How do the con dition numbers compare explain your

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routine for condition estimation. How do the con- dition numbers compare? Explain your results. ( b ) Using the best-conditioned basis found in part a , compute the polynomial interpolant to the pop- ulation data. Plot the resulting polynomial, us- ing Horner’s nested evaluation scheme to evaluate the polynomial at one-year intervals to obtain a smooth curve. Also plot the original data points on the same graph. ( c ) Use a routine for Hermite cubic interpolation, such as pchip from MATLAB or Netlib , to compute a monotone Hermite cubic interpolant to the pop- ulation data and again plot the resulting curve on the same graph. ( d ) Use a cubic spline routine to interpolate the population data and again plot the resulting curve on the same graph. ( e ) Extrapolate the population to 1990 using each of the polynomial, Hermite cubic, and cu- bic spline interpolants and compare the values ob- tained. How close are these to the true value of 248,709,873 according to the 1990 census? ( f ) Determine the Lagrange interpolant to the same nine data points and evaluate it at the same yearly intervals as in parts b and c . Compare the total execution time with those for Horner’s nested evaluation scheme and for evaluating the cubic spline. ( g ) Determine the Newton form of the polyno- mial interpolating the same nine data points. Now determine the Newton polynomial of one degree higher that also interpolates the additional data point for 1990 given in part d , without starting over from scratch (i.e., use the Newton polynomial of degree eight already computed to determine the new Newton polynomial). Plot both of the result- ing polynomials (of degree eight and nine) over the interval from 1900 to 1990. ( h ) Round the population data for each year to the nearest million and compute the correspond- ing polynomial interpolant of degree eight using the same basis as in part b . Compare the result- ing coefficients with those determined in part b . Explain your results. Downloaded 01/23/19 to 206.87.124.222. Redistribution subject to SIAM license or copyright; see

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