{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "Help Head ing" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }} {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 \+ " "6#\"\"!" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "infinity" "6#%)infinity G" }{TEXT -1 38 " at the endpoints) may cause trouble." }}{PARA 15 " " 0 "" {TEXT -1 126 "An exception to the requirements of continuity an d 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 "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 29 "This function is part of the \+ " }{TEXT 256 22 "Maple Advisor Database" }{TEXT -1 9 " library." }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "allsolve( tan(sin(x))=1, x = 0 .. 2*Pi );" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+36RL!*!#5$\"+VNDQA!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "allsolve((x-Pi)^2, x = 0 .. \+ 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+aEfTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "allsolve(sin(x) + x^2, x = -infinit y .. infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$!+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$\"+%HM(4^F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "allsolv e(Int(sin(s),s=0..x) = 1, x = 1 .. 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#$\"+Fjzq:!\"*" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "See als o:" }}{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 "gmin" 2 "gmin" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "greatestroot" 2 "greatestroot" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "leastroot" 2 "leastr oot" "" }}}{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 1998 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 1 10" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }