generation-diverse-questions-interpolation-exam.ipynb

{"metadata":{"kernelspec":{"name":"ir","display_name":"R","language":"R"},"language_info":{"name":"R","codemirror_mode":"r","pygments_lexer":"r","mimetype":"text/x-r-source","file_extension":".r","version":"4.0.5"},"kaggle":{"accelerator":"none","dataSources":[],"dockerImageVersionId":30618,"isInternetEnabled":true,"language":"r","sourceType":"notebook","isGpuEnabled":false}},"nbformat_minor":4,"nbformat":4,"cells":[{"cell_type":"markdown","source":"**This Notebook shows how to generate divers exam about a topic of interpolation by use five methods, namely Newton Forward, Newton Backward, Newton for divided differences, Gauss Forward, Gauss Backward**","metadata":{}},{"cell_type":"code","source":"# define the switch function\nmm<- function(A, type){\nswitch(type,\npolf=polyf(A[1,],A[2,]),\npolb=polyb(A[1,],A[2,]),\npold=polyd(A[1,],A[2,]),\ngaussf=gaussf(A[1,],A[2,]),\ngaussb=gaussb(A[1,],A[2,]))\n}","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:27:10.740431Z","iopub.execute_input":"2024-07-07T13:27:10.743757Z","iopub.status.idle":"2024-07-07T13:27:10.894906Z"},"trusted":true},"execution_count":1,"outputs":[]},{"cell_type":"markdown","source":"**function for interpolation of points by Gauss Forward Method**","metadata":{}},{"cell_type":"code","source":"gaussf<-function(xi,yi){\nli<-c()\nL<-length(xi)\nx0<-xi[3]\nh<-xi[2]-xi[1]\nx<-xi[L]\nk<-1\nyy<-yi\nd<-rep(0,5)\na<-rep(0,5)\ndd<-c()\nd[k]<-yy[3]\nwhile (length(yy)>1){\nfor (i in 1:(length(yy)-1)){\ndd<-cbind(dd,yy[i+1]-yy[i])\n}\nli<-c(li,dd)\nk<-k+1\nif (k==2){\nr=3}else if (k==3){\nr=2\n}else if (k==4){\nr=2}else if (k==5){\nr=1}\nd[k]<-dd[r]\nyy<-dd\ndd<-c()\n}\nvalu<-c(li[1],li[2],li[3],li[4],li[5],li[6],li[7],li[8],li[9],li[10])\nv<-d\na[1]=v[5]/(24*h^4)\na[2]=(v[4]/(6*h^3))-((4*v[5])/(24*h^4))*x0-((2*v[5])/(24*h^4))*h\na[3]=(v[3]/(2*h^2))-((3*v[4])/(6*h^3))*x0-((v[5])/(24*h^4))*h^2+((6*v[5])/(24*h^4))*x0*h+((6*v[5])/(24*h^4))*x0^2\na[4]=(v[2]/h)-(2*v[3]/(2*h^2))*x0-(v[3]/(2*h^2))*h+((3*v[4])/(6*h^3))*x0^2-((v[4])/(6*h^3))*h^2-(3*v[5]/(24*h^4))*x0^3-((v[5])/(24*h^4))*x0^3+((v[5])/(24*h^4))*x0*h^2-((6*v[5])/(24*h^4))*h*x0^2+((v[5])/(24*h^4))*x0*h^2+((2*v[5])/(24*h^4))*h^3\na[5]=v[1]-(v[2]/h)*x0+(v[3]/(2*h^2))*x0^2+(v[3]/(2*h^2))*h*x0-(v[4]/(6*h^3))*x0^3+(v[4]/(6*h^3))*x0*h^2-(v[5]/(24*h^4))*h^2*x0^2+(2*v[5]/(24*h^4))*h*x0^3-(2*v[5]/(24*h^4))*h^3*x0+(v[5]/(24*h^4))*x0^4\npx<-a[1]*x^4+a[2]*x^3+a[3]*x^2+a[4]*x+a[5]\nlis<-list(d=valu,a=a,px=px)\nreturn(lis)}","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:27:18.814678Z","iopub.execute_input":"2024-07-07T13:27:18.857012Z","iopub.status.idle":"2024-07-07T13:27:18.875705Z"},"trusted":true},"execution_count":2,"outputs":[]},{"cell_type":"markdown","source":"**function for interpolation of points by Gauss Backward Method**","metadata":{}},{"cell_type":"code","source":"gaussb<-function(xi,yi){\nli<-c()\nL<-length(xi)\nx0<-xi[3]\nh<-xi[2]-xi[1]\nx<-xi[L]\nk<-1\nyy<-yi\nd<-rep(0,5)\na<-rep(0,5)\ndd<-c()\nd[k]<-yy[3]\nwhile (length(yy)>1){\nfor (i in 1:(length(yy)-1)){\ndd<-cbind(dd,yy[i+1]-yy[i])\n}\nli<-c(li,dd)\nk<-k+1\nif (k==2){\nr=2}else if (k==3){\nr=2\n}else if (k==4){\nr=1}else if (k==5){\nr=1}\nd[k]<-dd[r]\nyy<-dd\ndd<-c()\n}\nvalu<-c(li[1],li[2],li[3],li[4],li[5],li[6],li[7],li[8],li[9],li[10])\nv<-d\na[1]=v[5]/(24*h^4);\na[2]=(v[4]/(6*h^3))-((4*v[5])/(24*h^4))*x0+((2*v[5])/(24*h^4))*h\na[3]=(v[3]/(2*h^2))-((3*v[4])/(6*h^3))*x0-((v[5])/(24*h^4))*h^2-((6*v[5])/(24*h^4))*x0*h+((6*v[5])/(24*h^4))*x0^2\na[4]=(v[2]/h)-(2*v[3]/(2*h^2))*x0+(v[3]/(2*h^2))*h+((3*v[4])/(6*h^3))*x0^2-((v[4])/(6*h^3))*h^2-(3*v[5]/(24*h^4))*x0^3-((v[5])/(24*h^4))*x0^3+((v[5])/(24*h^4))*x0*h^2+((6*v[5])/(24*h^4))*h*x0^2+((v[5])/(24*h^4))*x0*h^2-((2*v[5])/(24*h^4))*h^3\na[5]=v[1]-(v[2]/h)*x0+(v[3]/(2*h^2))*x0^2-(v[3]/(2*h^2))*h*x0-(v[4]/(6*h^3))*x0^3+(v[4]/(6*h^3))*x0*h^2-(v[5]/(24*h^4))*h^2*x0^2-(2*v[5]/(24*h^4))*h*x0^3+(2*v[5]/(24*h^4))*h^3*x0+(v[5]/(24*h^4))*x0^4\npx<-a[1]*x^4+a[2]*x^3+a[3]*x^2+a[4]*x+a[5]\nlis<-list(d=valu,a=a,px=px)\nreturn(lis)}","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:27:25.077080Z","iopub.execute_input":"2024-07-07T13:27:25.079417Z","iopub.status.idle":"2024-07-07T13:27:25.104926Z"},"trusted":true},"execution_count":3,"outputs":[]},{"cell_type":"markdown","source":"**function for interpolation of points by Newton Forward Method**","metadata":{}},{"cell_type":"code","source":"polyf<-function(xi,yi){\nli<-c()\nL<-length(xi)\nx0<-xi[1]\nh<-xi[2]-xi[1]\nx<-xi[L]\nk<-1\nyy<-yi\nd<-rep(0,5)\na<-rep(0,5)\ndd<-c()\nd[k]<-yy[1]\nwhile (length(yy)>1){\nfor (i in 1:(length(yy)-1)){\ndd<-cbind(dd,yy[i+1]-yy[i])\n}\nli<-c(li,dd)\nk<-k+1\nd[k]<-dd[1]\nyy<-dd\ndd<-c()\n}\nvalu<-c(li[1],li[2],li[3],li[4],li[5],li[6],li[7],li[8],li[9],li[10])\nv<-d\na[1]=v[5]/(24*h^4)\na[2]=(v[4]/(6*h^3))-((4*v[5])/(24*h^4))*x0-((6*v[5])/(24*h^4))*h\na[3]=(v[3]/(2*h^2))-((3*v[4])/(6*h^3))*x0-((3*h*v[4])/(6*h^3))+((6*v[5])/(24*h^4))*x0^2+((18*v[5])/(24*h^4))*x0*h+((11*v[5])/(24*h^4))*h^2\na[4]=(v[2]/h)-(v[3]/(h^2))*x0-(v[3]/(2*h^2))*h+((2*v[4])/(6*h^3))*x0^2+((6*v[4])/(6*h^3))*h*x0+(v[4]/(6*h^3))*x0^2+((2*v[4])/(6*h^3))*h^2-((22*v[5])/(24*h^4))*x0*h^2-((18*v[5])/(24*h^4))*h*x0^2-((4*v[5])/(24*h^4))*x0^3-((6*v[5])/(24*h^4))*h^3\na[5]=v[1]-(v[2]/h)*x0+(v[3]/(2*h^2))*x0^2+(v[3]/(2*h^2))*h*x0+((v[4]/(6*h^3))*(-3*h*x0^2-2*h^2*x0-x0^3))+((v[5]/(24*h^4))*(11*x0^2*h^2+6*h*x0^3+6*x0*h^3+x0^4))\npx<-a[1]*x^4+a[2]*x^3+a[3]*x^2+a[4]*x+a[5]\nlis<-list(d=valu,a=a,px=px)\nreturn(lis)}","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:27:50.715676Z","iopub.execute_input":"2024-07-07T13:27:50.717553Z","iopub.status.idle":"2024-07-07T13:27:50.732970Z"},"trusted":true},"execution_count":4,"outputs":[]},{"cell_type":"markdown","source":"**function for interpolation of points by Newton Backward Method**","metadata":{}},{"cell_type":"code","source":"polyb<-function(xi,yi){\nli<-c()\nL<-length(xi)\nx0<-xi[L]\nh<-xi[2]-xi[1]\nx<-xi[L]\nk<-1\nyy<-yi\nd<-rep(0,5)\na<-rep(0,5)\ndd<-c()\nd[k]<-yy[L]\nwhile (length(yy)>1){\nfor (i in 1:(length(yy)-1)){\ndd<-cbind(dd,yy[i+1]-yy[i])\n}\nli<-c(li,dd)\nk<-k+1\nLk<-length(dd)\nd[k]<-dd[Lk]\nyy<-dd\ndd<-c()\n}\nvalu<-c(li[1],li[2],li[3],li[4],li[5],li[6],li[7],li[8],li[9],li[10])\nv<-d\na[1]=v[5]/(24*h^4)\na[2]=((v[4]/(6*h^3))-((4*v[5])/(24*h^4))*x0+((6*v[5])/(24*h^4))*h)\na[3]=(v[3]/(2*h^2))-((3*v[4])/(6*h^3))*x0+((3*v[4])/(6*h^3))*h+((6*v[5])/(24*h^4))*x0^2-((18*v[5])/(24*h^4))*x0*h+((11*v[5])/(24*h^4))*h^2\na[4]=(v[2]/h)-(v[3]/h^2)*x0+(v[3]/(2*h^2))*h+((2*v[4])/(6*h^3))*x0^2-((6*v[4])/(6*h^3))*h*x0+(v[4]/(6*h^3))*x0^2+((2*v[4])/(6*h^3))*h^2-((22*v[5])/(24*h^4))*x0*h^2+((18*v[5])/(24*h^4))*h*x0^2-((4*v[5])/(24*h^4))*x0^3+((6*v[5])/(24*h^4))*h^3\na[5]=v[1]-(v[2]/h)*x0+(v[3]/(2*h^2))*x0^2-(v[3]/(2*h^2))*h*x0+((v[4]/(6*h^3))*(+3*h*x0^2-2*h^2*x0-x0^3))+((v[5]/(24*h^4))*(11*x0^2*h^2-6*h*x0^3-6*x0*h^3+x0^4));\npx<-a[1]*x^4+a[2]*x^3+a[3]*x^2+a[4]*x+a[5]\nlis<-list(d=valu,a=a,px=px)\nreturn(lis)}","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:28:01.524543Z","iopub.execute_input":"2024-07-07T13:28:01.526392Z","iopub.status.idle":"2024-07-07T13:28:01.542682Z"},"trusted":true},"execution_count":5,"outputs":[]},{"cell_type":"markdown","source":"**function for interpolation of points by Newton for divided differences Method**","metadata":{}},{"cell_type":"code","source":"polyd<-function(xi,yi){\nli<-c()\nL<-length(xi)\ny<-xi\nx<-xi[5]\nk<-1\nd<-rep(0,5)\na<-rep(0,5)\ndd<-c()\nyy<-yi\nd[k]<-yy[1]\nwhile (length(yy)>1){\nfor (i in 1:(length(yy)-1)){\ndd<-cbind(dd,(yy[i+1]-yy[i])/(y[i+k]-y[i]))\n}\nli<-c(li,dd)\nk<-k+1\nd[k]<-dd[1]\nyy<-dd\ndd<-c()\n}\nvalu<-c(li[1],li[2],li[3],li[4],li[5],li[6],li[7],li[8],li[9],li[10])\nv<-d\na[1]<-v[5]\na[2]<-v[4]-(v[5]*y[1])-(v[5]*y[2])-(v[5]*y[3])-(v[5]*y[4])\na[3]<-v[3]-(v[4]*y[1])-(v[4]*y[2])-(v[4]*y[3])+(v[5]*y[2]*y[1])+(v[5]*y[3]*y[1])+(v[5]*y[3]*y[2])+(v[5]*y[4]*y[1])+(v[5]*y[4]*y[2])+(v[5]*y[4]*y[3])\na[4]<-v[2]-(v[3]*y[1])-(v[3]*y[2])+(v[4]*y[2]*y[1])+(v[4]*y[3]*y[1])+(v[4]*y[3]*y[2])-(v[5]*y[3]*y[2]*y[1])-(v[5]*y[4]*y[2]*y[1])-(v[5]*y[4]*y[3]*y[1])-(v[5]*y[4]*y[3]*y[2])\na[5]<-v[1]-(v[2]*y[1])+(v[3]*y[2]*y[1])-(v[4]*y[3]*y[2]*y[1])+(v[5]*y[4]*y[3]*y[2]*y[1])\npx<-a[1]*x^4+a[2]*x^3+a[3]*x^2+a[4]*x+a[5]\nlis<-list(d=valu,a=a,px=px)\nreturn(lis)}\n","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:28:09.310412Z","iopub.execute_input":"2024-07-07T13:28:09.312487Z","iopub.status.idle":"2024-07-07T13:28:09.333504Z"},"trusted":true},"execution_count":6,"outputs":[]},{"cell_type":"markdown","source":"**The code for generate the questions**","metadata":{}},{"cell_type":"code","source":"NN=80 #the number of students \nxiyi<-c()\nyy<-c()\nlis1<-c()\nlis2<-c()\nans<-c()\npoi<-c()\nmessag<-c()\nallmsg<-c()\nk<-0\nj<-1\nfor (i in 1:NN){\nX<-c(2,4,6,8,10)\nvc<--4:5\nvv<-X\nmeth<-sample(c(\"polf\",\"polb\",\"gaussf\",\"gaussb\",\"pold\"),1)\ny1<-sample(vc,5)\npy1<-as.numeric(paste(abs(y1),collapse=\"\"))\nif (!(py1%in%yy) && j<=117){\nLs<-c()\nk<-k+1\nj<-j+1\nvv<-c(vv,y1)\nyy<-c(yy,py1)\nxiyi<-rbind(xiyi,c(k,y1))\nX<-rbind(X,y1)\nw<-mm(X,meth)\nvv<-c(vv,w\\$d)\nvv<-c(vv,w\\$a)\nvv<-as.numeric(vv)\nif (meth==\"polb\"){me=\"polynomial Backward\"\n}else if (meth==\"polf\"){me=\"polynomial forward\"\n}else if (meth==\"gaussf\"){me=\"Gauss forward\"\n}else if (meth==\"pold\"){me=\"Newton for divided differences\"\n}else if (meth==\"gaussb\"){me=\"Gauss backward\"}\np1<-paste(c(\"(\",as.character(vv[1]),\",\",as.character(vv[6]),\")\"),collapse=\"\")\np2<-paste(c(\"(\",as.character(vv[2]),\",\",as.character(vv[7]),\")\"),collapse=\"\")\np3<-paste(c(\"(\",as.character(vv[3]),\",\",as.character(vv[8]),\")\"),collapse=\"\")\np4<-paste(c(\"(\",as.character(vv[4]),\",\",as.character(vv[9]),\")\"),collapse=\"\")\np5<-paste(c(\"(\",as.character(vv[5]),\",\",as.character(vv[10]),\")\"),collapse=\"\")\nmsg<-paste(c(\"\\\\textcolor{red}{\",\nk,\"}\",\",\",\"\\\\hspace*{3mm}\",\n\"\\\\textcolor{blue}{\",\n\"Name:\",\"}\",\"\\\\\\\\ \\n\",\"Use the method of \", me,\" to interpolate the following points.\", \" The points are,\",p1,\",\",p2,\",\",p3,\",\",p4,\",\",p5,\". Then write the interpolated equation in the google form?\\n\",\n\"======================================\",\"\\n\"),collapse=\"\")\nmessag<-cbind(messag,msg)\nallmsg<-paste(c(allmsg,msg),collapse=\" \")\nif (floor(k/7)==k/7){allmsg<-paste(c(allmsg,\"\\\\newpage \\n\"),collapse=\" \")}\nans<-rbind(ans,c(k,vv[c(13,14,16,19,20,21:25)]))\npoi<-rbind(poi,vv[1:10])\nj<-dim(ans)[1]\nLs<-paste(c( \"points are \", p1,\",\",p2,\",\",p3,\",\",p4,\",\",p5, \"by\", me,\"\\n\",\"\\n\"),collapse=\" \")\nlis1<-paste(c(lis1,Ls),collapse=\" \")\n}\n}\nfinalmsg<- \"\\\\documentclass[16pt,a4paper]{report}\\n\n\\\\usepackage{listings}\\n\n\\\\usepackage{verbatim}\\n\n\\\\usepackage{tikz}\\n\n \\\\usepackage[top=2cm, bottom=0.2cm, outer=0.8cm, inner=0.2cm,left=1.5cm]{geometry}\\n\n\\\\usepackage{multirow}\\n\n\\\\usepackage{graphicx}\\n\n\\\\begin{document}\\n\n\\\\begin{flushleft}\\n\n\\\\begin{LARGE}\\n\"\nendmsg<-\"\\\\end{LARGE}\\n\n \\\\end{flushleft}\\n\n\\\\end{document}\\n\"\nfinal_latex_code<-paste(c(finalmsg,allmsg,endmsg),collapse=\" \")\n","metadata":{"execution":{"iopub.status.busy":"2024-07-07T13:28:28.401538Z","iopub.execute_input":"2024-07-07T13:28:28.403531Z","iopub.status.idle":"2024-07-07T13:28:29.850037Z"},"trusted":true},"execution_count":7,"outputs":[]},{"cell_type":"code","source":"writeLines(final_latex_code,con=\"/kaggle/working/exam2_interpolation.txt\")##to save the file in the directory /content to use it later in Latex software such as texmaker. you can also save it as .tex file ","metadata":{"execution":{"iopub.status.busy":"2024-07-07T14:11:37.418669Z","iopub.execute_input":"2024-07-07T14:11:37.421015Z","iopub.status.idle":"2024-07-07T14:11:37.441620Z"},"trusted":true},"execution_count":14,"outputs":[]},{"cell_type":"markdown","source":"### The latex code file now saved in /kaggle/working/exam2_interpolation.txt. Copy the text in this file and paste it into tex editor such as texmaker. The a pdf file will be generated contains the diverse questions. The generated latex code is as below","metadata":{}},{"cell_type":"markdown","source":"\\documentclass[16pt,a4paper]{report}\n\n\\usepackage{listings}\n\n\\usepackage{verbatim}\n\n\\usepackage{tikz}\n\n\\usepackage[top=2cm, bottom=0.2cm, outer=0.8cm, inner=0.2cm,left=1.5cm]{geometry}\n\n\\usepackage{multirow}\n\n\\usepackage{graphicx}\n\n\\begin{document}\n\n\\begin{flushleft}\n\n\\begin{LARGE}\n \\textcolor{red}{1},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,1),(4,0),(6,3),(8,-3),(10,-2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{2},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-4),(4,-3),(6,-1),(8,0),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{3},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,0),(4,1),(6,4),(8,3),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{4},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,0),(4,3),(6,4),(8,-4),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{5},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-2),(4,-1),(6,0),(8,-4),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{6},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,3),(4,4),(6,-4),(8,0),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{7},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,2),(4,1),(6,-1),(8,-3),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{8},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-2),(4,-1),(6,-4),(8,4),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{9},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,-3),(4,-4),(6,2),(8,1),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{10},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-1),(4,-4),(6,3),(8,0),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{11},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-1),(4,5),(6,-4),(8,0),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{12},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,0),(4,-3),(6,-1),(8,4),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{13},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-2),(4,3),(6,2),(8,1),(10,-4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{14},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,3),(4,2),(6,-4),(8,-3),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{15},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,0),(4,2),(6,-2),(8,-4),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{16},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,0),(4,-1),(6,1),(8,2),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{17},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-1),(4,-4),(6,4),(8,-2),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{18},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,1),(4,2),(6,-3),(8,-1),(10,0). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{19},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-1),(4,0),(6,-4),(8,-3),(10,-2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{20},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,1),(4,-1),(6,4),(8,-2),(10,-3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{21},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-3),(4,-4),(6,1),(8,3),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{22},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,5),(4,0),(6,1),(8,-1),(10,-4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{23},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,5),(4,4),(6,-2),(8,3),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{24},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,5),(4,3),(6,-3),(8,1),(10,-4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{25},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-4),(4,4),(6,5),(8,-3),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{26},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-3),(4,5),(6,-4),(8,4),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{27},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,0),(4,1),(6,-3),(8,-4),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{28},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-3),(4,-4),(6,0),(8,-1),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{29},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,0),(4,-4),(6,5),(8,4),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{30},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-4),(4,1),(6,0),(8,-2),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{31},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-3),(4,-4),(6,0),(8,-1),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{32},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-2),(4,2),(6,1),(8,0),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{33},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,0),(4,1),(6,3),(8,-1),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{34},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,4),(4,0),(6,-3),(8,-1),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{35},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,4),(4,-4),(6,1),(8,5),(10,0). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{36},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-4),(4,-1),(6,3),(8,1),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{37},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,5),(4,-3),(6,-4),(8,-1),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{38},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,1),(4,-1),(6,-2),(8,5),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{39},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-1),(4,-3),(6,5),(8,2),(10,-2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{40},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-2),(4,0),(6,3),(8,4),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{41},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-2),(4,5),(6,1),(8,-3),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{42},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-1),(4,2),(6,-2),(8,-3),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{43},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-1),(4,4),(6,-2),(8,5),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{44},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,0),(4,2),(6,3),(8,-4),(10,-2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{45},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-4),(4,0),(6,4),(8,1),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{46},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,-1),(4,4),(6,-4),(8,5),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{47},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,3),(4,-2),(6,1),(8,4),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{48},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-1),(4,1),(6,-3),(8,0),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{49},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,0),(4,-2),(6,-3),(8,4),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{50},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,5),(4,-4),(6,4),(8,0),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{51},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-4),(4,-1),(6,2),(8,-3),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{52},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-4),(4,1),(6,-1),(8,3),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{53},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-4),(4,-2),(6,4),(8,3),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{54},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,3),(4,0),(6,-3),(8,4),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{55},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-4),(4,-3),(6,-2),(8,2),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{56},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,1),(4,-3),(6,5),(8,-4),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{57},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,0),(4,-2),(6,4),(8,3),(10,5). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{58},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,-1),(4,5),(6,2),(8,3),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{59},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-4),(4,-1),(6,5),(8,0),(10,-3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{60},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,3),(4,2),(6,1),(8,-2),(10,0). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{61},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-2),(4,5),(6,0),(8,2),(10,-3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{62},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,1),(4,5),(6,0),(8,-3),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{63},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,-2),(4,3),(6,0),(8,1),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{64},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,4),(4,0),(6,-4),(8,3),(10,-3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{65},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Newton for divided differences to interpolate the following points. The points are,(2,-2),(4,2),(6,5),(8,1),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{66},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,5),(4,-2),(6,-4),(8,2),(10,1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{67},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-1),(4,3),(6,1),(8,-4),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{68},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,4),(4,-3),(6,-2),(8,5),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{69},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-4),(4,1),(6,-1),(8,-3),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{70},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,5),(4,-2),(6,-3),(8,1),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{71},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,4),(4,-4),(6,3),(8,0),(10,2). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{72},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss forward to interpolate the following points. The points are,(2,-4),(4,-1),(6,1),(8,-2),(10,-3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{73},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-3),(4,0),(6,-2),(8,-4),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{74},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial Backward to interpolate the following points. The points are,(2,3),(4,5),(6,-1),(8,2),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{75},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,5),(4,-4),(6,-1),(8,2),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{76},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of polynomial forward to interpolate the following points. The points are,(2,5),(4,3),(6,4),(8,2),(10,-1). Then write the interpolated equation in the google form?\n===============================================================\n \\textcolor{red}{77},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-1),(4,-3),(6,5),(8,-2),(10,3). Then write the interpolated equation in the google form?\n===============================================================\n \\newpage \n \\textcolor{red}{78},\\hspace*{3mm}\\textcolor{blue}{Name:}\\\\ \nUse the method of Gauss backward to interpolate the following points. The points are,(2,-4),(4,-2),(6,1),(8,-1),(10,4). Then write the interpolated equation in the google form?\n===============================================================\n \\end{LARGE}\n\n \\end{flushleft}\n\n\\end{document}\n\n","metadata":{}},{"cell_type":"code","source":"ans","metadata":{"execution":{"iopub.status.busy":"2024-07-07T14:14:53.910393Z","iopub.execute_input":"2024-07-07T14:14:53.913309Z","iopub.status.idle":"2024-07-07T14:14:54.071197Z"},"trusted":true},"execution_count":15,"outputs":[{"output_type":"display_data","data":{"text/html":"<table class=\"dataframe\">\n<caption>A matrix: 78 × 11 of type dbl</caption>\n<tbody>\n\t<tr><td> 1</td><td>-6.0</td><td> 1.0</td><td> -9.000</td><td> 16.00000000</td><td> 29.00000000</td><td> 0.075520833</td><td>-1.7812500</td><td> 14.3229167</td><td>-45.6250000</td><td>  48</td></tr>\n\t<tr><td> 2</td><td> 1.0</td><td> 2.0</td><td> -1.000</td><td>  2.00000000</td><td>  4.00000000</td><td> 0.010416667</td><td>-0.2500000</td><td>  2.0833333</td><td> -6.2500000</td><td>   2</td></tr>\n\t<tr><td> 3</td><td>-1.0</td><td> 2.0</td><td> -4.000</td><td>  7.00000000</td><td> 13.00000000</td><td> 0.033854167</td><td>-0.8020833</td><td>  6.4895833</td><td>-20.0416667</td><td>  20</td></tr>\n\t<tr><td> 4</td><td>-4.0</td><td> 1.5</td><td> -1.125</td><td>  0.41666667</td><td>  0.07031250</td><td> 0.070312500</td><td>-1.5520833</td><td> 11.3437500</td><td>-31.5416667</td><td>  29</td></tr>\n\t<tr><td> 5</td><td>-4.0</td><td> 5.0</td><td> -5.000</td><td> 14.00000000</td><td> 19.00000000</td><td> 0.049479167</td><td>-1.0937500</td><td>  8.1770833</td><td>-23.8750000</td><td>  21</td></tr>\n\t<tr><td> 6</td><td> 2.0</td><td> 2.5</td><td>  1.500</td><td> -0.22916667</td><td> -0.08333333</td><td>-0.083333333</td><td> 2.1041667</td><td>-18.0416667</td><td> 59.8333333</td><td> -60</td></tr>\n\t<tr><td> 7</td><td>-1.0</td><td> 3.5</td><td>  0.000</td><td>  0.18750000</td><td>  0.02083333</td><td> 0.020833333</td><td>-0.3958333</td><td>  2.5416667</td><td> -7.1666667</td><td>   9</td></tr>\n\t<tr><td> 8</td><td> 8.0</td><td>-2.0</td><td> 11.000</td><td>-21.00000000</td><td>-36.00000000</td><td>-0.093750000</td><td> 2.1875000</td><td>-17.3750000</td><td> 54.7500000</td><td> -58</td></tr>\n\t<tr><td> 9</td><td>-0.5</td><td> 2.0</td><td> -0.875</td><td>  0.25000000</td><td>  0.06770833</td><td> 0.067708333</td><td>-1.6458333</td><td> 13.8541667</td><td>-45.6666667</td><td>  45</td></tr>\n\t<tr><td>10</td><td>-3.0</td><td> 1.0</td><td>-10.000</td><td> 14.00000000</td><td> 34.00000000</td><td> 0.088541667</td><td>-2.1875000</td><td> 18.6458333</td><td>-62.7500000</td><td>  66</td></tr>\n\t<tr><td>11</td><td> 4.0</td><td> 1.0</td><td> 13.000</td><td>-16.00000000</td><td>-44.00000000</td><td>-0.114583333</td><td> 2.8750000</td><td>-24.9166667</td><td> 85.7500000</td><td> -94</td></tr>\n\t<tr><td>12</td><td> 5.0</td><td>-1.0</td><td>  3.000</td><td> -9.00000000</td><td> -7.00000000</td><td>-0.018229167</td><td> 0.3229167</td><td> -1.4270833</td><td>  0.2083333</td><td>   3</td></tr>\n\t<tr><td>13</td><td>-1.0</td><td>-5.0</td><td>  0.000</td><td> -4.00000000</td><td>-10.00000000</td><td>-0.026041667</td><td> 0.6458333</td><td> -5.8958333</td><td> 22.9166667</td><td> -29</td></tr>\n\t<tr><td>14</td><td> 1.0</td><td> 7.0</td><td>  7.000</td><td> -1.00000000</td><td>-13.00000000</td><td>-0.033854167</td><td> 0.9270833</td><td> -8.3645833</td><td> 27.7916667</td><td> -26</td></tr>\n\t<tr><td>15</td><td>-2.0</td><td> 5.0</td><td>  2.000</td><td>  5.00000000</td><td> -3.00000000</td><td>-0.007812500</td><td> 0.3229167</td><td> -3.8437500</td><td> 15.9583333</td><td> -19</td></tr>\n\t<tr><td>16</td><td> 0.5</td><td> 1.0</td><td> -0.125</td><td>  0.04166667</td><td>  0.01562500</td><td> 0.015625000</td><td>-0.3958333</td><td>  3.5625000</td><td>-12.6666667</td><td>  14</td></tr>\n\t<tr><td>17</td><td>-6.0</td><td> 7.0</td><td>-14.000</td><td> 27.00000000</td><td> 52.00000000</td><td> 0.135416667</td><td>-3.2291667</td><td> 26.5833333</td><td>-86.8333333</td><td>  90</td></tr>\n\t<tr><td>18</td><td> 2.0</td><td> 1.0</td><td>  7.000</td><td> -8.00000000</td><td>-21.00000000</td><td>-0.054687500</td><td> 1.3645833</td><td>-11.6562500</td><td> 38.7916667</td><td> -40</td></tr>\n\t<tr><td>19</td><td> 1.0</td><td> 1.0</td><td>  5.000</td><td> -5.00000000</td><td>-15.00000000</td><td>-0.039062500</td><td> 0.9895833</td><td> -8.5937500</td><td> 29.0416667</td><td> -32</td></tr>\n\t<tr><td>20</td><td>-6.0</td><td>-1.0</td><td>-11.000</td><td> 16.00000000</td><td> 34.00000000</td><td> 0.088541667</td><td>-2.1458333</td><td> 17.7708333</td><td>-58.1666667</td><td>  62</td></tr>\n\t<tr><td>21</td><td> 2.0</td><td> 1.0</td><td> -3.000</td><td>  2.00000000</td><td> 11.00000000</td><td> 0.028645833</td><td>-0.7604167</td><td>  7.0104167</td><td>-24.7083333</td><td>  24</td></tr>\n\t<tr><td>22</td><td>-2.0</td><td>-3.0</td><td> -3.000</td><td>  2.00000000</td><td> 11.00000000</td><td> 0.028645833</td><td>-0.7604167</td><td>  7.0104167</td><td>-26.7083333</td><td>  36</td></tr>\n\t<tr><td>23</td><td> 5.0</td><td>-1.0</td><td> 11.000</td><td>-17.00000000</td><td>-33.00000000</td><td>-0.085937500</td><td> 2.0520833</td><td>-16.6562500</td><td> 52.2916667</td><td> -48</td></tr>\n\t<tr><td>24</td><td> 4.0</td><td>-5.0</td><td> 10.000</td><td>-19.00000000</td><td>-33.00000000</td><td>-0.085937500</td><td> 2.0104167</td><td>-16.0312500</td><td> 49.2083333</td><td> -44</td></tr>\n\t<tr><td>25</td><td>-8.0</td><td> 2.0</td><td> -9.000</td><td> 19.00000000</td><td> 21.00000000</td><td> 0.054687500</td><td>-1.1354167</td><td>  7.2812500</td><td>-14.4583333</td><td>   4</td></tr>\n\t<tr><td>26</td><td> 8.0</td><td>-3.0</td><td> 17.000</td><td>-28.00000000</td><td>-62.00000000</td><td>-0.161458333</td><td> 3.9375000</td><td>-33.2291667</td><td>112.5000000</td><td>-124</td></tr>\n\t<tr><td>27</td><td>-1.0</td><td> 9.0</td><td>  3.000</td><td>  7.00000000</td><td> -1.00000000</td><td>-0.002604167</td><td> 0.2187500</td><td> -2.9895833</td><td> 12.6250000</td><td> -15</td></tr>\n\t<tr><td>28</td><td>-1.0</td><td> 6.0</td><td> -5.000</td><td> 12.00000000</td><td> 22.00000000</td><td> 0.057291667</td><td>-1.3541667</td><td> 11.1458333</td><td>-36.3333333</td><td>  35</td></tr>\n\t<tr><td>29</td><td>-0.5</td><td>-1.5</td><td> -1.250</td><td>  0.16666667</td><td>  0.08072917</td><td> 0.080729167</td><td>-2.0937500</td><td> 18.6770833</td><td>-65.1250000</td><td>  71</td></tr>\n\t<tr><td>30</td><td>-2.0</td><td> 6.0</td><td> -1.000</td><td>  9.00000000</td><td>  4.00000000</td><td> 0.010416667</td><td>-0.1041667</td><td> -0.5416667</td><td>  7.4166667</td><td> -16</td></tr>\n\t<tr><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr>\n\t<tr><td>49</td><td> 3.5</td><td>-0.5</td><td>  1.000</td><td> -0.3333333</td><td> -0.05989583</td><td>-0.059895833</td><td> 1.34375000</td><td>-10.0104167</td><td> 28.625000</td><td>-27</td></tr>\n\t<tr><td>50</td><td>-4.0</td><td>-1.0</td><td>-12.000</td><td> 15.0000000</td><td> 44.00000000</td><td> 0.114583333</td><td>-2.89583333</td><td> 25.4166667</td><td>-89.666667</td><td>104</td></tr>\n\t<tr><td>51</td><td>-5.0</td><td> 8.0</td><td> -8.000</td><td> 21.0000000</td><td> 29.00000000</td><td> 0.075520833</td><td>-1.67708333</td><td> 12.5729167</td><td>-36.041667</td><td> 30</td></tr>\n\t<tr><td>52</td><td> 4.0</td><td> 2.0</td><td>  6.000</td><td> -8.0000000</td><td>-21.00000000</td><td>-0.054687500</td><td> 1.36458333</td><td>-11.7812500</td><td> 41.541667</td><td>-50</td></tr>\n\t<tr><td>53</td><td>-1.0</td><td> 2.0</td><td> -7.000</td><td> 10.0000000</td><td> 21.00000000</td><td> 0.054687500</td><td>-1.32291667</td><td> 10.9062500</td><td>-33.958333</td><td> 30</td></tr>\n\t<tr><td>54</td><td> 7.0</td><td> 1.0</td><td> 10.000</td><td>-16.0000000</td><td>-26.00000000</td><td>-0.067708333</td><td> 1.56250000</td><td>-11.9791667</td><td> 34.750000</td><td>-30</td></tr>\n\t<tr><td>55</td><td> 4.0</td><td> 2.0</td><td>  3.000</td><td> -5.0000000</td><td> -8.00000000</td><td>-0.020833333</td><td> 0.47916667</td><td> -3.6666667</td><td> 11.583333</td><td>-16</td></tr>\n\t<tr><td>56</td><td>-9.0</td><td> 3.0</td><td>-17.000</td><td> 29.0000000</td><td> 58.00000000</td><td> 0.151041667</td><td>-3.62500000</td><td> 29.8958333</td><td>-98.000000</td><td>104</td></tr>\n\t<tr><td>57</td><td>-1.0</td><td> 2.0</td><td> -7.000</td><td> 10.0000000</td><td> 25.00000000</td><td> 0.065104167</td><td>-1.61458333</td><td> 13.8645833</td><td>-46.791667</td><td> 50</td></tr>\n\t<tr><td>58</td><td> 1.0</td><td>-2.0</td><td>  4.000</td><td> -7.0000000</td><td>-20.00000000</td><td>-0.052083333</td><td> 1.31250000</td><td>-11.6666667</td><td> 42.500000</td><td>-49</td></tr>\n\t<tr><td>59</td><td>-5.0</td><td>-3.0</td><td>-11.000</td><td> 13.0000000</td><td> 27.00000000</td><td> 0.070312500</td><td>-1.69791667</td><td> 13.7187500</td><td>-41.708333</td><td> 37</td></tr>\n\t<tr><td>60</td><td>-3.0</td><td> 2.0</td><td> -2.000</td><td>  7.0000000</td><td>  9.00000000</td><td> 0.023437500</td><td>-0.51041667</td><td>  3.7812500</td><td>-11.708333</td><td> 15</td></tr>\n\t<tr><td>61</td><td> 2.0</td><td>-5.0</td><td>  7.000</td><td>-14.0000000</td><td>-33.00000000</td><td>-0.085937500</td><td> 2.11458333</td><td>-18.2812500</td><td> 64.291667</td><td>-73</td></tr>\n\t<tr><td>62</td><td>-3.0</td><td> 6.0</td><td>  2.000</td><td>  7.0000000</td><td> -4.00000000</td><td>-0.010416667</td><td> 0.43750000</td><td> -5.3333333</td><td> 23.000000</td><td>-27</td></tr>\n\t<tr><td>63</td><td> 1.0</td><td> 3.0</td><td>  4.000</td><td> -2.0000000</td><td>-14.00000000</td><td>-0.036458333</td><td> 0.97916667</td><td> -9.1041667</td><td> 34.083333</td><td>-41</td></tr>\n\t<tr><td>64</td><td> 7.0</td><td>-6.0</td><td> 11.000</td><td>-24.0000000</td><td>-35.00000000</td><td>-0.091145833</td><td> 2.05208333</td><td>-15.5104167</td><td> 44.541667</td><td>-38</td></tr>\n\t<tr><td>65</td><td>-2.0</td><td>-1.0</td><td> -0.875</td><td>  0.1875000</td><td>  0.03906250</td><td> 0.039062500</td><td>-0.90625000</td><td>  6.8437500</td><td>-18.375000</td><td> 14</td></tr>\n\t<tr><td>66</td><td> 6.0</td><td>-1.0</td><td>  8.000</td><td>-15.0000000</td><td>-18.00000000</td><td>-0.046875000</td><td> 1.00000000</td><td> -6.6875000</td><td> 14.250000</td><td> -4</td></tr>\n\t<tr><td>67</td><td>-5.0</td><td> 8.0</td><td> -3.000</td><td> 16.0000000</td><td> 13.00000000</td><td> 0.033854167</td><td>-0.61458333</td><td>  3.2395833</td><td> -4.291667</td><td> -1</td></tr>\n\t<tr><td>68</td><td> 7.0</td><td>-3.0</td><td>  6.000</td><td>-16.0000000</td><td>-14.00000000</td><td>-0.036458333</td><td> 0.68750000</td><td> -3.6041667</td><td>  3.250000</td><td>  7</td></tr>\n\t<tr><td>69</td><td>-2.0</td><td> 7.0</td><td>  0.000</td><td>  9.0000000</td><td>  2.00000000</td><td> 0.005208333</td><td> 0.04166667</td><td> -1.8958333</td><td> 12.083333</td><td>-21</td></tr>\n\t<tr><td>70</td><td> 4.0</td><td> 2.0</td><td>  5.000</td><td> -7.0000000</td><td> -6.00000000</td><td>-0.015625000</td><td> 0.29166667</td><td> -1.1875000</td><td> -2.666667</td><td> 13</td></tr>\n\t<tr><td>71</td><td>-3.0</td><td> 2.0</td><td>-10.000</td><td> 15.0000000</td><td> 40.00000000</td><td> 0.104166667</td><td>-2.60416667</td><td> 22.7083333</td><td>-79.833333</td><td> 92</td></tr>\n\t<tr><td>72</td><td>-3.0</td><td>-1.0</td><td> -5.000</td><td>  7.0000000</td><td> 11.00000000</td><td> 0.028645833</td><td>-0.65625000</td><td>  4.8854167</td><td>-12.875000</td><td>  7</td></tr>\n\t<tr><td>73</td><td>-2.0</td><td> 7.0</td><td>  0.000</td><td>  9.0000000</td><td>  4.00000000</td><td> 0.010416667</td><td>-0.10416667</td><td> -0.4166667</td><td>  5.666667</td><td>-12</td></tr>\n\t<tr><td>74</td><td> 3.0</td><td> 2.0</td><td>  9.000</td><td>-10.0000000</td><td>-27.00000000</td><td>-0.070312500</td><td> 1.76041667</td><td>-15.0937500</td><td> 50.708333</td><td>-51</td></tr>\n\t<tr><td>75</td><td> 3.0</td><td> 1.0</td><td>  0.000</td><td> -2.0000000</td><td> 10.00000000</td><td> 0.026041667</td><td>-0.77083333</td><td>  8.1458333</td><td>-34.916667</td><td> 48</td></tr>\n\t<tr><td>76</td><td>-2.0</td><td>-3.0</td><td> -3.000</td><td>  2.0000000</td><td>  8.00000000</td><td> 0.020833333</td><td>-0.54166667</td><td>  4.7916667</td><td>-17.083333</td><td> 24</td></tr>\n\t<tr><td>77</td><td>-7.0</td><td> 5.0</td><td>-15.000</td><td> 27.0000000</td><td> 52.00000000</td><td> 0.135416667</td><td>-3.22916667</td><td> 26.4583333</td><td>-85.583333</td><td> 88</td></tr>\n\t<tr><td>78</td><td>-2.0</td><td> 5.0</td><td> -5.000</td><td> 12.0000000</td><td> 18.00000000</td><td> 0.046875000</td><td>-1.06250000</td><td>  8.1875000</td><td>-24.000000</td><td> 19</td></tr>\n</tbody>\n</table>\n","text/markdown":"\nA matrix: 78 × 11 of type dbl\n\n|  1 | -6.0 |  1.0 |  -9.000 |  16.00000000 |  29.00000000 |  0.075520833 | -1.7812500 |  14.3229167 | -45.6250000 |   48 |\n|  2 |  1.0 |  2.0 |  -1.000 |   2.00000000 |   4.00000000 |  0.010416667 | -0.2500000 |   2.0833333 |  -6.2500000 |    2 |\n|  3 | -1.0 |  2.0 |  -4.000 |   7.00000000 |  13.00000000 |  0.033854167 | -0.8020833 |   6.4895833 | -20.0416667 |   20 |\n|  4 | -4.0 |  1.5 |  -1.125 |   0.41666667 |   0.07031250 |  0.070312500 | -1.5520833 |  11.3437500 | -31.5416667 |   29 |\n|  5 | -4.0 |  5.0 |  -5.000 |  14.00000000 |  19.00000000 |  0.049479167 | -1.0937500 |   8.1770833 | -23.8750000 |   21 |\n|  6 |  2.0 |  2.5 |   1.500 |  -0.22916667 |  -0.08333333 | -0.083333333 |  2.1041667 | -18.0416667 |  59.8333333 |  -60 |\n|  7 | -1.0 |  3.5 |   0.000 |   0.18750000 |   0.02083333 |  0.020833333 | -0.3958333 |   2.5416667 |  -7.1666667 |    9 |\n|  8 |  8.0 | -2.0 |  11.000 | -21.00000000 | -36.00000000 | -0.093750000 |  2.1875000 | -17.3750000 |  54.7500000 |  -58 |\n|  9 | -0.5 |  2.0 |  -0.875 |   0.25000000 |   0.06770833 |  0.067708333 | -1.6458333 |  13.8541667 | -45.6666667 |   45 |\n| 10 | -3.0 |  1.0 | -10.000 |  14.00000000 |  34.00000000 |  0.088541667 | -2.1875000 |  18.6458333 | -62.7500000 |   66 |\n| 11 |  4.0 |  1.0 |  13.000 | -16.00000000 | -44.00000000 | -0.114583333 |  2.8750000 | -24.9166667 |  85.7500000 |  -94 |\n| 12 |  5.0 | -1.0 |   3.000 |  -9.00000000 |  -7.00000000 | -0.018229167 |  0.3229167 |  -1.4270833 |   0.2083333 |    3 |\n| 13 | -1.0 | -5.0 |   0.000 |  -4.00000000 | -10.00000000 | -0.026041667 |  0.6458333 |  -5.8958333 |  22.9166667 |  -29 |\n| 14 |  1.0 |  7.0 |   7.000 |  -1.00000000 | -13.00000000 | -0.033854167 |  0.9270833 |  -8.3645833 |  27.7916667 |  -26 |\n| 15 | -2.0 |  5.0 |   2.000 |   5.00000000 |  -3.00000000 | -0.007812500 |  0.3229167 |  -3.8437500 |  15.9583333 |  -19 |\n| 16 |  0.5 |  1.0 |  -0.125 |   0.04166667 |   0.01562500 |  0.015625000 | -0.3958333 |   3.5625000 | -12.6666667 |   14 |\n| 17 | -6.0 |  7.0 | -14.000 |  27.00000000 |  52.00000000 |  0.135416667 | -3.2291667 |  26.5833333 | -86.8333333 |   90 |\n| 18 |  2.0 |  1.0 |   7.000 |  -8.00000000 | -21.00000000 | -0.054687500 |  1.3645833 | -11.6562500 |  38.7916667 |  -40 |\n| 19 |  1.0 |  1.0 |   5.000 |  -5.00000000 | -15.00000000 | -0.039062500 |  0.9895833 |  -8.5937500 |  29.0416667 |  -32 |\n| 20 | -6.0 | -1.0 | -11.000 |  16.00000000 |  34.00000000 |  0.088541667 | -2.1458333 |  17.7708333 | -58.1666667 |   62 |\n| 21 |  2.0 |  1.0 |  -3.000 |   2.00000000 |  11.00000000 |  0.028645833 | -0.7604167 |   7.0104167 | -24.7083333 |   24 |\n| 22 | -2.0 | -3.0 |  -3.000 |   2.00000000 |  11.00000000 |  0.028645833 | -0.7604167 |   7.0104167 | -26.7083333 |   36 |\n| 23 |  5.0 | -1.0 |  11.000 | -17.00000000 | -33.00000000 | -0.085937500 |  2.0520833 | -16.6562500 |  52.2916667 |  -48 |\n| 24 |  4.0 | -5.0 |  10.000 | -19.00000000 | -33.00000000 | -0.085937500 |  2.0104167 | -16.0312500 |  49.2083333 |  -44 |\n| 25 | -8.0 |  2.0 |  -9.000 |  19.00000000 |  21.00000000 |  0.054687500 | -1.1354167 |   7.2812500 | -14.4583333 |    4 |\n| 26 |  8.0 | -3.0 |  17.000 | -28.00000000 | -62.00000000 | -0.161458333 |  3.9375000 | -33.2291667 | 112.5000000 | -124 |\n| 27 | -1.0 |  9.0 |   3.000 |   7.00000000 |  -1.00000000 | -0.002604167 |  0.2187500 |  -2.9895833 |  12.6250000 |  -15 |\n| 28 | -1.0 |  6.0 |  -5.000 |  12.00000000 |  22.00000000 |  0.057291667 | -1.3541667 |  11.1458333 | -36.3333333 |   35 |\n| 29 | -0.5 | -1.5 |  -1.250 |   0.16666667 |   0.08072917 |  0.080729167 | -2.0937500 |  18.6770833 | -65.1250000 |   71 |\n| 30 | -2.0 |  6.0 |  -1.000 |   9.00000000 |   4.00000000 |  0.010416667 | -0.1041667 |  -0.5416667 |   7.4166667 |  -16 |\n| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |\n| 49 |  3.5 | -0.5 |   1.000 |  -0.3333333 |  -0.05989583 | -0.059895833 |  1.34375000 | -10.0104167 |  28.625000 | -27 |\n| 50 | -4.0 | -1.0 | -12.000 |  15.0000000 |  44.00000000 |  0.114583333 | -2.89583333 |  25.4166667 | -89.666667 | 104 |\n| 51 | -5.0 |  8.0 |  -8.000 |  21.0000000 |  29.00000000 |  0.075520833 | -1.67708333 |  12.5729167 | -36.041667 |  30 |\n| 52 |  4.0 |  2.0 |   6.000 |  -8.0000000 | -21.00000000 | -0.054687500 |  1.36458333 | -11.7812500 |  41.541667 | -50 |\n| 53 | -1.0 |  2.0 |  -7.000 |  10.0000000 |  21.00000000 |  0.054687500 | -1.32291667 |  10.9062500 | -33.958333 |  30 |\n| 54 |  7.0 |  1.0 |  10.000 | -16.0000000 | -26.00000000 | -0.067708333 |  1.56250000 | -11.9791667 |  34.750000 | -30 |\n| 55 |  4.0 |  2.0 |   3.000 |  -5.0000000 |  -8.00000000 | -0.020833333 |  0.47916667 |  -3.6666667 |  11.583333 | -16 |\n| 56 | -9.0 |  3.0 | -17.000 |  29.0000000 |  58.00000000 |  0.151041667 | -3.62500000 |  29.8958333 | -98.000000 | 104 |\n| 57 | -1.0 |  2.0 |  -7.000 |  10.0000000 |  25.00000000 |  0.065104167 | -1.61458333 |  13.8645833 | -46.791667 |  50 |\n| 58 |  1.0 | -2.0 |   4.000 |  -7.0000000 | -20.00000000 | -0.052083333 |  1.31250000 | -11.6666667 |  42.500000 | -49 |\n| 59 | -5.0 | -3.0 | -11.000 |  13.0000000 |  27.00000000 |  0.070312500 | -1.69791667 |  13.7187500 | -41.708333 |  37 |\n| 60 | -3.0 |  2.0 |  -2.000 |   7.0000000 |   9.00000000 |  0.023437500 | -0.51041667 |   3.7812500 | -11.708333 |  15 |\n| 61 |  2.0 | -5.0 |   7.000 | -14.0000000 | -33.00000000 | -0.085937500 |  2.11458333 | -18.2812500 |  64.291667 | -73 |\n| 62 | -3.0 |  6.0 |   2.000 |   7.0000000 |  -4.00000000 | -0.010416667 |  0.43750000 |  -5.3333333 |  23.000000 | -27 |\n| 63 |  1.0 |  3.0 |   4.000 |  -2.0000000 | -14.00000000 | -0.036458333 |  0.97916667 |  -9.1041667 |  34.083333 | -41 |\n| 64 |  7.0 | -6.0 |  11.000 | -24.0000000 | -35.00000000 | -0.091145833 |  2.05208333 | -15.5104167 |  44.541667 | -38 |\n| 65 | -2.0 | -1.0 |  -0.875 |   0.1875000 |   0.03906250 |  0.039062500 | -0.90625000 |   6.8437500 | -18.375000 |  14 |\n| 66 |  6.0 | -1.0 |   8.000 | -15.0000000 | -18.00000000 | -0.046875000 |  1.00000000 |  -6.6875000 |  14.250000 |  -4 |\n| 67 | -5.0 |  8.0 |  -3.000 |  16.0000000 |  13.00000000 |  0.033854167 | -0.61458333 |   3.2395833 |  -4.291667 |  -1 |\n| 68 |  7.0 | -3.0 |   6.000 | -16.0000000 | -14.00000000 | -0.036458333 |  0.68750000 |  -3.6041667 |   3.250000 |   7 |\n| 69 | -2.0 |  7.0 |   0.000 |   9.0000000 |   2.00000000 |  0.005208333 |  0.04166667 |  -1.8958333 |  12.083333 | -21 |\n| 70 |  4.0 |  2.0 |   5.000 |  -7.0000000 |  -6.00000000 | -0.015625000 |  0.29166667 |  -1.1875000 |  -2.666667 |  13 |\n| 71 | -3.0 |  2.0 | -10.000 |  15.0000000 |  40.00000000 |  0.104166667 | -2.60416667 |  22.7083333 | -79.833333 |  92 |\n| 72 | -3.0 | -1.0 |  -5.000 |   7.0000000 |  11.00000000 |  0.028645833 | -0.65625000 |   4.8854167 | -12.875000 |   7 |\n| 73 | -2.0 |  7.0 |   0.000 |   9.0000000 |   4.00000000 |  0.010416667 | -0.10416667 |  -0.4166667 |   5.666667 | -12 |\n| 74 |  3.0 |  2.0 |   9.000 | -10.0000000 | -27.00000000 | -0.070312500 |  1.76041667 | -15.0937500 |  50.708333 | -51 |\n| 75 |  3.0 |  1.0 |   0.000 |  -2.0000000 |  10.00000000 |  0.026041667 | -0.77083333 |   8.1458333 | -34.916667 |  48 |\n| 76 | -2.0 | -3.0 |  -3.000 |   2.0000000 |   8.00000000 |  0.020833333 | -0.54166667 |   4.7916667 | -17.083333 |  24 |\n| 77 | -7.0 |  5.0 | -15.000 |  27.0000000 |  52.00000000 |  0.135416667 | -3.22916667 |  26.4583333 | -85.583333 |  88 |\n| 78 | -2.0 |  5.0 |  -5.000 |  12.0000000 |  18.00000000 |  0.046875000 | -1.06250000 |   8.1875000 | -24.000000 |  19 |\n\n","text/latex":"A matrix: 78 × 11 of type dbl\n\\begin{tabular}{lllllllllll}\n\t  1 & -6.0 &  1.0 &  -9.000 &  16.00000000 &  29.00000000 &  0.075520833 & -1.7812500 &  14.3229167 & -45.6250000 &   48\\\\\n\t  2 &  1.0 &  2.0 &  -1.000 &   2.00000000 &   4.00000000 &  0.010416667 & -0.2500000 &   2.0833333 &  -6.2500000 &    2\\\\\n\t  3 & -1.0 &  2.0 &  -4.000 &   7.00000000 &  13.00000000 &  0.033854167 & -0.8020833 &   6.4895833 & -20.0416667 &   20\\\\\n\t  4 & -4.0 &  1.5 &  -1.125 &   0.41666667 &   0.07031250 &  0.070312500 & -1.5520833 &  11.3437500 & -31.5416667 &   29\\\\\n\t  5 & -4.0 &  5.0 &  -5.000 &  14.00000000 &  19.00000000 &  0.049479167 & -1.0937500 &   8.1770833 & -23.8750000 &   21\\\\\n\t  6 &  2.0 &  2.5 &   1.500 &  -0.22916667 &  -0.08333333 & -0.083333333 &  2.1041667 & -18.0416667 &  59.8333333 &  -60\\\\\n\t  7 & -1.0 &  3.5 &   0.000 &   0.18750000 &   0.02083333 &  0.020833333 & -0.3958333 &   2.5416667 &  -7.1666667 &    9\\\\\n\t  8 &  8.0 & -2.0 &  11.000 & -21.00000000 & -36.00000000 & -0.093750000 &  2.1875000 & -17.3750000 &  54.7500000 &  -58\\\\\n\t  9 & -0.5 &  2.0 &  -0.875 &   0.25000000 &   0.06770833 &  0.067708333 & -1.6458333 &  13.8541667 & -45.6666667 &   45\\\\\n\t 10 & -3.0 &  1.0 & -10.000 &  14.00000000 &  34.00000000 &  0.088541667 & -2.1875000 &  18.6458333 & -62.7500000 &   66\\\\\n\t 11 &  4.0 &  1.0 &  13.000 & -16.00000000 & -44.00000000 & -0.114583333 &  2.8750000 & -24.9166667 &  85.7500000 &  -94\\\\\n\t 12 &  5.0 & -1.0 &   3.000 &  -9.00000000 &  -7.00000000 & -0.018229167 &  0.3229167 &  -1.4270833 &   0.2083333 &    3\\\\\n\t 13 & -1.0 & -5.0 &   0.000 &  -4.00000000 & -10.00000000 & -0.026041667 &  0.6458333 &  -5.8958333 &  22.9166667 &  -29\\\\\n\t 14 &  1.0 &  7.0 &   7.000 &  -1.00000000 & -13.00000000 & -0.033854167 &  0.9270833 &  -8.3645833 &  27.7916667 &  -26\\\\\n\t 15 & -2.0 &  5.0 &   2.000 &   5.00000000 &  -3.00000000 & -0.007812500 &  0.3229167 &  -3.8437500 &  15.9583333 &  -19\\\\\n\t 16 &  0.5 &  1.0 &  -0.125 &   0.04166667 &   0.01562500 &  0.015625000 & -0.3958333 &   3.5625000 & -12.6666667 &   14\\\\\n\t 17 & -6.0 &  7.0 & -14.000 &  27.00000000 &  52.00000000 &  0.135416667 & -3.2291667 &  26.5833333 & -86.8333333 &   90\\\\\n\t 18 &  2.0 &  1.0 &   7.000 &  -8.00000000 & -21.00000000 & -0.054687500 &  1.3645833 & -11.6562500 &  38.7916667 &  -40\\\\\n\t 19 &  1.0 &  1.0 &   5.000 &  -5.00000000 & -15.00000000 & -0.039062500 &  0.9895833 &  -8.5937500 &  29.0416667 &  -32\\\\\n\t 20 & -6.0 & -1.0 & -11.000 &  16.00000000 &  34.00000000 &  0.088541667 & -2.1458333 &  17.7708333 & -58.1666667 &   62\\\\\n\t 21 &  2.0 &  1.0 &  -3.000 &   2.00000000 &  11.00000000 &  0.028645833 & -0.7604167 &   7.0104167 & -24.7083333 &   24\\\\\n\t 22 & -2.0 & -3.0 &  -3.000 &   2.00000000 &  11.00000000 &  0.028645833 & -0.7604167 &   7.0104167 & -26.7083333 &   36\\\\\n\t 23 &  5.0 & -1.0 &  11.000 & -17.00000000 & -33.00000000 & -0.085937500 &  2.0520833 & -16.6562500 &  52.2916667 &  -48\\\\\n\t 24 &  4.0 & -5.0 &  10.000 & -19.00000000 & -33.00000000 & -0.085937500 &  2.0104167 & -16.0312500 &  49.2083333 &  -44\\\\\n\t 25 & -8.0 &  2.0 &  -9.000 &  19.00000000 &  21.00000000 &  0.054687500 & -1.1354167 &   7.2812500 & -14.4583333 &    4\\\\\n\t 26 &  8.0 & -3.0 &  17.000 & -28.00000000 & -62.00000000 & -0.161458333 &  3.9375000 & -33.2291667 & 112.5000000 & -124\\\\\n\t 27 & -1.0 &  9.0 &   3.000 &   7.00000000 &  -1.00000000 & -0.002604167 &  0.2187500 &  -2.9895833 &  12.6250000 &  -15\\\\\n\t 28 & -1.0 &  6.0 &  -5.000 &  12.00000000 &  22.00000000 &  0.057291667 & -1.3541667 &  11.1458333 & -36.3333333 &   35\\\\\n\t 29 & -0.5 & -1.5 &  -1.250 &   0.16666667 &   0.08072917 &  0.080729167 & -2.0937500 &  18.6770833 & -65.1250000 &   71\\\\\n\t 30 & -2.0 &  6.0 &  -1.000 &   9.00000000 &   4.00000000 &  0.010416667 & -0.1041667 &  -0.5416667 &   7.4166667 &  -16\\\\\n\t ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮\\\\\n\t 49 &  3.5 & -0.5 &   1.000 &  -0.3333333 &  -0.05989583 & -0.059895833 &  1.34375000 & -10.0104167 &  28.625000 & -27\\\\\n\t 50 & -4.0 & -1.0 & -12.000 &  15.0000000 &  44.00000000 &  0.114583333 & -2.89583333 &  25.4166667 & -89.666667 & 104\\\\\n\t 51 & -5.0 &  8.0 &  -8.000 &  21.0000000 &  29.00000000 &  0.075520833 & -1.67708333 &  12.5729167 & -36.041667 &  30\\\\\n\t 52 &  4.0 &  2.0 &   6.000 &  -8.0000000 & -21.00000000 & -0.054687500 &  1.36458333 & -11.7812500 &  41.541667 & -50\\\\\n\t 53 & -1.0 &  2.0 &  -7.000 &  10.0000000 &  21.00000000 &  0.054687500 & -1.32291667 &  10.9062500 & -33.958333 &  30\\\\\n\t 54 &  7.0 &  1.0 &  10.000 & -16.0000000 & -26.00000000 & -0.067708333 &  1.56250000 & -11.9791667 &  34.750000 & -30\\\\\n\t 55 &  4.0 &  2.0 &   3.000 &  -5.0000000 &  -8.00000000 & -0.020833333 &  0.47916667 &  -3.6666667 &  11.583333 & -16\\\\\n\t 56 & -9.0 &  3.0 & -17.000 &  29.0000000 &  58.00000000 &  0.151041667 & -3.62500000 &  29.8958333 & -98.000000 & 104\\\\\n\t 57 & -1.0 &  2.0 &  -7.000 &  10.0000000 &  25.00000000 &  0.065104167 & -1.61458333 &  13.8645833 & -46.791667 &  50\\\\\n\t 58 &  1.0 & -2.0 &   4.000 &  -7.0000000 & -20.00000000 & -0.052083333 &  1.31250000 & -11.6666667 &  42.500000 & -49\\\\\n\t 59 & -5.0 & -3.0 & -11.000 &  13.0000000 &  27.00000000 &  0.070312500 & -1.69791667 &  13.7187500 & -41.708333 &  37\\\\\n\t 60 & -3.0 &  2.0 &  -2.000 &   7.0000000 &   9.00000000 &  0.023437500 & -0.51041667 &   3.7812500 & -11.708333 &  15\\\\\n\t 61 &  2.0 & -5.0 &   7.000 & -14.0000000 & -33.00000000 & -0.085937500 &  2.11458333 & -18.2812500 &  64.291667 & -73\\\\\n\t 62 & -3.0 &  6.0 &   2.000 &   7.0000000 &  -4.00000000 & -0.010416667 &  0.43750000 &  -5.3333333 &  23.000000 & -27\\\\\n\t 63 &  1.0 &  3.0 &   4.000 &  -2.0000000 & -14.00000000 & -0.036458333 &  0.97916667 &  -9.1041667 &  34.083333 & -41\\\\\n\t 64 &  7.0 & -6.0 &  11.000 & -24.0000000 & -35.00000000 & -0.091145833 &  2.05208333 & -15.5104167 &  44.541667 & -38\\\\\n\t 65 & -2.0 & -1.0 &  -0.875 &   0.1875000 &   0.03906250 &  0.039062500 & -0.90625000 &   6.8437500 & -18.375000 &  14\\\\\n\t 66 &  6.0 & -1.0 &   8.000 & -15.0000000 & -18.00000000 & -0.046875000 &  1.00000000 &  -6.6875000 &  14.250000 &  -4\\\\\n\t 67 & -5.0 &  8.0 &  -3.000 &  16.0000000 &  13.00000000 &  0.033854167 & -0.61458333 &   3.2395833 &  -4.291667 &  -1\\\\\n\t 68 &  7.0 & -3.0 &   6.000 & -16.0000000 & -14.00000000 & -0.036458333 &  0.68750000 &  -3.6041667 &   3.250000 &   7\\\\\n\t 69 & -2.0 &  7.0 &   0.000 &   9.0000000 &   2.00000000 &  0.005208333 &  0.04166667 &  -1.8958333 &  12.083333 & -21\\\\\n\t 70 &  4.0 &  2.0 &   5.000 &  -7.0000000 &  -6.00000000 & -0.015625000 &  0.29166667 &  -1.1875000 &  -2.666667 &  13\\\\\n\t 71 & -3.0 &  2.0 & -10.000 &  15.0000000 &  40.00000000 &  0.104166667 & -2.60416667 &  22.7083333 & -79.833333 &  92\\\\\n\t 72 & -3.0 & -1.0 &  -5.000 &   7.0000000 &  11.00000000 &  0.028645833 & -0.65625000 &   4.8854167 & -12.875000 &   7\\\\\n\t 73 & -2.0 &  7.0 &   0.000 &   9.0000000 &   4.00000000 &  0.010416667 & -0.10416667 &  -0.4166667 &   5.666667 & -12\\\\\n\t 74 &  3.0 &  2.0 &   9.000 & -10.0000000 & -27.00000000 & -0.070312500 &  1.76041667 & -15.0937500 &  50.708333 & -51\\\\\n\t 75 &  3.0 &  1.0 &   0.000 &  -2.0000000 &  10.00000000 &  0.026041667 & -0.77083333 &   8.1458333 & -34.916667 &  48\\\\\n\t 76 & -2.0 & -3.0 &  -3.000 &   2.0000000 &   8.00000000 &  0.020833333 & -0.54166667 &   4.7916667 & -17.083333 &  24\\\\\n\t 77 & -7.0 &  5.0 & -15.000 &  27.0000000 &  52.00000000 &  0.135416667 & -3.22916667 &  26.4583333 & -85.583333 &  88\\\\\n\t 78 & -2.0 &  5.0 &  -5.000 &  12.0000000 &  18.00000000 &  0.046875000 & -1.06250000 &   8.1875000 & -24.000000 &  19\\\\\n\\end{tabular}\n","text/plain":"      [,1] [,2] [,3] [,4]    [,5]         [,6]         [,7]         [,8]       \n [1,]  1   -6.0  1.0  -9.000  16.00000000  29.00000000  0.075520833 -1.7812500 \n [2,]  2    1.0  2.0  -1.000   2.00000000   4.00000000  0.010416667 -0.2500000 \n [3,]  3   -1.0  2.0  -4.000   7.00000000  13.00000000  0.033854167 -0.8020833 \n [4,]  4   -4.0  1.5  -1.125   0.41666667   0.07031250  0.070312500 -1.5520833 \n [5,]  5   -4.0  5.0  -5.000  14.00000000  19.00000000  0.049479167 -1.0937500 \n [6,]  6    2.0  2.5   1.500  -0.22916667  -0.08333333 -0.083333333  2.1041667 \n [7,]  7   -1.0  3.5   0.000   0.18750000   0.02083333  0.020833333 -0.3958333 \n [8,]  8    8.0 -2.0  11.000 -21.00000000 -36.00000000 -0.093750000  2.1875000 \n [9,]  9   -0.5  2.0  -0.875   0.25000000   0.06770833  0.067708333 -1.6458333 \n[10,] 10   -3.0  1.0 -10.000  14.00000000  34.00000000  0.088541667 -2.1875000 \n[11,] 11    4.0  1.0  13.000 -16.00000000 -44.00000000 -0.114583333  2.8750000 \n[12,] 12    5.0 -1.0   3.000  -9.00000000  -7.00000000 -0.018229167  0.3229167 \n[13,] 13   -1.0 -5.0   0.000  -4.00000000 -10.00000000 -0.026041667  0.6458333 \n[14,] 14    1.0  7.0   7.000  -1.00000000 -13.00000000 -0.033854167  0.9270833 \n[15,] 15   -2.0  5.0   2.000   5.00000000  -3.00000000 -0.007812500  0.3229167 \n[16,] 16    0.5  1.0  -0.125   0.04166667   0.01562500  0.015625000 -0.3958333 \n[17,] 17   -6.0  7.0 -14.000  27.00000000  52.00000000  0.135416667 -3.2291667 \n[18,] 18    2.0  1.0   7.000  -8.00000000 -21.00000000 -0.054687500  1.3645833 \n[19,] 19    1.0  1.0   5.000  -5.00000000 -15.00000000 -0.039062500  0.9895833 \n[20,] 20   -6.0 -1.0 -11.000  16.00000000  34.00000000  0.088541667 -2.1458333 \n[21,] 21    2.0  1.0  -3.000   2.00000000  11.00000000  0.028645833 -0.7604167 \n[22,] 22   -2.0 -3.0  -3.000   2.00000000  11.00000000  0.028645833 -0.7604167 \n[23,] 23    5.0 -1.0  11.000 -17.00000000 -33.00000000 -0.085937500  2.0520833 \n[24,] 24    4.0 -5.0  10.000 -19.00000000 -33.00000000 -0.085937500  2.0104167 \n[25,] 25   -8.0  2.0  -9.000  19.00000000  21.00000000  0.054687500 -1.1354167 \n[26,] 26    8.0 -3.0  17.000 -28.00000000 -62.00000000 -0.161458333  3.9375000 \n[27,] 27   -1.0  9.0   3.000   7.00000000  -1.00000000 -0.002604167  0.2187500 \n[28,] 28   -1.0  6.0  -5.000  12.00000000  22.00000000  0.057291667 -1.3541667 \n[29,] 29   -0.5 -1.5  -1.250   0.16666667   0.08072917  0.080729167 -2.0937500 \n[30,] 30   -2.0  6.0  -1.000   9.00000000   4.00000000  0.010416667 -0.1041667 \n[31,] ⋮    ⋮    ⋮    ⋮       ⋮            ⋮            ⋮            ⋮          \n[32,] 49    3.5 -0.5   1.000  -0.3333333   -0.05989583 -0.059895833  1.34375000\n[33,] 50   -4.0 -1.0 -12.000  15.0000000   44.00000000  0.114583333 -2.89583333\n[34,] 51   -5.0  8.0  -8.000  21.0000000   29.00000000  0.075520833 -1.67708333\n[35,] 52    4.0  2.0   6.000  -8.0000000  -21.00000000 -0.054687500  1.36458333\n[36,] 53   -1.0  2.0  -7.000  10.0000000   21.00000000  0.054687500 -1.32291667\n[37,] 54    7.0  1.0  10.000 -16.0000000  -26.00000000 -0.067708333  1.56250000\n[38,] 55    4.0  2.0   3.000  -5.0000000   -8.00000000 -0.020833333  0.47916667\n[39,] 56   -9.0  3.0 -17.000  29.0000000   58.00000000  0.151041667 -3.62500000\n[40,] 57   -1.0  2.0  -7.000  10.0000000   25.00000000  0.065104167 -1.61458333\n[41,] 58    1.0 -2.0   4.000  -7.0000000  -20.00000000 -0.052083333  1.31250000\n[42,] 59   -5.0 -3.0 -11.000  13.0000000   27.00000000  0.070312500 -1.69791667\n[43,] 60   -3.0  2.0  -2.000   7.0000000    9.00000000  0.023437500 -0.51041667\n[44,] 61    2.0 -5.0   7.000 -14.0000000  -33.00000000 -0.085937500  2.11458333\n[45,] 62   -3.0  6.0   2.000   7.0000000   -4.00000000 -0.010416667  0.43750000\n[46,] 63    1.0  3.0   4.000  -2.0000000  -14.00000000 -0.036458333  0.97916667\n[47,] 64    7.0 -6.0  11.000 -24.0000000  -35.00000000 -0.091145833  2.05208333\n[48,] 65   -2.0 -1.0  -0.875   0.1875000    0.03906250  0.039062500 -0.90625000\n[49,] 66    6.0 -1.0   8.000 -15.0000000  -18.00000000 -0.046875000  1.00000000\n[50,] 67   -5.0  8.0  -3.000  16.0000000   13.00000000  0.033854167 -0.61458333\n[51,] 68    7.0 -3.0   6.000 -16.0000000  -14.00000000 -0.036458333  0.68750000\n[52,] 69   -2.0  7.0   0.000   9.0000000    2.00000000  0.005208333  0.04166667\n[53,] 70    4.0  2.0   5.000  -7.0000000   -6.00000000 -0.015625000  0.29166667\n[54,] 71   -3.0  2.0 -10.000  15.0000000   40.00000000  0.104166667 -2.60416667\n[55,] 72   -3.0 -1.0  -5.000   7.0000000   11.00000000  0.028645833 -0.65625000\n[56,] 73   -2.0  7.0   0.000   9.0000000    4.00000000  0.010416667 -0.10416667\n[57,] 74    3.0  2.0   9.000 -10.0000000  -27.00000000 -0.070312500  1.76041667\n[58,] 75    3.0  1.0   0.000  -2.0000000   10.00000000  0.026041667 -0.77083333\n[59,] 76   -2.0 -3.0  -3.000   2.0000000    8.00000000  0.020833333 -0.54166667\n[60,] 77   -7.0  5.0 -15.000  27.0000000   52.00000000  0.135416667 -3.22916667\n[61,] 78   -2.0  5.0  -5.000  12.0000000   18.00000000  0.046875000 -1.06250000\n      [,9]        [,10]       [,11]\n [1,]  14.3229167 -45.6250000   48 \n [2,]   2.0833333  -6.2500000    2 \n [3,]   6.4895833 -20.0416667   20 \n [4,]  11.3437500 -31.5416667   29 \n [5,]   8.1770833 -23.8750000   21 \n [6,] -18.0416667  59.8333333  -60 \n [7,]   2.5416667  -7.1666667    9 \n [8,] -17.3750000  54.7500000  -58 \n [9,]  13.8541667 -45.6666667   45 \n[10,]  18.6458333 -62.7500000   66 \n[11,] -24.9166667  85.7500000  -94 \n[12,]  -1.4270833   0.2083333    3 \n[13,]  -5.8958333  22.9166667  -29 \n[14,]  -8.3645833  27.7916667  -26 \n[15,]  -3.8437500  15.9583333  -19 \n[16,]   3.5625000 -12.6666667   14 \n[17,]  26.5833333 -86.8333333   90 \n[18,] -11.6562500  38.7916667  -40 \n[19,]  -8.5937500  29.0416667  -32 \n[20,]  17.7708333 -58.1666667   62 \n[21,]   7.0104167 -24.7083333   24 \n[22,]   7.0104167 -26.7083333   36 \n[23,] -16.6562500  52.2916667  -48 \n[24,] -16.0312500  49.2083333  -44 \n[25,]   7.2812500 -14.4583333    4 \n[26,] -33.2291667 112.5000000 -124 \n[27,]  -2.9895833  12.6250000  -15 \n[28,]  11.1458333 -36.3333333   35 \n[29,]  18.6770833 -65.1250000   71 \n[30,]  -0.5416667   7.4166667  -16 \n[31,] ⋮           ⋮           ⋮    \n[32,] -10.0104167  28.625000  -27  \n[33,]  25.4166667 -89.666667  104  \n[34,]  12.5729167 -36.041667   30  \n[35,] -11.7812500  41.541667  -50  \n[36,]  10.9062500 -33.958333   30  \n[37,] -11.9791667  34.750000  -30  \n[38,]  -3.6666667  11.583333  -16  \n[39,]  29.8958333 -98.000000  104  \n[40,]  13.8645833 -46.791667   50  \n[41,] -11.6666667  42.500000  -49  \n[42,]  13.7187500 -41.708333   37  \n[43,]   3.7812500 -11.708333   15  \n[44,] -18.2812500  64.291667  -73  \n[45,]  -5.3333333  23.000000  -27  \n[46,]  -9.1041667  34.083333  -41  \n[47,] -15.5104167  44.541667  -38  \n[48,]   6.8437500 -18.375000   14  \n[49,]  -6.6875000  14.250000   -4  \n[50,]   3.2395833  -4.291667   -1  \n[51,]  -3.6041667   3.250000    7  \n[52,]  -1.8958333  12.083333  -21  \n[53,]  -1.1875000  -2.666667   13  \n[54,]  22.7083333 -79.833333   92  \n[55,]   4.8854167 -12.875000    7  \n[56,]  -0.4166667   5.666667  -12  \n[57,] -15.0937500  50.708333  -51  \n[58,]   8.1458333 -34.916667   48  \n[59,]   4.7916667 -17.083333   24  \n[60,]  26.4583333 -85.583333   88  \n[61,]   8.1875000 -24.000000   19  "},"metadata":{}}]},{"cell_type":"markdown","source":"## The above (ans) is the required results, which the first 5 columns represent specific numbers from the differences table and the last 5 numbers is the polynomial coefficients. For example the interpolated polynomial can be written as\n$0.075520833x^4-1.7812500X^3+14.3229167x^2-45.625x+48$\n","metadata":{}}]}