hw2a97f - Homework#2 MEAM 501 Matrix/Analytical Methods in...

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1 Homework #2 MEAM 501 Matrix/Analytical Methods in Engineering ________________________________________________________________________ Kikuchi 97F Suppose that the following data have been obtained in a laboratory: Time (t) Response (x) Standard Deviation ( σ ) _____________________________________________________________ 0 7 0.02 0.1 4 0.2 -2 0.3 3 0.05 0.4 -1 0.5 0.07 0.6 0.7 0.8 1 0.9 1.0 0.03 1.1 1.2 2 1.3 1.4 1.5 8 1.6 12 1.7 18 0.01 1.8 1.9 2.0 Using the following basis functions: f 1 (t) = 1 f 2 (t) = t f 3 (t) = t 2 f 4 (t) = t 3 f 5 (t) = t 4 f 6 (t) = exp (-t) f 7 (t) = exp (-(t-1.7) 2 )
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2 f 8 (t) = cos (1.7 - t) we would like to make a curve fit of the data obtained as follows: xt c f t kk k (29 = (29 = 1 8 where c k , k = 1, . ..., 8, are the coefficients which must be determined. If a merit function is defined by Ξ 2 1 8 1 21 2 = - (29 = = xc f t ik k i k i i σ where (t i , x i , σ i ), i = 1, . .., 21, are the data set, then the coefficient c k , k = 1, . .., 8, are determined so as to minimize the merit function Ξ 2 . That is, the coefficients c k , k = 1, . .., 8, of the curve fit is the solution of the least squares problem . If a matrix A = [a i j ] and a vector b = {b i } are defined by a ft b x ij ji i i i i = = σσ and the merit function X 2 can be written as Ξ 2 1 2 =- - bA c bA c T . (a) Determine the coefficient matrix A and the vector b using the data given. (b) Obtain the range R( A ) of the matrix A . What is the dimension of R( A )? (c) Obtain the null space N( A ). What is the dimension of N( A (d) Using the Householder transformation, modify the matrix AA T to the form of the matrix T * such that its lower triangular portion becomes zero, that is,
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3 AA T ij ij (29 = * , 0 . (e) Make a LU decomposition of T . (f) Compute det T and rank T . (g) Compute det AA T and rank AA T . (h) Tridiagonalize the matrix T . (i) Compute the 2 norm || T || 2 of the matrix T . (j) Find the minimizer c of the merit function Ξ 2 . (k) Using c obtained in (j), plot the curve x(t) obtained. (l) If the singular value decomposition of the matrix A is obtained as AUV T , the covariance of the coefficient c k , k=1, . .., 8, is defined by Cov c c jk T jk , = ++ VV Σ Σ .
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hw2a97f - Homework#2 MEAM 501 Matrix/Analytical Methods in...

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