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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 116 "
allsolve - find list of floating-point approximations to all roots of
an equation or expression on a real interval." }}{PARA 0 "" 0 ""
{TEXT 26 17 "Calling sequence:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 28 "al
lsolve( 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 va
riable (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 8 "allsolve" }{TEXT -1 55 " computes numerically all the
solutions of an equation " }{MPLTEXT 0 21 4 "expr" }{TEXT -1 18 " in
one variable " }{MPLTEXT 0 21 2 "x " }{TEXT -1 21 "in the real interv
al " }{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 equ
ation " }{MPLTEXT 0 21 8 "expr = 0" }{TEXT -1 71 " is used. The resul
t is returned as a list, sorted in numerical order." }}{PARA 15 "" 0 "
" {TEXT -1 132 "Only one variable is allowed: the two sides of the equ
ation must evaluate to real constants when any constant value in the i
nterval " }{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 co
ntinuous 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 \+
" "\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "infinity" "I)infinityG6\"
" }{TEXT -1 38 " at the endpoints) may cause trouble." }}{PARA 15 ""
0 "" {TEXT -1 126 "An exception to the requirements of continuity and \+
differentiability is in the case of an expression defined piecewise, u
sing " }{MPLTEXT 0 21 9 "piecewise" }{TEXT -1 2 ", " }{MPLTEXT 0 21 6
"signum" }{TEXT -1 2 ", " }{MPLTEXT 0 21 9 "Heaviside" }{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 intervals. If this can't be \+
done, an error occurs." }}{PARA 15 "" 0 "" {TEXT -1 118 "Infinite endp
oints are allowed, but are not likely to work unless the expression ha
s finite limits at those endpoints." }}{PARA 15 "" 0 "" {TEXT -1 103 "
Since numerical techniques are used, the accuracy of the results is li
mited. 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 accuracy, or even missed entirely. In
such a case it may be better to use " }{MPLTEXT 0 21 7 "D(f)(x)" }
{TEXT -1 12 " instead of " }{MPLTEXT 0 21 4 "f(x)" }{TEXT -1 14 ". In
creasing " }{MPLTEXT 0 21 6 "Digits" }{TEXT -1 30 " should also improv
e accuracy." }}{PARA 15 "" 0 "" {TEXT -1 24 "In some difficult cases \+
" }{MPLTEXT 0 21 8 "allsolve" }{TEXT -1 137 " may take a very long tim
e. In particular, this will happen if the function is complicated or \+
changes direction rapidly in the interval." }}{PARA 15 "" 0 ""
{MPLTEXT 0 21 8 "allsolve" }{TEXT -1 6 " uses " }{MPLTEXT 0 21 5 "eval
r" }{TEXT -1 155 " to do interval arithmetic, 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 -1 106 "This function is part of the
Maple Advisor Database library, and must be loaded before use by the \+
command " }{MPLTEXT 0 21 18 "readlib(allsolve);" }{TEXT -1 1 "." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> "
0 "" {MPLTEXT 1 0 18 "readlib(allsolve):" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 41 "allsolve( tan(sin(x))=1, x = 0 .. 2*Pi );" }}{PARA
11 "" 1 "" {XPPMATH 20 "6#7$$\"+36RL!*!#5$\"+VNDQA!\"*" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allsolve((x-Pi)^2, x = 0 .. 4);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+aEfTJ!\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 50 "allsolve(sin(x) + x^2, x = -infinity .. infini
ty);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+a@En()!#5$\"(H%or!#B" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "allsolve(abs(sin(x)+x) = 1, \+
x = -2 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+%HM(4^!#5$\"+%HM
(4^F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "allsolve(Int(sin(s
),s=0..x) = 1, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+F
jzq:!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "See also:" }}{PARA
0 "" 0 "" {HYPERLNK 17 "convert(pwlist)" 2 "convert,pwlist" "" }{TEXT
-1 2 ", " }{HYPERLNK 17 "evalr" 2 "evalr" "" }{TEXT -1 2 ", " }
{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gm
ax" 2 "gmax" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gmin" 2 "gmin" "" }}}
{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Maple Advisor Database " }{TEXT
-1 14 "R. Israel 1998" }}}}{MARK "3 1 5" 2 }{VIEWOPTS 1 1 0 1 1 1803 }