcons) then\n ERR OR(Limit(f,x=a,right),`does not evaluate to a real constant`)\n fi; \n fb := traperror(evalf(fn(b)));\n if (fb = lasterror) or type( fb,undefined) then fb:= evalf(limit(f,x=b,left)) fi;\n if not type( fb, realcons) then\n ERROR(Limit(f,x=a,right),`does not evaluat e to a real constant`)\n fi;\n if fa < 0 then\n fa := -fa ;\n fb := -fb;\n fn := unapply(-f, x);\n fnp := u napply(-normal(diff(f, x)), x);\n fnpp := unapply(-normal(diff( f, x $ 2)), x)\n else\n fnp := unapply(normal(diff(f, x)), x );\n fnpp := unapply(normal(diff(f, x $ 2)), x)\n fi;\n ` allsolve/eps1`:= Float(1,5-Digits)*(abs(b)+abs(a));\n if has([fa,fb ],infinity) then `allsolve/eps2`:= Float(1,5-Digits)\n else `allsol ve/eps2` := Float(1, 5-Digits)*(abs(fa)+abs(fb)+.01)/2\n fi;\n i f b = a then `allsolve/eps3`:= 1\n else `allsolve/eps3`:= `allsolve /eps2`/(b-a)\n fi;\n res:=`allsolve/allsolve`(fn, fnp, fnpp, a, \+ b, fa, fb, flag);\n Digits:= Digits-5;\n if flag = 0 then sort([ op(evalf(\{res\}))])\n else evalf(res)\n fi;\nend;\n\n\n`allsolv e/allsolve` := proc(fn, fnp, fnpp, a, b, fa, fb, flag)\nlocal fr, fpp1 , fpp2, fp1, fp2, f1, f2, m, fm, v, w, fpa,\nm1, fm1, fbb, fppb, fpb, \+ m2, fm2, g1, q, res;\noptions \n`Maple Advisor Database 1.01 for Maple 6`,\n `Copyright (c) 1997 by Robert B. Israel. All rights reserved.` ;\n\n userinfo(4, allsolve, `Checking interval `, a .. b, `end valu es`,\n fa, fb);\n if b - a < `allsolve/eps1` then\n i f abs(fa + fb) < `allsolve/eps2` then RETURN(1/2*a + 1/2*b)\n e lse RETURN(NULL)\n fi\n fi;\n q:= fn(INTERVAL(a .. b));\n fr := traperror(evalf(evalr(q)));\n if (fr = lasterror) or hast ype(fr, undefined) then \n fr:= traperror(evalf(evalr(normal(q))) ) fi;\n if (fr = lasterror) or hastype(fr, undefined) then\n a ssume(w>=a,w<=b);\n if member(signum(0, fn(w), 0),\{-1,1\}) then \+ # cut off\n RETURN(NULL)\n else fr:= INTERVAL(-infinity .. infinity)\n fi\n fi;\n if type(fr, \{specfunc(range, INTER VAL), list(range)\}) then\n f1 := op(1, op(1,fr)); f2 := op(2, \+ op(-1,fr))\n elif type(fr, realcons) then f1 := fr; f2 := fr\n e lse ERROR(`evalr returned `, fr)\n fi;\n if has(f1,infinity) the n f1:= Float(-1,Digits+10) fi;\n if has(f2,infinity) then f2:= Floa t(1,Digits+10) fi;\n if signum(0, f1, 0) = 1 or signum(0, f2, 0) = \+ -1 then\n RETURN(NULL)\n fi;\n if abs(f1)+abs(f2) < .01*` allsolve/eps2` then \n fi;\n q:= fnp(INTERVAL(a .. b));\n fr \+ := traperror(evalf(evalr(q)));\n if (fr = lasterror) or hastype(fr, undefined) \n then fr:= traperror(evalf(evalr(normal(q)))) fi;\n \+ if (fr = lasterror) or hastype(fr,undefined) then \n assume(w> =a,w<=b);\n q:= fnp(w);\n if signum(0,q,1) = 1 then fr:= INT ERVAL(0 .. infinity)\n elif signum(0,q,-1) = -1 then fr:= INTERVA L(-infinity .. 0)\n else fr:= INTERVAL(-infinity .. infinity)\n \+ fi\n fi;\n if type(fr, \{specfunc(range, INTERVAL), list(ran ge)\}) then\n fp1 := op(1, op(1,fr)); fp2 := op(2, op(-1,fr))\n elif type(fr, realcons) then fp1 := fr; fp2 := fr\n else ERROR( `evalr returned `, fr)\n fi;\n if has(fp1,infinity) then fp1:= F loat(-1,Digits+10) fi;\n if has(fp2,infinity) then fp2:= Float(1,Di gits+10) fi;\n if fp1 >= -`allsolve/eps3` then ## nondecreasing; no te fa >= 0\n if fp2 <= `allsolve/eps3` then ## function is flat\n if fa = 0 then RETURN(INTERVAL(a..b))\n else RETURN(NUL L)\n fi\n elif fa = 0 and (flag <> -2 or a <> 0) then RETU RN(a)\n else RETURN(NULL)\n fi\n fi;\n if fp2 <= `alls olve/eps3` then ## nonincreasing with fa >= 0\n if fb <= 0 then \n RETURN(`allsolve/monotone`(fn, fnp, a, b, fa, fb, flag)) \n else RETURN(NULL)\n fi\n fi;\n## now not monotone, check for convex/concave\n q:= fnpp(INTERVAL(a .. b));\n fr := \+ traperror(evalf(evalr(q)));\n if (fr = lasterror) or hastype(fr,und efined) then\n fr:= traperror(evalf(evalr(normal(q)))) fi;\n i f (fr = lasterror) or hastype(fr,undefined) then \n assume(w > a, w < b);\n q:= fnpp(w);\n if signum(0,q,1) = 1 then f r:= INTERVAL(0 .. infinity)\n elif signum(0,q,-1) = -1 then fr:= \+ INTERVAL(-infinity .. 0)\n else fr:= INTERVAL(-infinity .. infin ity) \n fi\n fi;\n if type(fr, \{specfunc(range, INTERVAL), list(range)\}) then\n fpp1 := op(1, op(1,fr)); fpp2 := op(2 , op(-1,fr))\n elif type(fr, realcons) then fpp1 := fr; fpp2 := fr \n fi;\n if has(fpp1,infinity) then fpp1:= Float(-1,Digits+10) f i;\n if has(fpp2,infinity) then fpp2:= Float(1,Digits+10) fi;\n \+ fpa := traperror(evalf(fnp(a)));\n if (fpa = lasterror) or type(fpa ,undefined) \n then fpa:= evalf(limit(fnp(v),v=a,right)) fi;\n \+ if not type(fpa,realcons) then\n ERROR(Limit(fnp(v),v=a,right), `does not evaluate to a real constant`)\n fi;\n if fpa < 0 then \n if fpa + fpp2*(b - a) <= 0 then\n if fb <= 0 then \n RETURN(`allsolve/monotone`(fn, fnp, a, b, fa, fb, fl ag))\n else RETURN(NULL)\n fi\n fi;\n \+ m1 := a - fpa/fpp2;\n fm1 := evalf(fn(m1));\n if fm1 <= 0 then\n if flag = 0 then\n RETURN(`allsolve/m onotone`(fn, fnp, a, m1, fa, fm1, flag),\n `allsolve/al lsolve`(-fn, -fnp, -fnpp, m1, b, -fm1,\n -fb, flag))\n \+ elif flag <= -1 then\n res:= `allsolve/monotone`(f n, fnp, a, m1, fa, fm1, flag);\n if res <> NULL then RETURN (res) fi;\n RETURN(`allsolve/allsolve`(-fn, -fnp, -fnpp, m1 , b, -fm1,\n -fb, flag));\n else\n \+ res:= `allsolve/allsolve`(-fn, -fnp, -fnpp, m1, b, -fm1,\n \+ -fb, flag);\n if res <> NULL then RETURN(res) fi;\n \+ RETURN(`allsolve/monotone`(fn, fnp, a, m1, fa, fm1, flag));\n \+ fi; \n fi;\n else m1 := a; fm1 := fa\n fi; \n fpb:= traperror(evalf(fnp(b)));\n if (fpb = lasterror) or typ e(fpb,undefined) \n then fpa:= evalf(limit(fnp(x),x=b,left)) fi; \n if fb < 0 then fbb := -fb; fppb := -fpp1; fpb := -fpb\n else \+ fbb := fb; fppb := fpp2;\n fi;\n if 0 < fpb then\n if 0 < = fpb + fppb*(m1 - b) then\n if fb <= 0 then\n \+ RETURN(\n `allsolve/monotone`(fn, fnp, m1, b, fm1, f b,flag))\n elif fm1 < `allsolve/eps2` then RETURN(m1)\n \+ else RETURN(NULL)\n fi\n fi;\n m2 := \+ b - fpb/fppb;\n fm2 := evalf(fn(m2));\n if signum(0, fm2 , 0) <> signum(0, fb, 2) then\n if flag <= -1 then\n \+ g1:= `allsolve/allsolve`(fn, fnp, fnpp, m1, m2, fm1,\n \+ fm2, flag); \n if g1 <> NULL then RETURN(g1) fi;\n \+ if 0 <= fm2 then\n RETURN(`allsolve/monoto ne`(fn, fnp, m2, b, fm2, fb, flag))\n else \n \+ RETURN(`allsolve/monotone`(-fn, -fnp, m2, b, -fm2, -fb,flag))\n \+ fi;\n else \n if 0 <= fm2 then\n \+ g1 :=\n `allsolve/monotone`(fn, fnp, m2, \+ b, fm2, fb, flag)\n else g1 :=\n `allsolve/m onotone`(-fn, -fnp, m2, b, -fm2, -fb, flag)\n fi;\n \+ if flag >= 1 then\n if g1 <> NULL then RETURN(g1)\n \+ else RETURN(`allsolve/allsolve`(fn, fnp, fnpp, m1, m2, fm1 ,\n fm2, flag))\n fi;\n fi;\n \+ RETURN(\n `allsolve/allsolve`(fn, fnp, fnpp, \+ m1, m2, fm1,\n fm2, flag), g1)\n fi;\n \+ fi;\n else m2 := b; fm2 := fb\n fi;\n if -`allsolve/eps2`/(b- a)^2 < fpp1 then\n RETURN(`allsolve/convex`(fn, fnp, m1, m2, fm 1, fm2, flag))\n elif fpp2 < `allsolve/eps2`/(b-a)^2 then\n \+ if 0 < fm2 then \n if fm1 < `allsolve/eps2` then RETURN(m1)\n else RETURN(NULL)\n fi\n elif flag = 0 then \+ \n RETURN(`allsolve/reverse`([`allsolve/convex`(\n un apply(-fn(-v),v),unapply(fnp(-v),v),-m2,-m1,-fm2,-fm1,0)]));\n \+ else\n g1:= `allsolve/convex`(\n unapply(-fn(-v),v),u napply(fnp(-v),v),-m2,-m1,-fm2,-fm1,-flag);\n if g1 <> NULL t hen RETURN(-g1)\n else RETURN(NULL)\n fi;\n f i\n fi;\n if `allsolve/eps2` < fm1 then\n fpa := evalf(fn p(m1));\n if 0 <= fpa then\n m1 := m1 - (fpa + sqrt( fpa^2 - 2*fm1*fpp1))/fpp1\n else m1 := m1 - 2*fm1/(fpa - sqrt(f pa^2 - 2*fm1*fpp1))\n fi;\n fm1 := evalf(fn(m1));\n f i;\n if fm2 < 0 then fbb := -fm2; fppb := -fpp2; fpb := -evalf(fnp( m2))\n else fbb := fm2; fppb := fpp1; fpb := evalf(fnp(m2))\n fi ;\n if `allsolve/eps2` < fm2 then\n if fpb <= 0 then\n \+ m2 := m2 - (fpb - sqrt(fpb^2 - 2*fbb*fppb))/fppb\n else \+ m2 := m2 - 2*fbb/(fpb + sqrt(fpb^2 - 2*fbb*fppb))\n fi;\n \+ if m2 < m1 then RETURN(NULL) fi\n fi;\n m := 1/2*m1 + 1/2*m2; \n fm := evalf(fn(m));\n if flag = 0 then\n g1 := `allsolve /allsolve`(fn, fnp, fnpp, m1, m, fm1, fm,flag);\n if 0 <= fm then \n g1, `allsolve/allsolve`(fn, fnp, fnpp, m, m2, fm, fm2, flag) \n else\n g1,\n `allsolve/allsolve`(-fn, -fnp, -fnp p, m, m2, -fm, -fm2, flag)\n fi\n elif flag <= -1 then\n \+ g1 := `allsolve/allsolve`(fn, fnp, fnpp, m1, m, fm1, fm,flag);\n \+ if g1 <> NULL then RETURN(g1) fi;\n if 0 <= fm then\n `all solve/allsolve`(fn, fnp, fnpp, m, m2, fm, fm2, flag)\n else\n \+ `allsolve/allsolve`(-fn, -fnp, -fnpp, m, m2, -fm, -fm2, flag)\n \+ fi\n else\n if 0 <= fm then\n g1:=`allsolve/allsolve `(fn, fnp, fnpp, m, m2, fm, fm2, flag)\n else\n g1:=`allso lve/allsolve`(-fn, -fnp, -fnpp, m, m2, -fm, -fm2, flag)\n fi;\n \+ if g1 <> NULL then RETURN(g1) fi;\n `allsolve/allsolve`(fn, f np, fnpp, m1, m, fm1, fm,flag);\n fi; \nend;\n\n`allsolve/reve rse`:= proc(L) local n;\noptions\n`Maple Advisor Database 1.01 for Map le 6`,\n `Copyright (c) 1997 by Robert B. Israel. All rights reserved .`;\n seq(-L[nops(L)-n], n=0 .. nops(L)-1);\n end;\n\n`allso lve/convex` := proc(fn, fnp, a, b, fa, fb, flag)\nlocal fpa, fpb, v, m , fm, fmp, ap, bp, fap, fbp, res;\noptions `Maple Advisor Database 1. 01 for Maple 6`,\n `Copyright (c) 1997 by Robert B. Israel. All right s reserved.`;\n userinfo(5, allsolve, `Checking interval `, a .. b, `end values`,\n fa, fb);\n fpa := evalf(fnp(a));\n fpb : = evalf(fnp(b));\n if fpa > -`allsolve/eps3` then\n if `all solve/eps2` < fa then RETURN(NULL)\n elif `allsolve/eps2` < fb \+ then \n if flag <> -2 or a <> 0 then RETURN(a)\n els e RETURN(NULL)\n fi\n else RETURN(1/2*a + 1/2*b)\n \+ fi\n fi;\n if b <= a + `allsolve/eps1` then\n if abs (fa + fb) < `allsolve/eps2` then RETURN(1/2*a + 1/2*b)\n else R ETURN(NULL)\n fi\n fi;\n if fpb <= `allsolve/eps3` then\n if fb <= 0 then\n RETURN(`allsolve/monotone`(fn, fn p, a, b, fa, fb, flag))\n else RETURN(NULL)\n fi\n fi ;\n if fb < 0 then\n if fpa + fpb < 0 then m := b - 2*fpb*(b - a)/(fpb - fpa)\n else m := (b*fa - a*fb)/(fa - fb)\n \+ fi;\n fm := evalf(fn(m));\n if 0 <= fm then \n \+ ap:= m; fap:= fm; bp:= b; fbp:= fb;\n m:= (m+b)/2;\n \+ fm:= evalf(fn(m));\n if 0 <= fm then ap:= m; fap := fm \n else bp:= m; fbp:= fm;\n fi\n \+ else \n ap:= a; fap:= fa; bp:= m; fbp:= fm;\n \+ fmp:= evalf(fnp(m)); \n if 0 <= fmp then\n \+ m:= (a+m)/2;\n fm:= evalf(fn(m));\n \+ if 0 <= fm then ap:= m; fap:= fm\n else \+ bp:= m; fbp:= fm\n fi\n fi\n fi; \+ \n RETURN(`allsolve/convex`(fn, fnp, ap, bp, fap, fbp, flag))\n fi;\n ap := a - fa/fpa;\n bp := b - fb/fpb;\n if bp < ap \+ then\n RETURN(NULL)\n fi;\n fap := evalf(fn(ap));\n if fap < 0 then\n ap := .9*ap + .1*a; fap := evalf(fn(ap))\n f i;\n fbp := evalf(fn(bp));\n if fbp < 0 then\n bp := .9*b p + .1*b; fbp := evalf(fn(bp))\n fi;\n fpa := evalf(fnp(ap));\n \+ fpb := evalf(fnp(bp));\n if 0 <= fpa then\n if fap < `all solve/eps2` and (flag <> -2 or ap <> 0)\n then RETURN(ap) el se RETURN(NULL) fi\n fi;\n if fpb <= 0 then\n if fbp < `a llsolve/eps2` and (flag <> 2 or bp <> 0)\n then RETURN(bp) e lse RETURN(NULL) fi\n fi;\n m := (ap*fpb - bp*fpa)/(fpb - fpa); \n if m <= ap or bp <= m then m := 1/2*ap + 1/2*bp fi;\n fm := e valf(fn(m));\n if fm < 0 then\n if flag = 0 then\n `a llsolve/convex`(fn, fnp, ap, m, fap, fm, 0),\n `allsolve/rever se`([`allsolve/convex`(\n unapply(fn(-v),v),unapply(-fnp(-v), v),-bp,-m,fbp,fm, 0)])\n elif flag <= -1 then\n res:= `a llsolve/convex`(fn, fnp, ap, m, fap, fm, flag);\n if res <> NU LL then RETURN(res) fi;\n res:= `allsolve/convex`(\n \+ unapply(fn(-v),v),unapply(-fnp(-v),v),-bp,-m,fbp,fm, -flag);\n \+ if res <> NULL then -res\n else NULL\n fi\n els e\n res:= `allsolve/convex`(\n unapply(fn(-v),v),unap ply(-fnp(-v),v),-bp,-m,fbp,fm, -flag);\n if res <> NULL then R ETURN(-res) fi;\n `allsolve/convex`(fn, fnp, ap, m, fap, fm, f lag);\n fi \n else\n fmp := evalf(fnp(m));\n if abs(fmp) < `allsolve/eps3` and\n abs(fm) < `allsolve/eps2` t hen\n m\n elif 0 < fmp then\n `allsolve/c onvex`(fn, fnp, ap, m, fap, fm, flag)\n else\n `alls olve/convex`(fn, fnp, m, bp, fm, fbp, flag)\n fi\n fi\nend; \n\n\n\n`allsolve/monotone` := proc(fn, fnp, a, b, fa, fb, flag)\nloca l d, fpm, m1, fm1, m2, fm2, m, fm, res;\noptions `Maple Advisor Databa se 1.01 for Maple 6`,\n`Copyright (c) 1997 by Robert B. Israel. All r ights reserved.`;\n userinfo(5, allsolve, `Checking interval `, a . . b, `end values`,\n fa, fb);\n if fa = fb then RETURN(1/2*a + 1/2*b) fi;\n m1 := a;\n fm1 := fa;\n m2 := b;\n fm2 := \+ fb;\n d := m2 - m1;\n while `allsolve/eps2` < fm1 and fm2 < -`al lsolve/eps2` and\n `allsolve/eps1` < d do\n m := (fm1* m2 - fm2*m1)/(fm1 - fm2);\n if m1 < m and m < m2 then\n \+ fm := evalf(fn(m));\n if 0 < fm then m1 := m; fm1 := fm \n else m2 := m; fm2 := fm\n fi;\n fi;\n \+ fpm := evalf(fnp(m1));\n if fpm < 0 then\n m \+ := m1 - fm1/fpm;\n if m < m2 then\n fm := ev alf(fn(m));\n if 0 < fm then m1 := m; fm1 := fm\n \+ else m2 := m; fm2 := fm\n fi;\n fi \n fi;\n fpm := evalf(fnp(m2));\n if fpm < 0 then \n m := m2 - fm2/fpm;\n if m1 < m then\n \+ fm := evalf(fn(m));\n if 0 < fm then m1 := m; f m1 := fm\n else m2 := m; fm2 := fm\n fi; \n fi\n fi;\n if d < 2*m2 - 2*m1 then\n \+ m := 1/2*m1 + 1/2*m2;\n fm := evalf(fn(m));\n \+ if 0 < fm then m1 := m; fm1 := fm\n else m2 := m; fm2 : = fm\n fi;\n fi;\n d := m2 - m1\n od;\n \+ if -fm2 < fm1 then res:= m2 else res:= m1 fi;\n if abs(flag) <> 2 \+ or res <> 0 then res\n else NULL\n fi;\nend;" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 268 19 "Calling Sequence:\n " }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "allroo ts( expr, x = a .. b)" }}{PARA 0 "" 0 "" {TEXT -1 98 "leastroot( expr, x = a .. b)\ngreatestroot( expr, x = a .. b)\nleast_nonzero_root( exp r, x = a .. b)" }}{PARA 0 "" 0 "" {TEXT -1 40 "greatest_nonzero_root( \+ expr, x = a .. b)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }} {PARA 0 "" 0 "" {TEXT 23 3 " " }{TEXT -1 61 "expr - an equation \+ or expression in one variable, say x. " }}{PARA 0 "" 0 "" {TEXT -1 101 " If expr is an expression rather than an equa tion, the equation expr = 0 is used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 269 50 "a..b - the \+ interval to be searched for roots" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{TEXT 0 8 "allroots" } {TEXT -1 335 " computes numerically all the zeros in the real interval a..b of an expression expr, representing a function f(x) in one varia ble x (i.e. the solutions of f(x) = 0 are computed). If expr is an eq uation rather than an expression, then f(x) = lhs(expr)-rhs(expr) is u sed. The result is returned as a sequence, sorted in numerical order. " }}{PARA 15 "" 0 "" {TEXT -1 158 "Only one variable is allowed: the t wo sides of the equation must evaluate to real constants when any cons tant value in the interval a..b is substituted for x." }}{PARA 15 "" 0 "" {TEXT -1 217 "The expression and all subexpressions should have a t least two continuous derivatives on the interval. In particular, in finite limits at the endpoints, or indeterminate forms (such as f/g wh ere f and g both approach " }{XPPEDIT 18 0 "0 " "6#\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT -1 38 " at the endpoints) may cause trouble." }}{PARA 15 "" 0 "" {TEXT -1 126 "An e xception to the requirements of continuity and differentiability is in the case of an expression defined piecewise, using " }{TEXT 0 9 "piec ewise" }{TEXT -1 2 ", " }{TEXT 0 6 "signum" }{TEXT -1 2 ", " }{TEXT 0 9 "Heaviside" }{TEXT -1 2 ", " }{TEXT 0 3 "abs" }{TEXT -1 2 ", " } {TEXT 0 3 "min" }{TEXT -1 4 " or " }{TEXT 0 3 "max" }{TEXT -1 12 " as \+ long as " }{TEXT 0 19 "convert(...,pwlist)" }{TEXT -1 105 " can conver t it to a list of expressions on different intervals. If this can't b e done, an error occurs." }}{PARA 15 "" 0 "" {TEXT -1 118 "Infinite en dpoints are allowed, but are not likely to work unless the expression \+ has finite limits at those endpoints." }}{PARA 15 "" 0 "" {TEXT -1 292 "Since numerical techniques are used, the accuracy of the results \+ is limited. In particular, a root of f(x) that is also a root of f '( x)=D(f)(x) may be computed with poor accuracy, or even missed entirely . In such a case it may be better to use f '(x)=D(f)(x) instead of f (x). Increasing " }{TEXT 0 6 "Digits" }{TEXT -1 30 " should also impr ove accuracy." }}{PARA 15 "" 0 "" {TEXT -1 24 "In some difficult cases " }{TEXT 0 8 "allroots" }{TEXT -1 137 " may take a very long time. I n particular, this will happen if the function is complicated or chang es direction rapidly in the interval." }}{PARA 15 "" 0 "" {TEXT -1 12 "If possible " }{TEXT 0 8 "allroots" }{TEXT -1 6 " uses " }{TEXT 0 5 " evalr" }{TEXT -1 162 " to do interval arithmetic, and is therefore sub ject to the weaknesses of that procedure. In particular, it doesn't w ork with the two-variable version of arctan." }}{PARA 15 "" 0 "" {TEXT 0 5 "evalr" }{TEXT -1 288 " is not implemented for some special \+ functions such as the Bessel functions and the sine integral. In such \+ cases a method of estimation is used, which could lead to roots being \+ missed. The options \"numpoints\" and \"expansion\" can be adjusted to attempt to find any suspected missing roots." }}{PARA 15 "" 0 "" {TEXT -1 15 "The procedures " }{TEXT 0 9 "leastroot" }{TEXT -1 5 " and " }{TEXT 0 12 "greatestroot" }{TEXT -1 159 " compute the least and gr eatest solutions respectively of an equation expr in one variable x in the real interval a..b by a variation of the algorithm used by " } {TEXT 0 8 "allroots" }{TEXT -1 18 ". \nThe procedures " }{TEXT 0 18 "l east_nonzero_root" }{TEXT -1 5 " and " }{TEXT 0 21 "greatest_nonzero_r oot" }{TEXT -1 30 " do what their names suggest. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Options:" }{TEXT -1 1 "\n " }{TEXT 0 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "info=n o r info=true/false" }}{PARA 15 "" 0 "" {TEXT -1 46 "info=0: No inte rmediate output is printed." }}{PARA 15 "" 0 "" {TEXT -1 121 "info=1: \+ The various intervals being considered together with the sign of f (x) at each end of the interval are given. " }}{PARA 15 "" 0 "" {TEXT -1 108 "info=2: In addition to the previous information, the indiv idual roots are given as they are determined. " }}{PARA 15 "" 0 "" {TEXT -1 75 "info=3: Details concerning the computation of the roo ts is also given. " }}{PARA 15 "" 0 "" {TEXT -1 146 "info=4: Detai ls concerning the estimated range of values of f(x), f '(x) and f ''(x ) in the various intervals being considered are also given." }}{PARA 15 "" 0 "" {TEXT -1 74 "info=true is equivalent to info=2, and info=fa lse is equivalent to info=0." }}{PARA 0 "" 0 "" {TEXT -1 24 "The defau lt is \"info=0\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "width=d" }}{PARA 0 "" 0 "" {TEXT -1 47 "When the interval \+ a..b is -infinity..infinity, " }{TEXT 0 8 "allroots" }{TEXT -1 304 " w ill search for roots in the 'base interval' -d..d, but apply the trans formation x -> 1/x to the 'ends' -infinity..-d and d..infinity, befor e searching for roots of f(1/x) on the transformed intervals -1/d..0 a nd 0..1/d. Intervals which are infinite in only one direction are hand led in a similar way. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The default is \+ \"width=50\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "split=n/[" }{XPPEDIT 18 0 "x[0],x[1];" "6$&%\"xG6#\"\"!&F$ 6#\"\"\"" }{TEXT -1 8 ", . . . " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\" nG" }{TEXT -1 1 "]" }}{PARA 0 "" 0 "" {TEXT -1 130 "This option can be used to split up the interval x = a .. b into subintervals before ap plying the standard root-finding algorithm" }}{PARA 0 "" 0 "" {TEXT -1 158 "One can specify the actual points to be used for the subdivisi on, or, if equally spaced points are required, the number of intervals can be specified instead." }}{PARA 0 "" 0 "" {TEXT -1 61 "The default is to provide no subdivision, that is, \"split=1\"." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "use_evalr=true/false" } }{PARA 0 "" 0 "" {TEXT -1 29 "This option controls whether " }{TEXT 0 5 "evalr" }{TEXT -1 146 " is used to estimate the range of values on a n interval. With the option \"use_evalr=false\" the estimation is achi eved through sample evaluations. " }}{PARA 0 "" 0 "" {TEXT -1 120 "Thi s is useful for checking how the options \"maxpoints\", \"minpoints\" \+ and \"expansion\" affect the ability to detect roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "maxpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 129 "This option determines the maximum number of f unction evaluations to be used to estimate the range of values on an i nterval when " }{TEXT 0 5 "evalr" }{TEXT -1 163 " is not available.\nI t is the number of function evaluations on the original interval, and \+ is reduced on subintervals in proportion to the width of the subinterv al." }}{PARA 0 "" 0 "" {TEXT -1 31 "The default is \"maxpoints=20\". \+ " }}{PARA 0 "" 0 "" {TEXT -1 68 "Increasing \"maxpoints\" can help in \+ finding a suspected missing root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "minpoints=n" }}{PARA 0 "" 0 "" {TEXT -1 160 "This option determines the minimum number of function evaluations to be used to estimate the range of values on the original interval, \+ or any subinterval, when " }{TEXT 0 5 "evalr" }{TEXT -1 49 " is not av ailable. The default is \"minpoints=3\". " }}{PARA 0 "" 0 "" {TEXT -1 68 "Increasing \"minpoints\" can help in finding a suspected missing r oot." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "e xpansion=r" }}{PARA 0 "" 0 "" {TEXT -1 59 "The expansion factor for an estimated range of values when " }{TEXT 0 5 "evalr" }{TEXT -1 19 " is not available. " }}{PARA 0 "" 0 "" {TEXT -1 81 "The range of values i s expanded to provide a margin of safety in range estimates." }}{PARA 0 "" 0 "" {TEXT -1 98 "The default is \"expansion=5\". Increasing \"ex pansion\" can help in finding a suspected missing root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 156 "To ma ke the procedures active, open the subsection, place the cursor anywhe re after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT 26 50 "allroots, leastro ot, greatestroot: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16294 "# ORIGINAL CODE: Copyrigh t (c) 1997,2000 by Robert B. Israel. \n# israel@math.ubc.ca.\n# Modifi cations made by Peter Stone 2004\n# peter.stone@rmit.edu.au\n#\n# NOTE WITH ORIGINAL CODE: \n# This code is experimental, and is not guarant eed to work.\n# In particular,\"allroots\" uses \"evalr\" to do interv al arithmetic, and is\n# therefore subject to the weaknesses of \"eval r\". For example,\n# it doesn't work with the two-variable version of \+ \"arctan\".\n#\n# An idea for further investigation: use the interval \+ arithmetic\n# package of Connell and Corless in the share library inst ead of \"evalr\".\n\n\n`evalf/INTERVAL` := proc() map(evalf,INTERVAL(a rgs)) end:\n# Maple forgot this\n\nleastroot := proc(eq,r)\n if narg s<2 then\n error \"at least 2 arguments are required; the basic s yntax is: 'leastroot(f(x),x=a..b)'.\"\n end if;\n allroots(eq,r,-1 ,args[3..nargs]);\nend proc:\n\nleast_nonzero_root := proc(eq,r)\n i f nargs<2 then\n error \"at least 2 arguments are required; the b asic syntax is: 'least_nonzero_root(f(x),x=a..b)'.\"\n end if;\n a llroots(eq,r,-2,args[3..nargs]);\nend proc:\n \ngreatestroot := proc(e q,r)\n if nargs<2 then\n error \"at least 2 arguments are requi red; the basic syntax is: 'greatestroot(f(x),x=a..b)'.\"\n end if;\n allroots(eq,r,1,args[3..nargs]);\nend proc:\n\ngreatest_nonzero_roo t := proc(eq,r)\n if nargs<2 then\n error \"at least 2 argument s are required; the basic syntax is: 'greatest_nonzero_root(f(x),x=a.. b)'.\"\n end if;\n allroots(eq,r,2,args[3..nargs]);\nend proc:\n\n allroots := proc(eq,r)\n local x,a,b,fn,fnp,fnpp,fa,fb,f,fl,i,n,ai,b i,c,d,res,flag,extraDigits,eps,\n f0,inc0,pw,fl2,ee,sf,vars,prntf lg,Options,startopts,opts,optns,arg3,\n minpts,maxpts,usevalr,exp nd,dv,aa,bb,h,lims;\n global `allroots/evalr`,`allroots/evalrp`,`all roots/evalrpp`,`allroots/pts`,\n `allroots/minpts`,`allroots/expa nd`,`allroots/eps1`,`allroots/eps2`,\n `allroots/eps3`;\n\n ### argument checks and options (Peter S 2004)\n if nargs<2 then\n \+ error \"at least 2 arguments are required; the basic syntax is: 'allr oots(f(x),x=a..b)'.\"\n end if;\n \n if type(eq,algebraic) or ty pe(eq,equation) then\n if type(eq,equation) then\n ee := \+ lhs(eq)-rhs(eq);\n sf := traperror(simplify(ee));\n if sf<>lasterror then f := sf else f := ee end if;\n else\n \+ f := eq;\n end if;\n end if;\n vars := indets(f,name) minus \+ indets(f,realcons);\n if nops(vars)>1 then \n if not has(indets (f),\{'Int','Sum','int','sum'\}) then\n error \"the 1st argume nt, %1, is invalid .. it should be an expression or an equation which \+ depends only on a single variable\",eq;\n end if;\n end if;\n\n if type(r,name=realcons..realcons) then\n x := op(1,r);\n \+ if nops(vars)>0 and not member(x,vars) then\n error \"the 1st argument, %1, is invalid .. it should be an expression or an equation which depends only on the variable %2\",eq,x;\n end if;\n else \n error \"the 2nd argument, %1, is invalid .. it should have the form 'x=a..b', to provide the interval in which to search for roots\" ,r;\n end if;\n\n startopts := 3;\n ## third argument -1 for lea st root,1 for greatest root,0 for all roots\n ## -2 or 2 for least o r greatest not including 0\n flag := 0;\n if nargs>2 then\n a rg3 := args[3];\n if member(arg3,\{-2,-1,0,1,2\}) then\n \+ flag := arg3;\n startopts := startopts+1;\n elif not type (arg3,`=`) then\n error \"the 3rd argument, %1, is invalid .. \+ it should be an integer between -2 and 2 inclusive, or an option in th e form of an equation\",arg3;\n end if;\n end if;\n\n ## Get \+ the options.\n ## Set the default value to start with.\n prntflg : = 0;\n maxpts := 20;\n minpts := 3;\n expnd := 5;\n usevalr := true;\n d := 50;\n dv := 1;\n opts := args[startopts..nargs];\n if startopts<=nargs then\n optns := [args[startopts..nargs]]; \n if not type(optns,list(equation)) then\n error \"each \+ optional argument must be an equation\"\n end if;\n if hasop tion(optns,'split','dv','optns') then\n if not type(dv,posint) and not type(dv,list(realcons)) then\n error \"\\\"split\\ \" must be a positive integer or a list of real constants\" \n \+ end if;\n if hasoption(optns,'info','prntflg') then\n \+ if not member(prntflg,\{true,false,0,1,2,3,4\}) then\n \+ error \"\\\"info\\\" must be false <-> 0, true <-> 1,2,3 or 4\"\n \+ end if;\n if prntflg=false then prntflg := 0\n \+ elif prntflg=true then prntflg := 2 end if;\n end if; \n else\n if hasoption(optns,'use_evalr','usevalr','optns ') then\n if not usevalr then usevalr := false end if;\n \+ end if;\n if hasoption(optns,'maxpoints','maxpts','optns ') then\n if not type(maxpts,posint) or maxpts<3 then\n \+ error \"\\\"maxpoints\\\" must be a positive integer greate r than or equal to 3\"\n end if;\n end if;\n \+ if hasoption(optns,'minpoints','minpts','optns') then\n i f not type(minpts,posint) or minpts<3 then\n error \"\\ \"minpoints\\\" must be a positive integer greater than or equal to 3 \"\n end if;\n if minpts>maxpts then\n \+ error \"\\\"minpoints\\\" cannot be greater than \\\"maxpoints\\ \" which is %1\",maxpts;\n end if;\n end if;\n \+ if hasoption(optns,'expansion','expnd','optns') then\n \+ expnd := evalf(expnd);\n if not type(expnd,numeric) or expn d<1 then\n error \"\\\"expansion\\\" must evaluate to a \+ float which is greater than or equal to 1\"\n end if;\n \+ end if;\n if hasoption(optns,'width','d','optns') then\n \+ d := evalf(d);\n if not type(d,numeric) or d<=0 \+ then\n error \"\\\"width\\\" must evaluate to a float wh ich is greater than 0\"\n end if;\n end if;\n \+ if hasoption(optns,'info','prntflg','optns') then\n if n ot member(prntflg,\{true,false,0,1,2,3,4\}) then\n error \"\\\"info\\\" must be false <-> 0, true <-> 1,2,3 or 4\"\n \+ end if;\n if prntflg=false then prntflg := 0\n \+ elif prntflg=true then prntflg := 2 end if; \n end if;\n \+ if nops(optns)>0 then\n if type(op(1,optns),'name'='ran ge') or type(op(1,optns),'range') then\n error \"%1 is n ot a valid argument for %2 .. only one range is allowed as an argument \",op(1,optns),procname;\n else\n error \"%1 \+ is not a valid option for %2 .. the recognised options are \\\"split\\ \",\\\"use_evalr\\\",\\\"maxpoints\\\",\\\"minpoints\\\",\\\"expansion \\\",\\\"width\\\" and \\\"info\\\"\",op(1,optns),procname;\n \+ end if;\n end if;\n end if;\n end if;\n\n extraDig its := 8; ### (Peter S 2004)\n Digits := Digits+extraDigits;\n aa \+ := op(1,op(2,r));\n bb := op(2,op(2,r));\n a := evalf(aa);\n b : = evalf(bb);\n if a>=b then\n if a=b then return NULL\n el se\n error \"invalid interval %1\",aa..bb\n end if;\n e nd if;\n\n if dv<>1 then\n ## split given interval into subinte rvals (Peter S 2004)\n if type(dv,posint) then\n if prntf lg>0 then\n print(`Splitting the main interval into `||dv|| ` subintervals`);\n print('``');\n end if;\n \+ h := (b-a)/dv;\n lims := [seq(a+i*h,i=0..dv)];\n else\n lims := sort([op(\{a,op(evalf(dv)),b\})]);\n if op(1, lims)b then\n error \"intermediat e division points must lie in %1\",aa..bb;\n end if;\n \+ if prntflg>0 then\n print(`Splitting the main interval usi ng the points`);\n print(op(sort(evalf[Digits-extraDigits]( dv)))); \n print('``');\n end if;\n end if;\n \+ Digits := Digits-extraDigits;\n return seq(allroots(f,x=lims [i]..lims[i+1],flag,op(optns)),i=1..nops(lims)-1);\n end if;\n\n ## # apply transformations to handle infinities earlier (Peter S 2004)\n \+ if hastype(r,infinity) then\n if prntflg>0 then\n print (`Applying the transformation x -> 1/x to handle the infinite interval :`);\n end if;\n Digits := Digits-extraDigits;\n fn := \+ proc(t)\n local a,b; \n if t=0 then NULL \n el if type(t,specfunc(anything,INTERVAL)) then \n a := op([1,1 ],t);\n if a=0 then a := infinity else a := 1/a end if;\n \+ b := op([1,2],t);\n if b=0 then b := -infinity el se b := 1/b end if;\n INTERVAL(b..a)\n else 1/t \n \+ end if \n end proc:\n opts := args[startopts..nargs] ;\n \n if a=Float(-infinity) then\n if b=Float(infin ity) then ## two-sided infinite\n ## on (-infinity,-d] tran sform v=1/x,v=-1/d..0\n ## on [d, infinity) transform v=1/x ,v=0..1/d\n if prntflg>0 then\n print(-d..d,` is unaffected`);\n print(-infinity..-d,` is mapped to `,-1/d..0);\n print(d..infinity,` is mapped to `,0.. 1/d);\n end if;\n if flag=0 then\n \+ res := map(fn,\{allroots(subs(x=1/v,f),v=-1/d..0,0,opts),\n \+ allroots(subs(x=1/v,f),v=0..1/d,0,opts)\});\n r es := \{op(res),allroots(f,x=-d..d,0,opts)\};\n elif flag<= -1 then\n res := allroots(subs(x=1/v,f),v=-1/d..0,2,opts );\n if res<>NULL then return fn(res) end if;\n \+ res := allroots(f,x =-d..d,-1,opts);\n if res<>NUL L then return res end if;\n res := allroots(subs(x=1/v,f ),v=0..1/d,2,opts);\n if res<>NULL then return fn(res) e nd if;\n return NULL;\n else\n \+ res := allroots(subs(x=1/v,f),v=0..1/d,-2,opts);\n if re s<>NULL then return fn(res) end if;\n res := allroots(f, x=-d..d,1,opts);\n if res<>NULL then return res end if; \n res := allroots(subs(x=1/v,f),v=-1/d..0,-2,opts);\n \+ if res<>NULL then return fn(res) end if;\n \+ return NULL;\n end if;\n else ## one-sided left\n \+ c := -d-abs(b);\n if prntflg>0 then\n \+ print(c..b,` is unaffected`);\n print(-infinity..c, ` is mapped to `,1/c..0);\n end if;\n if flag= 0 then\n res := map(fn,\{allroots(subs(x=1/v,f),v=1/c..0 ,0,opts)\});\n res := \{op(res),allroots(f,x=c..b,0,opts )\};\n elif flag<=-1 then\n res := allroots(s ubs(x=1/v,f),v=1/c .. 0,2,opts);\n if res<>NULL then ret urn fn(res) end if;\n return allroots(f,x=c..b,-1,opts); \n else\n res := allroots(f,x=c..b,1,opts);\n if res<>NULL then return res end if;\n re s := allroots(subs(x=1/v,f),v=1/c..0,-2,opts);\n if res< >NULL then return fn(res) end if;\n return NULL;\n \+ end if;\n end if;\n else ## one-sided right\n \+ c := d+abs(a);\n if prntflg>0 then\n print(a..c, ` is unaffected`);\n print(c..infinity,` is mapped to `, 0..1/c);\n end if;\n if flag=0 then\n res : = map(fn,\{allroots(subs(x=1/v,f),v=0..1/c,0,opts)\});\n re s := \{op(res),allroots(f,x=a..c,0,opts)\};\n elif flag<=-1 th en\n res := allroots(f,x=a..c,-1,opts);\n if res <>NULL then return res end if;\n res := allroots(subs(x=1/v ,f),v=0..1/c,2,opts);\n if res<>NULL then return fn(res) en d if;\n return NULL\n else\n res := allr oots(subs(x=1/v,f),v=0..1/c,-2,opts);\n if res<>NULL then f n(res) end if;\n return allroots(f,x=a..c,1,opts);\n \+ end if;\n end if;\n return op(sort([op(res)]));\n end if ;\n\n ## Handle piecewise functions \n ## Split at 0 to avoid numb ers near 0 appearing as 'false' roots\n inc0 := evalb(a<0 and 02 and inc0) then\n if pw then\n fl := conver t(f,'pwlist',x);\n if has(fl,\{Heaviside,abs,signum,piecewise, min,max\}) then\n error \"could not convert to piecewise li st\"\n end if;\n n := (nops(fl)+1)/2;\n if inc 0 then ### split list at 0 (Peter S 2004)\n fl2 := [op(1,fl )];\n for i from 1 to n-1 do\n fl2 := [op(fl2 ),fl[2*i],fl[2*i+1]];\n if signum(0,fl[2*i],0)=-1 and si gnum(0,fl[2*i+2],0)=1 then\n fl2 := [op(fl2),0,fl[2*i +1]]\n end if;\n end do:\n if nops (fl2)>nops(fl) then\n fl := fl2;\n n := n+ 1;\n end if;\n end if;\n else\n fl := \+ [f,0,f]; ### split list at 0 (Peter S 2004)\n n := 2;\n e nd if;\n Digits:= Digits-extraDigits;\n res := NULL;\n \+ if flag=0 then\n for i from 1 to n do\n if i=1 then ai := a\n else ai := max(a,evalf(fl[2*i-2],Digits+extraDigits )) end if;\n if i=n then bi := b\n else bi := min(b ,evalf(fl[2*i],Digits+extraDigits)) end if;\n if ai>bi then next end if;\n res := res,allroots(fl[2*i-1],x=ai..bi,0,op ts);\n end do;\n ## avoid 0.,-0. both appearing (Peter S 2004)\n res := op(map(proc(t)`if`(t=0 and sign(t)=-1,-t,t) \+ end proc,[res]));\n return op(sort([op(\{res\})]));\n eli f flag<=-1 then\n for i to n do\n if i=1 then ai := a \n else ai := max(a,evalf(fl[2*i-2],Digits+extraDigits)) end if;\n if i=n then bi := b\n else bi := min( b,evalf(fl[2*i],Digits+extraDigits)) end if;\n if ai>bi the n next end if;\n res := allroots(fl[2*i-1],x=ai..bi,flag,op ts);\n if res<>NULL then return res end if;\n end d o;\n return res\n else\n for i from n to 1 by -1 \+ do\n if i=1 then ai := a\n else ai := max(a,eval f(fl[2*i-2],Digits+extraDigits)) end if;\n if i=n then bi : = b\n else bi := min(b,evalf(fl[2*i],Digits+extraDigits)) e nd if;\n if ai>bi then next end if;\n res := all roots(fl[2*i-1],x=ai..bi,flag,opts);\n if res<>NULL then re turn res end if;\n end do;\n return res\n end if; \n end if;\n\n fn := unapply(f,x);\n fa := traperror(evalf(fn(a) ));\n if fa=lasterror or type(fa,undefined) then\n fa := evalf( limit(f,x=a,'right'))\n end if;\n if not type(fa,realcons) then\n \+ error \"evaluation failed at %1\",a;\n end if;\n fb := traper ror(evalf(fn(b)));\n if fb=lasterror or type(fb,undefined) then\n \+ fb := evalf(limit(f,x=b,'left'))\n end if;\n if not type(fb,rea lcons) then\n error \"evaluation failed at %1\",b;\n end if;\n \+ if fa<0 then\n fa := -fa;\n fb := -fb;\n fn := unappl y(-f,x);\n fnp := unapply(-diff(f,x),x); ### removed \"normal\" ( Peter S 2004)\n fnpp := unapply(-diff(f,x$2),x);\n else\n \+ fnp := unapply(diff(f,x),x);\n fnpp := unapply(diff(f,x$2),x);\n \+ end if;\n ### changed 'epsilons', notably `allroots/eps2` \n ### to get more accurate results (Peter S 2004)\n eps := Float(1,extraD igits-Digits-2);\n `allroots/eps1` := eps*(abs(b)+abs(a));\n eps : = Float(1,2*(extraDigits-Digits));\n if hastype([fa,fb],infinity) th en `allroots/eps2` := eps ### \"hastype\" (Peter S 2004)\n else `all roots/eps2` := min(eps*(abs(fa)+abs(fb)+.01)/2,eps);\n end if;\n i f b=a then `allroots/eps3`:= 1\n else `allroots/eps3` := `allroots/e ps2`/(b-a)\n end if;\n if b=a then `allroots/pts` := 2\n else `a llroots/pts` := maxpts/(b-a) end if;\n `allroots/minpts` := minpts-2 ;\n if usevalr then\n `allroots/evalr` := true;\n `allroot s/evalrp` := true;\n `allroots/evalrpp` := true;\n else\n \+ `allroots/evalr` := false;\n `allroots/evalrp` := false;\n ` allroots/evalrpp` := false;\n end if;\n `allroots/expand` := expnd ;\n res := `allroots/allroots`(fn,fnp,fnpp,a,b,fa,fb,flag,prntflg); \n\n Digits := Digits-extraDigits;\n ### output is sequence rather than list (Peter S 2004)\n if flag=0 then\n ## avoid 0.,-0. both \+ appearing (Peter S 2004) \n res := op(map(proc(t)`if`(t=0 and sign(t )=-1,-t,t) end proc,[res]));\n res := op(sort([op(evalf(\{res\}))])) \n else res := evalf(res)\n end if;\nend proc: ## allroots" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14178 "`allroots/allroots` := proc(fn,fnp,fnpp,a,b,fa,fb,flag,prntflg) \n local fr,fpp1,fpp2,fp1,fp2,f1,f2,m,fm,v,w,fpa,m1,fm1,fbb,fppb,fpb ,\n m2,fm2,g1,q,res,jit,n,h,vals,i,t,mi,fni,fpi,fppa,fppi,ft1,ft2 ;\n global `allroots/evalr`,`allroots/evalrp`,`allroots/evalrpp`;\n \+ if prntflg>0 then\n print(`Checking the interval: `,a..b) \+ \n end if;\n if b-a<`allroots/eps1` then\n if abs(fa+fb)<`all roots/eps2` then return 1/2*a+1/2*b\n else return NULL\n end if\n end if;\n if `allroots/evalr` then\n q := fn(INTERVAL(a ..b));\n fr := traperror(evalf(evalr(q)));\n if fr=lasterror or hastype(fr,undefined) then\n fr := traperror(evalf(evalr(n ormal(q))))\n end if;\n if fr=lasterror or hastype(fr,undefi ned) then\n `allroots/evalr` := false;\n WARNING(\"can not use interval arithmetic for value ranges - estimating instead\"); \n end if;\n end if;\n if not `allroots/evalr` then \n # # make an estimate for fn(INTERVAL(a..b)) (Peter S 2004)\n jit := `plot/jitter`;\n n := max(trunc(`allroots/pts`*(b-a)),`allroots/ minpts`);\n h := (b-a)/(n+1);\n vals := fa;\n for i fro m 1 to n do\n mi := a+(i+jit[irem(i-1,101)+1])*h;\n fn i := traperror(evalf(fn(mi)));\n if not type(fni,realcons) the n\n error \"evaluation failed at %1\",mi;\n end if; \n vals := vals,fni;\n end do;\n vals := vals,fb;\n \+ ft1 := min(vals); \n ft2 := max(vals);\n ## provide a \+ margin of safety by expanding the interval\n t := `allroots/expan d`;\n f1 := t*ft1+(1-t)*ft2;\n f2 := t*ft2+(1-t)*ft1;\n \+ fr := INTERVAL(f1..f2); \n end if;\n if type(fr,\{ specfunc(range,INTERVAL),list(range)\}) then\n f1 := op(1,op(1,fr ));\n f2 := op(2,op(-1,fr))\n elif type(fr,realcons) then f1 := fr; f2 := fr\n else error \"evalr returned %1\",fr\n end if;\n \+ ### changed \"has\" to \"hastype\" (Peter S 2004)\n if hastype(f1,in finity) then f1 := Float(-1,Digits+10) end if;\n if hastype(f2,infin ity) then f2 := Float(1,Digits+10) end if;\n ## if the values do no \+ change sign and stay away from 0,\n ## there is no root in the inter val a..b. \n if signum(0,f1,0)=1 or signum(0,f2,0)=-1 then\n re turn NULL\n end if;\n if prntflg>3 then\n print(`estimated ra nge of values: `,f1..f2)\n end if;\n if `allroots/evalrp` then\n \+ q := fnp(INTERVAL(a..b));\n fr := traperror(evalf(evalr(q))); \n if fr=lasterror or hastype(fr,undefined) then \n fr := traperror(evalf(evalr(normal(q))))\n end if;\n if fr=laster ror or hastype(fr,undefined) then\n `allroots/evalrp` := false ;\n WARNING(\"cannot use interval arithmetic for derivative ra nges - estimating instead\");\n end if;\n end if;\n if not `a llroots/evalrp` then \n ## make an estimate for fnp(INTERVAL(a..b )) (Peter S 2004)\n jit := `plot/jitter`;\n n := min(max(tru nc(`allroots/pts`*(b-a)),`allroots/minpts`),101);\n h := (b-a)/(n +1);\n fpa := traperror(evalf(fnp(a)));\n if fpa=lasterror o r type(fpa,undefined) then \n fpa := evalf(limit(fnp(v),v=a,'r ight'))\n end if;\n if not type(fpa,realcons) then\n \+ error \"evaluation of derivative failed at %1\",a;\n end if;\n \+ vals := fpa;\n for i from 1 to n do\n mi := a+(i+jit[ irem(i-1,101)+1])*h;\n fpi := traperror(evalf(fnp(mi)));\n \+ if not type(fpi,realcons) then\n error \"evaluation of derivative failed at %1\",mi;\n end if;\n vals := val s,fpi;\n end do;\n fpb := traperror(evalf(fnp(b)));\n i f fpb=lasterror or type(fpb,undefined) then\n fpb := evalf(lim it(fnp(x),x=b,'left'));\n end if;\n if not type(fpb,realcons ) then\n error \"evaluation of derivative failed at %1\",b;\n \+ end if;\n vals := vals,fpb;\n ft1 := min(vals);\n \+ ft2 := max(vals);\n ## provide a margin of safety by expanding th e interval\n t := `allroots/expand`;\n fp1 := t*ft1+(1-t)*ft 2;\n fp2 := t*ft2+(1-t)*ft1;\n fr := INTERVAL(fp1..fp2); \n \+ end if;\n if type(fr,\{specfunc(range,INTERVAL),list(range)\}) th en\n fp1 := op(1,op(1,fr));\n fp2 := op(2,op(-1,fr))\n eli f type(fr,realcons) then fp1 := fr; fp2 := fr\n else error \"evalr r eturned %1\",fr\n end if;\n ### changed \"has\" to \"hastype\" (Pe ter S 2004)\n if hastype(fp1,infinity) then fp1 := Float(-1,Digits+1 0) end if;\n if hastype(fp2,infinity) then fp2 := Float(1,Digits+10) end if;\n if prntflg>3 then\n print(`estimated range for deriv ative: `,fp1..fp2)\n end if;\n if fp1>=-`allroots/eps3` then ## no ndecreasing with fa>=0\n if fp2<=`allroots/eps3` then ## function is flat\n if fa=0 then return INTERVAL(a..b)\n else r eturn NULL\n end if\n elif fa=0 and (flag<>-2 or a<>0) th en return a\n else return NULL\n end if\n end if;\n if f p2<=`allroots/eps3` then ## nonincreasing with fa>=0\n if fb<=0 t hen\n return `allroots/monotone`(fn,fnp,a,b,fa,fb,flag,prntflg )\n else return NULL\n end if\n end if;\n ## now not mon otone, check for convex/concave\n if `allroots/evalrpp` then\n \+ q := fnpp(INTERVAL(a..b));\n fr := traperror(evalf(evalr(q)));\n \+ if fr=lasterror or hastype(fr,undefined) then\n fr := tra perror(evalf(evalr(normal(q))))\n end if;\n if fr=lasterror \+ or hastype(fr,undefined) then\n `allroots/evalrpp` := false;\n WARNING(\"cannot use interval arithmetic for 2nd derivative r anges - estimating instead\");\n end if;\n end if;\n if not ` allroots/evalrpp` then \n ## make an estimate for fnpp(INTERVAL(a ..b)) (Peter S 2004)\n jit := `plot/jitter`;\n n := max(trun c(`allroots/pts`*(b-a)),`allroots/minpts`);\n h := (b-a)/(n+1);\n fppa := traperror(evalf(fnpp(a)));\n if fppa=lasterror or t ype(fppa,undefined) then \n fppa := evalf(limit(fnpp(v),v=a,'r ight'))\n end if;\n if not type(fppa,realcons) then\n \+ error \"evaluation of 2nd derivative failed at %1\",a;\n end if ;\n vals := fppa;\n for i from 1 to n do\n mi := a+( i+jit[irem(i-1,101)+1])*h;\n fppi := traperror(evalf(fnpp(mi)) );\n if not type(fppi,realcons) then\n error \"eval uation of 2nd derivative failed at %1\",mi;\n end if;\n \+ vals := vals,fppi;\n end do;\n fppb := traperror(evalf(fnp p(b)));\n if fppb=lasterror or type(fppb,undefined) then\n \+ fppb := evalf(limit(fnpp(x),x=b,'left'));\n end if;\n if n ot type(fppb,realcons) then\n error \"evaluation of 2nd deriva tive failed at %1\",b;\n end if;\n vals := vals,fppb;\n \+ ft1 := min(vals);\n ft2 := max(vals);\n ## provide a margin of safety by expanding the interval\n t := `allroots/expand`;\n \+ fpp1 := t*ft1+(1-t)*ft2;\n fpp2 := t*ft2+(1-t)*ft1;\n f r := INTERVAL(fpp1..fpp2);\n end if;\n if type(fr,\{specfunc(range ,INTERVAL),list(range)\}) then\n fpp1 := op(1,op(1,fr));\n f pp2 := op(2,op(-1,fr));\n elif type(fr,realcons) then fpp1 := fr; fp p2 := fr\n end if;\n if prntflg>3 then\n print(`estimated ran ge for 2nd derivative: `,fpp1..fpp2)\n end if;\n ### changed \"has \" to \"hastype\" (Peter S 2004)\n if hastype(fpp1,infinity) then fp p1 := Float(-1,Digits+10) end if;\n if hastype(fpp2,infinity) then f pp2 := Float(1,Digits+10) end if;\n if not assigned(fpa) then\n \+ fpa := traperror(evalf(fnp(a)));\n if fpa=lasterror or type(fpa, undefined) then \n fpa := evalf(limit(fnp(v),v=a,'right'))\n \+ end if;\n if not type(fpa,realcons) then\n error \"ev aluation of derivative failed at %1\",a;\n end if;\n end if;\n \+ if fpa<0 then\n if fpa+fpp2*(b-a)<=0 then ## nonincreasing with fa>=0\n if fb<=0 then return `allroots/monotone`(fn,fnp,a,b,f a,fb,flag,prntflg)\n else return NULL end if\n end if;\n \+ ## subdivide interval using Newton's method for derivative from l eft\n ## in an attempt to \"overshoot\" any root from the left\n \+ m1 := a-fpa/fpp2;\n fm1 := traperror(evalf(fn(m1)));\n \+ if fm1=lasterror or type(fm1,undefined) then \n error \"evalua tion failed at %1\",m1;\n end if;\n if fm1<=0 then ## root i n left interval a..m1\n if flag=0 then \n return ` allroots/monotone`(fn,fnp,a,m1,fa,fm1,flag,prntflg),\n `a llroots/allroots`(-fn,-fnp,-fnpp,m1,b,-fm1,-fb,flag,prntflg)\n \+ elif flag<=-1 then\n res := `allroots/monotone`(fn,fnp,a,m 1,fa,fm1,flag,prntflg);\n if res<>NULL then return res end \+ if;\n return `allroots/allroots`(-fn,-fnp,-fnpp,m1,b,-fm1,- fb,flag,prntflg);\n else\n res := `allroots/allroot s`(-fn,-fnp,-fnpp,m1,b,-fm1,-fb,flag,prntflg);\n if res<>NU LL then return res end if;\n return `allroots/monotone`(fn, fnp,a,m1,fa,fm1,flag,prntflg);\n end if; \n end if; \n else m1 := a; fm1 := fa ## m1 is new left boundary and fm1>0 \n \+ end if;\n if not assigned(fpb) then\n fpb := traperror(evalf(f np(b)));\n if fpb=lasterror or type(fpb,undefined) then\n \+ fpb := evalf(limit(fnp(x),x=b,'left'));\n end if;\n if not \+ type(fpb,realcons) then\n error \"evaluation of derivative fai led at %1\",b;\n end if;\n end if;\n if fb<0 then\n fppb := -fpp1; fpb := -fpb; ## not needed fbb := -fb; \n else\n fpp b := fpp2; ## not needed fbb := fb;\n end if;\n if 00 or nonincreasing with fb<0 \n if fb<=0 then return `al lroots/monotone`(fn,fnp,m1,b,fm1,fb,flag,prntflg)\n elif fm1<` allroots/eps2` then return m1 \n else return NULL\n en d if\n end if;\n ## subdivide interval using Newton's method for derivative from right\n ## in an attempt to \"overshoot\" an y root from the right\n m2 := b-fpb/fppb;\n fm2 := traperror (evalf(fn(m2)));\n if fm2=lasterror or type(fm2,undefined) then \+ \n error \"evaluation failed at %1\",m2;\n end if;\n \+ if signum(0,fm2,0)<>signum(0,fb,2) then ## root in right interval m2. .b\n if flag<=-1 then\n g1 := `allroots/allroots`(f n,fnp,fnpp,m1,m2,fm1,fm2,flag,prntflg); \n if g1<>NULL then return g1 end if;\n if 0<=fm2 then\n return \+ `allroots/monotone`(fn,fnp,m2,b,fm2,fb,flag,prntflg)\n else \n return `allroots/monotone`(-fn,-fnp,m2,b,-fm2,-fb,fl ag,prntflg)\n end if;\n else \n if 0 <=fm2 then g1 := `allroots/monotone`(fn,fnp,m2,b,fm2,fb,flag,prntflg) \n else g1 := `allroots/monotone`(-fn,-fnp,m2,b,-fm2,-fb,fl ag,prntflg)\n end if;\n if flag>=1 then\n \+ if g1<>NULL then return g1\n else return `allroo ts/allroots`(fn,fnp,fnpp,m1,m2,fm1,fm2,flag,prntflg)\n e nd if;\n end if;\n return `allroots/allroots`(fn ,fnp,fnpp,m1,m2,fm1,fm2,flag,prntflg),g1\n end if;\n end \+ if;\n else m2 := b; fm2 := fb; ## m2 is new right boundary \n end \+ if;\n ## Check for possible double root when 2nd derivative has smal l magnitude\n if -`allroots/eps2`/(b-a)^21 then print(`reflecting in the origin`) end if; \n return `allroots/reverse`([`allroo ts/convex`(\n unapply(-fn(-v),v),unapply(fnp(-v),v),-m2,-m1 ,-fm2,-fm1,0,prntflg)]);\n else\n g1 := `allroots/convex` (\n unapply(-fn(-v),v),unapply(fnp(-v),v),-m2,-m1,-fm2,-fm1 ,-flag,prntflg);\n if g1<>NULL then return -g1 else return NUL L end if;\n end if\n end if;\n ## shrink interval \n if `al lroots/eps2`NULL then return g1 end if;\n if 0<=fm then `allroots/allroo ts`(fn,fnp,fnpp,m,m2,fm,fm2,flag,prntflg)\n else `allroots/allroo ts`(-fn,-fnp,-fnpp,m,m2,-fm,-fm2,flag,prntflg)\n end if\n else \n if 0<=fm then g1 :=`allroots/allroots`(fn,fnp,fnpp,m,m2,fm,fm2 ,flag,prntflg)\n else g1 :=`allroots/allroots`(-fn,-fnp,-fnpp,m,m 2,-fm,-fm2,flag,prntflg)\n end if;\n if g1<>NULL then return g1 end if;\n `allroots/allroots`(fn,fnp,fnpp,m1,m,fm1,fm,flag,pr ntflg);\n end if; \nend proc: ## `allroots/allroots`" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7071 " `allroots/reverse` := proc(L) local n;\n seq(-L[nops(L)-n], n=0 .. n ops(L)-1);\nend proc: ## `allroots/reverse`\n\n`allroots/convex` := pr oc(fn,fnp,a,b,fa,fb,flag,prntflg)\n local fpa,fpb,v,m,fm,fmp,ap,bp,f ap,fbp,res;\n if prntflg>1 then\n print('``');\n print(`Ch ecking for a possible double root in the interval `,a..b); \n en d if;\n fpa := traperror(evalf(fnp(a)));\n if fpa=lasterror or typ e(fpa,undefined) then\n fpa := evalf(limit(fnp(x),x=a,'right')); \n end if;\n if not type(fpa,realcons) then \n error \"evalua tion of derivative failed at %1\",a;\n end if;\n fpb := traperror( evalf(fnp(b)));\n if fpb=lasterror or type(fpb,undefined) then\n \+ fpb := evalf(limit(fnp(x),x=b,'left'));\n end if;\n if not type( fpb,realcons) then \n error \"evaluation of derivative failed at \+ %1\",b;\n end if;\n if fpa>-`allroots/eps3` then\n if `allroo ts/eps2`-2 or a<>0 then res := a else return NULL end if\n \+ else res := 1/2*a+1/2*b\n end if;\n if prntflg>1 then\n \+ print(`double root -> `,res); \n print('``')\n end if;\n return res;\n end if;\n if b<=a+`allroots/eps1` then\n if abs(fa+fb)<`allroots/eps2` then\n res := 1/2*a+1/2*b; \n if prntflg>1 then\n print(`root -> `,res); \n print('``')\n end if;\n return res;\n \+ else return NULL\n end if;\n end if;\n if fpb<=`allroots/eps 3` then\n if fb<=0 then return `allroots/monotone`(fn,fnp,a,b,fa, fb,flag,prntflg)\n else return NULL\n end if\n end if;\n \+ if fb<0 then\n if fpa+fpb<0 then\n m := b-2*fpb*(b-a)/(f pb-fpa);\n if prntflg>2 then\n print(`accelerated d erivative based secant approximation -> `,m)\n end if;\n \+ else \n m := (b*fa-a*fb)/(fa-fb);\n if prntflg>2 the n\n print(`secant approximation -> `,m)\n end if; \n end if;\n fm := traperror(evalf(fn(m)));\n if fm=las terror or type(fm,undefined) then \n error \"evaluation failed at %1\",m;\n end if;\n if 0<=fm then \n ap := m; fa p := fm; bp := b; fbp := fb;\n m := 1/2*m+1/2*b;\n if \+ prntflg>2 then\n print(`bisection approximation -> `,m)\n end if;\n fm := traperror(evalf(fn(m)));\n if fm=lasterror or type(fm,undefined) then \n error \"evaluat ion failed at %1\",m;\n end if;\n if 0<=fm then ap := \+ m; fap := fm \n else bp := m; fbp := fm;\n end if\n \+ else \n ap := a; fap := fa; bp := m; fbp := fm;\n f mp := traperror(evalf(fnp(m)));\n if fmp=lasterror or type(fmp ,undefined) then \n error \"evaluation of derivative failed at %1\",m;\n end if; \n if 0<=fmp then\n \+ m := 1/2*a+1/2*m;\n if prntflg>2 then\n \+ print(`bisection approximation -> `,m)\n end if;\n \+ fm := traperror(evalf(fn(m)));\n if fm=lasterror \+ or type(fm,undefined) then \n error \"evaluation failed \+ at %1\",m;\n end if;\n if 0<=fm then ap := m; fa p := fm\n else bp:= m; fbp:= fm\n end if\n \+ end if\n end if; \n return `allroots/convex`(fn,fnp,ap,bp ,fap,fbp,flag,prntflg)\n end if;\n ap := a-fa/fpa;\n bp := b-fb/ fpb;\n if bp2 then\n print(`adjust ed Newton approximation from left -> `,ap)\n end if; \n fa p := traperror(evalf(fn(ap)));\n if fap=lasterror or type(fap,und efined) then \n error \"evaluation failed at %1\",ap;\n e nd if;\n end if;\n fbp := traperror(evalf(fn(bp)));\n if fbp=las terror or type(fbp,undefined) then \n error \"evaluation failed a t %1\",bp;\n end if;\n if fbp<0 then\n bp := .9*bp + .1*b;\n \+ if prntflg>2 then\n print(`adjusted Newton approximation \+ from right -> `,bp)\n end if;\n fbp := traperror(evalf(fn( bp)));\n if fbp=lasterror or type(fbp,undefined) then \n \+ error \"evaluation failed at %1\",bp;\n end if;\n end if;\n f pa := traperror(evalf(fnp(ap)));\n if fpa=lasterror or type(fpa,unde fined) then \n error \"evaluation of derivative failed at %1\",ap ;\n end if;\n fpb := traperror(evalf(fnp(bp)));\n if fpb=lasterr or or type(fpb,undefined) then \n error \"evaluation of derivativ e failed at %1\",bp;\n end if;\n if 0<=fpa then\n if fap<`all roots/eps2` and (flag<>-2 or ap<>0) then\n if prntflg>1 then\n print(`root -> `,ap); \n print('``')\n \+ end if;\n return ap \n else return NULL\n end if\n end if;\n if fpb<=0 then\n if fbp<`allroots/eps2` and (flag< >2 or bp<>0) then\n if prntflg>1 then\n print(`root -> `,bp); \n print('``')\n end if;\n ret urn bp\n else return NULL\n end if\n end if;\n m := (ap* fpb-bp*fpa)/(fpb-fpa);\n if m<=ap or bp<=m then\n m := 1/2*ap+1 /2*bp;\n if prntflg>2 then\n print(`bisection approximati on -> `,m)\n end if;\n else\n if prntflg>2 then\n \+ print(`derivative based secant approximation -> `,m)\n end if ;\n end if;\n fm := traperror(evalf(fn(m)));\n if fm=lasterror o r type(fm,undefined) then \n error \"evaluation failed at %1\",m; \n end if;\n if fm<0 then\n if flag=0 then\n `allroot s/convex`(fn,fnp,ap,m,fap,fm,0,prntflg),\n `allroots/reverse `([`allroots/convex`(\n unapply(fn(-v),v),unapply(-fnp(-v) ,v),-bp,-m,fbp,fm,0,prntflg)])\n elif flag<=-1 then\n res := `allroots/convex`(fn,fnp,ap,m,fap,fm,flag,prntflg);\n if r es<>NULL then return res end if;\n if prntflg>1 then print(`re flecting in the origin`) end if;\n res := `allroots/convex`(\n unapply(fn(-v),v),unapply(-fnp(-v),v),-bp,-m,fbp,fm,-flag ,prntflg);\n if res<>NULL then\n return -res else N ULL\n end if\n else\n if prntflg>1 then print(`re flecting in the origin`) end if;\n res := `allroots/convex`(\n unapply(fn(-v),v),unapply(-fnp(-v),v),-bp,-m,fbp,fm,-flag ,prntflg);\n if res<>NULL then\n return -res else N ULL\n end if;\n `allroots/convex`(fn,fnp,ap,m,fap,fm,f lag,prntflg);\n end if\n else\n fmp := traperror(evalf(fnp (m)));\n if fmp=lasterror or type(fmp,undefined) then \n \+ error \"evaluation of derivative failed at %1\",m;\n end if;\n \+ if abs(fmp)<`allroots/eps3` and abs(fm)<`allroots/eps2` then\n \+ if prntflg>1 then\n print(`double root -> `,m); \n \+ print('``')\n end if;\n return m\n elif \+ 0 " 0 "" {MPLTEXT 1 0 3719 "`allroots/monotone` := pr oc(fn,fnp,a,b,fa,fb,flag,prntflg)\n local d,fpm,m1,fm1,m2,fm2,m,fm,r es;\n if prntflg>1 then\n print('``');\n print(`Attempting to locate a single root in the interval `,a..b); \n end if;\n if \+ fa=fb then res := 1/2*a+1/2*b end if;\n if fa=0 then\n if a< >0 or flag<>-2 then res := a else res := NULL end if;\n end if;\n \+ if fb=0 then\n if b<>0 or flag<>2 then res := b else res := NU LL end if;\n end if;\n if assigned(res) then\n if prntflg>1 t hen\n print(`root -> `,res); \n print('``')\n e nd if;\n return res\n end if;\n m1 := a;\n fm1 := fa;\n m 2 := b;\n fm2 := fb;\n d := m2-m1;\n ### changed exit conditions from loop (Peter S 2004) \n while `allroots/eps1`2 then\n print(`secant approxim ation -> `,m)\n end if;\n fm := traperror(evalf(fn(m )));\n if fm=lasterror or type(fm,undefined) then \n \+ error \"evaluation failed at %1\",m;\n end if;\n if \+ 02 th en\n print(`Newton approximation from left -> `,m)\n \+ end if;\n fm := traperror(evalf(fn(m)));\n \+ if fm=lasterror or type(fm,undefined) then \n error \"evaluation failed at %1\",m;\n end if;\n if 0 2 then\n print(`Newton approximati on from right -> `,m)\n end if;\n fm := traper ror(evalf(fn(m)));\n if fm=lasterror or type(fm,undefined) \+ then \n error \"evaluation failed at %1\",m;\n \+ end if;\n if 02 then\n \+ print(`bisection approximation -> `,m)\n end if;\n \+ fm := traperror(evalf(fn(m)));\n if fm=lasterror or type(f m,undefined) then \n error \"evaluation failed at %1\",m;\n end if;\n if 01 then\n \+ print(`single root -> `,res); \n print('``');\n end if;\n r es;\nend proc: ## `allroots/monotone`" }}}{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." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 31 "allroo ts,leastroot,greatestroot" }{TEXT 26 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 " Example 1" }{TEXT 288 62 " .. a function with two nearby roots - using the \"info\" option" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the func tion " }{XPPEDIT 18 0 "f(x)=ln((x^2+1)/2)+3*sin(Pi*x)^2" "6#/-%\"fG6#% \"xG,&-%#lnG6#*&,&*$F'\"\"#\"\"\"F0F0F0F/!\"\"F0*&\"\"$F0*$-%$sinG6#*& %#PiGF0F'F0F/F0F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> ln((x^2+1)/2)+3*si n(Pi*x)^2:\n'f(x)'=f(x);\nplot(f(x),x=0..1.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%#lnG6#,&*&\"\"#!\"\"F'F.\"\"\"#F0F.F0 F0*&\"\"$F0)-%$sinG6#*&%#PiGF0F'F0F.F0F0" }}{PARA 13 "" 1 "" {GLPLOT2D 559 350 350 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"!F)$!3'GX*f 0=ZJp!#=7$$\"3\")******\\#HyI\"!#>$!3-bWm(fY\"zoF,7$$\"3h*******\\ech# F0$!3_sm36V^AnF,7$$\"3[****\\PPf`PF0$!3ix7`1+:-lF,7$$\"3-+++v*G:*[F0$! 3=\"37hy!o/iF,7$$\"3u******\\L)4X(F0$!33(f0Ko:@E&F,7$$\"3)******\\MSF+ \"F,$!3W6VvY7[^RF,7$$\"3#)****\\Fy:f7F,$!3Wp?WRMc>BF,7$$\"3')****\\d'* )o\\\"F,$!3#zB'[42b.bF07$$\"3w****\\(>ZIu\"F,$\"3k'pOM#ok*\\\"F,7$$\"3 u****\\xOi(*>F,$\"3'G_%fd>F.QF,7$$\"3#)****\\FPQ^AF,$\"3'yD-z&zLHiF,7$ $\"3/+++IrS7DF,$\"3o-t&))4fvz)F,7$$\"3p*****\\o;Bu#F,$\"3\"G8h\")4fo5 \"!#<7$$\"3*********QS6+$F,$\"3[wW1g3kd8Fio7$$\"3[******\\o-hKF,$\"3_* ywp$>9)f\"Fio7$$\"3(*******4cZ6NF,$\"36.-9=D`8=Fio7$$\"3S****\\xq!*QPF ,$\"3))=WGyB%4*>Fio7$$\"3&********3X$4SF,$\"3/'*R'HM*fu@Fio7$$\"3s**** **f:WQUF,$\"3?s*y?T%f.BFio7$$\"3f****\\<_$\\]%F,$\"3!y$)))GIS'>CFio7$$ \"3**)*****fs#3u%F,$\"3mI!)H*p[(*[#Fio7$$\"3S***\\(QFBq[F,$\"3&4g)y?Or 9DFio7$$\"3!)****\\<#Q'**\\F,$\"3=!eG+Rn*HDFio7$$\"3R*****\\$G'G7&F,$ \"37DYdt'Ha`#Fio7$$\"33++]_u3Y_F,$\"3W,EDu>2KDFio7$$\"3M++v8%fYP&F,$\" 3;&ylh=[#>DFio7$$\"3[*****\\PJK]&F,$\"3_*=Z*z>0(\\#Fio7$$\"3r***\\7d+8 i&F,$\"3e6\\^N?foCFio7$$\"3%*****\\n(p$RdF,$\"3#p+a>k6EV#Fio7$$\"3A*** **\\#p2%*fF,$\"3Xg_XHLhIBFio7$$\"3o****\\xgkeiF,$\"396&o\"RgE#>#Fio7$$ \"3g****\\-V&*)['F,$\"3Y-CI8$Q$[?Fio7$$\"3.+++&\\$pPnF,$\"3YnfEWpWs=Fi o7$$\"37******>am%*pF,$\"3I+l'oHKMn\"Fio7$$\"3?*****\\JigC(F,$\"3_?xHH #*>n9Fio7$$\"3G****\\Pi7Fio7$$\"3g*****\\kx$fxF,$ \"3Q[a0c#yZ.\"Fio7$$\"3\\)*****fG0-!)F,$\"3wg\">8gTQO)F,7$$\"35****** \\/;h#)F,$\"3Q$=g'\\XtnjF,7$$\"3M)***\\P/&f\\)F,$\"3$>1J0xDkr%F,7$$\"3 e******4zj_()F,$\"3DXm)p8B?8$F,7$$\"3u)***\\<3;%**)F,$\"33^!yR>kJ*=F,7 $$\"3]****\\Z=iY#*F,$\"3+vV44`in*)F07$$\"3Q***\\([_.q$*F,$\"3_)f`y:HDI &F07$$\"39)*****\\'[M\\*F,$\"3Qlrm/+FqCF07$$\"3-***\\P/^Ei*F,$\"3/d9Z] ^$yB%!#?7$$\"3y)***\\PM&=v*F,$!3$\\vM0OJoh'F_z7$$\"3w)**\\()p!Hw)*F,$! 3UVI&*\\CcTyF_z7$$\"3*******fzs++\"Fio$\"3#o3)>m/H&H(!#A7$$\"3()***\\K I)z75Fio$\"3qipDFm^kEdR2\"Fio$\"3f*[g$fx]HBF,7$$ \"3&*****\\o#R05\"Fio$\"3%y5(e)H5&)*QF,7$$\"3()*****>`9V7\"Fio$\"3Y&=S >7U!)e&F,7$$\"3)****\\<#Rm\\6Fio$\"3g]j&*)Rd#\\wF,7$$\"3%****\\A_ER<\" Fio$\"3^L\">/q8<$)*F,7$$\"3%**************>\"Fio$\"3bVF=n`KN7Fio-%'COL OURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fh^l-%%VIEWG6$;F($ \"#7Fc^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 13 "The zeros of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$ \"\"!\"\"\"" }{TEXT -1 29 " can be computed as follows. " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 22 " is clearly a zero of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 44 ", but there is al so a zero just less than 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := x -> ln((x^2+1)/2)+3*sin(Pi*x) ^2:\n'f(x)'=f(x);\nallroots(f(x),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%#lnG6#,&*&\"\"#!\"\"F'F.\"\"\"#F0F.F0 F0*&\"\"$F0)-%$sinG6#*&%#PiGF0F'F0F.F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+/OUl:!#5$\"+%))\\5m*F%$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Details concerning the ca lculation can be obtained via the option \"" }{TEXT 262 4 "info" } {TEXT -1 23 "\", which can be set to " }{TEXT 262 10 "true/false" } {TEXT -1 4 " or " }{TEXT 262 9 "0/1/2/3/4" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x -> exp(-cos(x))-exp(1)-2*cos (2*x)+2:\n'f(x)'=f(x);\nplot(f(x),x=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,*-%$expG6#,$-%$cosGF&!\"\"\"\"\"-F*6#F0F /*&\"\"#F0-F.6#,$*&F4F0F'F0F0F0F/F4F0" }}{PARA 13 "" 1 "" {GLPLOT2D 446 282 282 {PLOTDATA 2 "6%-%'CURVESG6$7_q7$$\"\"!F)$!3w-wG(Q-/N#!#<7$ $\"3Hjqb\"[W>r\"!#>$!3]*\\0%*Gw\"\\BF,7$$\"3eET6j*))QU$F0$!33ZD=Q$*\\X BF,7$$\"3_*=rYWLe8&F0$!3k*et#fbPRBF,7$$\"3;`#Gi#zxZoF0$!3$HA@nl63L#F,7 $$\"3\"zBM*)omr-\"!#=$!3J:S=POS1BF,7$$\"3i]cC&eb&p8FE$!3)*o@$H4#QsAF,7 $$\"3q5)>63x`'>FE$!3-S45lCs!>#F,7$$\"3YqR*pd)>hDFE$!3i*4^LD-:3#F,7$$\" 3Zs[E^dK,RFE$!3y#**4;V[Ju\"F,7$$\"3+9+\"[&4$ed%FE$!3Kg#foUe)H:F,7$$\"3 ab^NehL]_FE$!3'GCPY,&Q#H\"F,7$$\"3Ls#3kbN;#fFE$!39LUPwx&f.\"F,7$$\"35* QhW&\\$Hf'FE$!3hm4^A)zrj(FE7$$\"3]:5wGaJ:sFE$!3H57JS+j5]FE7$$\"3zS11.f pPyFE$!39U%>kMWHK#FE7$$\"3#e!R^M[8#[)FE$\"3-LP2Mmj[[F07$$\"3upr'fwtl7* FE$\"3_s*\\D2WtF$FE7$$\"3!Rbb2V`Iz*FE$\"31#Gzw]@j5'FE7$$\"3!QRa&4L&f/ \"F,$\"3g#F,7$$\"396:YIMRr:F,$\"3_iHMFSJ#G#F,7$$\"3\"Q:c%)[ 7ag\"F,$\"3iV$>$*3U@J#F,7$$\"3Z'z]kaJ%R;F,$\"3'[3A(3B*QL#F,7$$\"39RaW/ 1Xt;F,$\"3[1xc2IiZBF,7$$\"3!=3SCmpuq\"F,$\"3sH+Zs,V`BF,7$$\"3%z2'RGLDS F,$\"3K(yr&pF,7$$\"3Ua[#o!GC>AF,$\"3E[&=bLH?l\"F,7$$\"3-YwJf&y(eBF,$\"3w PC;28?.8F,7$$\"3oNrpV8H#[#F,$\"39@p;TR$)p)*FE7$$\"3EzY)QW/yh#F,$\"3cK' )))o#>ff'FE7$$\"35v)>8'\\%ou#F,$\"35tayD-w.RFE7$$\"3w$GCS?&[\")GF,$\"3 M(H%\\;QfYg%e1o%QIF,$\"3=1QG&)3_)z#F07$$\"3g`,ta\"4=2$F,$\" 3Y(e_f+URG\"F07$$\"3$zG9OC]^5$F,$\"3kb=x#[Qc]$!#?7$$\"3sA%)\\K8\\QJF,$ \"3$[gp['4-SD!#A7$$\"3#er1NNBJ<$F,$\"3)e%*)zpDjCEFiz7$$\"3#*3]^u`v2KF, $\"3cr>l57Na6F07$$\"3--L_&R(QUKF,$\"3;!*H2TA#Rn#F07$$\"3c&fJlT>qF$F,$ \"36JqA;2(Q\"[F07$$\"3e!=@(oRJPLF,$\"3U1k75YY()**F07$$\"39l2\"4_3wR$F, $\"35fI%*HRG$p\"FE7$$\"3MOnGd![y_$F,$\"3e4dy\">@hu$FE7$$\"3#*=4JU#)RiO F,$\"3S\"Hv(3@:GlFE7$$\"3%G,f%[$HSz$F,$\"3)\\sDKH))op*FE7$$\"3KoE+7#*Q @RF,$\"3)oDY!HPX&H\"F,7$$\"3y(f0ii+G1%F,$\"3u4uf0cH\\;F,7$$\"3;fORr]') *=%F,$\"3xf$RG3EE$>F,7$$\"3R&p%*z[LbK%F,$\"3)f6tI$Gnq@F,7$$\"3'epgGO,q Q%F,$\"3OS+3N(G'\\AF,7$$\"3M'pExBp%[WF,$\"3%QJ*yerc2BF,7$$\"3oPwB\\&p? 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" }}{PARA 0 "" 0 "" {TEXT -1 88 "A single root is associated with a sign change \+ and is not also a root of the derivative " }{XPPEDIT 18 0 "`f '`(x)" " 6#-%$f~'G6#%\"xG" }{TEXT -1 96 ", while a double root is not associate d with a sign change and is also a root of the derivative " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> \+ exp(-cos(x))-exp(1)-2*cos(2*x)+2:\n'f(x)'=f(x);\nallroots(f(x),x=0..2* Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,*-%$expG6#,$-%$ cosGF&!\"\"\"\"\"-F*6#F0F/*&\"\"#F0-F.6#,$*&F4F0F'F0F0F0F/F4F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+&>h3P)!#5$\"+aEfTJ!\"*$\"+)=*4YaF( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Some details concerning the calculation can be obtained via the option \" " }{TEXT 262 9 "info=true" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "f := x -> exp(-cos (x))-exp(1)-2*cos(2*x)+2:\n'f(x)'=f(x);\nallroots(f(x),x=0..2*Pi,info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,*-%$expG6#,$- %$cosGF&!\"\"\"\"\"-F*6#F0F/*&\"\"#F0-F.6#,$*&F4F0F'F0F0F0F/F4F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"\"!F&$\" 3['ezrI&=$G'!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~inte rval:~G;$\"3&4=jFPE[\"f!#=$\"3B$z*e`EfTJ!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%TAttempting~to~ locate~a~single~root~in~the~interval~G;$\"3&4=jFPE[\"f!#=$\"3U3QA(**>k f*F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2single~root~->~~~G$\"3()*y:[ >h3P)!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3U3QA(**>kf*!#=$\"3B$z*e` EfTJ!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G; $\"3*>5IE6bq&=!#<$\"3iZ*4J)QK*\\#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8Checking~the~interval:~G;$\"3iZ*4J)QK*\\#!#<$\"3B$z*e`EfTJF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3eoW_nw[ &*H!#<$\"3\"48d0;S&oIF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~ the~interval:~G;$\"3\"48d0;S&oI!#<$\"3B$z*e`EfTJF'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for ~a~possible~double~root~in~the~interval~G;$\"3>LMD_g$49$!#<$\"3B$z*e`E fTJF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%TAttempting~to~locate~a~single~root~in~the~interval~G; $\"3>LMD_g$49$!#<$\"3B$z*e`EfTJF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% +root~->~~~G$\"3B$z*e`EfTJ!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3B$z*e `EfTJ!#<$\"3PoK!*pEq\"p&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%TAttempting~to~locate~a~single~root~i n~the~interval~G;$\"3h0sX2LaB`!#<$\"3PoK!*pEq\"p&F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%2single~root~->~~~G$\"3[2!)p(=*4Ya!#<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checkin g~the~interval:~G;$\"3B$z*e`EfTJ!#<$\"3h0sX2LaB`F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3B$z*e`EfTJ!#<$\"3U*\\B 0)zcKUF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G ;$\"3B$z*e`EfTJ!#<$\"3Z)=$y_Mw(e$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%9reflecting~in~the~originG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root ~in~the~interval~G;$!30T4VxcPhK!#<$!3B$z*e`EfTJF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a ~possible~double~root~in~the~interval~G;$!3*z^qd4%RrJ!#<$!3B$z*e`EfTJF '" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~in~the~interval~G;$!3C$z*e `EfTJ!#<$!3B$z*e`EfTJF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%+root~->~~ ~G$!3C$z*e`EfTJ!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3Z)=$y_Mw(e$!#<$ \"3q$ew>DMR.%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~inte rval:~G;$\"3U*\\B0)zcKU!#<$\"3h0sX2LaB`F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"+&>h3P)!#5$\"+aEfTJ!\"*$\"+)=*4YaF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "We can also find the zeros " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 17 " i n the interval " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#Pi GF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := x -> exp(-cos(x))-exp(1)-2*cos(2*x)+ 2:\nDf := D(f):\n'`f '`(x)'=Df(x);\nplot(Df(x),x=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$f~'G6#%\"xG,&*&-%$sinGF&\"\"\"-%$expG6#, $-%$cosGF&!\"\"F,F,*&\"\"%F,-F+6#,$*&\"\"#F,F'F,F,F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 445 282 282 {PLOTDATA 2 "6%-%'CURVESG6$7^q7$$\"\"!F)F(7 $$\"3;`#Gi#zxZo!#>$\"3EUS!y>AMr&!#=7$$\"3i]cC&eb&p8F0$\"3%>C)e>NpK6!#< 7$$\"3q5)>63x`'>F0$\"3G,h9e$e`g\"F67$$\"3YqR*pd)>hDF0$\"33Jt;$[2o0#F67 $$\"3Y@%HT;i7B$F0$\"3?N=$\\T8=`#F67$$\"3Zs[E^dK,RF0$\"3A:Gsk\")pkHF67$ $\"3+9+\"[&4$ed%F0$\"3qg-,**zy]LF67$$\"3ab^NehL]_F0$\"3O=#GYm53o$F67$$ \"3Ls#3kbN;#fF0$\"3e8wG/8V[RF67$$\"35*QhW&\\$Hf'F0$\"3-(\\B&o@M^TF67$$ \"3]:5wGaJ:sF0$\"3S/.mH'z\"zUF67$$\"3zS11.fpPyF0$\"3%=T*oTCmZVF67$$\"3 'y&R#fj0))*zF0$\"31'p^Ar`cN%F67$$\"3'QF(yo`\"*f\")F0$\"3A&Q2D\"HkfVF67 $$\"3%)*e];5D5K)F0$\"3u%QEjNQ'fVF67$$\"3#e!R^M[8#[)F0$\"3u!QW4d_cN%F67 $$\"3yP0C+VN/))F0$\"3iK?1!Q3eL%F67$$\"3upr'fwtl7*F0$\"3FzB)40%H+VF67$$ \"3!Rbb2V`Iz*F0$\"3+)exoRd#yTF67$$\"3!QRa&4L&f/\"F6$\"3%zA.,)fR$*RF67$ $\"3YjXwg<#)y6F6$\"3cz;&=9\\]X$F67$$\"36qGW%H$\\:8F6$\"3ZyDNx6Z1FF67$$ \"3SH22vMov8F6$\"3Gy5N)[<(HBF67$$\"3$*)e)pbO(eV\"F6$\"36K`p2g_K>F67$$ \"3/]+3VNj.:F6$\"31Z@I#*)p'o9F67$$\"396:YIMRr:F6$\"3k)4EOv-#e**F07$$\" 3Z'z]kaJ%R;F6$\"3BQ.6c:26_F07$$\"3!=3SCmpuq\"F6$\"3!4P[)fK%ea&F-7$$\"3 3u?N%*p.tF6$!3Q-%f&)4:[W\"F67$$\"3#=uye<)G*4#F6$!3ehF!>T'o`?F67$$\"3! zz^8\\l#f@F6$!3g0X/@@oWAF67$$\"3Ua[#o!GC>AF6$!3m/msYF(GR#F67$$\"3W_![ \\uETD#F6$!3%HLHR@)*)eCF67$$\"3W]72$o5!*G#F6$!3&=jV7BK*4DF67$$\"3,[W>@ Y*QK#F6$!3Q1[\\!*p$fa#F67$$\"3-YwJf&y(eBF6$!3Q$\\wyo]pc#F67$$\"3w=DTbn l*Q#F6$!312d#4'>8tDF67$$\"31\"R2:&\\`?CF6$!3mjz*z*)yxc#F67$$\"3OjAgZJT ^CF6$!37Yqd*ee5b#F67$$\"3oNrpV8H#[#F6$!3t&>u\\!*yJ_#F67$$\"3C24z$*y/]D F6$!3nuHP;f:CCF67$$\"3EzY)QW/yh#F6$!3#eaU\")RgeF#F67$$\"35v)>8'\\%ou#F 6$!3]tS)*R4*>(=F67$$\"3w$GCS?&[\")GF6$!3?Bt`%y\"H78F67$$\"3Z%)='p(p70I F6$!3%=rF07$$\"3sA%)\\K8\\QJF6$!3i>o:$*)>!Q;F-7$$\"3c&fJlT>qF$ F6$\"3?OUiO#pb1(F07$$\"39l2\"4_3wR$F6$\"3kD@0+d_$H\"F67$$\"3MOnGd![y_$ F6$\"3$ybb^PP/%=F67$$\"3#*=4JU#)RiOF6$\"3![Gtp&)f#oAF67$$\"36m\\Q&z8#G PF6$\"3`RP(oD'*\\T#F67$$\"3%G,f%[$HSz$F6$\"3@!HkS!))[:DF67$$\"3%p#\\M9 $pe#QF6$\"3_tukYYeYDF67$$\"3eS3B!G4x&QF6$\"3A\"[B%pB'ec#F67$$\"3Aan6Y# \\&*)QF6$\"31wLqIk3tDF67$$\"3KoE+7#*Q@RF6$\"3u[#Qm2n!oDF67$$\"3o+Mbl?u cRF6$\"3KS!>H$*yza#F67$$\"31LT5>\\4#*RF6$\"3r/a@P+]7DF67$$\"3)\\'[lsxW FSF6$\"3!yY8;Y&ehCF67$$\"3y(f0ii+G1%F6$\"3E.*GYLq_R#F67$$\"3ZG'*z[GLET F6$\"3;'4LPQ9yB#F67$$\"3;fORr]')*=%F6$\"3N`N`CTSK?F67$$\"3R&p%*z[LbK%F 6$\"3U#Qy>gPYW\"F67$$\"3'epgGO,qQ%F6$\"352MI:0x=6F67$$\"3M'pExBp%[WF6$ \"35Z'pmyIch(F07$$\"3,z&[2')pc^%F6$\"3CRG6kn`5MF07$$\"3oh/x$[qGe%F6$!3 R!*y+$o'QP5F07$$\"3qQq'H.,hk%F6$!3'*3&)eIP%QP&F07$$\"3e;O;#eJ$4ZF6$!33 Swd0N)fy*F07$$\"3sU'G^sDax%F6$!3:u*GWOU+W\"F67$$\"3&)oO4o)>:%[F6$!3))y mEBdX$*=F67$$\"3;@9vt*Qh!\\F6$!3h6F#)er#4K#F67$$\"3Ou\"4%z!e2(\\F6$!3$ \\[&zJ_+DFF67$$\"3*zX#[4&eg5&F6$!3i>60jf?jMF67$$\"3n*e![/*ojB&F6$!3C=q &oB21*RF67$$\"3Dtq\\/&**HI&F6$!3!46@cs;i<%F67$$\"3#ob8X5I'p`F6$!3w&HI3 N#3*H%F67$$\"3@)>(3&QBK% fVF67$$\"37-jF=\"o_[&F6$!3?r&H=6%*[N%F67$$\"3PA\"GX$yy,bF6$!3cw,-Kv9YV F67$$\"3wxVy[u]ibF6$!31>O]Pd#yF%F67$$\"39L1/jqABcF6$!3/*HJTW qe;E`fF6$!3=m[4w8xxDF67$$\"3nKcu0ii>gF6$!3j:'Rr%eQ6@F67$$\"3G:=>Xb9$3' F6$!3]E%yXSRDj\"F67$$\"3w)*zj%)[mYhF6$!3-\\$=Msy\"H6F67$$\"3P***G'*3D \\@'F6$!3SN$HMhv`p&F07$$\"3)****>YH&=$G'F6$!3!G^Lq0r40\"!#C-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fhjl-%%VIEWG6$;F($\"+3`= $G'!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "f := x -> exp(-cos(x))-exp(1)-2*co s(2*x)+2:\nDf := D(f):\n'`f '`(x)'=Df(x);\nallroots(Df(x),x=0..2*Pi); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$f~'G6#%\"xG,&*&-%$sinGF&\"\"\" -%$expG6#,$-%$cosGF&!\"\"F,F,*&\"\"%F,-F+6#,$*&\"\"#F,F'F,F,F,F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6'$\"\"!F$$\"+(\\Cdr\"!\"*$\"+aEfTJF'$\" +53YnXF'$\"+2`=$G'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }{TEXT 289 65 " .. a function which has zeros which are cl ose together in pairs " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x) = tan(sin(x))+95/61;" "6#/-%\"fG6#%\"xG,&-%$tanG6 #-%$sinG6#F'\"\"\"*&\"#&*F/\"#h!\"\"F/" }{TEXT -1 46 " has zeros which are close together in pairs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f := x -> tan(sin(x))+95/61: \n'f(x)'=f(x);\nplot(f(x),x=0..6*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$tanG6#-%$sinGF&\"\"\"#\"#&*\"#hF." }}{PARA 13 " " 1 "" {GLPLOT2D 559 350 350 {PLOTDATA 2 "6%-%'CURVESG6$7]q7$$\"\"!F)$ \"3#yK!=\\qPd:!#<7$$\"3QNG(yPLV0#!#=$\"3'>9-)))eCkF,7$$\"3sV>PV78'*eF0$\"3C&[N*)*H#)y@F,7$$\"39;#)*4t&f$o(F 0$\"3KVWeD/3\"R#F,7$$\"3iM)3C\\'y$p*F0$\"3a*)Q[&o!))QEF,7$$\"3IX>QDxRq 6F,$\"3#\\*)3(396tGF,7$$\"3)p\"fW'G\\FP\"F,$\"3Wk%[*Q`z\\IF,7$$\"3)))) )4v%35v:F,$\"3i-nzQIv9JF,7$$\"3Y^i#pm!\\wF,$\"3))fy(ysif'GF,7$$\"3'y*GjGYfk@F,$\"3)o$e)\\qD#[EF,7$$\"3o \"=B4x38N#F,$\"3Sn#*3:y'yT#F,7$$\"3?r&f.XSYa#F,$\"3g+g!Q'**G(=#F,7$$\" 3sgfzH@(zt#F,$\"3U#*e8g3lr>F,7$$\"3_+HBHg\"z$HF,$\"3c,!Rg#GXi,BGlk`$F,$\"3/vn!*e^^_6F,7$$\"3z*3>Ges9u$F,$\"3e$\\vRw,3C*F07$$\"3_% )pL$))zk%RF,$\"3]tk`H&p(*y'F07$$\"37Y4AD/0FTF,$\"3C\")R.U[uhXF07$$\"3; 3\\5n4i2VF,$\"3AwgW[q2jCF07$$\"3-A(\\#H1!4^%F,$\"34!>EJWWgr'!#>7$$\"3w OXR\"H!=9ZF,$!3D/kWr&fz^#!#A7$$\"3HDGORYH=\\F,$\"3<_\"p`%=Y.qF[s7$$\"3 s96L()*3C7&F,$\"3g5O50<]>DF07$$\"3S4v1$)46>`F,$\"3\\iv`A+*Q#[F07$$\"34 /R!)yH\"e^&F,$\"3K3w6X#o([sF07$$\"3#R^?`BPWp&F,$\"3gpxOwgFp$*F07$$\"3! H7P=\\hI(eF,$\"3ZAE)4V)3O6F,7$$\"3=cLu4IY&3'F,$\"3gt&RDEu$e8F,7$$\"3[* e\\w_kyH'F,$\"3=O=\"ozc?d\"F,7$$\"3%49pSZ'zxkF,$\"35^q>Y%4Kv\"F,7$$\"3 _\"p)[?%Gxl'F,$\"3CJc400ZS>F,7$$\"3yA$G#\\?.noF,$\"35*)Qhri=s@F,7$$\"3 /az'znNj2(F,$\"3(4mM?Fz8U#F,7$$\"3K\"ePX&[ghsF,$\"3GKTd!=n*\\EF,7$$\"3 e3s5JS(oW(F,$\"3!Hu\\5bPf'GF,7$$\"3%e&*)Q\"oV,l(F,$\"3M=_wc:2YIF,7$$\" 35.2nJLT`yF,$\"3ZW(p1A%y9JF,7$$\"3w1R#y5up/)F,$\"3'>\")oR:pH0$F,7$$\"3 ?7r(R)[`S#)F,$\"3=XZsf7r()GF,7$$\"3>aT.[_\\U%)F,$\"35!)pw)\\'paEF,7$$ \"3<'>\"47cXW')F,$\"31U\\074o0CF,7$$\"3O\"*z\\r#=*H))F,$\"3yE(G!oN([=# F,7$$\"3w%y/4$4Q:!*F,$\"3iGe\"Q&4)y(>F,7$$\"3C;+@kuU:#*F,$\"3[y-kQjCo< F,7$$\"3sZ_^(*RZ:%*F,$\"3u!ojm9\"om:F,7$$\"3'oWlN7mKi*F,$\"3ALo3BNfd8F ,7$$\"3yZch\\#e5$)*F,$\"3--&*)Q7O-9\"F,7$$\"3hy%=1>%>,5!#;$\"3%evxQ[;v Q*F07$$\"3W#Rvib#G>5Fg[l$\"37\"zC\"3!R0C(F07$$\"3iH=t'[=)Q5Fg[l$\"3M#R 'GnO4K[F07$$\"3!oE)=zR\"p>TDF07$$\"3/P>%\\%p`y5Fg[l$ \"3[z')oelL)G(F[s7$$\"352cps%>()4\"Fg[l$\"3;4@FZN@e*)Fas7$$\"39jyhQTY= 6Fg[l$\"3qE=BLL)p$fF[s7$$\"3<>,a/)3#Q6Fg[l$\"3C%R\"z=g,qAF07$$\"3'zqrS y7t:\"Fg[l$\"3iip.A!*omWF07$$\"3s'H.Ow;k<\"Fg[l$\"3)3(>\\cbsAoF07$$\"3 =\"yLdZGw>\"Fg[l$\"3U4pijR'QN*F07$$\"3ilU'y=S)=7Fg[l$\"3K-(*y]biq6F,7$ $\"3L:Dka)**yB\"Fg[l$\"3iV'HDa9*o8F,7$$\"3/l2U@&fpD\"Fg[l$\"3!=0F,7$$\"3b,5')3/5;8Fg[l$\"3i*)**)*[Ch%=#F,7$$\"3xS3Kr2aM8Fg[l$\" 3*Rk*y7/6/CF,7$$\"3Q[uAe4qa8Fg[l$\"3!*=>HB[o_EF,7$$\"3+cS8X6'[P\"Fg[l$ \"33kX4wuh&)GF,7$$\"3UpI*)4.$QR\"Fg[l$\"3kfplZdF\\IF,7$$\"3%G3_YZ*z79F g[l$\"3T*\\$G\"pSY6$F,7$$\"3hK;aF,7$$\"3G\\x47TO^:Fg[l$\"3Aa2J:]\"H v\"F,7$$\"3]5vMr1\"4d\"Fg[l$\"3[>]S.OBc:F,7$$\"3%z\\_8&)**3f\"Fg[l$\"3 !)zna]h*\\N\"F,7$$\"3Q&[d8.*)3h\"Fg[l$\"3o^%>C4/g9\"F,7$$\"3J7(f3p72j \"Fg[l$\"37H<3mT3\\#*F07$$\"3CR>O]j`];Fg[l$\"3I\"Q\\\"f4l\")oF07$$\"3U a)QYB_(o;Fg[l$\"3F-?1!y8Rj%F07$$\"3epd\"*=\"opo\"Fg[l$\"3vNdQ#zd$4DF07 $$\"3Y+'ezdXyq\"Fg[l$\"3$R\"G0&[)))RmF[s7$$\"3NJ9+PIsG_F<'y)e!=Fg[ l$\"3;AP]%fWS'pF07$$\"3S<9cjO%\\#=Fg[l$\"34>)\\'=O.Q#*F07$$\"3'GJ&Rl%* *R%=Fg[l$\"3&4i!=LepO6F,7$$\"3ccE*o_xW'=Fg[l$\"3wfc%>Zu6N\"F,7$$\"3!** ***Q)eb\\)=Fg[l$\"3]n:a6qPd:F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESL ABELSG6$Q\"x6\"Q!Ffjl-%%VIEWG6$;F($\"+#fb\\)=!\")%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The zeros of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " in the interval \+ " }{XPPEDIT 18 0 "[0,6*Pi]" "6#7$\"\"!*&\"\"'\"\"\"%#PiGF'" }{TEXT -1 31 " can be calculated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "f := x -> tan(sin(x))+95/61 :\n'f(x)'=f(x);\nallroots(f(x),x=0..6*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$tanG6#-%$sinGF&\"\"\"#\"#&*\"#hF." } }{PARA 11 "" 1 "" {XPPMATH 20 "6($\"+Ip:3Z!\"*$\"+J5i;ZF%$\"+CU8*4\"! \")$\"+M1)**4\"F*$\"+aFXF " 0 "" {MPLTEXT 1 0 234 "evalf(evalf[18](fsolve(f(x),x=4.71)));\neval f(evalf[18](fsolve(f(x),x=4.72)));\nevalf(evalf[18](fsolve(f(x),x=10.9 9)));\nevalf(evalf[18](fsolve(f(x),x=11)));\nevalf(evalf[18](fsolve(f( x),x=17.27)));\nevalf(evalf[18](fsolve(f(x),x=17.28)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+Ip:3Z!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+J5i;Z!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+CU8*4\"!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M1)**4\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aFXF " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4 " }{TEXT 290 34 " .. a function with double roots " }}{PARA 0 "" 0 " " {TEXT -1 26 "The zeros of the function " }{XPPEDIT 18 0 "f(x) = tan( sin(x))+tan(1);" "6#/-%\"fG6#%\"xG,&-%$tanG6#-%$sinG6#F'\"\"\"-F*6#F/F /" }{TEXT -1 19 " are also zeros of " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$ f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f := x -> tan(sin(x))+tan(1) :\n'f(x)'=f(x);\nplot(f(x),x=0..4*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$tanG6#-%$sinGF&\"\"\"-F*6#F.F." }}{PARA 13 " " 1 "" {GLPLOT2D 559 350 350 {PLOTDATA 2 "6%-%'CURVESG6$7cr7$$\"\"!F)$ \"33-\\lCxSd:!#<7$$\"3o#e#[q66RF!#=$\"3v*zD([HqM=F,7$$\"3;4rAiTvIRF0$ \"3W'>f>OR.'>F,7$$\"3>O;(R:(RA^F0$\"3-wIR$zj44#F,7$$\"3ss#Q#GV_ikF0$\" 3:2CC8txWAF,7$$\"383\\]-:l-yF0$\"3+m^\"*F0$\"3 MNEnZe/sDF,7$$\"3o)on;Bn+0\"F,$\"34-#**\\FMmt#F,7$$\"3e%)yF6rK%=\"F,$ \"3%)3#>\"z'Hy)GF,7$$\"3^!3))3*pe=8F,$\"3/(zMH()*\\6IF,7$$\"33XyJQ]#3Q \"F,$\"3%eo9F]i[0$F,7$$\"3W4wu&3jIW\"F,$\"3'ypZ+Yds3$F,7$$\"3i\"\\i%4@ =u9F,$\"3)G!3)\\`b*)4$F,7$$\"3!QPxJ8,`]\"F,$\"3[iP1>a\\2JF,7$$\"3>cA*o :?k`\"F,$\"3]%fa\\]$z7JF,7$$\"3QQrg!=Rvc\"F,$\"3b(=4)ysz9JF,7$$\"3?*pz r7h(*f\"F,$\"3MCoj^&zL6$F,7$$\"3#)fAvtI)>j\"F,$\"3Ppi]5QU3JF,7$$\"3U?[ K?]?k;F,$\"3I/D&=y#)**4$F,7$$\"3-\"Q(*o'pU'p\"F,$\"3?&))>w?X\")3$F,7$$ \"3E-D/g3(3w\"F,$\"3g_b.v#*zaIF,7$$\"3rBw=`ZJD=F,$\"3:5F,$\"3K()3%H(QU')GF,7$$\"3%*G@5>m!>4#F,$\"3c%o.@&3=OFF,7$$ \"3gv=Jq]xCAF,$\"3oB?I,!*3uDF,7$$\"3EA;_@NkdBF,$\"3v79?Sc95CF,7$$\"39% \\*>b]J%\\#F,$\"3Ujk![l#)fC#F,7$$\"3flt())e')4j#F,$\"3%>5YU*\\8*3#F,7$ $\"3Q,F8]pO^FF,$\"3O4g/R5Gd>F,7$$\"3uO!)Q6turGF,$\"3D/&4!H**[I=F,7$$\" 3t=I\"4'oyUJF,$\"3GFy/!\\\"H'z$F,$\"3$3LS@aY4g)F07$ $\"3)QkqyKu`\"RF,$\"3UIdF(p:4<(F07$$\"3y\"Q2)R`(p0%F,$\"36t/nXt,DaF07$ $\"3C>Tu^jd)>%F,$\"3')zerflr*p$F07$$\"3O])*o#)4`=VF,$\"3'GY]dD7uM#F07$ $\"3e!eNOh&[QWF,$\"3&ed\"3'H7u?\"F07$$\"3ca<))*[`#3XF,$\"3Iiv(oF02*o!# >7$$\"3kFz7m8-yXF,$\"3gZz`bY,XIF[x7$$\"33:5D/`!Hh%F,$\"3;bQH3:z!o\"F[x 7$$\"3i,TPU#*yZYF,$\"33'p@\"o/&>7(!#?7$$\"3;)=(\\!=tEo%F,$\"3PMt'e`l7^ \"F[y7$$\"3gv-i=rb5[F,$\"3#*HN-n8$ \\i\"F[x7$$\"3Bs$**H!*p5%[F,$\"3yk#3j]miz#F[x7$$\"3b?*)=&HEG!\\F,$\"3t y+'QNeQ-'F[x7$$\"3')o%yto#ek\\F,$\"3(H&fg-V.L5F07$$\"3C\"elvy&4+^F,$\" 3Ac`;K2z#G#F07$$\"3i$p_x))3cB&F,$\"3LF]\"y!pX6QF07$$\"3Cyu=0%\\YO&F,$ \"3q%RlXL2gQ&F07$$\"3ujAiA**o$\\&F,$\"3;Wot#=D)ypF07$$\"3UViKl,LGcF,$ \"33&e\"o`P5*f)F07$$\"3)RAI!3/(Hw&F,$\"3RDR_))z!\\,\"F,7$$\"3yIu'4=7m) eF,$\"3')[wX*R=2:\"F,7$$\"3dPY!R&RD5gF,$\"3E'\\N99G6G\"F,7$$\"3'\\'o( \\m#)pF'F,$\"3#>F!oU]?^:F,7$$\"3qGB/L)QSb'F,$\"3=3orBU`J=F,7$$\"3996UP ziumF,$\"3!3;q))p'ee>F,7$$\"3[+**zTq@&z'F,$\"3q\"RQr!4t!4#F,7$$\"3Mda< ylXDpF,$\"34%*)H_Z+,C#F,7$$\"3?95b9hpbqF,$\"3%[mV+?thR#F,7$$\"3ooZd*HY ->(F,$\"3iv*yM0:?c#F,7$$\"3HA&)f%['zCtF,$\"3frcFOMbEFF,7$$\"3W(>Y2fFkX (F,$\"3#f1Qy^kh(GF,7$$\"3esQ*ope!)e(F,$\"3z![H\"4Vh+IF,7$$\"3O(\\l'G'Q '3$F,7$$\"3-!4%GXb* [0)F,$\"3+W6MLT*z/$F,7$$\"3/I`Q_7gD\")F,$\"3e^shP4'f*HF,7$$\"3S')Hd(pl ED)F,$\"3KfM$RAl\\(GF,7$$\"3aW1wU,tz$)F,$\"32+&=%[glIFF,7$$\"3B[7Of&)R :&)F,$\"3uR0YwY)[c#F,7$$\"3q`='f(p1^')F,$\"3'p2*R9wj(R#F,7$$\"3%QY$pDF +u()F,$\"3o^+.'>)4]AF,7$$\"3?s]Uv%Qp*))F,$\"3K^y_sD[3@F,7$$\"3[[<^n4ul \"*F,$\"3t9#4xy5$>=F,7$$\"3\"=D(HkJm=%*F,$\"3gRHBfD_j:F,7$$\"3O9l:O(RI o*F,$\"3Q5-uT`I'H\"F,7$$\"3U5;ZZzF7)*F,$\"37Df(R)pXg6F,7$$\"3p/nyeh^T* *F,$\"3bF8Vm>tm\"F[x7$$\"3QMLv*[,D6\"Fehl$\"3P=n%pH5*GGF[x7$$\" 3cS*y6Xt&=6Fehl$\"3oV$=z[no+'F[x7$$\"3uYXg7akC6Fehl$\"3k\"Fehl$\"3PTY>xm>S&)F07$$\"3A%HX6CDR?\"Fehl$\"3 Q%G$*>6Xr+\"F,7$$\"3g!\\M!4\"Hm@\"Fehl$\"3uGJK-q%p9\"F,7$$\"3!ooBp(HLH 7Fehl$\"3#)QWn_Y,\"G\"F,7$$\"3+++#*eqjc7Fehl$\"35Hd\\*p2ub\"F,-%'COLOU RG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Ffam-%%VIEWG6$;F($\"+ iqjc7!\")%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 13 "The zeros of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,4*Pi]" "6#7$ \"\"!*&\"\"%\"\"\"%#PiGF'" }{TEXT -1 29 " can be computed as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> tan(sin(x))+tan(1):\n'f(x)'=f(x);\nallroots(f(x),x=0 ..4*Pi,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$ tanG6#-%$sinGF&\"\"\"-F*6#F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Ch ecking~the~interval:~G;$\"\"!F&$\"3I@Nce8bja!#=$\"3Oqjdc=4C j!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\" 3G=pU2mL#=\"!#<$\"3$eclG2]x^$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8C hecking~the~interval:~G;$\"3$eclG2]x^$!#<$\"3Q8UIQN;`eF'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3*QTn9Gol9%!#<$\"3 +\"e%oG&*f#p%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~inte rval:~G;$\"3+\"e%oG&*f#p%!#<$\"38[~~~G$\"3G*o%Q!)*)Q7Z!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3G* o%Q!)*)Q7Z!#<$\"3s:\"\\#e3xdZF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9r eflecting~in~the~originG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~ in~the~interval~G;$!3ggEzY'\\Dr%!#<$!3G*o%Q!)*)Q7ZF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~fo r~a~possible~double~root~in~the~interval~G;$!3')*o%Q!)*)Q7Z!#<$!3G*o%Q !)*)Q7ZF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%+root~->~~~G$!3d*o%Q!)*) Q7Z!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3s:\"\\#e3xdZ!#<$\"3*p=W \")pfJ![F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval: ~G;$\"3Oqjdc=4Cj!#<$\"3')Q'Hx&G=57!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3[n!R\\_#ewo!#<$\"3WUU.M]r)H*F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3WUU.M]r )H*!#<$\"3uTHJaZ3s6!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~t he~interval:~G;$\"3K[c1e'y<'**!#<$\"3YUK`;qTi5!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3YUK`;qTi5!#;$\"34?*fs;c' G6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~in~the~interval~G ;$\"3\"[Zghk0A4\"!#;$\"3d&Q'oI5M.6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~dou ble~root~in~the~interval~G;$\"3AePT@`b*4\"!#;$\"3Iojt&*oW,6F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%UChecking~for~a~possible~double~root~in~the~interval~G;$\"3u%4&3vjb* 4\"!#;$\"3T(G=*Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~ in~the~interval~G;$\"3jFkvGub*4\"!#;$\"3T(G=*Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~ for~a~possible~double~root~in~the~interval~G;$\"3jFkvGub*4\"!#;$\"3_dt $)Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~in~the~interval ~G;$\"3jFkvGub*4\"!#;$\"3e#*ozGub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible ~double~root~in~the~interval~G;$\"3jFkvGub*4\"!#;$\"35gmxGub*4\"F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%UChecking~for~a~possible~double~root~in~the~interval~G;$\"3jFkvGub*4 \"!#;$\"3'Qam(Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~ in~the~interval~G;$\"3jFkvGub*4\"!#;$\"3u&[h(Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~ for~a~possible~double~root~in~the~interval~G;$\"3jFkvGub*4\"!#;$\"3oc* e(Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%UChecking~for~a~possible~double~root~in~the~interval ~G;$\"3jFkvGub*4\"!#;$\"3;#pd(Gub*4\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%UChecking~for~a~possible ~double~root~in~the~interval~G;$\"3jFkvGub*4\"!#;$\"3!*fqvGub*4\"F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%+root~->~~~G$\"3wVnvGub*4\"!#;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $$\"+!)*)Q7Z!\"*$\"+Hub*4\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Check " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf(evalf[12](3*Pi/2));\ne valf(evalf[12](7*Pi/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!)*)Q7 Z!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Hub*4\"!\")" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Alternatively, we \+ can look for zeros of " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" } {TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := x - > tan(sin(x))+tan(1):\nallroots(D(f)(x),x=0..4*Pi);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&$\"+Fjzq:!\"*$\"+!)*)Q7ZF%$\"+M;)R&yF%$\"+Hub*4\"!\") " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The \+ procedure " }{TEXT 0 9 "allbisect" }{TEXT -1 28 " can also be used for this. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := x -> tan(sin(x))+tan(1):\nallbisect(D(f)(x),x=0. .4*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$\"+Fjzq:!\"*$\"+!)*)Q7ZF%$ \"+M;)R&yF%$\"+Hub*4\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }{TEXT 312 34 " .. a function with a double roo t " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(x) = exp(x)-(x+1)^2*(1-x);" "6#/-%\"fG6#%\"xG,&-%$exp G6#F'\"\"\"*&,&F'F,F,F,\"\"#,&F,F,F'!\"\"F,F1" }{TEXT -1 25 " has a do uble root at 1. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "allroots(g(x),x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!++X#=o\"!#5$\"+!HM(4^F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(evalf[12](allro ots(g(x),x=-1..1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++X#=o\"!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(evalf[15](allroots(g(x),x=-1..1)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!++X#=o\"!#5$\"+%HM(4^F%" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The double root of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " is also a root o f " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allroots(D(g)(x),x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+ /\\^?c!#6$\"+%HM(4^!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }{TEXT 291 41 " .. a piecewise differentiable f unction " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "allroots" }{TEXT -1 28 " can cope with the function " }{XPPEDIT 18 0 "f(x)=abs(sin(x)+x)-1" "6#/-%\"fG6#%\"x G,&-%$absG6#,&-%$sinG6#F'\"\"\"F'F0F0F0!\"\"" }{TEXT -1 30 " which has no derivative when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". 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()\\t.\"F*$\"32Tw=.R#[U&F-7$$\"3f;H2qcZP5F*$\"3+)yF>9#Q=WF-7$$\"3*)*\\ 7.lQx.\"F*$\"3?ui%4%)Rad#F-7$$\"3T$3_0j,!Q5F*$\"3'y(H)yq[K<\"F-7$$\"3r m;z5YEQ5F*$\"3HQn@/MXUHFhv7$$\"3,]7.\"fF&Q5F*$!3zbg%\\qA;.\"F*7$$\"3=a QG-ZyQ5F*$\"3Y%fM1p$=REFhv7$$\"3Nek`8=/R5F*$\"3K+![<(QV(3\"F-7$$\"3_i! *yC*)HR5F*$\"3#[?*\\ur%HT#F-7$$\"3pm;/OgbR5F*$\"3emg%p)y)\\;%F-7$$\"3n *[o=i!pR5F*$\"3R=#[mzh'>_F-7$$\"3k7`p2_#)R5F*$\"3M@:!o)=!zM'F-7$$\"3hN @_$zf*R5F*$\"3>bxxk^>KvF-7$$\"3Pe*[$zV4S5F*$\"3[`3KEW=a()F-7$$\"38\"yv ^'*G-/\"F*$\"3UW\\'G_`\\***F-7$$\"35/E+^NOS5F*$\"3'[yh,1KN7\"FV7$$\"33 F%Ho8)\\S5F*$\"3[oguwMhX7FV7$$\"30]ilAFjS5F*$\"3'\\8>y:dQO\"FV7$$\"3[& Q.[Mi2/\"F*$\"3ck/^]4Ts9FV7$$\"3\"4_]p'>*3/\"F*$\"3G?X-*zmTd\"FV7$$\"3 Mcw4*e@5/\"F*$\"3yT6gD_nn;FV7$$\"3x\"zW7@^6/\"F*$\"3mpcx)Q1;v\"FV7$$\" 3ji!RbX59/\"F*$\"3%*>\\'HUEh)=FV7$$\"3FLL$)*pp;/\"F*$\"3Ul*z5QU,(>FV7$ $\"3,Fp^yftT5F*$\"3Ys$o@DzG)>FV7$$\"3'4_+sD-=/\"F*$\"3KGSM&yX>*>FV7$$ \"3q9T)e`o=/\"F*$\"37,v^#e4t*>FV7$$\"3V3xc9[$>/\"F*$\"3iN!He1_*)*>FV7$ $\"3<-8D$4,?/\"F*$\"3KMDdM&oo*>FV7$$\"3!f*[$>Pn?/\"F*$\"3]![e.FV7 $$\"3k*[=1lL@/\"F*$\"3qa'o0*Rd\")>FV7$$\"3P$3-$H**>U5F*$\"3S(G[.vA%o>F V7$$\"3Jek.W]YU5F*$\"3?N`RfLX!)=FV7$$\"3DL3xe,tU5F*$\"3%o&>kh)Q.u\"FV7 $$\"3s$4'\\=;'G/\"F*$\"3ek![P?(*Rl\"FV7$$\"3=a8AyI*H/\"F*$\"3'R#='fH> \"e:FV7$$\"3l9m%z`CJ/\"F*$\"35$3F4i5TX\"FV7$$\"3!\\(=n(*fDV5F*$\"3_!*e *oT$\\V8FV7$$\"3:NrRduQV5F*$\"3ZgF(HH%)yA\"FV7$$\"3i&RAr\"*=N/\"F*$\"3 J[oi?0(*36FV7$$\"33cw%oP]O/\"F*$\"3]Glc-N&[))*F-7$$\"3b;HdO=yV5F*$\"3' >amj9F=o)F-7$$\"3(=b2Wj-R/\"F*$\"3_u9w6!))Hf(F-7$$\"3S(=UAVBS/\"F*$\"3 (e0$4Q_\"Q`'F-7$$\"3%H#o2IU9W5F*$\"3#HB9#[DG#[Z/\"F* $\"3seeA-7U#[\"F-7$$\"3pT5::^-X5F*$\"3Ku'4<\\8rN%Fhv7$$\"3U$3_5,-`/\"F *$\"3)yp$=s4;YH!#A7$$\"3uVt-N7PX5F*$!3$3I!3DlD/!*F^w7$$\"3G/E+f/WX5F*$ \"3qC*\\%3/F!=#F*7$$\"3%['y(Ho4b/\"F*$\"3c&>$=tQgd#*F*7$$\"3;DJ&p!*yb/ \"F*$\"3y(f-Wah,.#Fhv7$$\"3!ek.\\N6but/\"F*$\"3aCJ*H<_8F\"FV7$$\"3719>zl]Z5F*$\"3!H;fuP\"*[Q\"FV7$ $\"3u?IE2'Qw/\"F*$\"3%G%[kk>#G\\\"FV7$$\"3MNYLN1xZ5F*$\"3Kf(y#4%)e$f\" FV7$$\"3%*\\iSjE!z/\"F*$\"3Qc#[NiSdo\"FV7$$\"3)[7G))Rb\"[5F*$\"3NDf\"4 'zTL=FV7$$\"3#)***\\U83%[5F*$\"32>#\\XFpq$>FV7$$\"3GP4'>]M&[5F*$\"3a\" zt#)f+1(>FV7$$\"3wu=np3m[5F*$\"3%3@CU#HI\"*>FV7$$\"3W7GQPsy[5F*$\"3?] \"[YY7*)*>FV7$$\"3#*\\P40O\"*[5F*$\"3?2?;Z$QL*>FV7$$\"3U7.#Q?&=\\5F*$ \"3:TBO@.OP>FV7$$\"3#\\(oa-oX\\5F*$\"3rLEfBweC=FV7$$\"33c,\">g#f\\5F*$ \"3aQlR+*Q*[m\"FV7$$\"3!)=nj+U')\\5 F*$\"3y/@=So)[c\"FV7$$\"3%*************\\5F*$\"33\"RM(=2Hf9FV-%'COLOUR G6&%$RGBG$\"#5!\"\"$\"\"!F\\cqF[cq-%+AXESLABELSG6$Q\"x6\"Q!Facq-%%VIEW G6$;$\"\"\"!\"$$\"$0\"!\"&%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "As an example the zeros of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " in the interval 0.00" } {XPPEDIT 18 0 "1<=x" "6#1\"\"\"%\"xG" }{XPPEDIT 18 0 "``<=0" "6#1%!G\" \"!" }{TEXT -1 35 ".00105 can be computed as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x \+ -> exp(-x)-sin(1/x):\n'f(x)'=f(x);\nallroots(f(x),x=0.001..0.00105);\n sols := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%$exp G6#,$F'!\"\"\"\"\"-%$sinG6#*&F.F.F'F-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "60$\"+(3tc+\"!#7$\"+*zjd+\"F%$\"+G!o?,\"F%$\"+*=g@,\"F%$\"+=[a=5F% $\"+g%Q'=5F%$\"+P]5D5F%$\"+%>+_-\"F%$\"+4.vJ5F%$\"+:q%=.\"F%$\"+(H#[Q5 F%$\"+*e!eQ5F%$\"+6FIX5F%$\"+DESX5F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "Check using " }{TEXT 0 6 "fsolve" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "op(map(u->fsolve(sin(1/x)=exp(-x),x=evalf[6](u )),sols));" }}{PARA 12 "" 1 "" {XPPMATH 20 "60$\"+(3tc+\"!#7$\"+*zjd+ \"F%$\"+G!o?,\"F%$\"+*=g@,\"F%$\"+=[a=5F%$\"+g%Q'=5F%$\"+P]5D5F%$\"+%> +_-\"F%$\"+4.vJ5F%$\"+:q%=.\"F%$\"+(H#[Q5F%$\"+*e!eQ5F%$\"+6FIX5F%$\"+ DESX5F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }{TEXT 301 73 " .. a cubic function with three zeros which are close t ogether near zero " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(x)=x^3-x/10^5" "6#/-%\"fG6#%\"xG,&*$F'\"\"$\"\"\"*&F' F+*$\"#5\"\"&!\"\"F0" }{TEXT -1 53 " has three zeros which are close t ogether near zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f := x -> x^3-x/10^5:\n'f(x)'=f(x);\nplot(f (x),x=-0.005..0.005);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x G,&*$)F'\"\"$\"\"\"F,*&#F,\"'++5F,F'F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 559 350 350 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!35+++++++]!#?$! 3I-++++++v!#D7$$!3yLLLe%G?y%F*$!37qY6^nT`hF-7$$!3[mmT&esBf%F*$!31G%>t# )))G4&F-7$$!3CLL$3s%3zVF*$!3aJ:QbeT=SF-7$$!3YLL$e/$QkTF*$!3Sm-U4f_dIF- 7$$!3)pm;/\"=q]RF*$!3]yIQs7d:AF-7$$!3iLL3_>f_PF*$!3#>6-b!zyJ:F-7$$!3Q+ +vo1YZNF*$!3QdbD.7Lo\"*!#E7$$!3>LL3-OJNLF*$!3=0e%p>\\*\\PFQ7$$!3!)*** \\P*o%Q7$F*$\"3Aat%>e2ma(!#F7$$!3Lmmm\"RFj!HF*$\"3r2=Rf_G9XFQ7$$!3iAr&3LQ*FQ7$$!3))* ****\\ixCG#F*$\"3wS\">*=VP$4\"F-7$$!3'******\\KqP2#F*$\"3'\\()HAbS>=\" F-7$$!3ILL3-TC%)=F*$\"3VzE7HmE:7F-7$$!3qmmm\"4z)e;F*$\"3%oi.')\\vB?\"F -7$$!3cmmmm`'zY\"F*$\"3C&yPOBI;:\"F-7$$!39++v=t)eC\"F*$\"39QqB-j\\_5F- 7$$!3'ommmh5$\\5F*$\"34]PL0)fxL*FQ7$$!3a)***\\(=[jL)!#@$\"3)\\><>;;qv( FQ7$$!3k)***\\iXg#G'F\\r$\"3KfMMBJiMgFQ7$$!3!*fmmT&Q(RTF\\r$\"3C\"R\"p SSzoSFQ7$$!3Ehm;/'=><#F\\r$\"3kM0ljJnh@FQ7$$!35WILLe*e$\\!#B$\"3&3iwz! Q)e$\\!#G7$$\"3#*pm;zRQb@F\\r$!3Yl=ra2PX@FQ7$$\"3Y,+](=>Y2%F\\r$!3-@4u M-(p+%FQ7$$\"3%Gnm;zXu9'F\\r$!34k*\\f6F^\"fFQ7$$\"3#e******\\y))G)F\\r $!3X6`&>!pQ>xFQ7$$\"3A++]i_QQ5F*$!3C(QLW**>UE*FQ7$$\"3R++D\"y%3T7F*$!3 _9iZb:#*\\5F-7$$\"35++]P![hY\"F*$!394qb6j)4:\"F-7$$\"3QLLL$Qx$o;F*$!3F T^nn6AF-7$$\"3uML$3dg6<%F*$\"35m[PHn1'3$ F-7$$\"3+ommmxGpVF*$\"3%oY8Pwx>(RF-7$$\"3+++D\"oK0e%F*$\"3PA%G\"H5,I]F -7$$\"3!4+](=5s#y%F*$\"3-)>hz2wu:'F-7$$\"35+++++++]F*$\"3I-++++++vF--% 'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fe[lFd[l-%+AXESLABELSG6$Q\"x6\"Q!Fj[l- %%VIEWG6$;$!\"&!\"$$\"\"&Fb\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 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 64 "f := x -> x^3-x/10 ^5:\n'f(x)'=f(x);\nallroots(f(x),x=-0.01..0.01);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&#F,\"'++5F,F'F,!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+gwFiJ!#7$\"\"!F'$\"+gwFiJF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f := x -> x^3-x/10^5:\n'f(x)'=f(x);\nsolve(f(x));\nevalf(evalf[1 4](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$ \"\"\"F,*&#F,\"'++5F,F'F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\" !,$*&\"%+5!\"\"\"#5#\"\"\"\"\"#F*,$*&F&F'F(F)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"\"!F$$\"+gwFiJ!#7$!+gwFiJF'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "f := x -> x^ 3-x/10^5:\n'f(x)'=f(x);\nfsolve(f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&#F,\"'++5F,F'F,!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%$!+gwFiJ!#7$\"\"!F'$\"+gwFiJF%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 9 "allbisect" }{TEXT -1 16 " succeeds here. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f \+ := x -> x^3-x/10^5:\n'f(x)'=f(x);\nallbisect(f(x),x=-0.01..0.01);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)F'\"\"$\"\"\"F,*&#F ,\"'++5F,F'F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+gwFiJ!#7$\"\" !F'$\"+gwFiJF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exam ple 9" }{TEXT 293 96 " .. a piecewise function based on the previous c ubic but which has other zeros, comparison with " }{TEXT 0 9 "allbisec t" }{TEXT 296 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 " Let " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 15 " be defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = PIECEWISE([x^3-x/100 000, abs(x) < 1],[99999/100000*sin(x)/sin(1), otherwise]);" "6#/-%\"gG 6#%\"xG-%*PIECEWISEG6$7$,&*$F'\"\"$\"\"\"*&F'F/\"'++5!\"\"F22-%$absG6# F'F/7$**\"&*****F/F1F2-%$sinG6#F'F/-F;6#F/F2%*otherwiseG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x )" "6#-%\"gG6#%\"xG" }{TEXT -1 37 " has the three zeros of the functio n " }{XPPEDIT 18 0 "f(x)=x^3-x/100000" "6#/-%\"fG6#%\"xG,&*$F'\"\"$\" \"\"*&F'F+\"'++5!\"\"F." }{TEXT -1 60 " considered in the previous exa mple, together with zeros of " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\" xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "g := x -> piecewise(abs(x)<1,x^3-x/10^5,99 999/100000*sin(x)/sin(1)):\n'g(x)'=g(x);\nplot(g(x),x=-7..7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&*$)F' \"\"$\"\"\"F0*&#F0\"'++5F0F'F0!\"\"2-%$absGF&F07$,$*&#\"&*****F3F0*&-% $sinGF&F0-F?6#F0F4F0F0%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 559 350 350 {PLOTDATA 2 "6%-%'CURVESG6$7cs7$$!\"(\"\"!$!3V&QE)3&=v!y!#=7$$ !3]LL$3#*>u%o!#<$!3Iy.j;KHkF1$!39PrS;Z[I\")F-7$$!3Q++vLcI#R&F1$\"3+m \"Q.]m6C*F-7$$!3)om;HtGOD&F1$\"3I\\7!fMG&=5F17$$!38LLeMo.5^F1$\"3O(H% \\C*ec4\"F17$$!3S***\\i$\\Wm\\F1$\"3Iqn9.tB]6F17$$!3olm\"HY$>#*[F1$\"3 [Eh_7\\Ap6F17$$!3'GL$e*)>%z\"[F1$\"3i^x><#p<=\"F17$$!3!pm;HD;3y%F1$\"3 'HTM)G@g&=\"F17$$!30++D;0pVZF1$\"3!Ra]')3,y=\"F17$$!3>LLezZc1ZF1$\"3E] K7mIO)=\"F17$$!3Mmm\"H/R%pYF1$\"3%\\_cnG(G(=\"F17$$!3U*****\\pDaf%F1$ \"3y9Hw*pj-=\"F17$$!3RLL3ZBT@XF1$\"3+6m'ygxn;\"F17$$!3Onm;***)RZWF1$\" 3Q&fKQ&G!p9\"F17$$!3W++D^cQtVF1$\"3g*[)Qh#[27\"F17$$!3Zm;z**>7@UF1$\"3 y))G)>7Sy/\"F17$$!3[KLL[$e)oSF1$\"3@kf!3Ebo]*F-7$$!3X***\\7WWZ$RF1$\"3 9-.+\"))R!o%)F-7$$!3Tmm;M0j+QF1$\"3i--(4#>9xsF-7$$!3!HL$eR+l\\OF1$\"3s &\\GRkd7y&F-7$$!3Q*****\\ap')\\$F1$\"3kTN&46RQ:%F-7$$!3i******4\"pqM$F 1$\"3P'p7tE,ZU#F-7$$!3')*****\\noa>$F1$\"3gY6')y!R%*R'!#>7$$!3y*****\\ ct$\\IF1$!3+CC;#\\jV4\"F-7$$!3q*****\\XyK!HF1$!3P>O=`XN0GF-7$$!35L$e*) 451x#F1$!3R\\uay8E3VF-7$$!3]mm\"HuTzj#F1$!3V\"=AV6]at&F-7$$!3u**\\iNi= ![#F1$!3qk*H0Bu$*H(F-7$$!3(HLL$G2VABF1$!3eoQ#yq;?o)F-7$$!3#GLL37\"z)=# F1$!3i$fPkO@eo*F-7$$!3oKLL8::b?F1$!3igzS'*)*o^5F17$$!3Am;zzop**=F1$!35 LOo^NoC6F17$$!3!)***\\iCUUu\"F1$!35U!ySj_0<\"F17$$!3?L3_+/Wv;F1$!3+VO% >`#)==\"F17$$!3im;za&Qmg\"F1$!3YrU>m*>w=\"F17$$!3W$3F>jPAd\"F1$!3m/rZ( )>Q)=\"F17$$!3.+D14n$y`\"F1$!3]BRG)yPx=\"F17$$!3i;z>'yNM]\"F1$!3!y++48 )o&=\"F17$$!3WLLLj[.p9F1$!3af))pSaB#=\"F17$$!3Nm;z/=1=8F1$!3Y^5h#)4j]6 F17$$!3C***\\iu)3n6F1$!31-Q'[7]G4\"F17$$!3;**\\P>x?&4\"F1$!3EG]Si/]c5F 17$$!33****\\#pEL-\"F1$!3uIOrOVp95F17$$!3$**)*\\ilcW^*F-$!3N\"H$*f%e&G h)F-7$$!3/*)**\\(QYcz)F-$!3s`:*RtAX!oF-7$$!3KB$3-GVc/)F-$!3<=Xi'eb!3_F -7$$!3cdm\"HF-7$$!3rEL3-+;=WF-$!3l([#3#RnQi)Fju7$$!3N FL$e/'oSIF-$!3#Hf?!4[/6GFju7$$!3#\\\\7y5Q#pEF-$!3>&eW^S?:!>Fju7$$!3]i; zp,z(H#F-$!35TW())emH@\"Fju7$$!33I3xJAME>F-$!3=.O;7OLYrFH7$$!3m(**\\PH %*[b\"F-$!3u7awJsqdPFH7$$!34GL3xT)*>\")Fju$!3A&R[w*>sX`!#@7$$!3/'zmm;a -\"pFH$!3u$*y)fQK(3E!#C7$$\"3DGLek[dDqFju$\"3KZ!G))*pqgMF``l7$$\"3gLLe 9v@u9F-$\"3Nvl&==oC?$FH7$$\"3+MekGv/g=F-$\"3W2fQb#*[LkFH7$$\"3SM$3Faxe C#F-$\"3`q&>YR)eK6Fju7$$\"3zM3xcvqJEF-$\"3B4v4IrUA=Fju7$$\"3=NL$3dPv,$ F-$\"3!*=8kOqKZFFju7$$\"3apmm;A+hVF-$\"3KV#o]LmMH)Fju7$$\"3!R++D'oY/dF -$\"3Yd0/>-Bc=F-7$$\"3uP$eR%oC:F1$\"3U(=M#***>r=\"F17$$\"3!4vV[hc,c\"F1$\"3ap\"4z&fJ)=\"F17$$\" 3%4+D1.Hcf\"F1$\"3=f#oX#o,)=\"F17$$\"3*4D1kW,6j\"F1$\"3Ak\"pb(HA'=\"F1 7$$\"3-,v=iQdm;F1$\"3#)f:'ymOH=\"F17$$\"33^(ozFY?q\"F1$\"3#p[Zh.i\"y6F 17$$\"37,+v$p=vt\"F1$\"3'4Bzw30><\"F17$$\"3Q+]7tH1&*=F1$\"3A&)H$)*>]k7 \"F17$$\"3k****\\_sg_?F1$\"3'\\'=PtV4`5F17$$\"3mKLe%znT>#F1$\"3W$*3Mp. m['*F-7$$\"3olmmO$GdL#F1$\"3[G`\\RkMt&)F-7$$\"3qLL$3Vuo[#F1$\"3@unXf. \\OsF-7$$\"3t,++D0-QEF1$\"3*=\"[\"*=,jMdF-7$$\"3An;/^8)\\x#F1$\"3OEU$H .2)fUF-7$$\"3qKL3x@%>\"HF1$\"3K&G)fw@?0FF-7$$\"3Mn;/OlnhIF1$\"3WJ,$ya \"*p[*Fju7$$\"3*>++]*3T6KF1$!3!3_d8]V.H)Fju7$$\"39M$3xD*H_LF1$!3y!*)R; a9b[#F-7$$\"3ImmT?w=$\\$F1$!3s;0Q!**RF4%F-7$$\"3RLLealXSOF1$!3+07:h0c& o&F-7$$\"3[++v)[Dxy$F1$!3=e!)f0YHbrF-7$$\"3sL$3-z2<$RF1$!3JXCa&4#oU%)F -7$$\"3'pmm;4!pvSF1$!3%Gmx(>tMb&*F-7$$\"3;L$3x@EkA%F1$!3'*)\\&ov*)z]5F 17$$\"3M***\\PMirP%F1$!3v2'y)*yKA7\"F17$$\"3+Le9Y2v\\WF1$!3O,$[CjJw9\" F17$$\"3_n;a[\"RB_%F1$!3KO5Brh)p;\"F17$$\"3/-v$4bF\\f%F1$!3o;'e/Z%>!= \"F17$$\"3oNLL`f^nYF1$!33^(\\^'p=(=\"F17$$\"3/Ne9;?j/ZF1$!3CbM?puM)=\" F17$$\"3GN$e*y![\"4ry=\"F17$$\"3_N3xTT')yZF1$!3)Gy-t[ed=\" F17$$\"3([L$e/-)f\"[F1$!33Kf_nD,#=\"F17$$\"3WM$3-L7-*[F1$!3)y/&=iOkp6F 17$$\"31ML$eXWW'\\F1$!3ch*)f,D$3:\"F17$$\"3N,]7[$z;6&F1$!3IS.Ge:!\\4\" F17$$\"3momTSU\"*e_F1$!3Y(=a>1y_,\"F17$$\"3CN$3_\"y?%R&F1$!3kLU$*fs$pA *F-7$$\"3&=+++R,&HbF1$!3E'eZYEeC8)F-7$$\"3uMLe%4jXo&F1$!3k\"3Fq]'f'p'F -7$$\"3lnm;*zC'ReF1$!3T]Z\"*eK/+^F-7$$\"3!3+]i$QJyfF1$!3%[Q*3enF-7$$\"39n;a8V([E'F1$!3A]\"*Qhh#f<#Fju7$$ \"3L++v`du7kF1$\"3)4E/*\\2PN:F-7$$\"3w+++!fxUb'F1$\"3CeWB\")**H#=$F-7$ $\"3=,+DE%4ep'F1$\"3e&f1'p+flZF-7$$\"3g+]78Z!z%oF1$\"3eUQwl^(*fjF-7$$ \"\"(F*$\"3V&QE)3&=v!yF--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fbfm-%+AXESLA BELSG6$Q\"x6\"Q!Fgfm-%%VIEWG6$;F(Fgem%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The zeros of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 17 " in the interval " } {XPPEDIT 18 0 "[-7,7]" "6#7$,$\"\"(!\"\"F%" }{TEXT -1 29 " can be comp uted as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "g := x -> piecewise(abs(x)<1,x^3-x/10^5,9999 9/100000*sin(x)/sin(1)):\n'g(x)'=g(x);\nallroots(g(x),x=-7..7);\nsols \+ := [%]:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG 6$7$,&*$)F'\"\"$\"\"\"F0*&#F0\"'++5F0F'F0!\"\"2-%$absGF&F07$,$*&#\"&** ***F3F0*&-%$sinGF&F0-F?6#F0F4F0F0%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$!+2`=$G'!\"*$!+aEfTJF%$!+gwFiJ!#7$\"\"!F,$\"+gwFiJF*$ \"+aEfTJF%$\"+2`=$G'F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Check using " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "op(map(u->fsolve(g(x),x=evalf[6](u)),sols));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6)$!+2`=$G'!\"*$!+aEfTJF%$!+gwFiJ!#7$\"\"!F,$\"+gwFiJF* $\"+aEfTJF%$\"+2`=$G'F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "allbisect" }{TEXT -1 47 " struggles to find all the zeros - the option \"" }{TEXT 262 13 "m initerations" }{TEXT -1 53 "\" must be increased, and many bisections \+ are needed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "g := x -> piecewise(abs(x)<1,x^3-x/10^5,99999/100 000*sin(x)/sin(1)):\n'g(x)'=g(x);\nallbisect(g(x),x=-4..4,miniter=12); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&* $)F'\"\"$\"\"\"F0*&#F0\"'++5F0F'F0!\"\"2-%$absGF&F07$,$*&#\"&*****F3F0 *&-%$sinGF&F0-F?6#F0F4F0F0%*otherwiseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+aEfTJ!\"*$!+gwFiJ!#7$\"\"!F*$\"+gwFiJF($\"+aEfTJF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 10" }{TEXT 305 88 " . . a degree 7 polynomial with five roots - three of which are close tog ether near zero " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "f(x) = x^7/(10^7)-x^3+x/(10^5);" "6#/-%\"fG6#%\"xG,(*&F '\"\"(*$\"#5F*!\"\"\"\"\"*$F'\"\"$F-*&F'F.*$F,\"\"&F-F." }{TEXT -1 62 " has five zeros, three of which are close together near zero. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The follo wing graph illustrates the 3 roots of least magnitude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f := x \+ -> x^7/10^7-x^3+x/10^5:\n'f(x)'=f(x);\nplot(f(x),x=-0.005..0.005);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\")+++5F+*$)F '\"\"(F+F+F+*$)F'\"\"$F+!\"\"*&#F+\"'++5F+F'F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!35+++++++]!#?$ \"3I-++++++v!#D7$$!3yLLLe%G?y%F*$\"37qY6^nT`hF-7$$!3[mmT&esBf%F*$\"31G %>t#)))G4&F-7$$!3CLL$3s%3zVF*$\"3aJ:QbeT=SF-7$$!3YLL$e/$QkTF*$\"3Sm-U4 f_dIF-7$$!3)pm;/\"=q]RF*$\"3]yIQs7d:AF-7$$!3iLL3_>f_PF*$\"3#>6-b!zyJ:F -7$$!3Q++vo1YZNF*$\"3QdbD.7Lo\"*!#E7$$!3>LL3-OJNLF*$\"3=0e%p>\\*\\PFQ7 $$!3!)***\\P*o%Q7$F*$!3Aat%>e2ma(!#F7$$!3Lmmm\"RFj!HF*$!3r2=Rf_G9XFQ7$ $!3iAr&3LQ*FQ7$$ !3))*****\\ixCG#F*$!3wS\">*=VP$4\"F-7$$!3'******\\KqP2#F*$!3'\\()HAbS> =\"F-7$$!3ILL3-TC%)=F*$!3VzE7HmE:7F-7$$!3qmmm\"4z)e;F*$!3%oi.')\\vB?\" F-7$$!3cmmmm`'zY\"F*$!3C&yPOBI;:\"F-7$$!39++v=t)eC\"F*$!39QqB-j\\_5F-7 $$!3'ommmh5$\\5F*$!34]PL0)fxL*FQ7$$!3a)***\\(=[jL)!#@$!3)\\><>;;qv(FQ7 $$!3k)***\\iXg#G'F\\r$!3KfMMBJiMgFQ7$$!3!*fmmT&Q(RTF\\r$!3C\"R\"pSSzoS FQ7$$!3Ehm;/'=><#F\\r$!3kM0ljJnh@FQ7$$!35WILLe*e$\\!#B$!3&3iwz!Q)e$\\! #G7$$\"3#*pm;zRQb@F\\r$\"3Yl=ra2PX@FQ7$$\"3Y,+](=>Y2%F\\r$\"3-@4uM-(p+ %FQ7$$\"3%Gnm;zXu9'F\\r$\"34k*\\f6F^\"fFQ7$$\"3#e******\\y))G)F\\r$\"3 X6`&>!pQ>xFQ7$$\"3A++]i_QQ5F*$\"3C(QLW**>UE*FQ7$$\"3R++D\"y%3T7F*$\"3_ 9iZb:#*\\5F-7$$\"35++]P![hY\"F*$\"394qb6j)4:\"F-7$$\"3QLLL$Qx$o;F*$\"3 FT^nn6AF-7$$\"3uML$3dg6<%F*$!35m[PHn1 '3$F-7$$\"3+ommmxGpVF*$!3%oY8Pwx>(RF-7$$\"3+++D\"oK0e%F*$!3PA%G\"H5,I] F-7$$\"3!4+](=5s#y%F*$!3-)>hz2wu:'F-7$$\"35+++++++]F*$!3I-++++++vF--%' COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fe[lFd[l-%+AXESLABELSG6$Q\"x6\"Q!Fj[l-% %VIEWG6$;$!\"&!\"$$\"\"&Fb\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The following graph illustrates the 2 roots of greatest magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f := x -> x^7/10^7-x^3+x/10 ^5:\n'f(x)'=f(x);\nplot(f(x),x=-60..60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\")+++5F+*$)F'\"\"(F+F+F+*$)F'\"\"$F+! \"\"*&#F+\"'++5F+F'F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 587 395 395 {PLOTDATA 2 "6%-%'CURVESG6$7_p7$$!#g\"\"!$!3%*******f+g$R'!#87$$!31++v oUInf!#;$!31m7rWcF%p&F-7$$!38++]P&3Y$fF1$!3,aeAbv4D]F-7$$!3>++D1G\">!f F1$!3M%o_x=m^Q%F-7$$!3E+++vq@peF1$!3i^@@.?gtPF-7$$!3Q++]7c#Q!eF1$!3(=' o+v`:KEF-7$$!3z******\\TVQdF1$!3+42_5-6%f\"F-7$$!3o**\\7)QP:o&F1$!3j?A *)*)G!4q(!#97$$!3E++DE1kCcF1$!3)f,1A\\]Sb\"!#:7$$!39+]PkQunbF1$\"3@g-p Z[zLnFR7$$!3,++]-r%3^&F1$\"3<897jLN+8F-7$$!3%****\\PQuGQ&F1$\"3]7q6mTB -DF-7$$!3))*****\\m,\\D&F1$\"3)4\"**y(=ieW$F-7$$!3^******f13E^F1$\"3%> 1>aWE$pTF-7$$!3')*****\\lfs*\\F1$\"3)HH,@\">)op%F-7$$!37++v840p[F1$\"3 !*RZ7r\")Qb]F-7$$!3n****\\s@%3u%F1$\"3_()H;mZws_F-7$$!3M*****\\@49o%F1 $\"3M,gE*\\w>L&F-7$$!3s****\\di(>i%F1$\"3W#3l*))Qun`F-7$$!35++++LaiXF1 $\"3AT(R-l0?Q&F-7$$!3x****\\U.6.XF1$\"3!R[(H(*pfw`F-7$$!3u****\\s:.!Q% F1$\"3LN2[m#G-J&F-7$$!3o****\\-G&pD%F1$\"3m!f8-;p4=&F-7$$!3l****\\iXmH TF1$\"3->DvsnV%*\\F-7$$!3i****\\AjP-SF1$\"3!*4PfQv=mZF-7$$!3)*****\\si h[PF1$\"3C8MF!z\\uA%F-7$$!3I+++qGf([$F1$\"3s1`@q(oWh$F-7$$!3))*****\\J $odKF1$\"3-2\"z9(p%y1$F-7$$!3))******4'f))*HF1$\"3=(3,^O.)yCF-7$$!3N++ +]J(*QFF1$\"3w()>)pzG\"R>F-7$$!3!********QC&)[#F1$\"33#pM!H5)>[\"F-7$$ !3'*****\\AH4hAF1$\"3#R%=09wxD6F-7$$!3%*******4\\l!*>F1$\"3aNE/e-^kxFR 7$$!3I+++S%e:w\"F1$\"3H/O\\X`j8aFR7$$!3W++]#yk]\\\"F1$\"3_0*o>K+^K$FR7 $$!3O+++SF9*>FR7$$!3#*****\\#yh.+\"F1$\"3\"*HJ%\\5$3+5 FR7$$!3u)****\\ZD\"Rv!#<$\"3%p1g=;MPG%FX7$$!3')******\\ion\\Fbw$\"3vf& 4;#e%eA\"FX7$$!3P%****\\K-jg#Fbw$\"3]>qzX8Sq$ \"3sXp+:&e?2#!#@7$$\"3/*****\\xgke#Fbw$!3'>jed&[FIam%**Fbw$!3zi:(oJ\"=J)*FX7$$\"3O+++:B1Y7F1$!3)*3TZ[31I>FR7$$\"3'*** **\\PF-7$$\"3!*****\\<3;%*HF1$!3)pb$zNs_oCF-7$$\"3_++]Z=iYKF1$!3<@#o#f /\">/$F-7$$\"39******\\'[M\\$F1$!3Oh*zj\")e%GOF-7$$\"3;++]PM&=v$F1$!36 `%yCK/[B%F-7$$\"3R+++gzs+SF1$!3=Y@8dk+jZF-7$$\"3h++]KI)z7%F1$!3y#R?`So ;*\\F-7$$\"3#3++]5Q_D%F1$!33f=j6`yy^F-7$$\"3z+]7t(Q$=VF1$!3'3q'z&ezCD& F-7$$\"3w++DT%R9Q%F1$!331U&4Qo8J&F-7$$\"3t+]P4,aWWF1$!3'yG#)3%Gr``F-7$ $\"3q++]x2k2XF1$!3#4(o'Q2iwP&F-7$$\"3%3+D\"QPilXF1$!3O2x([=l`9 V_F1$!3$*f%o.783_$F-7$$\"39++voA*)p`F1$!32>LHbU84EF-7$$\"3+,+]<#Rm\\&F 1$!3sH243zmZ9F-7$$\"3!4+](=dHdbF1$!3&RUKh1'\\IzFR7$$\"3\"3+++A_zh&F1$! 3WK5%)QAexoFX7$$\"3s++D@(3'ycF1$\"3%)Gc<%3()eH(FR7$$\"3i++]A_ERdF1$\"3 CQ%Q\"))3o1;F-7$$\"3Y+](o\"*[W!eF1$\"3X:Z$R4PDk#F-7$$\"3K++D6EjpeF1$\" 3IbiM6,?\"y$F-7$$\"3B+vVeWA-fF1$\"3_eLOb18\"R%F-7$$\"3;+]i0j\"[$fF1$\" 39C%ePjc#H]F-7$$\"33+D\"G:3u'fF1$\"3e4@'[Q\\kp&F-7$$\"#gF*$\"3%******* f+g$R'F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F_fl-%+AXESLABELSG6$Q\"x6\"Q! Fdfl-%%VIEWG6$;F(Fdel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "f := x -> x^7/10^7-x^3+x/10^5:\n'f(x)'=f(x);\nallroots(f(x),x=-60. .60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\")+ ++5F+*$)F'\"\"(F+F+F+*$)F'\"\"$F+!\"\"*&#F+\"'++5F+F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+ZKTBc!\")$!+gwFiJ!#7$\"\"!F*$\"+gwFiJF($\"+Z KTBcF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f := x -> x^7/10^7-x^3+x/10^5:\n'f(x)'=f(x);\nfsolve( f(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*&#\"\"\"\") +++5F+*$)F'\"\"(F+F+F+*$)F'\"\"$F+!\"\"*&#F+\"'++5F+F'F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+ZKTBc!\")$!+gwFiJ!#7$\"\"!F*$\"+gwFiJF($ \"+ZKTBcF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example \+ 11" }{TEXT 294 85 " .. a root-finding problem involving a fairly large number of roots, comparison with " }{TEXT 0 9 "allbisect" }{TEXT 295 1 " " }}{PARA 0 "" 0 "" {TEXT 273 8 "Question" }{TEXT -1 64 ": Find (c orrect to 10 digits) all the solutions of the equation " }{XPPEDIT 18 0 "4*sin(15*x)=1+sqrt(x)" "6#/*&\"\"%\"\"\"-%$sinG6#*&\"#:F&%\"xGF&F&, &F&F&-%%sqrtG6#F,F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 274 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot( [4*sin(15*x),1+sqrt(x)],x=0..10,color=[brown,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 801 273 273 {PLOTDATA 2 "6&-%'CURVESG6$7]fo7$$\"\"!F)F(7$ $\"3/LL3x1h6o!#?$\"3C`k'f4f)zS!#=7$$\"3gmmTN@Ki8!#>$\"3odgp-L;<\")F07$ $\"3#***\\7.K[V?F4$\"3chy))f,)p?\"!#<7$$\"3ALL$3FWYs#F4$\"3#\\;Ipga'*e \"F<7$$\"3[d&>F<7$$\"3%)***\\iSmp3%F4$\"3;N-w'Q U9I#F<7$$\"3[L$eRZF\"oZF4$\"3HbJDJ98BEF<7$$\"3WmmmT&)G\\aF4$\"3zr>Zb*f u\"HF<7$$\"3MKL3x1h6oF4$\"3cPAg'*G27MF<7$$\"3m****\\7G$R<)F4$\"3%)eZq> ))pkPF<7$$\"3±z%\\DO&*F4$\"3'41\\z&QmgRF<7$$\"3GLLL3x&)*3\"F0$\"3CO Se$H8=*RF<7$$\"3))**\\(=#**3E7F0$\"3!p+K7!4&o&QF<7$$\"3YmmTN@Ki8F0$\"3 3(=yy)GRhNF<7$$\"31L$e*[Vb)\\\"F0$\"3qnR%\\=Mx6$F<7$$\"3$*****\\ilyM;F 0$\"3!RMhx\"oLWDF<7$$\"3AL3FpE!Hq\"F0$\"3R9U_XvD;AF<7$$\"3!omTgx=5x\"F 0$\"3Q4Q*Hkh]'=F<7$$\"34+D\"G)[8R=F0$\"3;%[`?D7W\\\"F<7$$\"3RLLe*)4D2> F0$\"3:D4\"QUv\"36F<7$$\"3omTN'4n`(>F0$\"3<()))GQzz.rF07$$\"3)***\\7.K [V?F0$\"3=i,A:cu^IF07$$\"3GLe*)4$*f6@F0$!3*RI_)zy8K5F07$$\"3emmm;arz@F 0$!3xlnF\"ob_5&F07$$\"3/DJ&p(G)*QAF0$!3OHV%)[bT2')F07$$\"3A$eRsL]#)H#F 0$!3iDYY<#fT?\"F<7$$\"3STg_(z\\i)\\oX#F<7$$\"3;L$e*)4bQl#F0$! 31d_n4*3k(HF<7$$\"33]7`>+RsFF0$!3\"\\E,(G.7-MF<7$$\"3WmT5S\\#4*GF0$!3# Q'zQI7c?PF<7$$\"3#G3x1')f%4IF0$!3&p/7u,\"p@RF<7$$\"3v***\\7y%*z7$F0$!3 kO;Dpz;**RF<7$$\"3o;H#=qHlC$F0$!3Ac\\t*>\\0&RF<7$$\"3/LeRAY1lLF0$!3k4d V#ontx$F<7$$\"3S\\(oHa*f$[$F0$!3i\"zm_\"R3&[$F<7$$\"3Lm;ajW8-OF0$!3YO8 J[P\"H3$F<7$$\"3E$e9TQp1s$F0$!3=[Pc.y`$e#F<7$$\"3=+vo/V?RQF0$!3ky$=([; q-?F<7$$\"3keR(\\wr%)*QF0$!3MS2E9TP(o\"F<7$$\"35( e8F<7$$\"3dvoa&o1q,%F0$!3([>>\")3L$>5F<7$$\"3[LL$e9ui2%F0$!3Gve'))Hf*= nF07$$\"3ODc^.m#H9%F0$!3=$RoR&fk\\FF07$$\"3p;z>h!z&4UF0$\"3ETRDW#HrC\" F07$$\"3e3-))=:BwUF0$\"3Gvj-Y)[9B&F07$$\"3Z+DcwR)GM%F0$\"3Kkgx+)>N;*F0 7$$\"3!=zWUVO&4WF0$\"3fn9o)32/I\"F<7$$\"3q$3F>*))=wWF0$\"3_s*HH\\u9n\" F<7$$\"3ev$4'\\8%Ga%F0$\"31tI#*)=[e-#F<7$$\"3\"pm\"H2Q\\4YF0$\"3U48u** )))*fBF<7$$\"38]ilA()zUZF0$\"3yg*=U'zXaHF<7$$\"3OL3-QO5w[F0$\"38Z JMF<7$$\"38rPF<7$$\"3z***\\(oMrU^F0$\"3E%\\vhS84 'RF<7$$\"38\"HKk#fO4_F0$\"3QvHkv6\"o*RF<7$$\"3Y#e9TQ=gF&F0$\"3qdT,O6z# *RF<7$$\"3\"\\(ozT3nU`F0$\"3q@.4PM*)[RF<7$$\"3Cm\"z%*HB$4aF0$\"3c1vP.l blQF<7$$\"3,]P%[@GEa&F0$\"3Aiuw'zzUe$F<7$$\"3oK$3-8Lfn&F0$\"33p#zs')p, ;$F<7$$\"37D1*ye&eUdF0$\"35V\"ezbG'**GF<7$$\"3X;HdX!Q#4eF0$\"3]H!3pXF, h#F<7$$\"3y2_D.0*e(eF0$\"3'>'ecJzb%H#F<7$$\"3B+v$4'HaUfF0$\"3c#=Abqrg& >F<7$$\"3c\"z>'=a>4gF0$\"3aw=&4Q\\!)f\"F<7$$\"3!H3-j(y%e2'F0$\"3HS'R^n mSA\"F<7$$\"3MvV)RL+D9'F0$\"3#HB+J3(ey$)F07$$\"3nmmm\"z_\"4iF0$\"3a4MM %*p#GV%F07$$\"3ulT5lqCwiF0$\"3s,ZL&4qG;%F47$$\"3$fmT&Q8MVjF0$!3-&)y0he Y/OF07$$\"36m\"z>hN/T'F0$!3-=1%o+T()e(F07$$\"3ImmT&))HvZ'F0$!3+qmr(p@' \\6F<7$$\"3\\mT&)eTiWlF0$!3vYU.a\\tG:F<7$$\"3nm;HK%=seDF<7$ $\"3Sn;/Eb4!)oF0$!32kiG&>]<7$F<7$$\"3ynm\"H2%G9qF0$!3))e(=#zyseNF<7$$ \"39o;z>EZ[rF0$!3kMc\"*R1,_QF<7$$\"3Unmmm6m#G(F0$!3w)H#)eDc(*)RF<7$$\" 3nm;a8(\\oT(F0$!3c6y16GSmRF<7$$\"3/nmTg#Q5b(F0$!3wiG'[F$*Gy$F<7$$\"3Sn ;H2oA&o(F0$!3Wr.sEujYMF<7$$\"3ynm;a`T>yF0$!3onG\"Gm77(HF<7$$\"3'z;/wi4 l)yF0$!375+_%>6ro#F<7$$\"39o;/,Rg`zF0$!3A6mgld\"eP#F<7$$\"3Lo\"zW<)p?! )F0$!3'Rb6;tw//#F<7$$\"3_om\"zW#z(3)F0$!3#eY%*Gx([%o\"F<7$$\"3qoTN@n)[ :)F0$!3H374]:X68F<7$$\"3))o;z%*4)>A)F0$!3?ZBeYCV^#*F07$$\"33p\"H#o_2*G )F0$!3g9!3[`BZH&F07$$\"39ommT&phN)F0$!3w-N*=CJWG\"F07$$\"33xVtF\\%HU)F 0$\"3HamoL5x>FF07$$\"3#\\3-QJ?(*[)F0$\"3wbNPC'4np'F07$$\"3u#zp)*p&\\c& )F0$\"3#zg4u'=lg5F<7$$\"3e+v$f3rKi)F0$\"3+O!e]_+5W\"F<7$$\"3S3_+sk/!p) F0$\"3,z!RDA/p!=F<7$$\"3ORF<7$$\"3A,+v=ddC%*F0$\"3[Y<%f\")*****RF<7$$\"3+=a)3\\E \"e&*F0$\"3wT&Qb>[-#RF<7$$\"3wM3-jsn\"p*F0$\"3-wuK'>+Po$F<7$$\"3V]i:N! 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DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "There is a problem with " }{TEXT 0 8 "allroots" }{TEXT -1 46 " arising from the fact that the derivative of " }{XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6#%\"xG" }{TEXT -1 22 " tends to infinity as " } {TEXT 275 1 "x" }{TEXT -1 16 " tends to zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "allroots(4*s in(15*x)=1+sqrt(x),x=0..10);" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (i n allroots/allroots) evaluation failed at Float(infinity)\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "We can circumv ent this problem by replacing the left end of the interval with a smal l positive number. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "allroots(4*sin(15*x)=1+sqrt(x),x=1e-9..10); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6N$\"+.Z\\C>!#6$\"+!yM1&=!#5$\"+_ntv WF($\"+kr,xfF($\"+oX\"Rr)F($\"+;T<75!\"*$\"+iVA%H\"F1$\"+*edtU\"F1$\"+ .Rc;(F1$\"+k[&>D(F1$\"+*Qk8i(F1$\"+ @*4tm(F1$\"+G#zU/)F1$\"+]k;#3)F1$\"+4=8o%)F1$\"+,U3'\\)F1$\"+g&H\\*))F 1$\"+se02*)F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "allbisect(4*sin(15*x)=1+sqrt(x),x=0..10);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6N$\"+.Z\\C>!#6$\"+!yM1&=!#5$\"+_ntvWF($ \"+kr,xfF($\"+oX\"Rr)F($\"+;T<75!\"*$\"+iVA%H\"F1$\"+*edtU\"F1$\"+.Rc; (F1$\"+k[&>D(F1$\"+*Qk8i(F1$\"+@*4tm( F1$\"+G#zU/)F1$\"+]k;#3)F1$\"+4=8o%)F1$\"+,U3'\\)F1$\"+g&H\\*))F1$\"+s e02*)F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 12 " }{TEXT 277 37 " .. a difficult root-finding problem " }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 278 8 "Question" }{TEXT -1 60 ": Find (corre ct to 10 digits) all the zeros of the function " }{XPPEDIT 18 0 "f(x)= sin(x)^2+sin(100*x)^3/100" "6#/-%\"fG6#%\"xG,&*$-%$sinG6#F'\"\"#\"\"\" *&-F+6#*&\"$+\"F.F'F.\"\"$F3!\"\"F." }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> sin(x)^2+sin(100*x)^3/100:\n'f(x)'=f(x);\np lot(f(x),x=0..4,numpoints=200);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG,&*$)-%$sinGF&\"\"#\"\"\"F.*&#F.\"$+\"F.*$)-F,6#,$*&F1F.F'F .F.\"\"$F.F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 661 333 333 {PLOTDATA 2 " 6%-%'CURVESG6$7g]m7$$\"\"!F)F(7$$\"3@9Gc7D!)GE!#?$\"3;0J<0v#R#=!#@7$$ \"3UGc7D]gd_F-$\"3;A-$HyE$\"3!*>1v'GLVl'F-7$$\"3W)oPv]\"Gx:F>$\"3O>Zs@I\"[-\"F>7 $$\"3O^-05?/.@F>$\"3yvnXY5*)RoF-7$$\"3<_.29['z,$F>$\"3'*47\\o$eGH*F07$ $\"3i_/4=w)G$RF>$!3i[:QQ$)H_?F-7$$\"3gRyc8#>,>%F>$!3Y2z/wMXbZF-7$$\"3c E_/43NZWF>$!3S]z>gq.7qF-7$$\"3`8E_/Ce/ZF>$!3!pQiUw3uy(F-7$$\"3]++++S\" ='\\F>$!3)>l!>1x7SmF-7$$\"3WuZ&4>xiZ&F>$!3MMBPy[\"3q(F07$$\"3oZ&4>QS2* fF>$\"3w&e!e`T*\\M$F-7$$\"3m%*)yd:([EqF>$\"3NR&p(fi5G!)F-7$$\"3KU#['HR 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" }}{PARA 0 "" 0 "" {TEXT -1 86 "The following \+ picture gives the approximate location of the first two positive zeros . " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "g(x) =sin(x)^2" "6#/-%\"gG6#%\"xG*$-%$sinG6#F'\"\"#" }{TEXT -1 15 " is plot ted in " }{TEXT 261 3 "red" }{TEXT -1 20 " while the graph of " } {XPPEDIT 18 0 "h(x)=-sin(100*x)^3/100" "6#/-%\"hG6#%\"xG,$*&-%$sinG6#* &\"$+\"\"\"\"F'F/\"\"$F.!\"\"F1" }{TEXT -1 15 " is plotted in " } {TEXT 257 5 "green" }{TEXT -1 2 ". 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" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "op(ma p(u->fsolve(f(x),x=evalf[8](u)),sols));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,$\"\"!F$$\"+X?*3o$!#6$\"+bb%Ga&F'$\"+HQ@eI!\"*$\"+j$=%pIF,$\"+Co \\9JF,$\"+\\E\\TJF,$\"+aEfTJF,$\"+u:SyJF,$\"+46-(>$F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" }{TEXT -1 12 " is zero of " }{XPPEDIT 18 0 "` f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 25 ", it is a double root of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "f := x \+ -> sin(x)^2+sin(100*x)^3/100:\n'f(x)'=f(x);\nDiff('f(x)',x)=diff(f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)-%$sinGF&\" \"#\"\"\"F.*&#F.\"$+\"F.*$)-F,6#,$*&F1F.F'F.F.\"\"$F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#%\"xGF*,&*(\"\"#\"\"\"-%$sin GF)F.-%$cosGF)F.F.*(\"\"$F.)-F06#,$*&\"$+\"F.F*F.F.F-F.-F2F7F.F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "There is \+ another root which is very close to the double root " }{XPPEDIT 18 0 " x=Pi" "6#/%\"xG%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> sin(x)^2+sin(1 00*x)^3/100:\n'f(x)'=f(x);\nplot(f(x),x=3.14145..3.14165);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)-%$sinGF&\"\"#\"\"\"F.*&# F.\"$+\"F.*$)-F,6#,$*&F1F.F'F.F.\"\"$F.F.F." }}{PARA 13 "" 1 "" {GLPLOT2D 436 379 379 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"3&)*********\\ 99$!#<$!35lbt&)=2x')!#E7$$\"3Y;arz@XTJF*$!3SVP8h#QR)zF-7$$\"32L3VfVXTJ F*$!3$Gl>G(RL@tF-7$$\"3Ci&*)fDc99$F*$!3gQ7<*Rg\"pnF-7$$\"3U\"H[D:e99$F *$!3J=?0E1ARiF-7$$\"3E$e0$=CYTJF*$!3DX/<4FSE^F-7$$\"3:$3RBrm99$F*$!3Qu +mS#4I6%F-7$$\"3\\\"zjf)4ZTJF*$!3VrYEx\"Gh?$F-7$$\"3He4;[\\ZTJF*$!3S^^ 9*4\")=X#F-7$$\"3/Dmy]!z99$F*$!3wfcrrpdaYr)37#fJ9F-7$$\"3)[(3zMu`TJF*$\"3=Eh\\AjXl8F-7$$\"3e;H_?Jf'ej&Feo7$$\"3,DTO5TgTJF*$\"3/UQ&=kbHY\"F_o7$$\"3_mT9C# 3;9$F*$\"37$H!GF_o7$$\"3kC1*3`7;9$F*$\"36\"*p'pdNkt%F_o7$$\"3:L$* zymhTJF*$\"3')*yEorj)erF_o7$$\"3O$3N1#4iTJF*$\"3s&e=c%f)[-\"F-7$$\"3i \"HYt7D;9$F*$\"3=.$oaQ$*pR\"F-7$$\"3/+q(G**G;9$F*$\"3b'>yy#oT+=F-7$$\" 3q;9@BMjTJF*$\"3CV.+#f8)RBF-7$$\"3DL`v&QP;9$F*$\"3%)4')f%QLg*GF-7$$\"3 \"\\i`1hT;9$F*$\"3QKb8_+=qNF-7$$\"3![P?WlX;9$F*$\"3#f$)GE72zH%F-7$$\"3 $)*********\\;9$F*$\"3wYJ!)o*zW<&F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F ]\\lF\\\\l-%+AXESLABELSG6$Q\"x6\"Q!Fb\\l-%%VIEWG6$;$\"'XTJ!\"&$\"'lTJF j\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 49.000000 40.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "g := x -> sin(x)^2: h := x -> -sin(100*x)^3/100: \n'g(x)'=g(x); 'h(x)'=h(x); \nplot([g(x),h(x)],x=3.14145..3.14165,colo r=[red,COLOR(RGB,0,.7,0)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG 6#%\"xG*$)-%$sinGF&\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"hG6#%\"xG,$*&#\"\"\"\"$+\"F+*$)-%$sinG6#,$*&F,F+F'F+F+\"\"$F+F+!\"\" " }}{PARA 13 "" 1 "" {GLPLOT2D 459 477 477 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$\"3&)*********\\99$!#<$\"3:%y*GaY+N?!#D7$$\"32L3VfVXTJF*$\"3\"Qid !Gu_7>F-7$$\"3U\"H[D:e99$F*$\"3@*e)f0I04=F-7$$\"3E$e0$=CYTJF*$\"3Kaft: H7'p\"F-7$$\"3:$3RBrm99$F*$\"3/Vjla)>he\"F-7$$\"3\\\"zjf)4ZTJF*$\"39gf Q96I![\"F-7$$\"3He4;[\\ZTJF*$\"3?uNK\"zcaQ\"F-7$$\"3/Dmy]!z99$F*$\"3!f Z:Kof0H\"F-7$$\"3)z&zs$H$[TJF*$\"3'z#\\HLz&f>\"F-7$$\"3([7iI_([TJF*$\" 3D(>62yU_5\"F-7$$\"3Tm@Xt=\\TJF*$\"3zg(p0hjc,\"F-7$$\"3(H3y_q&\\TJF*$ \"3oRR-9^(*)R*!#E7$$\"3n*\\1!>+]TJF*$\"3UQ#Gu*f;\"e)F_o7$$\"3w*\\Z/N/: 9$F*$\"3[5bj=\\W(z(F_o7$$\"3s*\\$fC&3:9$F*$\"3;\\sl+!)oxqF_o7$$\"3.ez6 :B^TJF*$\"3=:/$H/qUX'F_o7$$\"3mm\"=C#o^TJF*$\"3#*)=Q#zlO]dF_o7$$\"3MmE pS1_TJF*$\"3#\\IWOMde=&F_o7$$\"3#\\i`A3D:9$F*$\"3iyBika)ec%F_o7$$\"3Om wy8!H:9$F*$\"39]4G4W-]SF_o7$$\"3+DOIFL`TJF*$\"3NRkQE#3'>NF_o7$$\"3)[(3 zMu`TJF*$\"3[Fn)3v:\"\\IF_o7$$\"3e;H_?)=#*HxTf#F_o7$$\"3Z \"zihlX:9$F*$\"3Fk1;V\"e(3AF_o7$$\"36$3#G,*\\:9$F*$\"3Bb&)>\"F _o7$$\"37$e\"*[Hi:9$F*$\"3R--Inb];#*!#F7$$\"3m**pvxlcTJF*$\"3+$RZsb!\\ *z'F`t7$$\"3z\\_qn2dTJF*$\"3K4rs@%>**y%F`t7$$\"3uCcp@[dTJF*$\"3i4w-'\\ k(zJF`t7$$\"3#)\\2'HKz:9$F*$\"3wDp#oqcqx\"F`t7$$\"3KmwanLeTJF*$\"32)o[ !Qy0B')!#G7$$\"30+v+'o(eTJF*$\"3k>J%RQ(onCFju7$$\"3.eR<*f\"fTJF*$\"3%[ 5WNv#z66!#H7$$\"3w*\\)HxefTJF*$\"3N&oZCyH#R5Fju7$$\"3m\"H!o-**fTJF*$\" 3+sFL08$\\D&Fju7$$\"3,DTO5TgTJF*$\"3,\"pM3ywDJ\"F`t7$$\"3_mT9C#3;9$F*$ \"3v^irM;UCCF`t7$$\"3kC1*3`7;9$F*$\"3C?:!H)32^RF`t7$$\"3:L$*zymhTJF*$ \"3m*\\cK$p5sdF`t7$$\"3O$3N1#4iTJF*$\"3'*)p3A\\e-*zF`t7$$\"3i\"HYt7D;9 $F*$\"35Odug[aa5F_o7$$\"3/+q(G**G;9$F*$\"3ZJp6%zV0K\"F_o7$$\"3q;9@BMjT JF*$\"3kJ\"*RY?;i;F_o7$$\"3DL`v&QP;9$F*$\"3:>RcKm'4+#F_o7$$\"3\"\\i`1h T;9$F*$\"3/HI5(4%z'R#F_o7$$\"3![P?WlX;9$F*$\"32!40(R!)34GF_o7$$\"3$)** *******\\;9$F*$\"3aiy(*f2h)G$F_o-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"! Fd[lFc[l-F$6$7S7$F($\"3mSL'G%=r-HF-7$F/$\"34*eR`#3mWEF-7$F4$\"3/\"z.#o ](HV#F-7$F9$\"3&))*Hl'=j(3AF-7$F>$\"3[]Bsy2U(*>F-7$FC$\"3HFC6KR\"4!=F- 7$FH$\"3M*3Q7!\\kI;F-7$FM$\"3)=/(Q!QyI\"F-7$FW$ \"3UF#)[hL)=;\"F-7$Ffn$\"31i![\"3_`B5F-7$F[o$\"3=yL9/&><6*F_o7$Fao$\"3 7a*['Gjy[zF_o7$Ffo$\"3g!eN>Y?^)oF_o7$F[p$\"3!o()ol`w@)* zz#F`t7$Fbt$\"3O^k7hx,t5')y.:Fju7$Ffw$!3u6/,8V&\\x$Fju7$F[x$!3?1ZlSpk`yFju7$F`x$!3 ?!HqNycnQ\"F`t7$Fex$!3%z:xR'4geAF`t7$Fjx$!3ppcAZ_[CMF`t7$F_y$!3w]EhP.t )z%F`t7$Fdy$!37<@,ca^wnF`t7$Fiy$!3)*3pM?vm]*)F`t7$F^z$!37.D.bfQt6F_o7$ Fcz$!3%euBH3>))[\"F_o7$Fhz$!3A%GD)3#pe)=F_o-%&COLORG6&F_[lFc[l$\"\"(! \"\"Fc[l-%+AXESLABELSG6$Q\"x6\"Q!Fgel-%%VIEWG6$;$\"'XTJ!\"&$\"'lTJF_fl %(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 82 "f := x -> sin(x)^2+sin(100*x)^3/100:\n'f(x)' =f(x);\nfsolve(f(x),x=3.14145..3.14152);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*$)-%$sinGF&\"\"#\"\"\"F.*&#F.\"$+\"F.*$)-F,6#,$ *&F1F.F'F.F.\"\"$F.F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\E\\TJ !\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 13 " }{TEXT 270 62 " .. finding the least and greatest roots in an interv al using " }{TEXT 0 9 "leastroot" }{TEXT 303 5 " and " }{TEXT 0 12 "gr eatestroot" }{TEXT 302 2 " " }}{PARA 0 "" 0 "" {TEXT 271 8 "Question " }{TEXT -1 79 ": Find (correct to 10 digits) the least and greatest s olutions of the equation " }{XPPEDIT 18 0 "3*sin(15*x) = ln(x+2);" "6# /*&\"\"$\"\"\"-%$sinG6#*&\"#:F&%\"xGF&F&-%#lnG6#,&F,F&\"\"#F&" }{TEXT -1 2 ". 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18 0 "[0,100]" "6#7$\"\"!\"$+\"" }{TEXT -1 2 ". 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However, the firs t and second derivatives can be handled. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := x -> Si(x)-Pi/2: \n'f(x)'=f(x);\nallroots(f(x),x=0..100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%#SiGF&\"\"\"*&\"\"#!\"\"%#PiGF+F." }}{PARA 7 " " 1 "" {TEXT -1 78 "Warning, cannot use interval arithmetic for value \+ ranges - estimating instead\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6B$\"+g wWE>!\"*$\"+`f$Q*[F%$\"+CEoszF%$\"+)z.$36!\")$\"+2!=1U\"F,$\"+4kcLbY&*)F,$\"+wm xo#*F,$\"+G1!He*F,$\"+ " 0 "" {MPLTEXT 1 0 78 "f := x -> Si(x)-Pi/2:\n'f(x)'=f(x); \nevalf(evalf[14](allbisect(f(x),x=0..100)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&-%#SiGF&\"\"\"*&\"\"#!\"\"%#PiGF+F." }} {PARA 12 "" 1 "" {XPPMATH 20 "6B$\"+gwWE>!\"*$\"+`f$Q*[F%$\"+CEoszF%$ \"+)z.$36!\")$\"+2!=1U\"F,$\"+4kcLbY&*)F,$\"+wmxo#*F,$\"+G1!He*F,$\"+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 16 " }{TEXT 300 54 " .. a root-finding problem involving Bessel functions " }}{PARA 0 "" 0 "" {TEXT 298 8 "Question" }{TEXT -1 64 ": Find (corr ect to 10 digits) all the solutions of the equation " }{XPPEDIT 18 0 " J[0](x)=J[1](x)" "6#/-&%\"JG6#\"\"!6#%\"xG-&F&6#\"\"\"6#F*" }{TEXT -1 27 " which lie in the interval " }{XPPEDIT 18 0 "[0,100]" "6#7$\"\"!\" $+\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "f := x -> BesselJ(0, x):\ng := x -> BesselJ(1,x):\n'f(x)'=f(x);\n'g(x)'=g(x);\nplot([f(x),g (x)],x=0..100,numpoints=75,color=[blue,coral]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(BesselJG6$\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%(BesselJG6$\"\"\"F'" }}{PARA 13 "" 1 " " {GLPLOT2D 801 273 273 {PLOTDATA 2 "6&-%'CURVESG6$7_]l7$$\"\"!F)$\"\" \"F)7$$\"+[OnMN!#5$\"+QE3!p*F/7$$\"+&HZ$pqF/$\"+#=&4*y)F/7$$\"+?ToO))F /$\"+C%e59)F/7$$\"+%4-/1\"!\"*$\"+@kO!Q(F/7$$\"+wd8P7F?$\"+33bClF/7$$ \"+f%pQT\"F?$\"+c[@$f&F/7$$\"+#\\Wwc\"F?$\"+u>(yt%F/7$$\"+C&>9s\"F?$\" +\\u'f&QF/7$$\"+cX>v=F?$\"+8ySiHF/7$$\"+)ep*G?F?$\"+?(>@2#F/7$$\"+?Yu# =#F?$\"+(*[w*>\"F/7$$\"+_'>lL#F?$\"+QDi%f$!#67$$\"+%o%H!\\#F?$!+[ExaVF do7$$\"+;(pSk#F?$!+Xxzs6F/7$$\"+H>%**)HF?$!+^$Gic#F/7$$\"+UT\"eL$F?$!+ d)[$>NF/7$$\"+[-v3NF?$!+;x88QF/7$$\"+ajo\"o$F?$!+i&[<)RF/7$$\"+2W:oPF? $!+ayT>SF/7$$\"+gCiaQF?$!+D&Ql-%F/7$$\"+904TRF?$!+\\ru.SF/7$$\"+n&ev-% F?$!+Ew&=&RF/7$$\"+YMsvVF?$!+/f\"4Z$F/7$$\"+C$))Qs%F?$!+A#[gi#F/7$$\"+ +K0s]F?$!+zuBP:F/7$$\"+!3=-U&F?$!+#R*HBMFdo7$$\"+!>Gnw&F?$\"+Wz`F\")Fd o7$$\"++$QK6'F?$\"+RMf1=F/7$$\"+5%[(fkF?$\"+KctODF/7$$\"+D&ei!oF?$\"+q %y\\$HF/7$$\"+S\"))o'pF?$\"+g7f(*HF/7$$\"+bx^FrF?$\"+?dZ#)HF/7$$\"+qt9 )G(F?$\"+59w\"*GF/7$$\"+&)px[uF?$\"+H3QHFF/7$$\"+:i.qxF?$\"+#>^N@#F/7$ $\"+WaH\"4)F?$\"+3Jo'\\\"F/7$$\"+l2%RU)F?$\"+3rWmiFdo7$$\"+!4'ec()F?$! 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$pp4)Fdo7$$\"+la8,(*Fix$\"+iiaDuFdo7$Fjcp$\"+d1d))eFdo7$Fddp$\"+$=Z1Z \"Fdo7$Fhep$!+s5Z&[$Fdo7$$\"++gZ'*)*Fix$!+\"Hm(=dFdo7$F]fp$!+ " 0 "" {MPLTEXT 1 0 106 "f := x -> BesselJ(0,x):\ng := x -> BesselJ( 1,x):\n'f(x)'=f(x);\n'g(x)'=g(x);\nallroots(f(x)=g(x),x=1e-9..100);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(BesselJG6$\"\"!F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%(BesselJG6$\"\"\"F'" }}{PARA 7 "" 1 "" {TEXT -1 78 "Warning, cannot use interval arithmetic for value ranges - estimating instead\n" }}{PARA 7 "" 1 "" {TEXT -1 83 "Warning, cannot use interval arithmetic for derivative ranges - es timating instead\n" }}{PARA 7 "" 1 "" {TEXT -1 87 "Warning, cannot use interval arithmetic for 2nd derivative ranges - estimating instead\n " }}{PARA 12 "" 1 "" {XPPMATH 20 "6B$\"+^cpM9!\"*$\"+aD5!o%F%$\"+NB+Oy F%$\"+hcJ)4\"!\")$\"+m\"pFT\"F,$\"+1-6Fcn#*F,$\"+uds\"e*F,$\"+ m#*)e*)*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 9 "allbisect" }{TEXT -1 19 " can also \+ be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "f := x -> BesselJ(0,x):\ng := x -> BesselJ(1,x):\n'f (x)'=f(x);\n'g(x)'=g(x);\nallbisect(f(x)=g(x),x=0..100);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%(BesselJG6$\"\"!F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%(BesselJG6$\"\"\"F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6B$\"+^cpM9!\"*$\"+aD5!o%F%$\"+NB+OyF%$\"+hcJ)4\" !\")$\"+m\"pFT\"F,$\"+1-6Fcn#*F,$\"+uds\"e*F,$\"+m#*)e*)*F," }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 17" }{TEXT 281 90 " .. a root-finding problem involving a function with a complic ated analytical description " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "The procedure allroots can be use to find nume rical approximations for all the zeros of the soliton wave function " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 36 " described in th e subsection below. " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 81 ": The formula for this function was given with Maple 4 as an examp le for Maple's " }{TEXT 0 4 "plot" }{TEXT -1 12 " procedure. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 33 "Function s(x) for a soli ton wave " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2060 "s := x -> sec(arctan(1/2*tan(arctan(1/5*tan(arctan( exp(-3*x-5/3))-arctan(exp(2*x+5/2))))-arctan(3*tan(arctan(exp(2*x+5/2) )-arctan(exp(x+5))))))+arctan(exp(2*x+5/2))-arctan (19*tan(arctan(3*ta n(arctan(exp(2*x+5/2))-arctan(exp(x+5))))+arctan(7/\n2*tan(arctan(exp( x+5))-arctan(exp(9/5*x+25/9))))))-arctan(exp(x+5)))^2*(1/2*\nsec(arcta n(1/5*tan(arctan(exp(-3*x-5/3))-arctan(exp(2*x+5/2))))-arctan(3*tan(\n arctan(exp(2*x+5/2))-arctan(exp(x+5)))))^2*(1/5*sec(arctan(exp(-3*x-5/ 3))-\narctan(exp(2*x+5/2)))^2*(-3*exp(-3*x-5/3)/(1+exp(-3*x-5/3)^2)-2* exp(2*x+5/2)/\n(1+exp(2*x+5/2)^2))/(1+1/25*tan(arctan(exp(-3*x-5/3))-a rctan(exp(2*x+5/2)))^2\n)-3*sec(arctan(exp(2*x+5/2))-arctan(exp(x+5))) ^2*(2*exp(2*x+5/2)/(1+exp(2*x +5/2)^2)-exp(x+5)/(1+exp(x+5)^2))/(1+9*t an(arctan(exp(2*x+5/2))-arctan (exp(x+5)))^2))/(1+1/4*tan(arctan(1/5*t an(arctan(exp(-3*x-5/3)) -arctan(exp(2*x+5/2))))-arctan(3*tan(arctan(e xp(2*x+5/2))-arctan(exp(x+5)))))^2)+2*exp(2*x+5/2)/(1+exp(2*x+5/2)^2)- 19*sec(arctan(3*tan(arctan(exp(2*x+5/2))-arctan(exp(x+5))))+arctan(7/ 2*tan (arctan(exp(x+5))-arctan(exp(9/5*x+25/9)))))^2*(3*sec(arctan(exp \n(2*x+5/2))-arctan(exp(x+5)))^2*(2*exp(2*x+5/2)/(1+exp(2*x+5/2)^2)-ex p(x+5)/(1\n+exp(x+5)^2))/(1+9*tan(arctan(exp(2*x+5/2))-arctan(exp(x+5) ))^2)+7/2*sec(\narctan(exp(x+5))-arctan(exp(9/5*x+25/9)))^2*(exp(x+5)/ (1+exp(x+5)^2)-9/5*exp(\n9/5*x+25/9)/(1+exp(9/5*x+25/9)^2))/(1+49/4*ta n(arctan(exp(x+5))-arctan(exp(9 /5*x+25/9)))^2))/(1+361*tan(arctan(3*t an(arctan(exp(2*x+5/2))-arctan(exp(x+5 ))))+arctan(7/2*tan(arctan(exp( x+5))-arctan(exp(9/5*x+25/9)))))^2)-exp(x+5)/(1+exp(x+5)^2))/(1+1/16*t an(arctan(1/2*tan(arctan(1/5*tan(arctan (exp(-3*x-5/3))-arctan(exp(2*x +5/2))))-arctan(3*tan(arctan(exp(2*x+5/2))-arctan(exp(x+5))))))+arctan (exp(2*x+5/2))-arctan(19*tan(arctan(3*tan(arctan (exp(2*x+5/2))-arctan (exp(x+5))))+arctan(7/2*tan(arctan(exp(x+5))-arctan (exp(9/5*x+25/9))) )))-arctan(exp(x+5)))^2)+12*sec(arctan(exp(2*x+5/2)) -arctan(exp(x+5)) )^2*(2*exp(2*x+5/2)/(1+exp(2*x+5/2)^2)-exp(x+5)/(1+exp(x+5)^2)) /(1+9* tan(arctan(exp(2*x+5/2))-arctan(exp(x+5)))^2):\n's(x)'=s(x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"xG,&*(-%$secG6#,*-%'arctanG6#,$* &#\"\"\"\"\"#F4-%$tanG6#,&-F/6#,$*&#F4\"\"&F4-F76#,&-F/6#-%$expG6#,&*& \"\"$F4F'F4!\"\"#F?FJFKF4-F/6#-FF6#,&*&F5F4F'F4F4#F?F5F4FKF4F4F4-F/6#, $*&FJF4-F76#,&FMFK-F/6#-FF6#,&F'F4F?F4F4F4F4F4F4F4FKFMFK-F/6#,$*&\"#>F 4-F76#,&FTFK-F/6#,$*&#\"\"(F5F4-F76#,&FenF4-F/6#-FF6#,&*(\"\"*F4F?FKF' F4F4#\"#DFapF4FKF4F4F4F4F4F4FenF4F5,**&F3F4*(-F+F8F5,&*&F>F4*(-F+FAF5, &*(FJF4FEF4,&F4F4*$)FEF5F4F4FKFK*(F5F4FOF4,&F4F4*$)FOF5F4F4FKFKF4,&F4F 4*&#F4FcpF4*$)F@F5F4F4F4FKF4F4**FJF4-F+FYF5,&*(F5F4FOF4FbqFKF4*&FgnF4, &F4F4*$)FgnF5F4F4FKFKF4,&F4F4*&FapF4)FXF5F4F4FKFKF4,&F4F4*&#F4\"\"%F4* $)F6F5F4F4F4FKF4F4*(F5F4FOF4FbqFKF4**F^oF4-F+F`oF5,&**FJF4F[rF5F\\rF4F brFKF4*&FfoF4*(-F+FioF5,&F^rF4*&#FapF?F4*&F]pF4,&F4F4*$)F]pF5F4F4FKF4F KF4,&F4F4*&#\"#\\FhrF4*$)FhoF5F4F4F4FKF4F4F4,&F4F4*&\"$h$F4)F_oF5F4F4F KFKF^rFKF4,&F4F4*&#F4\"#;F4*$)-F7F,F5F4F4F4FKF4**\"#7F4F[rF5F\\rF4FbrF KF4" }}}{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 67 "Because of the complicated nature of the analytical exp ression for " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" }{TEXT -1 149 " \+ (and therefore also its first and second derivatives) the best way to \+ handle this problem appears to be to avoid the use of interval arithme tic via " }{TEXT 0 5 "evalr" }{TEXT -1 18 " with the option \"" } {TEXT 262 15 "use_evalr=false" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "A graphical investigation suggests that all the roots lie in the interval " }{XPPEDIT 18 0 "-10 <=x" "6#1,$\"#5!\"\"%\"xG" }{XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" } {TEXT -1 2 ". 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$\"3N_Ge4R%Qz&!#<$\"3s()Gy#H`N#fF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %;secant~approximation~->~~~G$\"3V?\\Z2b(G&e!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%ENewton~approximation~from~left~->~~~G$\"38'Qi'4]PWe!# <" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%FNewton~approximation~from~right ~->~~~G$\"3b<\"y'4YrZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;secant~ approximation~->~~~G$\"3-TeB[wuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%ENewton~approximation~from~left~->~~~G$\"3+bYUPuuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%FNewton~approximation~from~right~->~~~G$\"3(4B h(QuuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;secant~approximation~- >~~~G$\"3&fGh(QuuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%FNewton~app roximation~from~right~->~~~G$\"3)eGh(QuuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2single~root~->~~~G$\"3)eGh(QuuZe!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~th e~interval:~G;$\"3s()Gy#H`N#f!#<$\"3\\&Q?qR?r@'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3\\&Q?qR?r@'!#<$\"3i=zX%) oRSmF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$ \"3#*[*G:VH0k'!#<$\"3V[g>)f'*\\n(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8Checking~the~interval:~G;$\"3NY)yr'>@Er!#<$\"3g#QDb7$R!=(F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3g#QDb7$ R!=(!#<$\"3%)=>(QGuXB(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking ~the~interval:~G;$\"3V[g>)f'*\\n(!#<$\"3Xw(4.S'e****F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8Checking~the~interval:~G;$\"3O))R%R@e8.*!#<$\"3v ')=F%$\"+;\\-JEF%$\"+RuuZeF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 9 "allbisect" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "allbise ct(s(x),x=-10..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&$!+kG6l\\!\"*$! +')=F%$\"+<\\-JEF%$\"+QuuZeF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "Example 18" }{TEXT 306 49 " .. a degree 50 Taylor polyn omial with 14 roots " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 40 " be the degree 50 Taylor polynomial for " }{XPPEDIT 18 0 "cos(x )" "6#-%$cosG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=0" "6#/%\"x G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The following graph indicates that " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 20 " has 14 real roots. 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3#eLGVv$f8UYXa\"F1$!3=Q'[`a_ul*Feo7$$\"3;]7GQ!*eb:F 1$!3A>5eJFg%))*Feo7$$\"37DJ?b;jm:F1$!3O_(4CqW8***Feo7$$\"3C+]7sUnx:F1$ !3cv*QQ%)yj(**Feo7$$\"39vVB!H@ye\"F1$!3_/,'=^ca&)*Feo7$$\"3=]PM3$ozf\" F1$!3$p*f(*H7:L'*Feo7$$\"3DDJXE`63;F1$!3Jy[4U8v6$*Feo7$$\"38+DcWBE=;F1 $!3?qGi/^c%*))Feo7$$\"3E]7y!Qc&Q;F1$!3a')3=%=`4z(Feo7$$\"3-+++Y=F1$\"3;#)HJ%41.=*Feo7$$\"3I ]7.y:Gd=F1$\"3$yy+F2ON^*Feo7$$\"3+vo/O>Po=F1$\"3i]?'oBk&=(*Feo7$$\"31+ D1%Hi%z=F1$\"3U'=(4'y^*)y*Feo7$$\"35D\"y?l_0*=F1$\"30K*HyZ\"f=(*Feo7$$ \"3\")\\P45Ik,>F1$\"3Z)f>vs[8]*Feo7$$\"3([P4\"oLt7>F1$\"3kezRyRcI\"*Fe o7$$\"3$***\\7EP#Q#>F1$\"3u:+F+(z$)f)Feo7$$\"3/+Dc,N0X>F1$\"3u,$>sXI04 (Feo7$$\"39+++xKGm>F1$\"3%y'zt>>LT[Feo7$$\"3E+vV_I^()>F1$\"3?$*fZ(>!GZ ;Feo7$$\"3O+](y#Gu3?F1$!3OND`9alkGFeo7$$\"3SD1Ru*\\,-#F1$!3-oL`&zM(\\g Feo7$$\"3X]i!47d:.#F1$!3Q)>oS7tc&**Feo7$$\"3\\v=UnU'H/#F1$!3_P=D^$f\"z 9F-7$$\"3=+v$RTrV0#F1$!3G'GE&\\1o$3#F-7$$\"3q7`>()\\2g?F1$!3Jo*)[A'fCW #F-7$$\"3)[7`/cyd1#F1$!3AQ&))zO`i%GF-7$$\"3SP4rL@[r?F1$!3')[a\"y>*f,LF -7$$\"3#*\\(opq&=x?F1$!3vL5MES+;QF-7$$\"3WilA!G*)G3#F1$!3UQA/(o!3)R%F- 7$$\"3'\\P%[`Gf)3#F1$!3oSmQVKqd]F-7$$\"3S\"G8,kW94#F1$!3[d%HH_n+U&F-7$ $\"3[(=UnU'H%4#F1$!3`09VcG=1eF-7$$\"3c$4rL@[r4#F1$!3,HLR91s " 0 "" {MPLTEXT 1 0 78 "st := time():\nallroots(p(x) ,x=-infinity..infinity,use_evalr=false);\ntime()-st;" }}{PARA 12 "" 1 "" {XPPMATH 20 "60$!+j/:'*>!\")$!+M6!zs\"F%$!+$p;PT\"F%$!+Hub*4\"F%$!+ M;)R&y!\"*$!+!)*)Q7ZF.$!+Fjzq:F.$\"+Fjzq:F.$\"+!)*)Q7ZF.$\"+M;)R&yF.$ \"+Hub*4\"F%$\"+$p;PT\"F%$\"+M6!zs\"F%$\"+j/:'*>F%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"%OU!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The computation takes a long time if interval a rithmetic is used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "st := time():\nallroots(p(x),x=-20..20);\nt ime()-st;" }}{PARA 12 "" 1 "" {XPPMATH 20 "60$!+j/:'*>!\")$!+M6!zs\"F% $!+$p;PT\"F%$!+Hub*4\"F%$!+M;)R&y!\"*$!+!)*)Q7ZF.$!+Fjzq:F.$\"+Fjzq:F. $\"+!)*)Q7ZF.$\"+M;)R&yF.$\"+Hub*4\"F%$\"+$p;PT\"F%$\"+M6!zs\"F%$\"+j/ :'*>F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'95Y!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Check" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fsolve( p(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "60$!+j/:'*>!\")$!+M6!zs\"F%$!+ $p;PT\"F%$!+Hub*4\"F%$!+M;)R&y!\"*$!+!)*)Q7ZF.$!+Fjzq:F.$\"+Fjzq:F.$\" +!)*)Q7ZF.$\"+M;)R&yF.$\"+Hub*4\"F%$\"+$p;PT\"F%$\"+M6!zs\"F%$\"+j/:'* >F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 19 " }{TEXT 307 36 " .. a function with a sharp \"spike\" " }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)=9 9999/100000-exp(-100000*(x-1)^2)" "6#/-%\"fG6#%\"xG,&*&\"&*****\"\"\" \"'++5!\"\"F+-%$expG6#,$*&F,F+*$,&F'F+F+F-\"\"#F+F-F-" }{TEXT -1 48 " \+ has two zeros close together near 1. Note that " }{XPPEDIT 18 0 "f(1)= -1/100000" "6#/-%\"fG6#\"\"\",$*&F'F'\"'++5!\"\"F+" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "f := x -> (100000-1)/100000-exp(-100000*(x-1)^2):\n'f(x)'=f(x); \nplot(f(x),x=0..2,numpoints=60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"fG6#%\"xG,&#\"&*****\"'++5\"\"\"-%$expG6#,$*&F+F,),&F'F,F,!\"\"\" \"#F,F4F4" }}{PARA 13 "" 1 "" {GLPLOT2D 589 310 310 {PLOTDATA 2 "6%-%' CURVESG6$7bq7$$\"\"!F)$\"3Y+++++!*****!#=7$$\"3yQ.A$fbma$!#>F*7$$\"3T \\p,h;eKmF0F*7$$\"3+W'=r#GI55F,F*7$$\"3)*GPUDzkf8F,F*7$$\"3(pidukKtq\" F,F*7$$\"3eBa_\"*3oH?F,F*7$$\"3%Rk=r#QXjBF,F*7$$\"3;riduAk3FF,F*7$$\"3 %[30$)pBF0$F,F*7$$\"3F?KfNPl1MF,F*7$$\"3fXZ30bR=PF,F*7$$\"3k:\\p,HMpSF ,F*7$$\"3H)*Q.A8t@WF,F*7$$\"3Y\\p,hGKhZF,F*7$$\"3%oSk=^/(p]F,F*7$$\"3m )*Q.A,SOaF,F*7$$\"3)3$)*Q.%Rqu&F,F*7$$\"3w>KfNnQ3hF,F*7$$\"3esBa_.CGkF ,F*7$$\"3iznSk/#y(F,F*7$$\"3&)*Q.A$>cF\")F,F*7$$\"3y%p,hw*H'[)F,F*7$$\"3oXZ30 @e)z)F,F*7$$\"3vy1W'ebe8*F,F*7$$\"3;`_\"\\4#H%[*F,F*7$$\"3Q4mzn1ta'*F, F*7$$\"3glznS#p^#)*F,$\"3mo%********)****F,7$$\"3IV'oe*fZN)*F,$\"3QVC) ******)****F,7$$\"35A$f5v#yX)*F,$\"33!)4`*****)****F,7$$\"3!)***\\i]*3 c)*F,$\"3C\"zo)*)***)****F,7$$\"3]x1WhiRm)*F,$\"3?=c/B)**)****F,7$$\"3 Ic8j;Iqw)*F,$\"3!o#G&3](*)****F,7$$\"3+M?#=x4q))*F,$\"3o6\\2g9()****F, 7$$\"3q6F,FlJ(*)*F,$\"3))=FNhkj****F,7$$\"3]!R.AGBw!**F,$\"3I_O[mA$z** *F,7$$\"3?oSRP+$z\"**F,$\"3?5X3P*>!))**F,7$$\"3!fu%e#zO#G**F,$\"3q&4lL ^.>%**F,7$$\"3qCaxZNaQ**F,$\"3=7V#*eM'4x*F,7$$\"3S-h'HI])[**F,$\"3??4& *pk;p#*F,7$$\"35!yc\"eq:f**F,$\"3^GV49e$R6)F,7$$\"3!*euM8QYp**F,$\"3e& QbP&=0kgF,7$$\"3gO\"Q&o0xz**F,$\"3))[H_TZGeLF,7$$\"3I9)GPKx+***F,$\"3/ Bm()o4wv$*F07$$\"34\\>*ySQ++\"!#<$\"35DScjt0v8!#@7$$\"3(p,6M3p5+\"Fgv$ \"3A6O,0F%*z5F,7$$\"3&[3I*e(*4-5Fgv$\"3oM%fw]@`c$F,7$$\"3t_\"\\WVIJ+\" Fgv$\"39cDnIjkYiF,7$$\"3h?#o*46;/5Fgv$\"3[@#p8<%oH#)F,7$$\"3F)G([&y\"> 05Fgv$\"3Ke..t#*yC$*F,7$$\"3$fN15YAi+\"Fgv$\"3iwz>vEr\"z*F,7$$\"3\"QUD l8`s+\"Fgv$\"3'z!p*QW))z%**F,7$$\"3p\"\\W?\"QG35Fgv$\"3]?_/KMV*)**F,7$ $\"3NfNc([9$45Fgv$\"35l,#Gi$>)***F,7$$\"3,FE3j^M55Fgv$\"3mMW/b]n****F, 7$$\"3*[p,'QeP65Fgv$\"3[-`FAg()****F,7$$\"3xi279lS75Fgv$\"3'\\gML$z*)* ***F,7$$\"3lI)R'*=PM,\"Fgv$\"3g>n'f&)**)****F,7$$\"3`)*)e^'yY95Fgv$\"3 !)yJ)=****)****F,7$$\"3>mznS&)\\:5Fgv$\"3aQ,j*****)****F,7$$\"3iDl6IHk ;5Fgv$\"3ga1*******)****F,7$$\"3%[3b&>ty<5Fgv$\"3Q=)********)****F,7$$ \"30WO**3<$*=5Fgv$\"3A)**********)****F,7$$\"3E.AV)4w+-\"FgvF*7$$\"3\" >K4t([OA5FgvF*7$$\"3bSk=cOlC5FgvF*7$$\"3&ynSR@J#H5FgvF*7$$\"39:\\pr(3Q .\"FgvF*7$$\"3t*Q.s)Q'H/\"FgvF*7$$\"3Kk=r-!>@0\"FgvF*7$$\"3!o,h'f:do5F gvF*7$$\"3Gp,h;T-&3\"FgvF*7$$\"3]UD:\\t:?6FgvF*7$$\"3%o,h'zK*>:\"FgvF* 7$$\"3=mzng$)z'=\"FgvF*7$$\"3FD:\\Hsa>7FgvF*7$$\"33A$fNBzPD\"FgvF*7$$ \"33ridusC(G\"FgvF*7$$\"3i1W'=@&GA8FgvF*7$$\"3c30$)44.c8FgvF*7$$\"3#** *****f2a!R\"FgvF*7$$\"3\"3W'=^[wC9FgvF*7$$\"3mp,hOK@c9FgvF*7$$\"3CPUDN mD#\\\"FgvF*7$$\"3+mzn!G%\\C:FgvF*7$$\"3AridMj')e:FgvF*7$$\"3K.A$fxkFgvF*7$$\"3MUD:4Skk>FgvF*7$$\"\"#F)F*-%'COLOURG6&%$RGBG $\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F\\dl-%%VIEWG6$;F(F_cl%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The pr ocedure " }{TEXT 0 8 "allroots" }{TEXT -1 101 " can find the two zeros when operating in the default mode whereby interval arithmetic using \+ Maple's " }{TEXT 0 5 "evalr" }{TEXT -1 43 " is employed to estimate ra nges of values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 33 "evalf[20](allroots(f(x),x=0..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"5ek)**\\(*****)****!#?$\"5a8+]-++,+5!#>" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "When usi ng sampling to estimate ranges of values, the zeros are missed unless \+ the maximum number of sampling points to be used is increased. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "evalf[20](allroots(f(x),x=0..2,use_evalr=false,maxpoints=260));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"5ek)**\\(*****)****!#?$\"5a8+]-++, +5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "This is not necessary if a narrower starting interval is used. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "evalf[20](allroots(f(x),x=.95..1.05,use_evalr=false));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"5ek)**\\(*****)****!#?$\"5a8+]-++,+5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Check \+ using " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "evalf[20](fsolve(f (x),x=0..1));\nevalf[20](fsolve(f(x),x=1..2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5ek)**\\(*****)****!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5a8+]-++,+5!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 20" }{TEXT 304 47 " .. a function with a number o f sharp \"spikes\" " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Thi s example is again concerned with the function " }{XPPEDIT 18 0 "f(x)= 99999/100000-exp(-100000*(x-1)^2)" "6#/-%\"fG6#%\"xG,&*&\"&*****\"\"\" \"'++5!\"\"F+-%$expG6#,$*&F,F+*$,&F'F+F+F-\"\"#F+F-F-" }{TEXT -1 36 " \+ considered in the previous example." }}{PARA 0 "" 0 "" {TEXT -1 32 "Co nsider the auxiliary function " }{XPPEDIT 18 0 "h(x)=100*sin(3*x)/(20+ x^2)" "6#/-%\"hG6#%\"xG*(\"$+\"\"\"\"-%$sinG6#*&\"\"$F*F'F*F*,&\"#?F** $F'\"\"#F*!\"\"" }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "g(x)" "6#-%\" gG6#%\"xG" }{TEXT -1 20 " be the composition " }{XPPEDIT 18 0 "g(x)=f( h(x))" "6#/-%\"gG6#%\"xG-%\"fG6#-%\"hG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#% \"xG" }{TEXT -1 33 " has the value 1 at 10 values of " }{TEXT 308 1 "x " }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"x G" }{XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "h := x \+ -> 100*sin(3*x)/(20+x^2):\n'h(x)'=h(x);\nplot([h(x),1],x=0..10,color=[ red,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG,$*(\"$+ \"\"\"\"-%$sinG6#,$*&\"\"$F+F'F+F+F+,&\"#?F+*$)F'\"\"#F+F+!\"\"F+" }} {PARA 13 "" 1 "" {GLPLOT2D 527 212 212 {PLOTDATA 2 "6&-%'CURVESG6$7[v7 $$\"\"!F)F(7$$\"3WmmmT&)G\\a!#>$\"31'[g\"!#<7$$\"3$*****\\ilyM;F0$\"3io!)f$z4>N#F67$$\"3emmm ;arz@F0$\"3(fW&zrCGMIF67$$\"3;L$e*)4bQl#F0$\"39Ms&QyD3c$F67$$\"3v***\\ 7y%*z7$F0$\"39s\\`4!GM,%F67$$\"3Lm;ajW8-OF0$\"3!z(odUD'GQ%F67$$\"3[LL$ e9ui2%F0$\"3eR`g%>+yF0$\"3oimZLO/mMF67$$\"3 9ommT&phN)F0$\"3gs\\rL*G['GF67$$\"37M$3-js.*))F0$\"3wW%GxHQv>#F67$$\"3 A,+v=ddC%*F0$\"3/\"fNn%)f'z9F67$$\"3?n;H2)y(e**F0$\"3:(eq%[ZJ0tF07$$\" 3KLLe*=)H\\5F6$!3O9q\"F6$!3'=m3z#=%y.#F67$$\"3smm \"z/3uC\"F6$!3QJ]kWa$=i#F67$$\"31++vo3p)H\"F6$!3#))y!zvq@eJF67$$\"3ULL e*ot*\\8F6$!3X2p$\\C[Jh$F67$$\"3xmmT5lD,9F6$!3#3$yf;UnwRF67$$\"3!**** \\7LRDX\"F6$!3&f)3#p757C%F67$$\"3YLLe9x0z9F6$!3?GL>L\\SPVF67$$\"3!om;z 4wb]\"F6$!3a5l`m%3`S%F67$$\"3ZLLe*GN)=:F6$!3u)pt'G%p&GWF67$$\"39++D\"[ %4K:F6$!3F,5[ULoWWF67$$\"3#om;Hn``a\"F6$!3in/e%3]OX%F67$$\"3[LLekGhe:F 6$!3oytc)G\"[bWF67$$\"39++Dc?(=d\"F6$!3svQ+z-?]WF67$$\"3#om;zCJ^e\"F6$ !3C)>de)>%yV%F67$$\"3[LLeR/R)f\"F6$!3S14iRGX=WF67$$\"3;++DJ'\\;h\"F6$! 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "The functio n " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 15 " has the val ue " }{XPPEDIT 18 0 "-1/100000" "6#,$*&\"\"\"F%\"'++5!\"\"F'" }{TEXT -1 18 " at each of these " }{TEXT 309 1 "x" }{TEXT -1 10 " values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "rts := [allroots(h(x)=1,x=0..10)];\nf := x -> (100000-1)/100000 -exp(-100000*(x-1)^2):\n'f(x)'=f(x);\ng := unapply(f(h(x)),x):\n'g(x)' =g(x);\nmap(g,rts);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$rtsG7,$\"+LS Y8n!#6$\"+YsGo(*!#5$\"+)yNx<#!\"*$\"+z'>E/$F.$\"+)y[6K%F.$\"+W%yu2&F.$ \"+OLe2lF.$\"+MW'=2(F.$\"+CQ_N))F.$\"+eP@F*)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&#\"&*****\"'++5\"\"\"-%$expG6#,$*&F+F,) ,&F'F,F,!\"\"\"\"#F,F4F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#% \"xG,&#\"&*****\"'++5\"\"\"-%$expG6#,$*&F+F,),&*(\"$+\"F,-%$sinG6#,$*& \"\"$F,F'F,F,F,,&\"#?F,*$)F'\"\"#F,F,!\"\"F,F,FAF@F,FAFA" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7,$!'++5!#5F$F$F$F$F$F$F$F$F$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "For most of the time th roughout the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=10" "6#1%!G\"#5" }{TEXT -1 14 " the function " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 39 " has the value 1, but near each of the " }{TEXT 311 1 "x" }{TEXT -1 14 " values where \+ " }{XPPEDIT 18 0 "h(x)=1" "6#/-%\"hG6#%\"xG\"\"\"" }{TEXT -1 41 ", the re is a sharp spike in the graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6 #%\"xG" }{TEXT -1 40 " where the graph dips briefly below the " } {TEXT 310 1 "x" }{TEXT -1 48 " axis. To see all ten spikes in a Maple \+ plot of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 235 " it i s necessary to increase the number of function evaluations used for th e plot rather drastically. (Since the memory requirement when the plot is constructed in this manner is quite large, the next command has be en left unexecuted.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(g(x),x=0..10,numpoints=6000,resoluti on=1000);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "A more efficient way to obtain the same picture (in terms of t he memory requirement) is to construct separate plots of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 59 " over suitable subinterval s, and then patch them together. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "d := .01:\nfor i to nops(rt s) do\n p||i := plot(g(x),x=rts[i]-d..rts[i]+d);\nend do:\nq0 := plo t(g(x),x=0..rts[1]-d):\nfor i to nops(rts)-1 do\n q||i := plot(g(x), x=rts[i]+d..rts[i+1]-d);\nend do:\nq||(nops(rts)) := plot(g(x),x=rts[n ops(rts)]+d..10):\nplots[display](p||(1..nops(rts)),q||(0..nops(rts))) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 730 242 242 {PLOTDATA 2 "69-%'CURVESG6 $7[q7$$\"3++++LSY8d!#>$\"3Y+++++!*****!#=7$$\"3WLLLT$eqv&F*F+7$$\"3smm \"f^*)\\z&F*F+7$$\"3.LL$))3Zw$eF*F+7$$\"3MLL$QU(e!)eF*F+7$$\"3Xmm\"4nB L#fF*F+7$$\"37LLeUc%H'fF*F+7$$\"3x***\\#**=(R+'F*F+7$$\"3QLLe78SYgF*F+ 7$$\"33++DaYp)3'F*F+7$$\"3emmma&)>KhF*F+7$$\"3cLL$Q\"o^qhF*F+7$$\"3=++ +)4aO@'F*F+7$$\"3m+++3&opD'F*F+7$$\"3c+++o*4()H'F*F+7$$\"3)QL$e7_hOjF* F+7$$\"3/nmm9#)o\"Q'F*F+7$$\"3rmmmf4()>kF*F+7$$\"3c***\\#plGkkF*F+7$$ \"3^nmm4>g.lF*F+7$$\"3))***\\#pqtYlF*F+7$$\"3c++vT>\"ye'F*$\"3!\\)**** *****)****F-7$$\"3uK$e>gS#4mF*$\"3_^IM*****)****F-7$$\"3Imm;i#p1j'F*$ \"3mA7?Ti*)****F-7$$\"3gmT&=O30k'F*$\"3E.Nswsz****F-7$$\"3!pmT:YZ.l'F* $\"3;*4JA;_!)***F-7$$\"31+K***F-7$$\"3?n\"H7c'=gmF *$\"3-7BB&)*=!y**F-7$$\"3MR^Dz*=cF-7$$\"3WP4rc5B&p'F*$\"3TFa?(f^&=^F-7$$\"3u$e*[cw b'p'F*$\"3%*y-_?:6-YF-7$$\"3-I#oiD%)yp'F*$\"3exc-U>>wSF-7$$\"3%\\(o/c3 @*p'F*$\"3)pR#)=p8#[NF-7$$\"3C@b#eXP0q'F*$\"3VtEM;pLEIF-7$$\"3anTgbS'= q'F*$\"3gDF-7$$\"3ue9;bs^/nF*$\"3M^'*ozws&e\"F-7$$\"3&*\\(=ZXq rq'F*$\"3UUKfYhD)=)F*7$$\"3bUgFaO#)4nF*$\"3a%GV(3l9F-7$$\"3,sn&G`y[s'F*$\"3e$[BJie(\\CF-7$$\"384F>(\\ cis'F*$\"3#zcXfG^Q(HF-7$$\"3kZ'G:YMws'F*$\"3Xni,I1*[^$F-7$$\"3w%ekeU7! 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" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "allroots(g(x),x=0..10);\nsols := [%]:" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$\"+XfR8n!#6$\"+A@`8nF%$\"+],Go(*!#5$\"+UVHo(*F*$\"+C\\ tx@!\"*$\"+`mtx@F/$\"+!o=E/$F/$\"+x1iUIF/$\"+Xt9@VF/$\"+J-:@VF/$\"+!zw u2&F/$\"+(4![x]F/$\"+@0e2lF/$\"+]he2lF/$\"+o8'=2(F/$\"++v'=2(F/$\"+L** \\N))F/$\"+FxaN))F/$\"+!f*=F*)F/$\"+8zBF*)F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The solutions can be chec ked using " }{TEXT 0 6 "fsolve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "op(map(u->fs olve(g(x),x=evalf[8](u)),sols));" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$ \"+XfR8n!#6$\"+A@`8nF%$\"+],Go(*!#5$\"+UVHo(*F*$\"+C\\tx@!\"*$\"+`mtx@ F/$\"+!o=E/$F/$\"+x1iUIF/$\"+Xt9@VF/$\"+J-:@VF/$\"+!zwu2&F/$\"+(4![x]F /$\"+@0e2lF/$\"+]he2lF/$\"+o8'=2(F/$\"++v'=2(F/$\"+L**\\N))F/$\"+FxaN) )F/$\"+!f*=F*)F/$\"+8zBF*)F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "By splitting up the interval " }{TEXT 0 8 "allroots" }{TEXT -1 71 " can obtain all the zeros using sampling to estimate ranges of values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "n := 200;\nw := 10./n;\nseq( allroots(g(x),x=w*(i-1)..w*i,use_evalr=false,maxpoints=100),i=1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"$+#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG$\"+++++]!#6" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$ \"+XfR8n!#6$\"+A@`8nF%$\"+],Go(*!#5$\"+UVHo(*F*$\"+C\\tx@!\"*$\"+`mtx@ F/$\"+!o=E/$F/$\"+x1iUIF/$\"+Xt9@VF/$\"+J-:@VF/$\"+!zwu2&F/$\"+(4![x]F /$\"+@0e2lF/$\"+]he2lF/$\"+o8'=2(F/$\"++v'=2(F/$\"+L**\\N))F/$\"+FxaN) )F/$\"+!f*=F*)F/$\"+8zBF*)F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "This can also be achieved by using the op tion \"" }{TEXT 262 9 "split=200" }{TEXT -1 3 "\". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "allroots(g (x),x=0..10,use_evalr=false,maxpoints=100,split=200);" }}{PARA 12 "" 1 "" {XPPMATH 20 "66$\"+XfR8n!#6$\"+A@`8nF%$\"+],Go(*!#5$\"+UVHo(*F*$ \"+C\\tx@!\"*$\"+`mtx@F/$\"+!o=E/$F/$\"+x1iUIF/$\"+Xt9@VF/$\"+J-:@VF/$ \"+!zwu2&F/$\"+(4![x]F/$\"+@0e2lF/$\"+]he2lF/$\"+o8'=2(F/$\"++v'=2(F/$ \"+L**\\N))F/$\"+FxaN))F/$\"+!f*=F*)F/$\"+8zBF*)F/" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 58 "Find (correct to 10 digits) all the \+ zeros of the function " }{XPPEDIT 18 0 "f(x) = sin(exp(sqrt(x+1)));" " 6#/-%\"fG6#%\"xG-%$sinG6#-%$expG6#-%%sqrtG6#,&F'\"\"\"F2F2" }{TEXT -1 27 " which lie in the interval " }{XPPEDIT 18 0 "[0, 13];" "6#7$\"\"! \"#8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "________________ __________________________" }}{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 42 "_______________________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 62 "Find (correc t to 10 digits) all the solutions of the equation " }{XPPEDIT 18 0 "2* exp(sin(x))=cos(53*x)" "6#/*&\"\"#\"\"\"-%$expG6#-%$sinG6#%\"xGF&-%$co sG6#*&\"#`F&F-F&" }{TEXT -1 27 " which lie in the interval " } {XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 42 "___________________________________ _______" }}{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 42 "__________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 62 "Find (correct to 10 di gits) all the solutions of the equation " }{XPPEDIT 18 0 "sin(arctan(x /10)+x)=cos(x)" "6#/-%$sinG6#,&-%'arctanG6#*&%\"xG\"\"\"\"#5!\"\"F-F,F --%$cosG6#F," }{TEXT -1 27 " which lie in the interval " }{XPPEDIT 18 0 "[0,2*Pi]" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 \+ " }}{PARA 0 "" 0 "" {TEXT -1 63 "Find the greatest and least positive \+ solutions of the equation " }{XPPEDIT 18 0 "sqrt(2*x+1) = 5*sin(4*x); " "6#/-%%sqrtG6#,&*&\"\"#\"\"\"%\"xGF*F*F*F**&\"\"&F*-%$sinG6#*&\"\"%F *F+F*F*" }{TEXT -1 22 " correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }}{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 42 "__ ________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }} {PARA 0 "" 0 "" {TEXT -1 63 "Find the greatest and least positive solu tions of the equation " }{XPPEDIT 18 0 "sqrt(4-ln(x^2+1))=2*cos(x^2)" "6#/-%%sqrtG6#,&\"\"%\"\"\"-%#lnG6#,&*$%\"xG\"\"#F)F)F)!\"\"*&F0F)-%$c osG6#*$F/F0F)" }{TEXT -1 22 " correct to 10 digits." }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "sqrt(4-ln(x^2+1) )" "6#-%%sqrtG6#,&\"\"%\"\"\"-%#lnG6#,&*$%\"xG\"\"#F(F(F(!\"\"" } {TEXT -1 16 " is defined for " }{XPPEDIT 18 0 "-sqrt(exp(4)-1)<=x" "6# 1,$-%%sqrtG6#,&-%$expG6#\"\"%\"\"\"F-!\"\"F.%\"xG" }{XPPEDIT 18 0 "``< =sqrt(exp(4)-1)" "6#1%!G-%%sqrtG6#,&-%$expG6#\"\"%\"\"\"F-!\"\"" } {TEXT -1 5 ", or " }{XPPEDIT 18 0 "-7.32107574 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "____________ ______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 " " {TEXT -1 63 "Find the greatest and least positive solutions of the e quation " }{XPPEDIT 18 0 "sqrt(ln(x^2+1))=2*sin(x^2)" "6#/-%%sqrtG6#-% #lnG6#,&*$%\"xG\"\"#\"\"\"F.F.*&F-F.-%$sinG6#*$F,F-F." }{TEXT -1 22 " \+ correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 42 "________________ __________________________" }}{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 42 "_______________________________________ ___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find (correc t to 10 digits) all the zeros of the cosine integral " }{XPPEDIT 18 0 "Ci(x);" "6#-%#CiG6#%\"xG" }{TEXT -1 27 " which lie in the interval " }{XPPEDIT 18 0 "[0,100]" "6#7$\"\"!\"$+\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Ci(x)" "6#-%#CiG6#%\"xG" } {TEXT -1 15 " is defined by " }{XPPEDIT 18 0 "Ci(x) = gamma + ln(x) \+ + int((cos(t)-1)/t, t=0..x)" "6#/-%#CiG6#%\"xG,(%&gammaG\"\"\"-%#lnG6# F'F*-%$intG6$*&,&-%$cosG6#%\"tGF*F*!\"\"F*F6F7/F6;\"\"!F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Hint" }{TEXT -1 28 ": Replace 0 in the interval " }{XPPEDIT 18 0 "[0,100]" "6#7$\"\"!\"$+\"" } {TEXT -1 42 " by a small positive number such as 1e-9. " }}{PARA 0 "" 0 "" {TEXT -1 42 "__________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 42 "__ ________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Find (cor rect to 10 digits) all the solutions of the equation " }{XPPEDIT 18 0 "Y[0](x) = Y[1](x);" "6#/-&%\"YG6#\"\"!6#%\"xG-&F&6#\"\"\"6#F*" } {TEXT -1 27 " which lie in the interval " }{XPPEDIT 18 0 "[0,100]" "6# 7$\"\"!\"$+\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "Y[n](x)" "6#-&% \"YG6#%\"nG6#%\"xG" }{TEXT -1 8 " is the " }{TEXT 287 1 "n" }{TEXT -1 55 " th Bessel function of the 2nd kind, given in Maple by " }{TEXT 262 12 "BesselY(n,x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 " __________________________________________" }}{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 42 "____________________ ______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }