mx4x3x2xonessizexy m 16 8 4 2 1 26 1 1 1 1 1 2 1 1 1

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Unformatted text preview: 128 Note that the first column of the augmented matrix (9) is crafted by raising every element of the vector x to the fourth power. The second column is built by raising each element of the vector x to the third power, and so on. The last column of the augmented matrix (9) is a vector of ones that has the same size as the vector x. Matlab’s elementwise operators make it extremely easy to build the augmented matrix. >> M=[x.ˆ4,x.ˆ3,x.ˆ2,x,ones(size(x)),y] M= 16 -8 4 -2 1 26 1 -1 1 -1 1 -2 1 1 1 1 1 -4 16 8 4 2 1 -2 256 64 16 4 1 128 Next, place the augmented matrix in reduced row echelon form. The Interpolating Polynomial title page contents previous page next page back print doc close exit >> R=rref(M) R= 1 0 0 1 0 0 0 0 0 0 The 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 -2 0 1 -4 Interpolating Polynomial Hence, a = 1, b = −2, c = 0, d = 1, and e = −4. Substitute these values in the general form y = ax 4 + bx 3 + cx 2 + dx + e and the interpolating polynomial is y = 1x 4 − 2x 3 + 0x 2 + 1x − 4, or p(x) = x 4 − 2x 3 + x − 4. Enter this polynomial at the Matlab prompt as follows: >> p=[1 -2 0 1 -4]’ p= 1 -2 0 1 -4 Again, if the interpolating polynomial does not pass through each of the original data points, then our answer is wrong. So, let’s plot the data points, then determine whether our solution passes through each data point. Recall that the original data are still stored in the vectors x and y. We need only create a set of data points satisfying our polynomial. The minimum x -value in our data set is −2 and the maximum x -value is 4. Let’s plot the polynomial on the interval [−3, 5]. This will insure that each of our data points is visible in the final plot. You can use column vectors with the plot command as easily as row vectors. The following commands produce an image similar to that in Figure 5. Because the polynomial passes through each data point, it is highly likely that we have found the correct interpolating polynomial. title page contents previous page next page back print doc close exit >> xp=(linspace(-3...
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This document was uploaded on 02/14/2014.

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