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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "Function:" }{TEXT -1 94 " \+
gmin - find floating-point approximations to the minimum of an express
ion on a real interval." }}{PARA 0 "" 0 "" {TEXT 26 17 "Calling sequen
ce:" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 55 "gmin( expr, x = a .. b);\ngmi
n( 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 minimum is attained." }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 ""
{MPLTEXT 0 21 4 "gmin" }{TEXT -1 57 " computes numerically the minimum
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 minimum 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\"\"\"%)infinityGF$" }{TEXT -1 11 " or finite." }}{PARA 15 "" 0 ""
{TEXT -1 285 "Since numerical techniques are used, the accuracy of the
results is limited. In particular, a minimum where the second and t
hird derivatives of the expression are 0 may be hard to locate (the mi
nimum value should be accurate, but the location of the minimum may no
t be). Increasing " }{MPLTEXT 0 21 6 "Digits" }{TEXT -1 117 " should \+
also improve accuracy. Also, if the minimum value is attained at seve
ral points, roundoff error may prevent " }{MPLTEXT 0 21 4 "gmin" }
{TEXT -1 64 " from recognizing that the values at these points are the
same. " }}{PARA 15 "" 0 "" {TEXT -1 24 "In some difficult cases " }
{MPLTEXT 0 21 4 "gmin" }{TEXT -1 137 " may take a very long time. In \+
particular, this will happen if the function is complicated or changes
direction rapidly in the interval." }}{PARA 15 "" 0 "" {MPLTEXT 0 21
4 "gmin" }{TEXT -1 6 " uses " }{MPLTEXT 0 21 5 "evalr" }{TEXT -1 155 "
to do interval arithmetic, and is therefore subject to the weaknesses
of that procedure. In particular, it doesn't work with the two-varia
ble version of " }{MPLTEXT 0 21 6 "arctan" }{TEXT -1 1 "." }}{PARA 15
"" 0 "" {TEXT -1 106 "This function is part of the Maple Advisor Datab
ase library, and must be loaded before use by the command " }{MPLTEXT
0 21 14 "readlib(gmin);" }{TEXT -1 1 "." }}}{SECT 0 {PARA 3 "" 0 ""
{TEXT 26 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rea
dlib(gmin):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gmin( x^2/4 \+
+ cos(x), x = -Pi .. Pi );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\5-
#z&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "gmin( x^2/4 + cos
(x), x = -Pi .. Pi, 'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\5-
#z&!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<$$\"+nU\\&*=!\"*$!+nU\\&*=F&" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 51 "gmin(sin(x) + x^2, x = -infinity .. infin
ity,'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+_dlCB!#5" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 "" {XPPMATH 20
"6#<#$!+8h$=]%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gmin(x
^2,x=2..3,'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"%\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "" 1 ""
{XPPMATH 20 "6#<#$\"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 29 "gmin(1/x,x=0..infinity,'x0');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "x0;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<#%)infinityG" }}}}{SECT 0 {PARA 3 "" 0 ""
{TEXT 26 10 "See also: " }}{PARA 0 "" 0 "" {HYPERLNK 17 "allsolve" 2 "
allsolve" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "convert(pwlist)" 2 "conve
rt,pwlist" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "evalr" 2 "evalr" "" }
{TEXT -1 2 ", " }{HYPERLNK 17 "fsolve" 2 "fsolve" "" }{TEXT -1 2 ", "
}{HYPERLNK 17 "gmax" 2 "gmax" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26
23 "Maple Advisor Database " }{TEXT -1 15 " R. Israel 1998" }}}}{MARK
"4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }