The Gram-Schmidt Procedure and Orthogonal Projections 1 ...

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Unfortunately, the naive Gram-Schmidt procedure is numerically unstable, which meansthat it is highly sensitive to numerical errors. For example, try running NAIVEGS on the7th order Hilbert matrix:H7 = [ 1/1 1/2 1/3 1/4 1/5 1/6 1/7;1/2 1/3 1/4 1/5 1/6 1/7 1/8;1/3 1/4 1/5 1/6 1/7 1/8 1/9;1/4 1/5 1/6 1/7 1/8 1/9 1/10;1/5 1/6 1/7 1/8 1/9 1/10 1/11;1/6 1/7 1/8 1/9 1/10 1/11 1/12;1/7 1/8 1/9 1/10 1/11 1/12 1/13 ];C = NAIVEGS(H7);Z = abs(C * C’ - eye(7))In this computation, eye(7) denotes the 7 × 7 identity matrix, and abs computes theabsolute value of each matrix component.Ideally, what should Z be?What is the actual largest entry in Z?Fortunately, the Gram-Schmidt procedure can be made numerically stable via a simplemodification. Instead of computing∑i−1u i = v i − proj wk(v i )as in the naive Gram-Schmidt algorithm, compute u i in a series of steps:k=1u (1)i = v i − proj w1(v i )u (2)i = u (1)i − proj w2(u (1)i )u (3)i = u (2)i − proj w3(u (2)i ). .u i = u (i−2)i − proj wi−1(u (i−2)i )Copy your file NAIVEGS.m to the file GS.m and modify GS.m to implement the modifiedGram-Schmidt procedure as the functionfunction w = GS(v)4
Using your new modified GS procedure, try running:D = GS(H7);Z = abs(D * D’ - eye(7))Now what is the largest entry in Z?3 Orthogonal ProjectionsLet P be a finite-dimensional subspace of an inner-product space V . Once an orthonormalbasis {w 1 , w 2 , . . . , w m } for P is obtained, the orthogonal projection of an arbitrary vectorv ∈ V on P is easily computed asˆv =m∑〈v, w i 〉w i .i=1The vector ˆv is the orthogonal projection of v on P and the components of the m-tuple(〈v, w 1 〉, 〈v, w 2 〉, . . . , 〈v, w m 〉)are the coordinates of ˆv with respect to the given orthonormal basis.Let B be an m × l matrix whose rows form an orthonormal basis for subspace P . (Forexample, B might be produced by your GS function.) Let v be an arbitrary 1 × l rowvector, and let ˆv be the orthogonal projection of v on P .• Give a Matlab or Octave expression (in terms of B and v) for the coordinatesof ˆv with respect to basis given in the rows of B.• Give a Matlab or Octave expression (in terms of B and v) for ˆv.5
Page 2 and 3: Show that proj v (u) is the orthogo
Page 6 and 7: For example, let P be the subspace
Page 8: hold off; plot(f); hold; plot(g);Ex

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