{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 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 }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 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 }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 }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 94 " \+ gmax - find floating-point approximations to the maximum of an express ion on a real interval." }}{PARA 0 "" 0 "" {TEXT 26 17 "Calling sequen ce:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 55 "gmax( expr, x = a .. b);\ngma x( expr, x = a .. b, 'x0');" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters :" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 5 "expr " }{TEXT -1 40 "- the expression, involving one variable" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 1 "x" }{TEXT -1 33 " - the var iable (a name)" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 4 "a, b" }{TEXT -1 47 " - endpoints of the interval (real constants)." }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{MPLTEXT 0 21 2 "x0" }{TEXT -1 56 " \+ - (optional) a name to use for saving the set of " }{MPLTEXT 0 21 1 "x" }{TEXT -1 41 " values at which the maximum is attained." }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {MPLTEXT 0 21 4 "gmax" }{TEXT -1 57 " computes numerically the maximum value of an expression " }{MPLTEXT 0 21 4 "expr" }{TEXT -1 18 " in o ne variable " }{MPLTEXT 0 21 2 "x " }{TEXT -1 21 "on the real interval " }{MPLTEXT 0 21 6 "a .. b" }{TEXT -1 23 " (including endpoints)." }} {PARA 15 "" 0 "" {TEXT -1 95 "If the optional third argument is includ ed, it must be a name. It will be assigned the set of " }{MPLTEXT 0 21 1 "x" }{TEXT -1 57 " values at which the maximum is attained. Usin g quotes (" }{MPLTEXT 0 21 4 "'x0'" }{TEXT -1 58 ") to delay evaluatio n ensures that this will work even if " }{MPLTEXT 0 21 2 "x0" }{TEXT -1 38 " has previously been assigned a value." }}{PARA 15 "" 0 "" {TEXT -1 118 "Only one variable is allowed: the expression must evalua te to a real constant when any constant 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 "T he expression and all subexpressions should have at least two continuo us 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 limits of the expression at those endpoints are " }{XPPEDIT 18 0 "``-infinity" ",&% !G\"\"\"%)infinityG!\"\"" }{TEXT -1 11 " or finite." }}{PARA 15 "" 0 " " {TEXT -1 285 "Since numerical techniques are used, the accuracy of t he results is limited. In particular, a maximum where the second and third derivatives of the expression are 0 may be hard to locate (the \+ maximum value should be accurate, but the location of the maximum may \+ not be). Increasing " }{MPLTEXT 0 21 6 "Digits" }{TEXT -1 117 " shoul d also improve accuracy. Also, if the maximum value is attained at se veral points, roundoff error may prevent " }{MPLTEXT 0 21 4 "gmax" } {TEXT -1 64 " from recognizing that the values at these points are the same. " }}{PARA 15 "" 0 "" {TEXT -1 24 "In some difficult cases " } {MPLTEXT 0 21 4 "gmax" }{TEXT -1 137 " may take a very long time. In \+ particular, this will happen if the function is complicated or changes direction rapidly in the interval." }}{PARA 15 "" 0 "" {MPLTEXT 0 21 4 "gmax" }{TEXT -1 6 " uses " }{MPLTEXT 0 21 5 "evalr" }{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-varia ble version of " }{MPLTEXT 0 21 6 "arctan" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 106 "This function is part of the Maple Advisor Datab ase library, and must be loaded before use by the command " }{MPLTEXT 0 21 14 "readlib(gmin);" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rea dlib(gmin):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gmax( sin(x) +x -x^2/2 , x = -1 .. 1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&)4 ZT8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gmax( sin(x) + x -x^2/2 , x = -1 .. 1, 'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+& )4ZT8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "gmax(sin(x) + x - x^2/2, x = -infinity .. infinity,'x 0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ut#)=9!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#$\"+U(GMG\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gma x(x^2-x^4,x=-2..2,'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++D! #5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$$\"+7y1rq!#5$!+7y1rqF&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "gmax(1/x,x=0..infinity,'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 10 "See also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "all solve" 2 "allsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "convert(pwlist) " 2 "convert,pwlist" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalr" 2 "eval r" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "gmin" 2 "gmin" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 23 "Maple Advisor Database " }{TEXT -1 15 " R. Israel 1998" } }}}{MARK "4 0 1" 15 }{VIEWOPTS 1 1 0 1 1 1803 }