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Type 'q()' to quit R. > # runge1.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*(2*(1:n)-n-1)/(n-1) # equally spaced interpolation pts + 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 7.1 + } [,1] [,2] [,3] [,4] [,5] [1,] 1 -4 16 -64 256 [2,] 1 -2 4 -8 16 [3,] 1 0 0 0 0 [4,] 1 2 4 8 16 [5,] 1 4 16 64 256 x s [1,] -4.10000000 0.05614823 0.171365882 [2,] -4.01717172 0.05835099 0.076862439 [3,] -3.93434343 0.06068308 -0.005399751 [4,] -3.85151515 0.06315456 -0.076058581 [5,] -3.76868687 0.06577649 -0.135738655 [6,] -3.68585859 0.06856099 -0.185051288 [7,] -3.60303030 0.07152141 -0.224594505 [8,] -3.52020202 0.07467244 -0.254953041 [9,] -3.43737374 0.07803028 -0.276698344 [10,] -3.35454545 0.08161283 -0.290388569 [11,] -3.27171717 0.08543990 -0.296568584 [12,] -3.18888889 0.08953343 -0.295769967 [13,] -3.10606061 0.09391777 -0.288511006 [14,] -3.02323232 0.09861997 -0.275296701 [15,] -2.94040404 0.10367017 -0.256618761 [16,] -2.85757576 0.10910195 -0.232955607 [17,] -2.77474747 0.11495279 -0.204772368 [18,] -2.69191919 0.12126461 -0.172520886 [19,] -2.60909091 0.12808435 -0.136639714 [20,] -2.52626263 0.13546459 -0.097554112 [21,] -2.44343434 0.14346438 -0.055676055 [22,] -2.36060606 0.15215001 -0.011404225 [23,] -2.27777778 0.16159601 0.034875983 [24,] -2.19494949 0.17188618 0.082792465 [25,] -2.11212121 0.18311477 0.131986405 [26,] -2.02929293 0.19538769 0.182112279 [27,] -1.94646465 0.20882387 0.232837850 [28,] -1.86363636 0.22355658 0.283844170 [29,] -1.78080808 0.23973471 0.334825584 [30,] -1.69797980 0.25752386 0.385489723 [31,] -1.61515152 0.27710699 0.435557508 [32,] -1.53232323 0.29868449 0.484763150 [33,] -1.44949495 0.32247292 0.532854151 [34,] -1.36666667 0.34870205 0.579591300 [35,] -1.28383838 0.37760914 0.624748676 [36,] -1.20101010 0.40942911 0.668113648 [37,] -1.11818182 0.44437915 0.709486875 [38,] -1.03535354 0.48263551 0.748682304 [39,] -0.95252525 0.52430017 0.785527173 [40,] -0.86969697 0.56935520 0.819862008 [41,] -0.78686869 0.61760330 0.851540624 [42,] -0.70404040 0.66859539 0.880430128 [43,] -0.62121212 0.72155044 0.906410915 [44,] -0.53838384 0.77527965 0.929376669 [45,] -0.45555556 0.82813618 0.949234364 [46,] -0.37272727 0.87802046 0.965904262 [47,] -0.28989899 0.92247397 0.979319918 [48,] -0.20707071 0.95888468 0.989428173 [49,] -0.12424242 0.98479847 0.996189159 [50,] -0.04141414 0.99828781 0.999576296 [51,] 0.04141414 0.99828781 0.999576296 [52,] 0.12424242 0.98479847 0.996189159 [53,] 0.20707071 0.95888468 0.989428173 [54,] 0.28989899 0.92247397 0.979319918 [55,] 0.37272727 0.87802046 0.965904262 [56,] 0.45555556 0.82813618 0.949234364 [57,] 0.53838384 0.77527965 0.929376669 [58,] 0.62121212 0.72155044 0.906410915 [59,] 0.70404040 0.66859539 0.880430128 [60,] 0.78686869 0.61760330 0.851540624 [61,] 0.86969697 0.56935520 0.819862008 [62,] 0.95252525 0.52430017 0.785527173 [63,] 1.03535354 0.48263551 0.748682304 [64,] 1.11818182 0.44437915 0.709486875 [65,] 1.20101010 0.40942911 0.668113648 [66,] 1.28383838 0.37760914 0.624748676 [67,] 1.36666667 0.34870205 0.579591300 [68,] 1.44949495 0.32247292 0.532854151 [69,] 1.53232323 0.29868449 0.484763150 [70,] 1.61515152 0.27710699 0.435557508 [71,] 1.69797980 0.25752386 0.385489723 [72,] 1.78080808 0.23973471 0.334825584 [73,] 1.86363636 0.22355658 0.283844170 [74,] 1.94646465 0.20882387 0.232837850 [75,] 2.02929293 0.19538769 0.182112279 [76,] 2.11212121 0.18311477 0.131986405 [77,] 2.19494949 0.17188618 0.082792465 [78,] 2.27777778 0.16159601 0.034875983 [79,] 2.36060606 0.15215001 -0.011404225 [80,] 2.44343434 0.14346438 -0.055676055 [81,] 2.52626263 0.13546459 -0.097554112 [82,] 2.60909091 0.12808435 -0.136639714 [83,] 2.69191919 0.12126461 -0.172520886 [84,] 2.77474747 0.11495279 -0.204772368 [85,] 2.85757576 0.10910195 -0.232955607 [86,] 2.94040404 0.10367017 -0.256618761 [87,] 3.02323232 0.09861997 -0.275296701 [88,] 3.10606061 0.09391777 -0.288511006 [89,] 3.18888889 0.08953343 -0.295769967 [90,] 3.27171717 0.08543990 -0.296568584 [91,] 3.35454545 0.08161283 -0.290388569 [92,] 3.43737374 0.07803028 -0.276698344 [93,] 3.52020202 0.07467244 -0.254953041 [94,] 3.60303030 0.07152141 -0.224594505 [95,] 3.68585859 0.06856099 -0.185051288 [96,] 3.76868687 0.06577649 -0.135738655 [97,] 3.85151515 0.06315456 -0.076058581 [98,] 3.93434343 0.06068308 -0.005399751 [99,] 4.01717172 0.05835099 0.076862439 [100,] 4.10000000 0.05614823 0.171365882 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 1 -4.000000 16.000000 -64.000000 256.000000 -1024.000000 4096.000000 [2,] 1 -2.666667 7.111111 -18.962963 50.567901 -134.847737 359.593964 [3,] 1 -1.333333 1.777778 -2.370370 3.160494 -4.213992 5.618656 [4,] 1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 [5,] 1 1.333333 1.777778 2.370370 3.160494 4.213992 5.618656 [6,] 1 2.666667 7.111111 18.962963 50.567901 134.847737 359.593964 [7,] 1 4.000000 16.000000 64.000000 256.000000 1024.000000 4096.000000 x s [1,] -4.10000000 0.05614823 -0.234861300 [2,] -4.01717172 0.05835099 0.014480706 [3,] -3.93434343 0.06068308 0.207155081 [4,] -3.85151515 0.06315456 0.350688578 [5,] -3.76868687 0.06577649 0.451999443 [6,] -3.68585859 0.06856099 0.517425950 [7,] -3.60303030 0.07152141 0.552754311 [8,] -3.52020202 0.07467244 0.563246003 [9,] -3.43737374 0.07803028 0.553664465 [10,] -3.35454545 0.08161283 0.528301205 [11,] -3.27171717 0.08543990 0.491001288 [12,] -3.18888889 0.08953343 0.445188228 [13,] -3.10606061 0.09391777 0.393888263 [14,] -3.02323232 0.09861997 0.339754030 [15,] -2.94040404 0.10367017 0.285087628 [16,] -2.85757576 0.10910195 0.231863080 [17,] -2.77474747 0.11495279 0.181748180 [18,] -2.69191919 0.12126461 0.136125741 [19,] -2.60909091 0.12808435 0.096114231 [20,] -2.52626263 0.13546459 0.062587804 [21,] -2.44343434 0.14346438 0.036195722 [22,] -2.36060606 0.15215001 0.017381174 [23,] -2.27777778 0.16159601 0.006399484 [24,] -2.19494949 0.17188618 0.003335713 [25,] -2.11212121 0.18311477 0.008121657 [26,] -2.02929293 0.19538769 0.020552232 [27,] -1.94646465 0.20882387 0.040301260 [28,] -1.86363636 0.22355658 0.066936638 [29,] -1.78080808 0.23973471 0.099934913 [30,] -1.69797980 0.25752386 0.138695234 [31,] -1.61515152 0.27710699 0.182552713 [32,] -1.53232323 0.29868449 0.230791169 [33,] -1.44949495 0.32247292 0.282655265 [34,] -1.36666667 0.34870205 0.337362045 [35,] -1.28383838 0.37760914 0.394111859 [36,] -1.20101010 0.40942911 0.452098678 [37,] -1.11818182 0.44437915 0.510519810 [38,] -1.03535354 0.48263551 0.568585004 [39,] -0.95252525 0.52430017 0.625524945 [40,] -0.86969697 0.56935520 0.680599147 [41,] -0.78686869 0.61760330 0.733103239 [42,] -0.70404040 0.66859539 0.782375637 [43,] -0.62121212 0.72155044 0.827803616 [44,] -0.53838384 0.77527965 0.868828776 [45,] -0.45555556 0.82813618 0.904951892 [46,] -0.37272727 0.87802046 0.935737167 [47,] -0.28989899 0.92247397 0.960815873 [48,] -0.20707071 0.95888468 0.979889385 [49,] -0.12424242 0.98479847 0.992731612 [50,] -0.04141414 0.99828781 0.999190812 [51,] 0.04141414 0.99828781 0.999190812 [52,] 0.12424242 0.98479847 0.992731612 [53,] 0.20707071 0.95888468 0.979889385 [54,] 0.28989899 0.92247397 0.960815873 [55,] 0.37272727 0.87802046 0.935737167 [56,] 0.45555556 0.82813618 0.904951892 [57,] 0.53838384 0.77527965 0.868828776 [58,] 0.62121212 0.72155044 0.827803616 [59,] 0.70404040 0.66859539 0.782375637 [60,] 0.78686869 0.61760330 0.733103239 [61,] 0.86969697 0.56935520 0.680599147 [62,] 0.95252525 0.52430017 0.625524945 [63,] 1.03535354 0.48263551 0.568585004 [64,] 1.11818182 0.44437915 0.510519810 [65,] 1.20101010 0.40942911 0.452098678 [66,] 1.28383838 0.37760914 0.394111859 [67,] 1.36666667 0.34870205 0.337362045 [68,] 1.44949495 0.32247292 0.282655265 [69,] 1.53232323 0.29868449 0.230791169 [70,] 1.61515152 0.27710699 0.182552713 [71,] 1.69797980 0.25752386 0.138695234 [72,] 1.78080808 0.23973471 0.099934913 [73,] 1.86363636 0.22355658 0.066936638 [74,] 1.94646465 0.20882387 0.040301260 [75,] 2.02929293 0.19538769 0.020552232 [76,] 2.11212121 0.18311477 0.008121657 [77,] 2.19494949 0.17188618 0.003335713 [78,] 2.27777778 0.16159601 0.006399484 [79,] 2.36060606 0.15215001 0.017381174 [80,] 2.44343434 0.14346438 0.036195722 [81,] 2.52626263 0.13546459 0.062587804 [82,] 2.60909091 0.12808435 0.096114231 [83,] 2.69191919 0.12126461 0.136125741 [84,] 2.77474747 0.11495279 0.181748180 [85,] 2.85757576 0.10910195 0.231863080 [86,] 2.94040404 0.10367017 0.285087628 [87,] 3.02323232 0.09861997 0.339754030 [88,] 3.10606061 0.09391777 0.393888263 [89,] 3.18888889 0.08953343 0.445188228 [90,] 3.27171717 0.08543990 0.491001288 [91,] 3.35454545 0.08161283 0.528301205 [92,] 3.43737374 0.07803028 0.553664465 [93,] 3.52020202 0.07467244 0.563246003 [94,] 3.60303030 0.07152141 0.552754311 [95,] 3.68585859 0.06856099 0.517425950 [96,] 3.76868687 0.06577649 0.451999443 [97,] 3.85151515 0.06315456 0.350688578 [98,] 3.93434343 0.06068308 0.207155081 [99,] 4.01717172 0.05835099 0.014480706 [100,] 4.10000000 0.05614823 -0.234861300 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [1,] 1 -4 16 -64 256 -1024 4096 -16384 65536 [2,] 1 -3 9 -27 81 -243 729 -2187 6561 [3,] 1 -2 4 -8 16 -32 64 -128 256 [4,] 1 -1 1 -1 1 -1 1 -1 1 [5,] 1 0 0 0 0 0 0 0 0 [6,] 1 1 1 1 1 1 1 1 1 [7,] 1 2 4 8 16 32 64 128 256 [8,] 1 3 9 27 81 243 729 2187 6561 [9,] 1 4 16 64 256 1024 4096 16384 65536 x s [1,] -4.10000000 0.05614823 0.76747954 [2,] -4.01717172 0.05835099 0.15835691 [3,] -3.93434343 0.06068308 -0.24928062 [4,] -3.85151515 0.06315456 -0.49864620 [5,] -3.76868687 0.06577649 -0.62637989 [6,] -3.68585859 0.06856099 -0.66325917 [7,] -3.60303030 0.07152141 -0.63485671 [8,] -3.52020202 0.07467244 -0.56214811 [9,] -3.43737374 0.07803028 -0.46207174 [10,] -3.35454545 0.08161283 -0.34804319 [11,] -3.27171717 0.08543990 -0.23042641 [12,] -3.18888889 0.08953343 -0.11696393 [13,] -3.10606061 0.09391777 -0.01316806 [14,] -3.02323232 0.09861997 0.07732447 [15,] -2.94040404 0.10367017 0.15243275 [16,] -2.85757576 0.10910195 0.21134458 [17,] -2.77474747 0.11495279 0.25426011 [18,] -2.69191919 0.12126461 0.28216442 [19,] -2.60909091 0.12808435 0.29662709 [20,] -2.52626263 0.13546459 0.29962706 [21,] -2.44343434 0.14346438 0.29340099 [22,] -2.36060606 0.15215001 0.28031347 [23,] -2.27777778 0.16159601 0.26274744 [24,] -2.19494949 0.17188618 0.24301328 [25,] -2.11212121 0.18311477 0.22327492 [26,] -2.02929293 0.19538769 0.20549168 [27,] -1.94646465 0.20882387 0.19137433 [28,] -1.86363636 0.22355658 0.18235393 [29,] -1.78080808 0.23973471 0.17956229 [30,] -1.69797980 0.25752386 0.18382266 [31,] -1.61515152 0.27710699 0.19564954 [32,] -1.53232323 0.29868449 0.21525638 [33,] -1.44949495 0.32247292 0.24257007 [34,] -1.36666667 0.34870205 0.27725125 [35,] -1.28383838 0.37760914 0.31871935 [36,] -1.20101010 0.40942911 0.36618143 [37,] -1.11818182 0.44437915 0.41866398 [38,] -1.03535354 0.48263551 0.47504682 [39,] -0.95252525 0.52430017 0.53409824 [40,] -0.86969697 0.56935520 0.59451074 [41,] -0.78686869 0.61760330 0.65493664 [42,] -0.70404040 0.66859539 0.71402295 [43,] -0.62121212 0.72155044 0.77044490 [44,] -0.53838384 0.77527965 0.82293758 [45,] -0.45555556 0.82813618 0.87032534 [46,] -0.37272727 0.87802046 0.91154833 [47,] -0.28989899 0.92247397 0.94568602 [48,] -0.20707071 0.95888468 0.97197723 [49,] -0.12424242 0.98479847 0.98983650 [50,] -0.04141414 0.99828781 0.99886652 [51,] 0.04141414 0.99828781 0.99886652 [52,] 0.12424242 0.98479847 0.98983650 [53,] 0.20707071 0.95888468 0.97197723 [54,] 0.28989899 0.92247397 0.94568602 [55,] 0.37272727 0.87802046 0.91154833 [56,] 0.45555556 0.82813618 0.87032534 [57,] 0.53838384 0.77527965 0.82293758 [58,] 0.62121212 0.72155044 0.77044490 [59,] 0.70404040 0.66859539 0.71402295 [60,] 0.78686869 0.61760330 0.65493664 [61,] 0.86969697 0.56935520 0.59451074 [62,] 0.95252525 0.52430017 0.53409824 [63,] 1.03535354 0.48263551 0.47504682 [64,] 1.11818182 0.44437915 0.41866398 [65,] 1.20101010 0.40942911 0.36618143 [66,] 1.28383838 0.37760914 0.31871935 [67,] 1.36666667 0.34870205 0.27725125 [68,] 1.44949495 0.32247292 0.24257007 [69,] 1.53232323 0.29868449 0.21525638 [70,] 1.61515152 0.27710699 0.19564954 [71,] 1.69797980 0.25752386 0.18382266 [72,] 1.78080808 0.23973471 0.17956229 [73,] 1.86363636 0.22355658 0.18235393 [74,] 1.94646465 0.20882387 0.19137433 [75,] 2.02929293 0.19538769 0.20549168 [76,] 2.11212121 0.18311477 0.22327492 [77,] 2.19494949 0.17188618 0.24301328 [78,] 2.27777778 0.16159601 0.26274744 [79,] 2.36060606 0.15215001 0.28031347 [80,] 2.44343434 0.14346438 0.29340099 [81,] 2.52626263 0.13546459 0.29962706 [82,] 2.60909091 0.12808435 0.29662709 [83,] 2.69191919 0.12126461 0.28216442 [84,] 2.77474747 0.11495279 0.25426011 [85,] 2.85757576 0.10910195 0.21134458 [86,] 2.94040404 0.10367017 0.15243275 [87,] 3.02323232 0.09861997 0.07732447 [88,] 3.10606061 0.09391777 -0.01316806 [89,] 3.18888889 0.08953343 -0.11696393 [90,] 3.27171717 0.08543990 -0.23042641 [91,] 3.35454545 0.08161283 -0.34804319 [92,] 3.43737374 0.07803028 -0.46207174 [93,] 3.52020202 0.07467244 -0.56214811 [94,] 3.60303030 0.07152141 -0.63485671 [95,] 3.68585859 0.06856099 -0.66325917 [96,] 3.76868687 0.06577649 -0.62637989 [97,] 3.85151515 0.06315456 -0.49864620 [98,] 3.93434343 0.06068308 -0.24928062 [99,] 4.01717172 0.05835099 0.15835691 [100,] 4.10000000 0.05614823 0.76747954 > rm(list=ls()) > q()