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8. Gram-Schmidt Orthogonalization

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8. Gram-Schmidt Orthogonalization Chapter 8 Gram-Schmidt Orthogonalization (September 8, 2010) _______________________________________________________________________ Jørgen Pedersen Gram (June 27, 1850 – April 29, 1916) was a Danish actuary and mathematician who was born in Nustrup, Duchy of Schleswig, Denmark and died in Copenhagen, Denmark. Important papers of his include On series expansions determined by the methods of least squares, and Investigations of the number of primes less than a given number. The process that bears his name, the Gram–Schmidt process, was first published in the former paper, in 1883. http://en.wikipedia.org/wiki/J%C3%B8rgen_Pedersen_Gram ________________________________________________________________________ Erhard Schmidt (January 13, 1876 – December 6, 1959) was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia (now Estonia). His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations. Together with David Hilbert he made important contributions to functional analysis. http://en.wikipedia.org/wiki/File:Erhard_Schmidt.jpg ________________________________________________________________________ 8.1 Gram-Schmidt Procedure I Gram-Schmidt orthogonalization is a method that takes a non-orthogonal set of linearly independent function and literally constructs an orthogonal set over an arbitrary interval and with respect to an arbitrary weighting function. Here for convenience, all functions are assumed to be real. un(x) linearly independent non-orthogonal un-normalized functions Here we use the following notations. n un (x) x (n = 0, 1, 2, 3, …..). n (x) linearly independent orthogonal un-normalized functions n (x) linearly independent orthogonal normalized functions with b ( x) (x)w(x)dx i j i , j . a Starting with n = 0, let 0 (x) u0 (x) , (x) (x) 0 . 0 b 2w(x)dx 0 a For n = 1, 1(x) u1(x) a100 (x) . We demand that 1(x) be orthogonal to 0 (x). b b b (x) (x)w(x)dx 0 u (x) (x)w(x)dx a (x)2 w(x)dx 1 0 1 0 10 0 a a a or b a u (x) (x)w(x)dx . 10 1 0 a Normalizing, we define (x) (x) 1 . 1 b 2w(x)dx 1 a Generalizing, we have (x) (x) i , i b 2w(x)dx i a where i (x) ui (x) ai0 0 (x) ai11 (x) ai 2 2 (x) ... ai,i 1i 1 (x) . The coefficient (aij) are given by b a u (x) (x)w(x)dx . ij i j a since b (x) (x)w(x)dx 0 i j a b {u (x) a (x) a (x) a (x) ... a (x)} (x)w(x)dx i i0 0 i1 1 i2 2 i,i1 i1 j a b b 0 u (x) (x)w(x)dx a [ (x)]2 w(x)dx . i j ij j a a 8.2 Example: Hermite polynomials The Hermite polynomial Hn(x) can be derived from the orthogonalization, n un (x) x (n = 0,1,2,3, ), with the weight function 2 w(x) e x . Starting with n = 0, let u ( x) 1 ( x) 0 0 1 / 4 2 e x dx For n = 1, let 1(x) u1(x) a100 (x) , 2 2 0 (x)e x (x)dx (x)e x [u (x) a (x)]dx 0 1 0 1 10 0 or 2 0 (x)e x u (x)dx a 0 1 10 or 2 1 2 a (x)e x u (x)dx xe x dx 0 10 0 1 1/ 4 Then we have 1(x) u1(x) . u (x) 2x (x) 1 . 1 1/ 4 2 x 2ex dx For n = 2, let 2 (x) u 2 (x) a20 0 (x) a211 (x) , 2 2 0 (x)e x (x) (x)e x [u (x) a (x)]dx , 0 2 0 2 20 0 or 2 2 1 1 a u (x)e x (x)dx x2e x dx , 20 2 0 1/ 4 1/ 4 2 2 2 2x a u (x)e x (x)dx x2e x dx 0, 21 2 1 1/ 4 1 1 1 (x) x2 x2 . 2 2 1/ 4 1/ 4 2 Normalizing, (x) (x) 2 2 2 (x)e x (x)dx 2 2 1 x2 2 1 2 (x4 x )e x dx 4 or 2 1 x 2 2x 1 (x) 2 . 2 1 1/ 4 ( 1/ 2 )1/ 2 2 2 For n = 3, let 3 (x) u3 (x) a300 (x) a311(x) a322 (x) , 2 2 0 (x)e x (x) (x)e x [u (x) a (x)]dx , 0 3 0 3 30 0 or 2 2 1 a u (x)e x (x)dx x3e x dx 0 , 30 3 0 1/ 4 2 2 2 3 a u (x)ex (x)dx x4ex dx 2 1/ 4 , 31 3 1 1/ 4 4 2 1 x x 2 3 x 2 2 a32 u3 (x)e 2 (x)dx x e dx 0 , 1 1/ 2 1/ 2 ( ) 2 3 1/ 4 2x 3 (x) x3 2 x3 x . 3 4 1/ 4 2 Normalizing, 3 (x) 3 (x) 2 (x)ex (x)dx 3 3 3 3 x x 1/ 4 2 (2x3 3x) 3 2 3 (x3 x)2 ex dx 2 We find that n (x) is proportional to the Hermite polynomial. The Hermite polynomials are given by n Hn(x) 01 12x 2 2 4x2 3 12 x 8x3 412 48 x2 16 x4 5120x 160 x3 32 x5 6 120 720 x2 480 x4 64 x6 7 1680 x 3360 x3 1344 x5 128 x7 81680 13 440 x2 13 440 x4 3584 x6 256 x8 930240x 80 640 x3 48 384 x5 9216 x7 512 x9 2 4 6 8 10 10 30 240 302 400 x 403 200 x 161 280 x 23 040 x 1024 x n 8.3 Independence of un(x) = x ; Wronskian n We show the independence of {un(x) = x } using the Wrtonskian determinant. ((Mathematica)) Wronskian G k_ : Table xn, n,0,k W k_, m_ : D G k , x, m WR k_ : Table W k, m , m, 0, k G 6 1, x, x2,x3,x4,x5,x6 WR 6 1, x, x2,x3,x4,x5,x6 , 0, 1, 2 x, 3 x2,4x3,5x4,6x5 , 2 3 4 2 3 0, 0, 2, 6 x, 12 x ,20x,30x , 0, 0, 0, 6, 24 x, 60 x , 120 x , 2 0, 0, 0, 0, 24, 120 x, 360 x , 0, 0, 0, 0, 0, 120, 720 x , 0, 0, 0, 0, 0, 0, 720 WR 6 MatrixForm 1x x2 x3 x4 x5 x6 2 3 4 5 0 1 2x 3x 4x 5x 6x 00 2 6x12x2 20 x3 30 x4 00 0 6 24x60x2 120 x3 0 0 0 0 24 120 x 360 x2 0 0 0 0 0 120 720 x 00 0 0 0 0 720 Table Det WR i , i,1,8 1, 2, 12, 288, 34 560, 24 883 200, 125 411 328 000, 5 056 584 744 960 000 8.4 Gram-Schmidt orthogonalization procedure II We consider a family of functions u1(x), u2(x), u3(x), ……., un(x),…. where {un{x}} is independent function and b (u ,u ) a u (x)w(x)u (x)dx i j ij i j a Suppose that the relation is always valid for the range a x b . a1u1(x) a2u2 (x) ... anun (x) 0 Then we have, a1 = a2 = …= an = 0. We now construct a linear combination of the functions a11 a12 a1n a21 a22 a2n n (x) an1,1 an1,2 an1,n u1(x) u2 (x) un (x) This function is always orthogonal to ui(x). Thus n (x) is orthogonal to 1(x),2 (x),...,n1(x) . We introduce a Gram determinant, An. A0 = 1 and a11 a12 a1n a21 a22 a2n An an1,1 an1,2 an1,n an,1 an,2 an,n a11 a12 a1n a21 a22 a2n (n ,n ) (un ,n )An1 an1,1 an1,2 an1,n 0 0 (un ,n ) a11 a12 a1n a11 a12 a1n a21 a22 a2n a21 a22 a2n (un ,n ) An an1,1 an1,2 an1,n an1,1 an1,2 an1,n (un ,u1) (un ,u2 ) (un ,un ) an,1 an,2 an,n Then we have (n ,n ) An An1 or Then we have the final form of the orthogonal (normalized) functions, n n (n = 1, 2, …) An1An where A0 = 1. 8.5 Legendre polynomials Gram-Schmidt orthogonalization procedure for the Legendre polynomials. We use the method discussed in 8.4. n w(x) = 1 and un(x) = x , for -1≤x≤1. ((Mathematica)) Clear "Global`" weight 1; u i_, x_ : xi; a i_,j_ : Integrate weight u i, x u j, x , x, 1, 1 , GenerateConditions False gra m n_ : Det Table a i, j , i,0,n , j,0, n ; gram 1 1; n_, x_ : Det Append Table a i, j , i,0,n 1 , j,0,n , x ^Range 0, n Simplify; n, x n_, x_ : ; gram n 1 gram n Table n, n, x , LegendreP n, x , n, 0, 5 Simplify TableForm 0 1 1 2 1 3 xx 2 2 1 5 1 3x2 1 1 3x2 2 2 2 3 1 7 x 3 5x2 1 x 3 5x2 2 2 2 3 330 x235 x4 4 1 3 30 x2 35 x4 8 2 8 5 1 11 x 15 70 x2 63 x4 1 x 15 70 x2 63 x4 8 2 8 fn x Legendre n = 4 2 1 n = 1 n = 0 n = 2 x -1.0 -0.5 0.5 1.0 -1 n = 3 n = 5 -2 Fig.
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