Question

Year    Population
1900     200507
1910     430980
1920     732016
1930    1265258
1940    1394711
1950    1451277
1960    1424815
1970    1471701
1980    1168972
1990    1203789
2000    1332650
2010    1385108

the census populations of the Bronx from 1900 to 2010 is given above. Estimate the Bronx's 2005 population using the following methods:

(a) Linear interpolation interp1 (year, population, 2005)) (b) Polynomial interpolation of just the last three censuses (polyval and polyfit) (c) Polynomial interpolation of all the censuses (d) Second-degree polynomial regression (not interpolation; polyfit(year, population, 2)) of all the censuses (e) Cubic spline interpolation (not-a-knot boundary conditions) using all the censuses (spline) (f) Cubic spline interpolation (natural boundary conditions) using all the censuses ( csape with the 'variational' option) (g) Piecewise cubic Hermite interpolation using all the censuses (pchip)

Answer 1

Matlab code for interpolation clear all close all all year and populations year-1900 10:2010; pop [200507 430980 732016 1265258 1394711 1451277 1424815 1471701 1168972 1203789 1332650 1385108; %linear interpolation interp 1 popl-interpl (year, pop, 2005); fprintf("\n\ta.Línear interpolation (interp! for population of 2005) %f\n' ,pop!) %polynomial interpolation last three censuses nn length (year); plp2-polyfit(year 10:12) ,pop (10:12),1); pop2-polyval (plp2, 2005); fprintfntb.polynomial interpolation last three censuses (polyfit and polyval)=%f\n", pop2) polynomial interpolation all censuses plp3-polyfit(year, pop,1); pop3-polyval (plp3,2005); fprintf nltc-polynomial interpolation of all censuses (polyfit and polyva1 )=%f\n ' , pop3 ) 2nd degree polynomial regression plp4-polyfit(year, pop, 2); pop4-polyval (plp4,2005); fprint f ( "\n\td . 2nd degree n",pop4) polynomial regression of all =% f censuses %cubic spline interpolation using all cenc sus pop5-spline (year, pop, 2005) fprintf( .\n\te.cubic spline interpolation using all cenc sus-%f n' ,pop5) Cubic spline interpolation using all cencsus using csape pop6- csapi(year,pop,2005); fprintfnltf.Cubic spline interpolation using all cencsus using %f\n',pop6) csape- %Piecewise cubic Hermite polynom.aï using all census pop7pchip(year, pop, 2005); fprint f ( "\n\tg .Piecewise cubic Hermite polynomial using all census n' pop?) f 88%88888888888888 End of Code 8888888888888888옹

a.Linear interpolation (interpl for population of 2005) 1358879.000000 b.polynomial interpolation last three censuses (polyfit and polyval) -1352512.083333 c.polynomial interpolation of all censuses (polyfit and polyval)-1544387.536131 values, described Warning: Polynomial is badly conditioned. Add points with distinct x reduce the degree of the polynomial, or try centering and scaling as in HELP POLYFIT. d. 2nd degree polynomial regression of al1 censuses -1243981.596528 e.cubic spline interpolation using all cencsus -1372131.834779 f.cubic spline interpolation using all cencsus using csape 1372131.834779 g.Piecewise cubic Hermite polynomial using all census 1366417.238100 Published with MATLABE R2018a

%Matlab code for interpolation
clear all
close all
%all year and populations
year=1900:10:2010;
pop=[200507 430980 732016 1265258 1394711 1451277 1424815 1471701 1168972 1203789 1332650 1385108];

%linear interpolation interp1
pop1=interp1(year,pop,2005);
fprintf('\n\ta.Linear interpolation (interp1 for population of 2005) = %f\n',pop1)

%polynomial interpolation last three censuses
nn=length(year);
plp2=polyfit(year(10:12),pop(10:12),1);
pop2=polyval(plp2,2005);

fprintf('\n\tb.polynomial interpolation last three censuses (polyfit and polyval)=%f\n',pop2)

%polynomial interpolation all censuses

plp3=polyfit(year,pop,1);
pop3=polyval(plp3,2005);

fprintf('\n\tc.polynomial interpolation of all censuses (polyfit and polyval)=%f\n',pop3)

%2nd degree polynomial regression

plp4=polyfit(year,pop,2);
pop4=polyval(plp4,2005);
fprintf('\n\td.2nd degree polynomial regression of all censuses =%f\n',pop4)

%Cubic spline interpolation using all cencsus
pop5=spline(year,pop,2005);
fprintf('\n\te.Cubic spline interpolation using all cencsus =%f\n',pop5)

%Cubic spline interpolation using all cencsus using csape
pop6= csapi(year,pop,2005);
fprintf('\n\tf.Cubic spline interpolation using all cencsus using csape = %f\n',pop6)

%Piecewise cubic Hermite polynomial using all census
pop7 = pchip(year,pop,2005);
fprintf('\n\tg.Piecewise cubic Hermite polynomial using all census =%f\n',pop7)

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