R version 2.9.0 (2009-04-17) Copyright (C) 2009 The R Foundation for Statistical Computing ISBN 3-900051-07-0 R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > # runge2.r Observe Runge phenomenon with interpolation > # > npts = 100 # points for plotting > for ( n in(c(5,7,9))) { # how many points for interpolation + xi <- 4*cos((2*(1:n)-1)*pi/(2*n))# Chebyshev interpolation points + b <- 1/(1+xi*xi) # function values + A <- outer(xi,(0:(n-1)),FUN="^") # vandermonde matrix + print(A) + b <- solve(A,b) # solution + # now get ready to plot + x <- 4.1*(2*(1:npts)-npts-1)/(npts-1) + s <- 1/(1+x*x) # true function + y <- outer(x,(0:(n-1)),FUN="^") %*% b # interpolated values + print(cbind(x,s,y)) + plot(c(x,x),c(s,y)) # Figure 7.2 + } [,1] [,2] [,3] [,4] [,5] [1,] 1 3.804226e+00 1.447214e+01 5.505528e+01 2.094427e+02 [2,] 1 2.351141e+00 5.527864e+00 1.299679e+01 3.055728e+01 [3,] 1 2.449294e-16 5.999039e-32 1.469341e-47 3.598847e-63 [4,] 1 -2.351141e+00 5.527864e+00 -1.299679e+01 3.055728e+01 [5,] 1 -3.804226e+00 1.447214e+01 -5.505528e+01 2.094427e+02 x s [1,] -4.10000000 0.05614823 0.302634653 [2,] -4.01717172 0.05835099 0.223102053 [3,] -3.93434343 0.06068308 0.153871497 [4,] -3.85151515 0.06315456 0.094406144 [5,] -3.76868687 0.06577649 0.044180339 [6,] -3.68585859 0.06856099 0.002679609 [7,] -3.60303030 0.07152141 -0.030599336 [8,] -3.52020202 0.07467244 -0.056148599 [9,] -3.43737374 0.07803028 -0.074449101 [10,] -3.35454545 0.08161283 -0.085970578 [11,] -3.27171717 0.08543990 -0.091171580 [12,] -3.18888889 0.08953343 -0.090499477 [13,] -3.10606061 0.09391777 -0.084390451 [14,] -3.02323232 0.09861997 -0.073269501 [15,] -2.94040404 0.10367017 -0.057550443 [16,] -2.85757576 0.10910195 -0.037635907 [17,] -2.77474747 0.11495279 -0.013917339 [18,] -2.69191919 0.12126461 0.013224997 [19,] -2.60909091 0.12808435 0.043422023 [20,] -2.52626263 0.13546459 0.076315846 [21,] -2.44343434 0.14346438 0.111559756 [22,] -2.36060606 0.15215001 0.148818226 [23,] -2.27777778 0.16159601 0.187766916 [24,] -2.19494949 0.17188618 0.228092668 [25,] -2.11212121 0.18311477 0.269493509 [26,] -2.02929293 0.19538769 0.311678651 [27,] -1.94646465 0.20882387 0.354368487 [28,] -1.86363636 0.22355658 0.397294599 [29,] -1.78080808 0.23973471 0.440199749 [30,] -1.69797980 0.25752386 0.482837885 [31,] -1.61515152 0.27710699 0.524974140 [32,] -1.53232323 0.29868449 0.566384830 [33,] -1.44949495 0.32247292 0.606857454 [34,] -1.36666667 0.34870205 0.646190698 [35,] -1.28383838 0.37760914 0.684194430 [36,] -1.20101010 0.40942911 0.720689704 [37,] -1.11818182 0.44437915 0.755508757 [38,] -1.03535354 0.48263551 0.788495009 [39,] -0.95252525 0.52430017 0.819503066 [40,] -0.86969697 0.56935520 0.848398719 [41,] -0.78686869 0.61760330 0.875058941 [42,] -0.70404040 0.66859539 0.899371890 [43,] -0.62121212 0.72155044 0.921236909 [44,] -0.53838384 0.77527965 0.940564523 [45,] -0.45555556 0.82813618 0.957276445 [46,] -0.37272727 0.87802046 0.971305567 [47,] -0.28989899 0.92247397 0.982595971 [48,] -0.20707071 0.95888468 0.991102918 [49,] -0.12424242 0.98479847 0.996792856 [50,] -0.04141414 0.99828781 0.999643418 [51,] 0.04141414 0.99828781 0.999643418 [52,] 0.12424242 0.98479847 0.996792856 [53,] 0.20707071 0.95888468 0.991102918 [54,] 0.28989899 0.92247397 0.982595971 [55,] 0.37272727 0.87802046 0.971305567 [56,] 0.45555556 0.82813618 0.957276445 [57,] 0.53838384 0.77527965 0.940564523 [58,] 0.62121212 0.72155044 0.921236909 [59,] 0.70404040 0.66859539 0.899371890 [60,] 0.78686869 0.61760330 0.875058941 [61,] 0.86969697 0.56935520 0.848398719 [62,] 0.95252525 0.52430017 0.819503066 [63,] 1.03535354 0.48263551 0.788495009 [64,] 1.11818182 0.44437915 0.755508757 [65,] 1.20101010 0.40942911 0.720689704 [66,] 1.28383838 0.37760914 0.684194430 [67,] 1.36666667 0.34870205 0.646190698 [68,] 1.44949495 0.32247292 0.606857454 [69,] 1.53232323 0.29868449 0.566384830 [70,] 1.61515152 0.27710699 0.524974140 [71,] 1.69797980 0.25752386 0.482837885 [72,] 1.78080808 0.23973471 0.440199749 [73,] 1.86363636 0.22355658 0.397294599 [74,] 1.94646465 0.20882387 0.354368487 [75,] 2.02929293 0.19538769 0.311678651 [76,] 2.11212121 0.18311477 0.269493509 [77,] 2.19494949 0.17188618 0.228092668 [78,] 2.27777778 0.16159601 0.187766916 [79,] 2.36060606 0.15215001 0.148818226 [80,] 2.44343434 0.14346438 0.111559756 [81,] 2.52626263 0.13546459 0.076315846 [82,] 2.60909091 0.12808435 0.043422023 [83,] 2.69191919 0.12126461 0.013224997 [84,] 2.77474747 0.11495279 -0.013917339 [85,] 2.85757576 0.10910195 -0.037635907 [86,] 2.94040404 0.10367017 -0.057550443 [87,] 3.02323232 0.09861997 -0.073269501 [88,] 3.10606061 0.09391777 -0.084390451 [89,] 3.18888889 0.08953343 -0.090499477 [90,] 3.27171717 0.08543990 -0.091171580 [91,] 3.35454545 0.08161283 -0.085970578 [92,] 3.43737374 0.07803028 -0.074449101 [93,] 3.52020202 0.07467244 -0.056148599 [94,] 3.60303030 0.07152141 -0.030599336 [95,] 3.68585859 0.06856099 0.002679609 [96,] 3.76868687 0.06577649 0.044180339 [97,] 3.85151515 0.06315456 0.094406144 [98,] 3.93434343 0.06068308 0.153871497 [99,] 4.01717172 0.05835099 0.223102053 [100,] 4.10000000 0.05614823 0.302634653 [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 3.899712e+00 1.520775e+01 5.930584e+01 2.312757e+02 9.019085e+02 [2,] 1 3.127326e+00 9.780167e+00 3.058577e+01 9.565168e+01 2.991340e+02 [3,] 1 1.735535e+00 3.012082e+00 5.227573e+00 9.072635e+00 1.574588e+01 [4,] 1 2.449294e-16 5.999039e-32 1.469341e-47 3.598847e-63 8.814633e-79 [5,] 1 -1.735535e+00 3.012082e+00 -5.227573e+00 9.072635e+00 -1.574588e+01 [6,] 1 -3.127326e+00 9.780167e+00 -3.058577e+01 9.565168e+01 -2.991340e+02 [7,] 1 -3.899712e+00 1.520775e+01 -5.930584e+01 2.312757e+02 -9.019085e+02 [,7] [1,] 3.517183e+03 [2,] 9.354894e+02 [3,] 2.732752e+01 [4,] 2.158962e-94 [5,] 2.732752e+01 [6,] 9.354894e+02 [7,] 3.517183e+03 x s [1,] -4.10000000 0.05614823 -0.15310605 [2,] -4.01717172 0.05835099 -0.04588572 [3,] -3.93434343 0.06068308 0.03485411 [4,] -3.85151515 0.06315456 0.09298539 [5,] -3.76868687 0.06577649 0.13205263 [6,] -3.68585859 0.06856099 0.15528842 [7,] -3.60303030 0.07152141 0.16562873 [8,] -3.52020202 0.07467244 0.16572784 [9,] -3.43737374 0.07803028 0.15797292 [10,] -3.35454545 0.08161283 0.14449826 [11,] -3.27171717 0.08543990 0.12719926 [12,] -3.18888889 0.08953343 0.10774599 [13,] -3.10606061 0.09391777 0.08759646 [14,] -3.02323232 0.09861997 0.06800956 [15,] -2.94040404 0.10367017 0.05005765 [16,] -2.85757576 0.10910195 0.03463886 [17,] -2.77474747 0.11495279 0.02248900 [18,] -2.69191919 0.12126461 0.01419319 [19,] -2.60909091 0.12808435 0.01019711 [20,] -2.52626263 0.13546459 0.01081798 [21,] -2.44343434 0.14346438 0.01625514 [22,] -2.36060606 0.15215001 0.02660035 [23,] -2.27777778 0.16159601 0.04184772 [24,] -2.19494949 0.17188618 0.06190337 [25,] -2.11212121 0.18311477 0.08659467 [26,] -2.02929293 0.19538769 0.11567921 [27,] -1.94646465 0.20882387 0.14885345 [28,] -1.86363636 0.22355658 0.18576095 [29,] -1.78080808 0.23973471 0.22600041 [30,] -1.69797980 0.25752386 0.26913323 [31,] -1.61515152 0.27710699 0.31469082 [32,] -1.53232323 0.29868449 0.36218162 [33,] -1.44949495 0.32247292 0.41109765 [34,] -1.36666667 0.34870205 0.46092086 [35,] -1.28383838 0.37760914 0.51112911 [36,] -1.20101010 0.40942911 0.56120177 [37,] -1.11818182 0.44437915 0.61062505 [38,] -1.03535354 0.48263551 0.65889697 [39,] -0.95252525 0.52430017 0.70553201 [40,] -0.86969697 0.56935520 0.75006541 [41,] -0.78686869 0.61760330 0.79205715 [42,] -0.70404040 0.66859539 0.83109562 [43,] -0.62121212 0.72155044 0.86680091 [44,] -0.53838384 0.77527965 0.89882779 [45,] -0.45555556 0.82813618 0.92686841 [46,] -0.37272727 0.87802046 0.95065456 [47,] -0.28989899 0.92247397 0.96995970 [48,] -0.20707071 0.95888468 0.98460062 [49,] -0.12424242 0.98479847 0.99443873 [50,] -0.04141414 0.99828781 0.99938111 [51,] 0.04141414 0.99828781 0.99938111 [52,] 0.12424242 0.98479847 0.99443873 [53,] 0.20707071 0.95888468 0.98460062 [54,] 0.28989899 0.92247397 0.96995970 [55,] 0.37272727 0.87802046 0.95065456 [56,] 0.45555556 0.82813618 0.92686841 [57,] 0.53838384 0.77527965 0.89882779 [58,] 0.62121212 0.72155044 0.86680091 [59,] 0.70404040 0.66859539 0.83109562 [60,] 0.78686869 0.61760330 0.79205715 [61,] 0.86969697 0.56935520 0.75006541 [62,] 0.95252525 0.52430017 0.70553201 [63,] 1.03535354 0.48263551 0.65889697 [64,] 1.11818182 0.44437915 0.61062505 [65,] 1.20101010 0.40942911 0.56120177 [66,] 1.28383838 0.37760914 0.51112911 [67,] 1.36666667 0.34870205 0.46092086 [68,] 1.44949495 0.32247292 0.41109765 [69,] 1.53232323 0.29868449 0.36218162 [70,] 1.61515152 0.27710699 0.31469082 [71,] 1.69797980 0.25752386 0.26913323 [72,] 1.78080808 0.23973471 0.22600041 [73,] 1.86363636 0.22355658 0.18576095 [74,] 1.94646465 0.20882387 0.14885345 [75,] 2.02929293 0.19538769 0.11567921 [76,] 2.11212121 0.18311477 0.08659467 [77,] 2.19494949 0.17188618 0.06190337 [78,] 2.27777778 0.16159601 0.04184772 [79,] 2.36060606 0.15215001 0.02660035 [80,] 2.44343434 0.14346438 0.01625514 [81,] 2.52626263 0.13546459 0.01081798 [82,] 2.60909091 0.12808435 0.01019711 [83,] 2.69191919 0.12126461 0.01419319 [84,] 2.77474747 0.11495279 0.02248900 [85,] 2.85757576 0.10910195 0.03463886 [86,] 2.94040404 0.10367017 0.05005765 [87,] 3.02323232 0.09861997 0.06800956 [88,] 3.10606061 0.09391777 0.08759646 [89,] 3.18888889 0.08953343 0.10774599 [90,] 3.27171717 0.08543990 0.12719926 [91,] 3.35454545 0.08161283 0.14449826 [92,] 3.43737374 0.07803028 0.15797292 [93,] 3.52020202 0.07467244 0.16572784 [94,] 3.60303030 0.07152141 0.16562873 [95,] 3.68585859 0.06856099 0.15528842 [96,] 3.76868687 0.06577649 0.13205263 [97,] 3.85151515 0.06315456 0.09298539 [98,] 3.93434343 0.06068308 0.03485411 [99,] 4.01717172 0.05835099 -0.04588572 [100,] 4.10000000 0.05614823 -0.15310605 [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 3.939231e+00 1.551754e+01 6.112718e+01 2.407941e+02 9.485435e+02 [2,] 1 3.464102e+00 1.200000e+01 4.156922e+01 1.440000e+02 4.988306e+02 [3,] 1 2.571150e+00 6.610815e+00 1.699740e+01 4.370287e+01 1.123667e+02 [4,] 1 1.368081e+00 1.871644e+00 2.560560e+00 3.503053e+00 4.792459e+00 [5,] 1 2.449294e-16 5.999039e-32 1.469341e-47 3.598847e-63 8.814633e-79 [6,] 1 -1.368081e+00 1.871644e+00 -2.560560e+00 3.503053e+00 -4.792459e+00 [7,] 1 -2.571150e+00 6.610815e+00 -1.699740e+01 4.370287e+01 -1.123667e+02 [8,] 1 -3.464102e+00 1.200000e+01 -4.156922e+01 1.440000e+02 -4.988306e+02 [9,] 1 -3.939231e+00 1.551754e+01 -6.112718e+01 2.407941e+02 -9.485435e+02 [,7] [,8] [,9] [1,] 3.736532e+03 1.471906e+04 5.798179e+04 [2,] 1.728000e+03 5.985968e+03 2.073600e+04 [3,] 2.889116e+02 7.428351e+02 1.909941e+03 [4,] 6.556470e+00 8.969779e+00 1.227138e+01 [5,] 2.158962e-94 5.287933e-110 1.295170e-125 [6,] 6.556470e+00 -8.969779e+00 1.227138e+01 [7,] 2.889116e+02 -7.428351e+02 1.909941e+03 [8,] 1.728000e+03 -5.985968e+03 2.073600e+04 [9,] 3.736532e+03 -1.471906e+04 5.798179e+04 x s [1,] -4.10000000 0.05614823 0.24664314 [2,] -4.01717172 0.05835099 0.12832355 [3,] -3.93434343 0.06068308 0.05744936 [4,] -3.85151515 0.06315456 0.02194112 [5,] -3.76868687 0.06577649 0.01180474 [6,] -3.68585859 0.06856099 0.01888816 [7,] -3.60303030 0.07152141 0.03665668 [8,] -3.52020202 0.07467244 0.05998620 [9,] -3.43737374 0.07803028 0.08497354 [10,] -3.35454545 0.08161283 0.10876282 [11,] -3.27171717 0.08543990 0.12938732 [12,] -3.18888889 0.08953343 0.14562578 [13,] -3.10606061 0.09391777 0.15687260 [14,] -3.02323232 0.09861997 0.16302099 [15,] -2.94040404 0.10367017 0.16435840 [16,] -2.85757576 0.10910195 0.16147351 [17,] -2.77474747 0.11495279 0.15517411 [18,] -2.69191919 0.12126461 0.14641509 [19,] -2.60909091 0.12808435 0.13623594 [20,] -2.52626263 0.13546459 0.12570712 [21,] -2.44343434 0.14346438 0.11588467 [22,] -2.36060606 0.15215001 0.10777236 [23,] -2.27777778 0.16159601 0.10229095 [24,] -2.19494949 0.17188618 0.10025385 [25,] -2.11212121 0.18311477 0.10234871 [26,] -2.02929293 0.19538769 0.10912436 [27,] -1.94646465 0.20882387 0.12098261 [28,] -1.86363636 0.22355658 0.13817440 [29,] -1.78080808 0.23973471 0.16079987 [30,] -1.69797980 0.25752386 0.18881182 [31,] -1.61515152 0.27710699 0.22202217 [32,] -1.53232323 0.29868449 0.26011098 [33,] -1.44949495 0.32247292 0.30263769 [34,] -1.36666667 0.34870205 0.34905406 [35,] -1.28383838 0.37760914 0.39871865 [36,] -1.20101010 0.40942911 0.45091228 [37,] -1.11818182 0.44437915 0.50485434 [38,] -1.03535354 0.48263551 0.55971948 [39,] -0.95252525 0.52430017 0.61465456 [40,] -0.86969697 0.56935520 0.66879540 [41,] -0.78686869 0.61760330 0.72128329 [42,] -0.70404040 0.66859539 0.77128089 [43,] -0.62121212 0.72155044 0.81798732 [44,] -0.53838384 0.77527965 0.86065231 [45,] -0.45555556 0.82813618 0.89858924 [46,] -0.37272727 0.87802046 0.93118681 [47,] -0.28989899 0.92247397 0.95791938 [48,] -0.20707071 0.95888468 0.97835567 [49,] -0.12424242 0.98479847 0.99216590 [50,] -0.04141414 0.99828781 0.99912720 [51,] 0.04141414 0.99828781 0.99912720 [52,] 0.12424242 0.98479847 0.99216590 [53,] 0.20707071 0.95888468 0.97835567 [54,] 0.28989899 0.92247397 0.95791938 [55,] 0.37272727 0.87802046 0.93118681 [56,] 0.45555556 0.82813618 0.89858924 [57,] 0.53838384 0.77527965 0.86065231 [58,] 0.62121212 0.72155044 0.81798732 [59,] 0.70404040 0.66859539 0.77128089 [60,] 0.78686869 0.61760330 0.72128329 [61,] 0.86969697 0.56935520 0.66879540 [62,] 0.95252525 0.52430017 0.61465456 [63,] 1.03535354 0.48263551 0.55971948 [64,] 1.11818182 0.44437915 0.50485434 [65,] 1.20101010 0.40942911 0.45091228 [66,] 1.28383838 0.37760914 0.39871865 [67,] 1.36666667 0.34870205 0.34905406 [68,] 1.44949495 0.32247292 0.30263769 [69,] 1.53232323 0.29868449 0.26011098 [70,] 1.61515152 0.27710699 0.22202217 [71,] 1.69797980 0.25752386 0.18881182 [72,] 1.78080808 0.23973471 0.16079987 [73,] 1.86363636 0.22355658 0.13817440 [74,] 1.94646465 0.20882387 0.12098261 [75,] 2.02929293 0.19538769 0.10912436 [76,] 2.11212121 0.18311477 0.10234871 [77,] 2.19494949 0.17188618 0.10025385 [78,] 2.27777778 0.16159601 0.10229095 [79,] 2.36060606 0.15215001 0.10777236 [80,] 2.44343434 0.14346438 0.11588467 [81,] 2.52626263 0.13546459 0.12570712 [82,] 2.60909091 0.12808435 0.13623594 [83,] 2.69191919 0.12126461 0.14641509 [84,] 2.77474747 0.11495279 0.15517411 [85,] 2.85757576 0.10910195 0.16147351 [86,] 2.94040404 0.10367017 0.16435840 [87,] 3.02323232 0.09861997 0.16302099 [88,] 3.10606061 0.09391777 0.15687260 [89,] 3.18888889 0.08953343 0.14562578 [90,] 3.27171717 0.08543990 0.12938732 [91,] 3.35454545 0.08161283 0.10876282 [92,] 3.43737374 0.07803028 0.08497354 [93,] 3.52020202 0.07467244 0.05998620 [94,] 3.60303030 0.07152141 0.03665668 [95,] 3.68585859 0.06856099 0.01888816 [96,] 3.76868687 0.06577649 0.01180474 [97,] 3.85151515 0.06315456 0.02194112 [98,] 3.93434343 0.06068308 0.05744936 [99,] 4.01717172 0.05835099 0.12832355 [100,] 4.10000000 0.05614823 0.24664314 > rm(list=ls()) > q()