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Unformatted text preview: CAS708/CES700 07. 1 Assignment 2 Due . In class Oct. 5 (Friday) 1. The following figures from the Census Bureau give the population of the United States: Year Population 1900 75,994,575 1910 91,972,266 1920 105,710,620 1930 122,775,046 1940 131,669,275 1950 150,697,361 1960 179,323,175 1970 203,235,298 • Since there are eight points, there is a unique polynomial of degree 7 which interpolates the data. However, some of the ways of representing this polynomial are computationally more satisfactory than others. Here are four possibilities, each with t ranging over the interval 1900 ≤ t ≤ 1970: 7 summationdisplay j =0 a j t j , 7 summationdisplay j =0 b j ( t- 1900) j , 7 summationdisplay j =0 c j ( t- 1935) j , 7 summationdisplay j =0 d j parenleftbigg t- 1935 35 parenrightbigg j . In each case, the coefficients are found by solving an 8-by-8 Vandermond system, but the matrices of various systems are quite different. Set up each of the four matrices, and find the estimate of its condition using Matlab function cond() . Then use Matlab operator “ \ ” to find the coefficients. Check each of the representations to see how well it reproduces the original data. • Interpolate the data by a 7th-degree polynomial, using the best conditioned representa- tion found above, and by the natural cubic spline using ncspline.m . Graph the resulting functions at one-year intervals over the period from 1900 to 1980. Find the 1980 census data. Which approach gives more accurate prediction? Solution The condition numbers: model a model b model c model d cond 1.239e+32 1.785e+13 7.891e+10 5.354e+2 CAS708/CES700 07. 2 Relative errors of the reproduced data by evaluating the polynomials using the Horner’s rule: model a model b model c model d relative error 4.0- 3 4.7e- 14 2.3e- 16 3.3e- 16 Predictions for 1980: model d spline real prediction 402.33 million 227.15 million 226.44 million 2. In the natural spline, we set σ 1 = σ n = 0. The following is an alternative way of setting σ 1 and σ n . Let c 1 ( x ) and c n ( x ) be the unique cubics which pass through the first four and the last four data points, respectively. The two end conditions match the third derivatives of s ( x ) to the third derivatives of these cubics, namely s ′′′ ( x 1 ) = c ′′′ 1 and s ′′′ ( x n ) = c ′′′ n . The constants c ′′′ 1 and c ′′′ n can be determined directly from the data without actually finding c 1 ( x ) and c n ( x ). We have already introduced the quantities Δ i = y i +1- y i x i +1- x i , which are approximations to the first derivatives. Let Δ (2) i =...
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This note was uploaded on 11/03/2011 for the course ENGINEERIN 708 taught by Professor Qiao during the Spring '11 term at McMaster University.
- Spring '11