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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "A bisection method for determinin
g all roots in an interval" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter St
one, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  2
4.3.2007" }}{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 procedure
s" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 273 7 "roo
ts.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 9 
"allbisect" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "
" {TEXT -1 122 "It can be read into a Maple session by a command simil
ar to the one that follows, 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 49 "A method for determining all roots in an interval" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 14 "The procedure " }{TEXT 0 9 "allbisect" }{TEXT -1 43 " given in \+
this worksheet works as follows:\n" }}{PARA 0 "" 0 "" {TEXT -1 56 "The
 idea is to first locate all the roots of a function " }{XPPEDIT 18 0 
"f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 372 " roughly by finding small inte
rvals in which they are located. This is achieved by successively bise
cting the whole interval into progressively smaller subintervals, doub
ling the number of subintervals with each iteration. The sign of the f
unction is determined at all the subdivision points, and the position \+
of any subinterval in which a sign change occurs is recorded. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 190 "The proc
edure can fail to find roots which are located close together in pairs
. If there is any possibility of this happening, the sensitivity of th
e procedure can be increased in two ways.\n" }}{PARA 15 "" 0 "" {TEXT 
-1 66 "The minimum number of bisections to be performed can be specifi
ed." }}{PARA 15 "" 0 "" {TEXT -1 238 "The procedure counts the number \+
of roots detected at each bisection iteration, by means of sign differ
ences across subintervals, and terminates when the number of roots det
ected has remained constant for a specified number of iterations.\n" }
}{PARA 0 "" 0 "" {TEXT -1 118 "When the roots have been pinned down to
 small subintervals, they are computed accurately using the built-in p
rocedure " }{TEXT 0 6 "fsolve" }{TEXT -1 248 ".\n\nSince the procedure
 uses sign changes to detect roots, it cannot find roots where there i
s no sign change, such as would occur if a local maximum or minimum fo
r the function has second coordinate zero. It can also be confused by \+
discontinuities." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 64 "A bisection procedure for determining all roots in
 an interval: " }{TEXT 0 9 "allbisect" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 16 "allbisect: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}
{PARA 0 "" 0 "" {TEXT -1 25 "   allbisect( eqn, rng ) " }{TEXT 275 1 "
\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT 23 10 "   e
qn  - " }{TEXT -1 63 "  an equation or expression involving a single v
ariable, say x," }}{PARA 0 "" 0 "" {TEXT -1 22 "                      \+
" }{TEXT 262 2 "OR" }{TEXT -1 73 " a function of the form x -> f(x), w
here f(x) evaluates to a real number," }}{PARA 0 "" 0 "" {TEXT -1 22 "
                      " }{TEXT 262 2 "OR" }{TEXT -1 71 " a numerical p
rocedure which evaluates to a real floating point number." }}{PARA 0 "
" 0 "" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 23 
8 "rng  -  " }{TEXT 276 93 "a range x=a..b (or simply a..b when the1st
 argument is a procedure) where a and b are the end" }}{PARA 0 "" 0 "
" {TEXT 278 190 "                     points of the interval [a,b] to \+
be searched for roots, and x is the variable appearing in the 1st argu
ment.\n                     a and b may or may not bracket the root." 
}{TEXT -1 0 "" }{TEXT 277 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 9 "allbisect" }{TEXT -1 420 " attemp
ts to finds all numerical roots 'x' of the equation f(x)=0 in a given \+
interval [a,b]. The interval [a,b] is successively bisected and roots \+
are located by detecting sign changes. As a consequence, the procedure
 will only detect roots where a sign change occurs, and cannot deal wi
th singularities. When a root is pinned down to a particular, possibly
 small, interval, it is located accurately using the procedure " }
{TEXT 0 6 "fsolve" }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT 261 8 "O
ptions:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 15 "miniterations
=n" }}{PARA 0 "" 0 "" {TEXT -1 213 "The minimum number of bisections t
o be performed before concluding that there are some or no roots in th
e current subinterval. \"miniterations\" must be between 2 and 20 incl
usive. The default is \"miniterations=6\". " }}{PARA 0 "" 0 "" {TEXT 
-1 253 "\nsensitivity=n\nIf there are some roots, bisections continue \+
until no further roots are detected for a specified number of further \+
bisections as determined by this option. \"sensitivity\" must be betwe
en 2 and 12 inclusive. The default is \"sensitivity=2\". " }}{PARA 0 "
" 0 "" {TEXT 0 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "info
=n or info=true/false" }}{PARA 15 "" 0 "" {TEXT -1 46 "info=0:     No \+
intermediate output is printed." }}{PARA 15 "" 0 "" {TEXT -1 148 "info
=1:     The number of roots detected at the current bisection level is
 printed. This gives a general picture of the progress of the computat
ion." }}{PARA 15 "" 0 "" {TEXT -1 429 "info=2:     This gives a more c
omplete picture of the progress of the computation. During the initial
 search for roots the number of roots detected, the list of signs at t
he evaluation points and the locations of the sign changes in the list
 of signs are printed. During computation of the roots each root is pr
inted immediately after it has been calculated along with the ends of \+
the bisected subinterval in which it is located." }}{PARA 15 "" 0 "" 
{TEXT -1 73 "info=true is equivalent to info=2 and info=false is equiv
alent to info=0." }}{PARA 0 "" 0 "" {TEXT -1 93 "\nThe default values \+
of these options are \"miniterations = 5\", \"sensitivity = 2\" and \"
info=0\"." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
262 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the
 procedure active, open the subsection, place the cursor anywhere afte
r the prompt [ >  and press [Enter].\nYou can then close up the subsec
tion." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "allbisect: implementatio
n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "allroots" 
{MPLTEXT 1 0 7938 "allbisect := proc(ff,rng)\n   local Options,rslt,fo
undroot,finished,rcrd,sp,i,i2,j,x,\n      numroots,lft,rgt,r,a,b,c,sns
tvy,minit,div,sgn,sn,numrtslist,\n      newdiv,pp,prntflg,f,fn,rg,xx,l
mr,sf,insert,drop,append,signz,\n      proctype,vars,fa,fb,fxx;\n\nins
ert := proc(alist,x,i)\n      [op(1..i-1,alist),x,op(i..nops(alist),al
ist)]  \nend proc: # of insert\n\ndrop := proc(alist,i)\n   [op(1..i-1
,alist),op(i+1..nops(alist),alist)]\nend proc: # of drop\n\nappend := \+
proc(alist,x) [op(alist),x] end proc: # of append\n\nsignz := proc(x)
\n  if x>0 then 1 elif x<0 then -1 else 0 end if;\nend proc: # of sign
z\n\n   # start of main procedure allbisect\n   if type(ff,procedure) \+
then\n      if nops([op(1,eval(ff))])<>1 then\n         error \"the 1s
t argument, %1, is invalid .. it should be a procedure with a single a
rgument\",ff;\n      end if;\n      proctype := true;\n      if type(r
ng,realcons..realcons) then\n         rg := rng\n      else\n         \+
error \"the 2nd argument, %1, is invalid .. when the 1st argument is a
 procedure, the 2nd argument should be a range of real constants\",rng
;\n      end if;\n   elif type(ff,algebraic) or type(ff,equation) then
\n      if type(ff,equation) then\n         lmr := lhs(ff)-rhs(ff);\n \+
        sf := traperror(simplify(lmr));\n         if sf<>lasterror the
n\n            f := sf;\n         else\n            f := lmr;\n       \+
  end if;\n      else\n         f := ff;\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\}) then\n            error \+
\"the 1st argument, %1, is invalid .. it should be an expression or an
 equation which depends only on a single variable\",ff;\n         end \+
if;\n      end if;\n      if type(rng,name=realcons..realcons) then\n \+
        proctype := false;\n         x := op(1,rng);\n         if not \+
member(x,vars) then\n            error \"the 1st argument, %1, is inva
lid .. it should be an expression or an equation which depends only on
 the variable %2\",ff,x;\n         end if;\n         rg := op(2,rng);
\n      else\n         error \"the 2nd argument, %1, is invalid .. it \+
should have the form 'x=a..b', to provide two real number starting val
ues\",rng;\n      end if;\n   else\n      error \"the 1st argument, %1
, is invalid .. it should be an algebraic expression in a single varia
ble, an equation in a single variable, or a procedure with a single re
al argument\",ff;\n   end if;\n\n   # Get the options \"sensitivity\",
\"miniterations\",\"evaluation\" and \"info\".\n   # Set the default v
alues to start with.\n   snstvy := 2;\n   minit := 6;\n   prntflg := 0
;\n   if nargs>2 then\n      Options:=[args[3..nargs]];\n      if not \+
type(Options,list(equation)) then\n         error \"each optional argu
ment must be an equation\"\n      end if;\n      if hasoption(Options,
'sensitivity','snstvy','Options') then\n         if not type(snstvy,po
sint) or snstvy<2 or snstvy>12 then\n            error \"\\\"sensitivi
ty\\\" must be a positive integer between 2 and 12 inclusive\"\n      \+
   end if;\n      end if;\n      if hasoption(Options,'miniterations',
'minit','Options') then\n         if not type(minit,posint) or minit<2
 or\n               minit>20 then\n            error \"\\\"miniteratio
ns\\\" must be a positive integer between 2 and 20 inclusive\"\n      \+
   end if;\n      elif hasoption(Options,'miniter','minit','Options') \+
then\n         if not type(minit,posint) or minit<2 or\n              \+
 minit>20 then\n            error \"\\\"miniter\\\" must be a positive
 integer between 2 and 20 inclusive\"\n         end if;\n      end if;
\n      if hasoption(Options,'info','prntflg','Options') then\n       \+
  if not member(prntflg,\{true,false,0,1,2\}) then\n            error \+
\"\\\"info\\\" must be false <-> 0, 1, or true <-> 2\"\n         end i
f;\n         if prntflg=false then prntflg := 0\n         elif prntflg
=true then prntflg := 2 end if; \n      end if;\n      if nops(Options
)>0 then\n         error \"%1 is not a valid option for %2\",op(1,Opti
ons),procname;\n            error \"%1 is not a valid option for %2 ..
 the recognised options are \\\"sensitivity\\\", \\\"miniterations\\\"
,(or \\\"miniter\\\") and \\\"info\\\"\",op(1,Options),procname;\n    \+
  end if;\n   end if;\n     \n   a := evalf(min(op(rg)));\n   b := eva
lf(max(op(rg)));\n   if a=b then\n      error \"distinct starting valu
es are required\"\n   end if;\n  \n   if proctype then\n      fn := ff
;\n   else\n      f := evalf(f);\n      fn := subs(_fx=f,proc(x) evalf
(_fx) end proc);\n      if not type([fn(a),fn(.7101449275*a+.289855072
5*b),\n           fn(.381966011*a+.618033989*b),fn(b)],list(numeric)) \+
then\n         error \"function does not evaluate to a numeric\"\n    \+
  end if;\n   end if;\n\n   rslt := []; # list to record roots     \n \+
  div := 1; # number of subintervals\n\n   # list for no. of roots loc
ated after recent successive bisections\n   numrtslist := [seq(0,i=1..
snstvy)];\n     \n   # Get the signs of the function at the ends of th
e interval.\n   # If we hit a root, add it to the list.\n   fa := trap
error(fn(a));\n   if fa=lasterror or not type(fa,numeric) then\n      \+
error \"a non-numeric value occurred\"\n   end if;   \n   sgn := signz
(fa);\n   if evalb(sgn=0) then rslt := append(rslt,a) end if;\n   sn :
= [sgn];\n   fb := traperror(fn(b));\n   if fb=lasterror or not type(f
b,numeric) then\n      error \"a non-numeric value occurred\"\n   end \+
if;\n   sgn := signz(fb);\n   if evalb(sgn=0) then rslt := append(rslt
,b) end if;\n   sn := append(sn,sgn);\n     \n   finished := false;\n \+
  foundroot := false; \n   for j from 1 to 30 while not finished\n   d
o\n      rcrd := [];\n        \n      # Double the number of evaluatio
n points.\n      newdiv := div+div;\n        \n      # Determine the s
pacing of the evaluation points.\n      sp := (b-a)/newdiv;\n   \n    \+
  # Determine sign of function at new evaluation points. \n      # If \+
we hit a root, add it to the list.\n      for i from 1 to div do\n    \+
     i2 := i+i;\n         xx := a+(i2-1)*sp;\n         fxx := traperro
r(fn(xx));\n         if fxx=lasterror or not type(fxx,numeric) then\n \+
           error \"a non-numeric value occurred\"\n         end if;\n \+
        sgn := signz(fxx);\n         if evalb(sgn=0) then rslt := appe
nd(rslt,xx) end if;\n           \n         # Insert the new signs in t
he list of signs.\n         sn := insert(sn,sgn,i2);\n      end do;\n \+
       \n      # Record where sign changes occur in multiples of the c
urrent\n      # spacing of the evaluation points.\n      for i from 1 \+
to newdiv do\n         if evalb(sn[i]*sn[i+1]<0) then rcrd := append(r
crd,i) end if;\n      end do;\n      if prntflg = 2 then\n         pri
nt(`The signs at the bisection points are:`);\n         print(sn);\n  \+
       print(`The sign changes occur at the locations:`);\n         pr
int(rcrd);\n      end if;\n      numroots := nops(rcrd);\n\n      # Re
tain number of roots found in the last \"snstvy\" iterations.\n      n
umrtslist := drop(append(numrtslist,numroots),1);\n      div := newdiv
; \n      if prntflg > 0 then\n         print(`Number of roots detecte
d =`,numroots+nops(rslt))\n      end if;\n        \n      # Has no. of
 roots detected in last 2 iterations changed?\n      # Have we detecte
d at least one root?\n      # Have we performed the min number of iter
ations?\n      finished := evalb(convert(numrtslist,set)=\n           \+
     \{numroots\} and j>=minit);\n      foundroot := evalb(numroots>0)
;\n   end do;\n\n   # Determine the roots to the required precision.\n
   if prntflg=2  and foundroot then\n      print(``);\n      print(`Th
e roots with their associated intervals are:`);\n      print(``);\n   \+
end if;\n   if foundroot then\n      for i from 1 to numroots do\n    \+
     lft := a + (rcrd[i]-1)*sp;\n         rgt := a + rcrd[i]*sp;\n    \+
     r := traperror(fsolve(fn,lft..rgt));\n         if r=lasterror or \+
not type(r,realcons) then\n            error \"fsolve could not find a
 root in one of the subintervals\"\n         end if;\n         if prnt
flg=2 then print(op(map(evalf,[lft,r,rgt]))) end if;\n         rslt :=
 append(rslt,r);\n      end do;\n   end if;\n   if prntflg>0 then prin
tf(\"\\n\") end if;\n   op(sort(map(evalf,rslt)));\nend proc: # of all
bisect" }}}{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 9 "allbisect" }{TEXT -1 10 ": examples
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 280 8 "Question" }
{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 65 "Find ( correct to 10 \+
digits ) all the solutions of the equation  " }{XPPEDIT 18 0 "3*x^2*si
n(x)^3-3*sin(x)^3-x^3+x = 0;" "6#/,**(\"\"$\"\"\"*$%\"xG\"\"#F'-%$sinG
6#F)F&F'*&F&F'*$-F,6#F)F&F'!\"\"*$F)F&F2F)F'\"\"!" }{TEXT -1 29 " whic
h lie in the interval  [" }{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " ]. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution
" }{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 60 "plot(3*x^2*sin(x)^3-3*sin(x)^3-x^3+x,x=-Pi.
.Pi,y=-2.5..2.5);" }}{PARA 13 "" 1 "" {GLPLOT2D 441 282 282 {PLOTDATA 
2 "6%-%'CURVESG6$7Y7$$!3*****4tk#fTJ!#<$\"3kf_q%Qoky#!#;7$$!3w\"*pr)3P
Y+$F*$\"3.r%*owo%fS#F-7$$!3?3E[*ysa)GF*$\"3e+Y+CL:y?F-7$$!3gX&))>2g9v#
F*$\"3m4bP(\\G%*p\"F-7$$!3)4577.flh#F*$\"3vAO]')G')38F-7$$!3u\"QM::*H#
[#F*$\"37Xe?EB%QD*F*7$$!3y\\AhcI#yN#F*$\"3Jr4i^0!y$fF*7$$!3&e=d-FN*GAF
*$\"3y$['Hg#G,&HF*7$$!3u%*GBP$Rc4#F*$\"3q1?N))42J^!#=7$$!3mCj&f)3xi>F*
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "allbisect(3*x^2*sin(x)^3-3*sin(x)^3
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}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Check, u
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14](allbisect(3*x^2*sin(x)^3-3*sin(x)^3-x^3+x,x=-Pi..Pi)));" }}{PARA 
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(x)^3 = x/3;" "6#/*$-%$sinG6#%\"xG\"\"$*&F(\"\"\"F)!\"\"" }{TEXT -1 
18 "  in the interval " }{XPPEDIT 18 0 "[-3, 3]" "6#7$,$\"\"$!\"\"F%" 
}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 283 8 "Solution" }{TEXT -1 4 
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{MPLTEXT 1 0 46 "plot([sin(x)^3,x/3],x=-3..3,color=[red,blue]);" }}
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" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"$" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"$" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"&" }}{PARA 6 "
" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$!0FwW:y21#!#9$
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Check, using higher preci
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2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve
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{MPLTEXT 1 0 59 "allbisect(abs(sin(5*x)+x)=4*x^2+x-x^4,x=-2..2.2,info=
true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GThe~signs~at~the~bisection
~points~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\"F$F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locati
ons:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%GThe~signs~at~the~bisection~points~are:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7'\"\"\"!\"\"F$F%F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locations:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%GThe~signs~at~the~bisection~points~are:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7+\"\"\"!\"\"F%F$F$F%F%F%F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locations:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&\"\"\"\"\"$\"\"&\"\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%GThe~signs~at~the~bisection~points~are:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#73\"\"\"F$!\"\"F%F%F%F$F$F$F$F%F%F%F%F%F%F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locati
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{PARA 11 "" 1 "" {XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%GThe~signs~at~the~bisection~points~ar
e:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7C\"\"\"F$F$F$!\"\"F%F%F%F%F%F%
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{XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locations:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&\"\"%\"#6\"#>\"#J" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%GThe~signs~at~the~bisection~points~are:G" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#7]o\"\"\"F$F$F$F$F$F$F$!\"\"F%F%F%F%F%F%F%F%F%F%
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F%F%F%F%F%F%F%F%F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%IThe~sign~c
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\"\")\"#A\"#Q\"#i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;Number~of~roots
~detected~=G\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GThe~signs~at~th
e~bisection~points~are:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7]s\"\"\"F
$F$F$F$F$F$F$F$F$F$F$F$F$F$F$!\"\"F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%F%
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{XPPMATH 20 "6#%IThe~sign~changes~occur~at~the~locations:G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7&\"#;\"#V\"#v\"$C\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%;Number~of~roots~detected~=G\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%OThe~roots~with
~their~associated~intervals~are:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!++D\"y]\"!\"*$!+kW!**\\\"F%$!+
+++v9F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!*+](=i!\"*$!+l<^Wg!#5$!*+
D1*eF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"*+]7G%!\"*$\"+>(*>9X!#5$
\"*+v$4YF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$\"++v$f.#!\"*$\"+.q$>/#
F%$\"+++vo?F%" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&$!+kW!**\\\"!\"*$!+l<^Wg!#5$\"+>(*>9XF($\"+.q$>/#F%" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The solu
tion " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 42 " has been mi
ssed since no sign change of  " }{XPPEDIT 18 0 "abs(sin(5*x)+x) -x-4*x
^2+x^4" "6#,*-%$absG6#,&-%$sinG6#*&\"\"&\"\"\"%\"xGF-F-F.F-F-F.!\"\"*&
\"\"%F-*$F.\"\"#F-F/*$F.F1F-" }{TEXT -1 14 " occurs here. " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" 
{TEXT 267 8 "Question" }{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 
43 "Find ( correct to 30 digits ) all zeros of " }{XPPEDIT 18 0 "f(x) \+
= sin(1/x);" "6#/-%\"fG6#%\"xG-%$sinG6#*&\"\"\"F,F'!\"\"" }{TEXT -1 
37 "  in the interval [0.001,  0.00102]. " }}{PARA 0 "" 0 "" {TEXT 
270 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "[evalf[30](allbisect(sin(1/x),x=0.001..0.00102))];" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#7($\"?r,4=Xjk(oa[[u4+\"!#K$\"?^*pl^*G'[*=2L@8/5F&$
\"?dzEt'Hf3S/kw4t+\"F&$\"?D\"H\\3]_qj'=vv]55F&$\"?a%Qz4*yHxw<_ds85F&$
\"?%z-'G1m>q**f#\\kp,\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 262 4 "Note" }{TEXT -1 32 ": The solutions are of the fo
rm " }{XPPEDIT 18 0 "1/(n*Pi);" "6#*&\"\"\"F$*&%\"nGF$%#PiGF$!\"\"" }
{TEXT -1 32 " for suitable integer values of " }{TEXT 272 1 "n" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 87 "sort([seq(evalf[30](1/(n*Pi)),n=ceil(evalf(10000
0/(102*Pi)))..floor(evalf(1000/Pi)))]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#7($\"?r,4=Xjk(oa[[u4+\"!#K$\"?^*pl^*G'[*=2L@8/5F&$\"?dzEt'Hf3S/k
w4t+\"F&$\"?D\"H\\3]_qj'=vv]55F&$\"?a%Qz4*yHxw<_ds85F&$\"?%z-'G1m>q**f
#\\kp,\"F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5
" }}{PARA 0 "" 0 "" {TEXT 264 8 "Question" }{TEXT -1 4 ":   " }}{PARA 
0 "" 0 "" {TEXT -1 61 "Find ( correct to 15 digits ) all solutions of \+
the equation  " }{XPPEDIT 18 0 "sin(1/x) = exp(-x);" "6#/-%$sinG6#*&\"
\"\"F(%\"xG!\"\"-%$expG6#,$F)F*" }{TEXT -1 36 "  in the interval [0.00
1,  0.00102]." }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 4 ":
   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "I
n this example there are pairs of roots which are close together. In o
rder to detect these roots, the option " }{TEXT 273 13 "miniterations
" }{TEXT -1 13 " is set at 8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "plot([sin(1/x),exp(-x)],x=0.
001..0.00102,color=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 
270 {PLOTDATA 2 "6&-%'CURVESG6$7[t7$$\"1+++++++5!#=$\"1E+K0azo#)!#;7$$
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hw$\"3kqlHh&=)*)**F-7$F]x$\"3[9(f)>&=)*)**F-7$Fbx$\"3aXX[x%=)*)**F-7$F
gx$\"3sj-YN%=)*)**F-7$F\\y$\"3G8V%oR=)*)**F-7$Fay$\"3Kxge_$=)*)**F-7$F
fy$\"3W&)4+8$=)*)**F-7$F[z$\"3s;]zq#=)*)**F-7$F`z$\"3!R_)RI#=)*)**F-7$
Fez$\"3)G(p)p==)*)**F--Fjz6&F\\[lF`[lF`[lF][l-%+AXESLABELSG6$Q\"x6\"Q!
F^el-%%VIEWG6$;$\"&&=5!\"($\"&(=5Ffel;$\"%&)**!\"%$\"&,+\"F\\fl" 1 2 
0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2
" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "evalf[15](allbisect(sin(1/x)=exp(-x),x=0.001..0.00102
,miniterations=8));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"00&)p3tc+\"!
#<$\"01p')zjd+\"F%$\"0?X$G!o?,\"F%$\"0tt*)=g@,\"F%$\"0eCz\"[a=5F%$\"0b
2,YQ'=5F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 33 "We check a couple of these roots." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "evalf[15](fsolve(sin(
1/x)=exp(-x),x=.00101863846));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0
b2,YQ'=5!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 51 "evalf[15](fsolve(sin(1/x)=exp(-x),x=.00100567309))
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0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 0 "" 0 "" 
{TEXT 266 8 "Question" }{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" {TEXT -1 
64 "Find ( correct to 10 digits ) all the solutions of the equation " 
}{XPPEDIT 18 0 "tan(x) = x;" "6#/-%$tanG6#%\"xGF'" }{TEXT -1 28 "  whi
ch lie in the interval " }{XPPEDIT 18 0 "[-Pi/2, 9*Pi/2];" "6#7$,$*&%#
PiG\"\"\"\"\"#!\"\"F)*(\"\"*F'F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT 269 8 "Solution" }{TEXT -1 4 ":   " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 235 "p1 := plot(
[tan(x),x],x=-1.6..14.2,y=-15..20,color=[red,blue],discont=true):\np2 \+
:= plots[implicitplot](\{x=-Pi/2,x=Pi/2,x=3*Pi/2,x=5*Pi/2,x=7*Pi/2,x=9
*Pi/2\},\n     x=-1.6..14.2,y=-15..20,color=black,linestyle=3):\nplots
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"Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "allbisect(sin(x)=x*cos(x),x=-Pi/2..9*Pi/2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"\"!F$$\"+e%4M\\%!\"*$\"+P=DDxF'$\"+
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{TEXT -1 9 "Example 7" }{TEXT 285 24 " .. Legendre Polynomials" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The Legen
dre polynomial " }{XPPEDIT 18 0 "P[k](x)" "6#-&%\"PG6#%\"kG6#%\"xG" }
{TEXT -1 25 " is availble in Maple as " }{TEXT 263 14 "LegendreP(k,x)
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 140 "n := 7:\n[seq(LegendreP(i,x),i=1..n)];\nmap(e
xpand,%);\nplot(%,x=-1.2..1.2,y=-1.2..1.2,color=[seq(COLOR('HUE',(i-1)
/n),i=1..n)],numpoints=100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)%\"x
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1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+
4" "Curve 5" "Curve 6" "Curve 7" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 14 "The zeros of  " }{XPPEDIT 18 0 "P[5](x) =
 63/8*x^5-35/4*x^3+15/8*x" "6#/-&%\"PG6#\"\"&6#%\"xG,(*(\"#j\"\"\"\"\"
)!\"\"F*F(F.*(\"#NF.\"\"%F0F*\"\"$F0*(\"#:F.F/F0F*F.F." }{TEXT -1 25 "
 can be found as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "expand(LegendreP(5,x));\nallbisect(
%,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"#j\"\")\"\"\"*$)
%\"xG\"\"&F(F(F(*&#\"#N\"\"%F(*$)F+\"\"$F(F(!\"\"*&#\"#:F'F(F+F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6'$!+f%)zh!*!#5$!+,Jp%Q&F%$\"\"!F)$\"+,J
p%Q&F%$\"+f%)zh!*F%" }}}{PARA 0 "" 0 "" {TEXT -1 9 "\nBecause " }
{XPPEDIT 18 0 "P[5](x);" "6#-&%\"PG6#\"\"&6#%\"xG" }{TEXT -1 17 " is a
 polynomial " }{TEXT 0 6 "fsolve" }{TEXT -1 29 " can find all of these
 roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "expand(LegendreP(5,x));\nfsolve(%,x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,(*&#\"#j\"\")\"\"\"*$)%\"xG\"\"&F(F(F(*&#\"#N\"\"
%F(*$)F+\"\"$F(F(!\"\"*&#\"#:F'F(F+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6'$!+f%)zh!*!#5$!+,Jp%Q&F%$\"\"!F)$\"+,Jp%Q&F%$\"+f%)zh!*F%" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }{TEXT 286 
29 " .. Zeros of Bessel functions" }}{PARA 0 "" 0 "" {TEXT -1 115 "\nT
he zeros of Bessel functions play an important role in the generalised
 Fourier series involving Bessel functions." }}{PARA 0 "" 0 "" {TEXT 
-1 64 "The following picture shows the graphs of the Bessel functions \+
 " }{XPPEDIT 18 0 "J[0](x),J[1](x),` . . . `,J[10](x)" "6&-&%\"JG6#\"
\"!6#%\"xG-&F%6#\"\"\"6#F)%(~.~.~.~G-&F%6#\"#56#F)" }{TEXT -1 5 " for \+
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G\"#:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "nn := 10:\nplot([seq(BesselJ(i,x),
i=0..nn)],x=0..15,numpoints=80,\n    color=[seq(COLOR('HUE',i/(nn+1)),
i=0..nn)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 601 317 317 {PLOTDATA 2 "6/-
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1 "" {XPPMATH 20 "6#7+$\"\"!F%$\"+p+bZ9!\")$\"+njMV=F($\"+O&)p/AF($\"+
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" 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 46 "(a) Plot the graphs of the B
essel Y functions " }{XPPEDIT 18 0 "Y[n](x);" "6#-&%\"YG6#%\"nG6#%\"xG
" }{TEXT -1 6 "  for " }{XPPEDIT 18 0 "n=0,1,2,3,4,5" "6(/%\"nG\"\"!\"
\"\"\"\"#\"\"$\"\"%\"\"&" }{TEXT -1 5 " and " }{TEXT 274 1 "x" }{TEXT 
-1 17 " in the interval " }{XPPEDIT 18 0 "[0,20]" "6#7$\"\"!\"#?" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "    These are obtained \+
in Maple as functions " }{TEXT 273 12 "BesselY(n,x)" }{TEXT -1 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 75 "    You will need to include a restrict
ion on the dependent variable, say  " }{TEXT 273 11 "y = -1.5..1" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "(b) Find (correct to 10
 digits) all the zeros of the functions " }{XPPEDIT 18 0 "Y[n](x);" "6
#-&%\"YG6#%\"nG6#%\"xG" }{TEXT -1 6 "  for " }{XPPEDIT 18 0 "n=0,1,2,3
,4,5" "6(/%\"nG\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&" }{TEXT -1 27 " which l
ie in the interval " }{XPPEDIT 18 0 "[0,40]" "6#7$\"\"!\"#S" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "______________________________
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0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
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" {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 9 "Find the " }{TEXT 
279 17 "all the solutions" }{TEXT -1 18 " of the equation  " }
{XPPEDIT 18 0 "2*arctan(x)=3*sin(x*sqrt(x))" "6#/*&\"\"#\"\"\"-%'arcta
nG6#%\"xGF&*&\"\"$F&-%$sinG6#*&F*F&-%%sqrtG6#F*F&F&" }{TEXT -1 44 " co
rrect to 10 digits by a numerical method." }}{PARA 0 "" 0 "" {TEXT 
262 4 "Hint" }{TEXT -1 64 ": In order to find all the solutions in a s
ingle application of " }{TEXT 0 9 "allbisect" }{TEXT -1 52 " you will \+
need to make use of either of the options " }{TEXT 273 13 "miniteratio
ns" }{TEXT -1 4 " or " }{TEXT 273 11 "sensitivity" }{TEXT -1 1 "." }}
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