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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 109 "
 leastroot - find floating-point approximation to least root of an equ
ation or expression on a real interval." }}{PARA 0 "" 0 "" {TEXT 26 
17 "Calling sequence:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 29 "leastroot( \+
expr, x = a .. b);" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }{MPLTEXT 0 21 5 "expr " }{TEXT -1 52 
"- the equation or expression, involving one variable" }}{PARA 0 "" 0 
"" {TEXT -1 2 "  " }{MPLTEXT 0 21 1 "x" }{TEXT -1 24 " - the variable \+
(a name)" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{MPLTEXT 0 21 4 "a, b" }
{TEXT -1 46 " - endpoints of the interval (real constants)." }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" 
{MPLTEXT 0 21 23 "leastroot(expr, x=a..b)" }{TEXT -1 56 " computes num
erically the least solution of an equation " }{MPLTEXT 0 21 4 "expr" }
{TEXT -1 18 "  in one variable " }{MPLTEXT 0 21 2 "x " }{TEXT -1 21 "i
n the real interval " }{MPLTEXT 0 21 6 "a .. b" }{TEXT -1 6 ".  If " }
{MPLTEXT 0 21 4 "expr" }{TEXT -1 56 " is an expression rather than an \+
equation, the equation " }{MPLTEXT 0 21 8 "expr = 0" }{TEXT -1 73 " is
 used.  If there is no solution in this interval, nothing is returned.
" }}{PARA 15 "" 0 "" {TEXT -1 132 "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 " }{MPLTEXT 0 21 6 "a .. b" }{TEXT -1 20 " \+
is substituted for " }{MPLTEXT 0 21 1 "x" }{TEXT -1 1 "." }}{PARA 15 "
" 0 "" {TEXT -1 186 "The expression and all subexpressions should have
 at least two continuous derivatives on the interval.  In particular, \+
infinite limits at the endpoints, or  indeterminate forms (such as " }
{MPLTEXT 0 21 4 "f/g " }{TEXT -1 6 "where " }{MPLTEXT 0 21 1 "f" }
{TEXT -1 5 " and " }{MPLTEXT 0 21 1 "g" }{TEXT -1 15 " both approach \+
" }{XPPEDIT 18 0 "0 " "6#\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "inf
inity" "6#%)infinityG" }{TEXT -1 38 " at the endpoints)  may cause tro
uble." }}{PARA 15 "" 0 "" {TEXT -1 126 "An exception to the requiremen
ts of continuity and differentiability is in the case of an expression
 defined piecewise, using " }{MPLTEXT 0 21 9 "piecewise" }{TEXT -1 2 "
, " }{MPLTEXT 0 21 6 "signum" }{TEXT -1 2 ", " }{MPLTEXT 0 21 9 "Heavi
side" }{TEXT -1 2 ", " }{MPLTEXT 0 21 3 "abs" }{TEXT -1 2 ", " }
{MPLTEXT 0 21 3 "min" }{TEXT -1 4 " or " }{MPLTEXT 0 21 3 "max" }
{TEXT -1 14 ", as long as  " }{MPLTEXT 0 21 19 "convert(...,pwlist)" }
{TEXT -1 105 " can convert it to a list of expressions on different in
tervals.  If this can't be done, an error occurs." }}{PARA 15 "" 0 "" 
{TEXT -1 118 "Infinite endpoints are allowed, but are not likely to wo
rk unless the expression has finite limits at those endpoints." }}
{PARA 15 "" 0 "" {TEXT -1 103 "Since numerical techniques are used, th
e accuracy of the results is limited.  In particular, a root of " }
{MPLTEXT 0 21 4 "f(x)" }{TEXT -1 24 " that is also a root of " }
{MPLTEXT 0 21 7 "D(f)(x)" }{TEXT -1 102 " may be computed with poor ac
curacy, or even missed entirely.  In such a case it may be better to u
se " }{MPLTEXT 0 21 7 "D(f)(x)" }{TEXT -1 12 " instead of " }{MPLTEXT 
0 21 4 "f(x)" }{TEXT -1 14 ".  Increasing " }{MPLTEXT 0 21 6 "Digits" 
}{TEXT -1 30 " should also improve accuracy." }}{PARA 15 "" 0 "" 
{TEXT -1 24 "In some difficult cases " }{MPLTEXT 0 21 9 "leastroot" }
{TEXT -1 137 " may take a very long time.  In particular, this will ha
ppen if the function is complicated or changes direction rapidly in th
e interval." }}{PARA 15 "" 0 "" {MPLTEXT 0 21 9 "leastroot" }{TEXT -1 
6 " uses " }{MPLTEXT 0 21 5 "evalr" }{TEXT -1 155 " to do interval ari
thmetic, and is therefore subject to the weaknesses of that procedure.
  In particular, it doesn't work with the two-variable version of " }
{MPLTEXT 0 21 6 "arctan" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 
29 "This function is part of the " }{TEXT 256 22 "Maple Advisor Databa
se" }{TEXT -1 9 " library." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Exa
mples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "leastroot( tan(sin
(x))=1, x = 0 .. 2*Pi );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+36RL!*
!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "leastroot((x-Pi)^2, \+
x = 0 .. 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "leastroot(sin(x) + x^2, x = \+
-infinity .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+a@En()!#
5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "leastroot(abs(sin(x)+x
) = 1, x = -2 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+%HM(4^!#5" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "leastroot(Int(sin(s),s=0.
.x) = 1, x = 1 .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Fj
zq:!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "See also:" }}{PARA 0 
"" 0 "" {HYPERLNK 17 "allsolve" 2 "allsolve" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "convert(pwlist)" 2 "convert,pwlist" "" }{TEXT -1 2 ", " 
}{HYPERLNK 17 "evalr" 2 "evalr" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fso
lve" 2 "fsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gmin" 2 "gmin" "" }
{TEXT -1 2 ", " }{HYPERLNK 17 "greatestroot" 2 "greatestroot" "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" 
}{TEXT 26 23 "Maple Advisor Database " }{TEXT -1 15 "R. Israel 2000 " 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 0 2" 14 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }