{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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 172 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE " " -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 35 "The Golden Section Searc h Algorithm" }}{PARA 19 "" 0 "" {TEXT -1 195 "Dr. William P. Fox, Depa rtment of Mathematics, Francis Marion University, Florence, SC 29501 \nDr. Margie Witherspoon, Department of Computer Science, Francis Mari on University, Florence, SC 29501" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 102 "This program performs the Golden Section Search a lgorithm to find the maximum of a unimodal function, " }{TEXT 277 4 "f (x)" }{TEXT -1 20 ", over an interval," }{TEXT 272 2 " a" }{TEXT 273 1 "<" }{TEXT 274 3 " x " }{TEXT 275 1 "<" }{TEXT 276 2 " b" }{TEXT -1 104 ". The program calculates the number of iterations required to in sure the final interval is within the " }{TEXT 278 14 "user-specified " }{TEXT -1 272 " tolerance. This is found by solving for the smallest value of k that makes this inequality true: (b-a)(0.618^k)< tolerance . The Golden section concept involves placing two experiments between [a,b] using the Golden section ratios. One experiment is placed at po sition, " }{TEXT 279 12 "a+0.382(b-a)" }{TEXT -1 30 ", and the other \+ at position, " }{TEXT 280 12 "a+0.618(b-a)" }{TEXT -1 611 ". The func tion, to be maximized, is evaluated at these two points and the functi onal values are compared. We want to keep the larger functional value \+ (in our maximization problem) and its corresponding opposite end -poin t. At the end of the required iterations, the final interval is the an swer. At times when the final answer must be a single point and not an interval, the convention of selecting the midpoint is provided. This \+ program works when the function is not differentiable and we are looki ng for a solution. If you need to minimize a function, then multiple t he function by (-1) and find the maximum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "To utilize the Golden section rout ine you need the following:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "The user enters the function " }{TEXT 281 1 "f " }{TEXT -1 6 " using" }}{PARA 0 "" 0 "" {TEXT 282 33 "f:=x->" }}{PARA 0 "" 0 "" {TEXT -1 5 "Type " }{TEXT 283 22 "GOLD(f,a,b,tolerance) " }{TEXT -1 23 "for specific values of " } {TEXT 284 5 "a, b," }{TEXT -1 9 " and the " }{TEXT 285 9 "tolerance" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "The output is the iterat ive process (each step)." }}{PARA 0 "" 0 "" {TEXT -1 80 "The last outp ut provided is the midpoint of the final interval and the value of " } {TEXT 286 5 "f(x) " }{TEXT -1 14 "at that point." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "The Algorithm Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "GOLD:=proc(f::procedure,a::numeric,b::numeric,T::numeric)" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "local x1,x2, N, val;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "N := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# The first 2 experimental endpoi nts x1 and x2 are determined:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x1 :=a+0.382*(b-a);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x2:=a+0.618*(b- a);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "# The specified interval and tolerance level is print ed:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "printf(\"The interval [a,b] is [% 4.2f,% 4.2f]and user specified tolerance level is% 6.5f.\\n\",a ,b,T);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# The first 2 experimental endpoints x1 and x2 are pr inted" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# followed by 2 blank line s." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "printf(\"The first 2 experime ntal endpoints are x1= % 6.3f and x2 = % 6.3f. \\n\",x1,x2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "# The number of iterations i s determined:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "N:=ceil((ln(T/(b-a ))/ln(0.618)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# The column headings for the calculations in th e procedure" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# iterate are printe dl" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "printf(\" Iteration x(1) \+ x(2) f(x1) f(x2) Interval \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# The procedure iterate is called and parameters are passed to" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "# the procedure." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "iterate(f,a,b,N,x1,x2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# The value f(m idpoint) is found. This midpoint is the midpoint" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 45 "# of the interval found in the Nth iteration." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "val:=f(mdpt);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# Two blank li nes are printed along with the midpointand " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# f(midpoint)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\ n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "printf(\"The midpoint of t he final interval is% 9.6f and f(midpoint) = % 7.3f. \\n\",mdpt, val); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# Two blank lines are printed followed by the maximum value of " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# of the input function." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "printf(\"The maximum of the function is % 7.3f and th e x value = % 9.6f, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "the %s poin t of interval number % 3.0f.\\n\",fkeep,xkeep,pos,N);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "printf(\" \\n\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# The procedure iterate follows:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 84 "iterate:=proc(f::procedure,a::numeric,b::numeric, N ::posint,x1::numeric,x2::numeric)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# Local variable declared" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "local x1n,x2n,an,bn,fx1,fx2,j;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "# Global variables declared" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " global mdpt,fkeep,xkeep,pos;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# The first experimental endpoint i s assigned to x1n(1) and " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# the \+ second experimental endpoint is assigned to x2n(1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "x1n(1):=x1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " x2n(1):=x2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 64 "# Assigning the original endpoints to variables an( 1) and bn(1)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "an(1):=a;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "bn(1):=b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# The following loops run s from 2 to N(the number of iterations" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# for the length of the last interval to be less than the" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# tolerance specified." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "for j from 2 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fx1(j-1):=f(x1n(j-1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fx2(j-1):=f(x2n(j-1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "i f fx1(j-1)<=fx2(j-1) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "an(j): =x1n(j-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "bn(j):=bn(j-1);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x1n(j):=x2n(j-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x2n(j):=an(j)+.618*(bn(j)-an(j));" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "an(j ):=an(j-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "bn(j):=x2n(j-1);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x2n(j):=x1n(j-1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "x1n(j):=an(j)+.382*(bn(j)-an(j));" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# The following displays the iterat ion, each new x(1), x(2)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# f(x1 ), f(x2), and the calculated intervals." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "printf(\" % 3.0f % 11.4f % 10.4f % 10.4f %10.4f [% 6.4f , % 6.4f]\\n\",j,x1n(j),x2n(j),fx1(j-1),fx2(j-1),an(j),bn(j));" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# For the Nth interval a midpoint is found. After finding " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# the midpoint, an error check is p erformed to ensure the" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# midpoin t of the interval within the specified tolerance" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# is the max value for the given function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "if (j = N) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "mdpt := (an(j) + bn(j))/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "if (f(an(j)) > f(bn(j))) and (f(an(j)) > f(mdpt)) the n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " fkeep := f(an(j)); xkeep \+ := an(j); pos := \"left end\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " e lse" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " if (f(bn(j)) > f(mdpt)) a nd (f(bn(j)) < f(an(j))) then" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 " \+ fkeep := f(bn(i)); xkeep := bn(i); pos := \"right end\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " else" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " fkeep := f(mdpt); xkeep := (an(j) + bn(j))/2; pos := \"midpoin t\";" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e nd proc:" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f:= x->1-exp(-x)+(1/(1+x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowG F(,(\"\"\"F--%$expG6#,$9$!\"\"F3*&F-F-,&F-F-F2F-F3F-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "GOLD(f,0,20,.001);" }}{PARA 6 "" 1 "" {TEXT -1 81 "The interval [a,b] is [ 0.00, 20.00]and user specified tolerance level is .00100." }}{PARA 6 "" 1 "" {TEXT -1 68 "The first \+ 2 experimental endpoints are x1= 7.640 and x2 = 12.360. " }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 71 " Iteration x(1) x(2) f(x1) f(x2) \+ Interval " }}{PARA 6 "" 1 "" {TEXT -1 75 " 2 4.7215 7.6400 1.1153 1.0748 [ 0.0000, 12.3600]" }}{PARA 6 " " 1 "" {TEXT -1 74 " 3 2.9185 4.7215 1.1659 1.115 3 [ 0.0000, 7.6400]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 4 1 .8036 2.9185 1.2012 1.1659 [ 0.0000, 4.7215]" }} {PARA 6 "" 1 "" {TEXT -1 74 " 5 2.9185 3.6069 1.1920 \+ 1.2012 [ 1.8036, 4.7215]" }}{PARA 6 "" 1 "" {TEXT -1 74 " \+ 6 2.4925 2.9185 1.2012 1.1899 [ 1.8036, 3.6069] " }}{PARA 6 "" 1 "" {TEXT -1 74 " 7 2.2295 2.4925 1.2 036 1.2012 [ 1.8036, 2.9185]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 8 2.4925 2.6553 1.2021 1.2036 [ 2.2295, 2.9 185]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 9 2.3921 2.4925 \+ 1.2036 1.2033 [ 2.2295, 2.6553]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 10 2.4925 2.5548 1.2034 1.2036 [ 2.3921, \+ 2.6553]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 11 2.4543 2.4925 1.2036 1.2036 [ 2.3921, 2.5548]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 12 2.4925 2.5164 1.2036 1.2036 [ 2.4543, 2.5548]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 13 2.5164 \+ 2.5310 1.2036 1.2036 [ 2.4925, 2.5548]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 14 2.5072 2.5164 1.2036 1.2036 \+ [ 2.4925, 2.5310]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 15 2.5 164 2.5219 1.2036 1.2036 [ 2.5072, 2.5310]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 16 2.5128 2.5164 1.2036 1. 2036 [ 2.5072, 2.5219]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 17 \+ 2.5107 2.5128 1.2036 1.2036 [ 2.5072, 2.5164]" }} {PARA 6 "" 1 "" {TEXT -1 74 " 18 2.5128 2.5142 1.2036 \+ 1.2036 [ 2.5107, 2.5164]" }}{PARA 6 "" 1 "" {TEXT -1 74 " \+ 19 2.5120 2.5128 1.2036 1.2036 [ 2.5107, 2.5142] " }}{PARA 6 "" 1 "" {TEXT -1 74 " 20 2.5128 2.5134 1.2 036 1.2036 [ 2.5120, 2.5142]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 21 2.5125 2.5128 1.2036 1.2036 [ 2.5120, 2.5 134]" }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 74 "The midpoint of the final interval \+ is 2.512705 and f(midpoint) = 1.204. " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 68 " The maximum of the function is 1.204 and the x value = 2.512705, " }}{PARA 6 "" 1 "" {TEXT -1 42 "the midpoint point of interval number \+ 21." }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:= x->x-exp(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$ex pG6#F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GOLD(f ,-1,3,.1);" }}{PARA 6 "" 1 "" {TEXT -1 80 "The interval [a,b] is [-1.0 0, 3.00]and user specified tolerance level is .10000." }}{PARA 6 "" 1 "" {TEXT -1 67 "The first 2 experimental endpoints are x1= .528 and \+ x2 = 1.472. " }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 71 " Iteration x(1) \+ x(2) f(x1) f(x2) Interval " }}{PARA 6 "" 1 "" {TEXT -1 74 " 2 -.0557 .5280 -1.1675 -2.8859 [ -1.0000, 1.4720]" }}{PARA 6 "" 1 "" {TEXT -1 73 " 3 -.4163 \+ -.0557 -1.0015 -1.1675 [-1.0000, .5280]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 4 -.0557 .1673 -1.0758 -1.0015 \+ [-.4163, .5280]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 5 -.1934 \+ -.0557 -1.0015 -1.0148 [-.4163, .1673]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 6 -.0557 .0295 -1.0175 -1.0015 \+ [-.1934, .1673]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 7 .0295 \+ .0821 -1.0015 -1.0004 [-.0557, .1673]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 8 -.0031 .0295 -1.0004 -1.0035 \+ [-.0557, .0821]" }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 " " {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 74 "The midpoint of the f inal interval is .013202 and f(midpoint) = -1.000. " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 " " {TEXT -1 68 "The maximum of the function is -1.000 and the x value \+ = .013202, " }}{PARA 6 "" 1 "" {TEXT -1 42 "the midpoint point of in terval number 8." }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 " " {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:= x-> -3*x^2+21.6*x+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6 $%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"!\"$*&$\"$;#!\"\"F1F/F1F1F1F1F (F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GOLD(f,0,25,.1);" }}{PARA 6 "" 1 "" {TEXT -1 81 "The interval [a,b] is [ 0.00, 25.00]and user specified tolerance level is .10000." }}{PARA 6 "" 1 "" {TEXT -1 68 "The first 2 experimental endpoints are x1= 9.550 and x2 = 15. 450. " }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 71 " Iteration x(1) x(2) \+ f(x1) f(x2) Interval " }}{PARA 6 "" 1 "" {TEXT -1 75 " \+ 2 5.9019 9.5500 -66.3275 -381.3875 [ 0.0000, 15.4 500]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 3 3.6481 5.9019 \+ 23.9838 -66.3275 [ 0.0000, 9.5500]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 4 2.2545 3.6481 39.8731 23.9838 [ 0.0000, \+ 5.9019]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 5 3.6481 4.5086 34.4491 39.8731 [ 2.2545, 5.9019]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 6 3.1156 3.6481 39.8731 37.4033 [ 2.2545, 4.5086]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 7 3.6481 \+ 3.9765 39.1760 39.8731 [ 3.1156, 4.5086]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 8 3.4444 3.6481 39.8731 39.4548 \+ [ 3.1156, 3.9765]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 9 3.6 481 3.7732 39.8074 39.8731 [ 3.4444, 3.9765]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 10 3.5700 3.6481 39.8731 39. 7900 [ 3.4444, 3.7732]" }}{PARA 6 "" 1 "" {TEXT -1 74 " 11 \+ 3.5222 3.5700 39.8773 39.8731 [ 3.4444, 3.6481]" }} {PARA 6 "" 1 "" {TEXT -1 74 " 12 3.5700 3.6000 39.8619 \+ 39.8773 [ 3.5222, 3.6481]" }}{PARA 6 "" 1 "" {TEXT -1 2 " " } }{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 74 "The mi dpoint of the final interval is 3.585170 and f(midpoint) = 39.879. " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }} {PARA 6 "" 1 "" {TEXT -1 68 "The maximum of the function is 39.879 an d the x value = 3.585170, " }}{PARA 6 "" 1 "" {TEXT -1 42 "the midpoi nt point of interval number 12." }}{PARA 6 "" 1 "" {TEXT -1 2 " " }} {PARA 6 "" 1 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f:= x->-(abs(2-x)+abs(5-4*x)+abs(8-9*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrowGF(,(-%$absG6#,&\" \"#\"\"\"9$!\"\"F4-F.6#,&\"\"&F2*&\"\"%F2F3F2F4F4-F.6#,&\"\")F2*&\"\"* F2F3F2F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "GOLD(f, 0,3,.1);" }}{PARA 6 "" 1 "" {TEXT -1 80 "The interval [a,b] is [ 0.00, 3.00]and user specified tolerance level is .10000." }}{PARA 6 "" 1 " " {TEXT -1 67 "The first 2 experimental endpoints are x1= 1.146 and x 2 = 1.854. " }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 71 " Iteration x(1) \+ x(2) f(x1) f(x2) Interval " }}{PARA 6 "" 1 "" {TEXT -1 74 " 2 .7082 1.1460 -3.5840 -11.2480 [ 0.0000, 1.8540]" }}{PARA 6 "" 1 "" {TEXT -1 73 " 3 1.1460 \+ 1.4163 -5.0848 -3.5840 [ .7082, 1.8540]" }}{PARA 6 "" 1 "" {TEXT -1 73 " 4 .9787 1.1460 -3.5840 -5.9958 \+ [ .7082, 1.4163]" }}{PARA 6 "" 1 "" {TEXT -1 73 " 5 .8755 .9787 -2.9149 -3.5840 [ .7082, 1.1460]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 6 .8116 .8755 -2.7436 -2.9149 \+ [ .7082, .9787]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 7 .875 5 .9149 -3.6382 -2.7436 [ .8116, .9787]" }}{PARA 6 "" 1 "" {TEXT -1 72 " 8 .9149 .9393 -2.7436 -2.6594 \+ [ .8755, .9787]" }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 74 "The midpoint of th e final interval is .927087 and f(midpoint) = -2.708. " }}{PARA 6 " " 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 68 "The maximum of the function is -2.708 and the x val ue = .927087, " }}{PARA 6 "" 1 "" {TEXT -1 42 "the midpoint point of interval number 8." }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " \+ f:= x->-2*x^2+4*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6 \"6$%)operatorG%&arrowGF(,&*$)9$\"\"#\"\"\"!\"#*&\"\"%F1F/F1F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GOLD(f,-1,2,.4);" }} {PARA 6 "" 1 "" {TEXT -1 80 "The interval [a,b] is [-1.00, 2.00]and us er specified tolerance level is .40000." }}{PARA 6 "" 1 "" {TEXT -1 67 "The first 2 experimental endpoints are x1= .146 and x2 = .854. " }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 71 " Iteration x(1) x(2) f(x1 ) f(x2) Interval " }}{PARA 6 "" 1 "" {TEXT -1 73 " \+ 2 .8540 1.2918 .5414 1.9574 [ .1460, 2.0000]" }}{PARA 6 "" 1 "" {TEXT -1 73 " 3 .5837 .8540 1.957 4 1.8297 [ .1460, 1.2918]" }}{PARA 6 "" 1 "" {TEXT -1 73 " \+ 4 .8540 1.0213 1.6534 1.9574 [ .5837, 1.2918] " }}{PARA 6 "" 1 "" {TEXT -1 73 " 5 1.0213 1.1245 1.9 574 1.9991 [ .8540, 1.2918]" }}{PARA 6 "" 1 "" {TEXT -1 2 " \+ " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 74 "The midpoint of the final interval is 1.072886 and f(midpoint) = 1.989. " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}{PARA 6 "" 1 "" {TEXT -1 2 " \+ " }}{PARA 6 "" 1 "" {TEXT -1 68 "The maximum of the function is 1.98 9 and the x value = 1.072886, " }}{PARA 6 "" 1 "" {TEXT -1 42 "the mi dpoint point of interval number 5." }}{PARA 6 "" 1 "" {TEXT -1 2 " \+ " }}{PARA 6 "" 1 "" {TEXT -1 2 " " }}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }