module modamerica implicit none ! The objective is to draw a map of America in which the countries ! appear with areas that are proportional to a target value (real ! area, population, PIB, etc). The unknowns are 132 points in R^2, ! which define the boundaries of 17 "countries". Each point is ! assigned to the boundary of one or more country. The computed area ! of each country is calculated as a function of its boundary points ! using Green's formula. The constraints of the problem are: ! ! computed area = (scaled) target value ! ! for each country. As initial approximation we took the coordinates ! of the 132 points in the New York Times map of America which, of ! course, do not fit with target values. The objective function is ! ! 1/2 \sum_{j=1}^{132} \|P_j - Q_j\|_2^2, ! ! where P_1,...,P_132 are the unknowns and Q_1,...Q_132 are the ! initial approximation. ! ! Considered "countries" are: ! ! 1 Cuba ! 2 Canada ! 3 USA (excluding Alaska that is considered in separate) ! 4 Mexico ! 5 America Central (Guatemala, Hunduras, Nicaragua, El Salvador, ! Costa Rica, Panama, and Belize) ! 6 Colombia ! 7 Venezuela ! 8 Guianas (Guyana, Suriname, and French Guiana) ! 9 Brasil ! 10 Ecuador ! 11 Peru ! 12 Bolivia ! 13 Paraguay ! 14 Chile ! 15 Argentina ! 16 Alaska ! 17 Uruguay ! Number of states and number of points integer :: nstates,np parameter ( nstates = 17 ) parameter ( np = 132 ) ! Number of boundary points for each state integer :: nbord(nstates) data nbord(1:nstates) / 4, 21, 22, 26, 8, 9, 9, 5, 22, 4, 10, 7, 6, 16, 20, 8, 5 / ! Indices of the boundary points per state integer pbord(np,nstates) data pbord(1: 4, 1) / 49, 50, 51, 52 /, & pbord(1:21, 2) / 83, 35, 64, 63, 84, 36, 99,100, 37, 38, 39,101,102, & 40, 41, 42,127, 81, 82, 43, 46 /, & pbord(1:22, 3) / 33,108,109,110,111,112, 34, 94, 62, 98, 61, 60, 96, & 97, 36, 84, 63, 64, 35,120, 56,121 /, & pbord(1:26, 4) / 86,113,114, 31, 32, 87,115, 88,103, 89, 85, 34,112, & 111,110,109,108, 33,118,104,116,119,105,106,117,107 /, & pbord(1: 8, 5) / 30, 29, 91, 90, 32, 31, 92, 93 /, & pbord(1: 9, 6) / 28, 25, 23, 22, 55, 95, 24, 29, 30 /, & pbord(1: 9, 7) / 22, 21, 20, 19,122,123, 24, 95, 55 /, & pbord(1: 5, 8) / 20,126, 18, 17, 19 /, & pbord(1:22, 9) / 53, 16, 15, 10, 8, 7, 6, 65, 5, 68, 76, 48,129, & 47,125, 17, 18,126, 20, 21, 22, 23 /, & pbord(1: 4,10) / 27, 26, 25, 28 /, & pbord(1:10,11) / 128, 59, 12, 13, 16, 53, 23, 25, 26, 27 /, & pbord(1: 7,12) / 14, 9, 10, 15, 16, 13, 12 /, & pbord(1: 6,13) / 54, 7, 8, 10, 9, 70 /, & pbord(1:16,14) / 1, 11,132,131, 14, 12, 59,130, 57, 58, 2, 73, 77, & 71, 75, 74 /, & pbord(1:20,15) / 1, 2, 73, 71, 72, 75, 74, 69, 67, 66, 4, 6, 7, & 54, 70, 9, 14,131,132, 11 /, & pbord(1: 8,16) / 43, 80, 44, 78,124, 45, 79, 46 /, & pbord(1: 5,17) / 4, 3, 5, 65, 6 / ! Putative coordinates of each boundary point (initial approximation) real(kind=8) :: put(1:2,np) data put(1:2, 1) / 11.70d0, 5.70d0 /, & put(1:2, 2) / 12.10d0, 5.60d0 /, & put(1:2, 3) / 13.50d0, 8.30d0 /, & put(1:2, 4) / 13.10d0, 8.40d0 /, & put(1:2, 5) / 13.70d0, 8.60d0 /, & put(1:2, 6) / 13.20d0, 9.00d0 /, & put(1:2, 7) / 13.60d0, 9.40d0 /, & put(1:2, 8) / 13.40d0, 10.00d0 /, & put(1:2, 9) / 12.50d0, 10.00d0 /, & put(1:2, 10) / 13.10d0, 10.40d0 /, & put(1:2, 11) / 11.70d0, 6.40d0 /, & put(1:2, 12) / 11.60d0, 10.60d0 /, & put(1:2, 13) / 11.60d0, 11.20d0 /, & put(1:2, 14) / 12.00d0, 9.90d0 /, & put(1:2, 15) / 12.70d0, 11.20d0 /, & put(1:2, 16) / 11.80d0, 11.60d0 /, & put(1:2, 17) / 13.80d0, 13.50d0 /, & put(1:2, 18) / 13.50d0, 13.20d0 /, & put(1:2, 19) / 12.90d0, 14.10d0 /, & put(1:2, 20) / 12.80d0, 13.70d0 /, & put(1:2, 21) / 12.40d0, 13.30d0 /, & put(1:2, 22) / 11.90d0, 13.30d0 /, & put(1:2, 23) / 11.70d0, 12.70d0 /, & put(1:2, 24) / 11.60d0, 14.60d0 /, & put(1:2, 25) / 11.10d0, 12.80d0 /, & put(1:2, 26) / 10.80d0, 12.50d0 /, & put(1:2, 27) / 10.50d0, 12.50d0 /, & put(1:2, 28) / 10.70d0, 13.20d0 /, & put(1:2, 29) / 10.90d0, 14.20d0 /, & put(1:2, 30) / 10.80d0, 14.10d0 /, & put(1:2, 31) / 9.20d0, 15.00d0 /, & put(1:2, 32) / 9.40d0, 15.30d0 /, & put(1:2, 33) / 6.60d0, 17.60d0 /, & put(1:2, 34) / 8.40d0, 16.60d0 /, & put(1:2, 35) / 6.40d0, 20.00d0 /, & put(1:2, 36) / 12.50d0, 19.50d0 /, & put(1:2, 37) / 13.80d0, 20.60d0 /, & put(1:2, 38) / 11.80d0, 22.40d0 /, & put(1:2, 39) / 11.20d0, 20.50d0 /, & put(1:2, 40) / 10.00d0, 22.20d0 /, & put(1:2, 41) / 11.50d0, 23.00d0 /, & put(1:2, 42) / 11.00d0, 24.60d0 /, & put(1:2, 43) / 5.80d0, 23.80d0 /, & put(1:2, 44) / 3.50d0, 23.80d0 /, & put(1:2, 45) / 3.00d0, 21.50d0 /, & put(1:2, 46) / 5.50d0, 22.00d0 /, & put(1:2, 47) / 15.70d0, 12.20d0 /, & put(1:2, 48) / 15.30d0, 10.80d0 /, & put(1:2, 49) / 10.40d0, 16.20d0 /, & put(1:2, 50) / 11.00d0, 15.80d0 /, & put(1:2, 51) / 11.30d0, 16.00d0 /, & put(1:2, 52) / 10.40d0, 16.30d0 /, & put(1:2, 53) / 11.60d0, 12.00d0 /, & put(1:2, 54) / 13.00d0, 9.30d0 /, & put(1:2, 55) / 12.00d0, 13.80d0 /, & put(1:2, 56) / 6.00d0, 18.70d0 /, & put(1:2, 57) / 11.50d0, 5.70d0 /, & put(1:2, 58) / 11.80d0, 5.50d0 /, & put(1:2, 59) / 11.40d0, 10.60d0 /, & put(1:2, 60) / 10.70d0, 17.40d0 /, & put(1:2, 61) / 10.60d0, 16.60d0 /, & put(1:2, 62) / 10.40d0, 17.20d0 /, & put(1:2, 63) / 11.00d0, 19.10d0 /, & put(1:2, 64) / 10.00d0, 20.00d0 /, & put(1:2, 65) / 13.50d0, 8.80d0 /, & put(1:2, 66) / 13.20d0, 7.70d0 /, & put(1:2, 67) / 12.60d0, 7.70d0 /, & put(1:2, 68) / 14.20d0, 9.70d0 /, & put(1:2, 69) / 12.30d0, 6.70d0 /, & put(1:2, 70) / 13.10d0, 9.60d0 /, & put(1:2, 71) / 12.20d0, 5.20d0 /, & put(1:2, 72) / 12.50d0, 5.30d0 /, & put(1:2, 73) / 12.20d0, 5.40d0 /, & put(1:2, 74) / 12.10d0, 5.60d0 /, & put(1:2, 75) / 12.20d0, 5.40d0 /, & put(1:2, 76) / 15.00d0, 10.00d0 /, & put(1:2, 77) / 11.90d0, 5.30d0 /, & put(1:2, 78) / 2.70d0, 23.00d0 /, & put(1:2, 79) / 4.00d0, 22.00d0 /, & put(1:2, 80) / 4.50d0, 24.00d0 /, & put(1:2, 81) / 9.00d0, 23.50d0 /, & put(1:2, 82) / 7.30d0, 23.70d0 /, & put(1:2, 83) / 6.00d0, 21.00d0 /, & put(1:2, 84) / 12.00d0, 19.50d0 /, & put(1:2, 85) / 8.50d0, 16.00d0 /, & put(1:2, 86) / 7.70d0, 15.80d0 /, & put(1:2, 87) / 9.30d0, 15.50d0 /, & put(1:2, 88) / 9.90d0, 16.00d0 /, & put(1:2, 89) / 9.00d0, 15.60d0 /, & put(1:2, 90) / 10.20d0, 15.30d0 /, & put(1:2, 91) / 10.20d0, 14.50d0 /, & put(1:2, 92) / 9.70d0, 14.70d0 /, & put(1:2, 93) / 10.00d0, 14.40d0 /, & put(1:2, 94) / 9.30d0, 17.20d0 /, & put(1:2, 95) / 11.50d0, 14.00d0 /, & put(1:2, 96) / 11.30d0, 17.80d0 /, & put(1:2, 97) / 11.40d0, 18.40d0 /, & put(1:2, 98) / 10.50d0, 16.60d0 /, & put(1:2, 99) / 13.10d0, 19.50d0 /, & put(1:2,100) / 12.40d0, 19.90d0 /, & put(1:2,101) / 11.20d0, 20.20d0 /, & put(1:2,102) / 10.00d0, 21.50d0 /, & put(1:2,103) / 9.50d0, 16.00d0 /, & put(1:2,104) / 7.10d0, 16.30d0 /, & put(1:2,105) / 6.80d0, 17.50d0 /, & put(1:2,106) / 6.90d0, 17.50d0 /, & put(1:2,107) / 7.80d0, 15.90d0 /, & put(1:2,108) / 6.80d0, 17.60d0 /, & put(1:2,109) / 7.40d0, 17.40d0 /, & put(1:2,110) / 7.70d0, 17.50d0 /, & put(1:2,111) / 8.10d0, 17.10d0 /, & put(1:2,112) / 8.30d0, 17.30d0 /, & put(1:2,113) / 8.50d0, 15.30d0 /, & put(1:2,114) / 8.90d0, 15.30d0 /, & put(1:2,115) / 9.70d0, 15.60d0 /, & put(1:2,116) / 7.20d0, 16.30d0 /, & put(1:2,117) / 7.40d0, 16.60d0 /, & put(1:2,118) / 6.80d0, 16.80d0 /, & put(1:2,119) / 7.00d0, 16.80d0 /, & put(1:2,120) / 6.10d0, 19.50d0 /, & put(1:2,121) / 6.20d0, 18.20d0 /, & put(1:2,122) / 12.50d0, 14.50d0 /, & put(1:2,123) / 12.10d0, 14.50d0 /, & put(1:2,124) / 2.75d0, 22.15d0 /, & put(1:2,125) / 14.60d0, 12.80d0 /, & put(1:2,126) / 13.00d0, 13.20d0 /, & put(1:2,127) / 10.20d0, 23.30d0 /, & put(1:2,128) / 10.70d0, 11.25d0 /, & put(1:2,129) / 15.30d0, 11.30d0 /, & put(1:2,130) / 11.40d0, 7.50d0 /, & put(1:2,131) / 11.65d0, 9.00d0 /, & put(1:2,132) / 11.60d0, 7.25d0 / ! 'Center' of each country real(kind=8) :: w(2,nstates) ! Target values ! Population taken from ! http://en.wikipedia.org/wiki/List_of_countries_by_population, ! accessed on May 21th, 2013. ! Area in km^2 taken from ! https://en.wikipedia.org/wiki/List_of_countries_and_dependencies_by_area ! accessed on May 21th, 2013. ! GDP in millions of $US taken from: ! http://en.wikipedia.org/wiki/List_of_countries_by_GDP_%28nominal%29 ! accessed on May 21th, 2013. ! Particular cases: ! ! Alaska population and area: https://en.wikipedia.org/wiki/Alaska, ! accessed on May 21th, 2013. ! Alaska GDP from: http://en.wikipedia.org/wiki/List_of_U.S._states_by_GDP, ! accessed on May 21th, 2013. ! French Guiana population and GDP: https://en.wikipedia.org/wiki/French_Guiana, ! accessed on May 21th, 2013. ! Population: ! ! 1 Cuba 11,163,934 ! 2 Canada 35,056,064 ! 3 USA 315,150,551 (USA 315,882,000 - Alaska 731,449) ! 4 Mexico 112,336,538 ! 5 America Central 44,463,381 (Guatemala 15,438,384, Hunduras 8,385,072, Nicaragua 6,071,045, El Salvador 6,183,000, ! Costa Rica 4,667,096, Panama 3,405,813, and Belize 312,971) ! 6 Colombia 47,059,000 ! 7 Venezuela 28,946,101 ! 8 Guianas 1,548,123 (Guyana 784,894, Suriname 534,189, and French Guiana 229,040) ! 9 Brasil 193,946,886 ! 10 Ecuador 15,489,300 ! 11 Peru 30,475,144 ! 12 Bolivia 10,389,913 ! 13 Paraguay 6,672,631 ! 14 Chile 16,634,603 ! 15 Argentina 40,117,096 ! 16 Alaska 731,449 ! 17 Uruguay 3,286,314 !!$ real(kind=8) :: tarval(nstates) !!$ data tarval(1:nstates) / 11163934.0d0, & !!$ 35056064.0d0, & !!$ 315150551.0d0, & !!$ 112336538.0d0, & !!$ 44463381.0d0, & !!$ 47059000.0d0, & !!$ 28946101.0d0, & !!$ 1548123.0d0, & !!$ 193946886.0d0, & !!$ 15489300.0d0, & !!$ 30475144.0d0, & !!$ 10389913.0d0, & !!$ 6672631.0d0, & !!$ 16634603.0d0, & !!$ 40117096.0d0, & !!$ 731449.0d0, & !!$ 3286314.0d0 / ! Areas: ! ! 1 Cuba 109,884 ! 2 Canada 9,984,670 ! 3 USA 7,911,237 ( USA 9,629,091 - Alaska 1,717,854 ) ! 4 Mexico 1,964,375 ! 5 America Central 476,278 (Guatemala 108,889, Hunduras 112,492, Nicaragua 130,373, El Salvador 21,041, ! Costa Rica 51,100, Panama 75,417, and Belize 22,966) ! 6 Colombia 1,141,748 ! 7 Venezuela 912,050 ! 8 Guianas 462,323 (Guyana 214,969, Suriname 163,820, and French Guiana 83,534) ! 9 Brasil 8,514,877 ! 10 Ecuador 256,369 ! 11 Peru 1,285,216 ! 12 Bolivia 1,098,581 ! 13 Paraguay 406,752 ! 14 Chile 756,102 ! 15 Argentina 2,780,400 ! 16 Alaska 1,717,854 ! 17 Uruguay 181,034 real(kind=8) :: tarval(nstates) data tarval(1:nstates) / 109884.0d0, & 9984670.0d0, & 7911237.0d0, & 1964375.0d0, & 476278.0d0, & 1141748.0d0, & 912050.0d0, & 462323.0d0, & 8514877.0d0, & 256369.0d0, & 1285216.0d0, & 1098581.0d0, & 406752.0d0, & 756102.0d0, & 2780400.0d0, & 1717854.0d0, & 181034.0d0 / !!$ ! Original areas in a previous version of this problem (unkonwn source) !!$ !!$ real(kind=8) :: tarval(nstates) !!$ data tarval(1:nstates) / 115.0d0, 9922.0d0, 7885.0d0, 1973.0d0, 543.0d0, 1139.0d0, & !!$ 912.0d0, 470.0d0, 8512.0d0, 461.0d0, 1285.0d0, 1099.0d0, & !!$ 407.0d0, 752.0d0, 2767.0d0, 1478.0d0, 187.0d0 / ! GDP (gross domestic product) in millions of $US: ! ! 1 Cuba 68,715 ! 2 Canada 1,736,869 ! 3 USA 14,991,300 ! 4 Mexico 1,155,206 ! 5 America Central 167,854 (Guatemala 46,898, Hunduras 17,447, Nicaragua 7,297, El Salvador 23,054, ! Costa Rica 41,007, Panama 30,677, and Belize 1,474) ! 6 Colombia 333,185 ! 7 Venezuela 315,893 ! 8 Guianas 12,126 (Guyana 2,577, Suriname 4,610, and French Guiana 4,939) ! 9 Brasil 2,476,651 ! 10 Ecuador 66,381 ! 11 Peru 180,464 ! 12 Bolivia 23,949 ! 13 Paraguay 22,890 ! 14 Chile 248,592 ! 15 Argentina 485,416 ! 16 Alaska 45,600 ! 17 Uruguay 46,710 !!$ real(kind=8) :: tarval(nstates) !!$ data tarval(1:nstates) / 68715.0d0, & !!$ 1736869.0d0, & !!$ 14991300.0d0, & !!$ 1155206.0d0, & !!$ 167854.0d0, & !!$ 333185.0d0, & !!$ 315893.0d0, & !!$ 12126.0d0, & !!$ 2476651.0d0, & !!$ 66381.0d0, & !!$ 180464.0d0, & !!$ 23949.0d0, & !!$ 22890.0d0, & !!$ 248592.0d0, & !!$ 485416.0d0, & !!$ 45600.0d0, & !!$ 46710.0d0 / ! Colors for the graphical representation character * 6 color(nstates) data color( 1) /'yellow'/, & color( 2) /'yellow'/, & color( 3) /'purple'/, & color( 4) /'green '/, & color( 5) /'coral '/, & color( 6) /'orange'/, & color( 7) /'green '/, & color( 8) /'purple'/, & color( 9) /'yellow'/, & color(10) /'yellow'/, & color(11) /'coral '/, & color(12) /'orange'/, & color(13) /'purple'/, & color(14) /'yellow'/, & color(15) /'green '/, & color(16) /'purple'/, & color(17) /'orange'/ end module modamerica