HW3Solns - 12.2 Suppose n-vector of data is D and the n n...

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Unformatted text preview: 12.2 Suppose n-vector of data is D and the n n Vandermonde matrix of x j is Vx . 1 x1 d0 d 1 x2 1 D Vx d n1 1 xn x1n1 n x2 1 n xn 1 Then the degree n 1 polynomial interpolant of the data is VxC D of which c0 c C 1 . cn1 Since A maps D to an m-vector of sampled values { p( y j )} , we have AD Vy C n 1 y1 1 y2 Vy 1 ym 1 y1n1 n y2 1 m n1 ym 1 From above, we can attain that C Vx D and A Vy CD . Therefore we have A VyVx 1 . (b) 10 Semilog of Infinity Norm of A 10 5 10 0 10 0 10 20 30 (c) With the condition, we have D 1 1 column of Vx is 1 1 1 . Since VxC D and the first T 1 , it can be concluded that except the first element T of C being 1, others are zero. In particular, C 1 0 1 and D 1 1 AD Vy C 1 1 T 1 and F ( D) ( AD1 ) D1 J ( D) T 1 . If we regard AD as T F ( D) , which is AD F ( D) , we have D 0 . Then 0 AD1 D1 ( AD2 ) D2 AD2 D2 ( ADn ) Dn ADn Dn A . 1. In addition, According to 12.6, Therefore, J ( x) f ( x) / x . Then J ( D) F ( D) A VyVx 1 . The condition numbers n=1 1 n=2 1 n=3 1.25000000000000 n=4 1.62500000000000 n=5 2.17187500000000 n=6 2.99218750000000 n=7 4.26367187500000 n=8 6.29394531250000 n=9 9.61932373046876 n=10 15.1834411621094 n=11 24.6609878540046 n=12 41.0473136901850 n=13 69.7373995780222 n=14 120.509180783711 n=15 211.184145869046 n=16 374.409046122295 n=17 670.263206750751 n=18 1209.77020252113 n=19 2198.87390756793 n=20 4020.91399819004 for each n are as following J ( D) /D A . n=21 7391.69458492794 n=22 13651.7216227416 n=23 25318.1403960481 n=24 47129.2705501207 n=25 88025.1241302036 n=26 164909.552835531 n=27 309807.653886070 n=28 583512.246276936 n=29 1101542.90295973 n=30 2084427.64366012 (d) Figure 11.1 with x linearly spaced from -1 to 1 6 4 2 0 -2 -1 -0.5 0 0.5 1 From (c) we know that when n 11, 24.66 . In Figure 11.1, the infinity condition number is 63657.407 which is much higher than . Therefore the result is not close to the implicit bound. Complete code is as follows clc; clear; close all; n=30; infnA=zeros(1,30); f or n=1:30 m=2*n-1; x=linspace(-1,1,n); y=linspace(-1,1,m); Vx=fliplr(vander(x')); Vy=fliplr(vander(y')); Vy=Vy(:,1:n); A=Vy/Vx; if n==11 x0=x; Vx0=Vx; A0=A; end infnA(n)=norm(A,Inf); end semilogy(infnA); title('Semilog of Infinity Norm of A'); grid on; z=[0,0,0,1,1,1,0,0,0,0,0]; p=polyfit(x0,z,10); x1=linspace(-1,1,1000); y1=polyval(p,x1); figure; plot(x0,z,'bx',x1,y1,'r'); title('Figure 11.1 with x linearly spaced from -1 to 1'); grid on; f printf('Bound of Fig 11.1 is %f\n',cond(Vx0,Inf)); 13.2 (a) First it is sensible to shed light on the formula 13.2 that is x (m / ) . Since t the sign decides whether x is positive or negative and e e decides on the range of x, we can drop them and only observe m / . From the lecture, we know that t 1 m t or equivalently t 1 m t 1. Therefore, it can be concluded that t and t 1 belong to F. But t +1 does not belong to F because it needs t+1 %Part A for k = 1:5; [U,R1] = qr(randn(50)); [V,R1] = qr(randn(50)); S = diag(sort(rand(50,1), 'descend')); A = U*S*V'; [U2,S2,V2] = svd(A); [norm(U-U2) norm(V-V2) norm(A-U2*S2*V2')] end % norms of U-U2 and V-V2 are about 2, since the columns are % nearly the same up to a factor of -1. Thus an SVD [u,s,v] of U-U2 % will be essentially u = U2, s is diagonal with entries nearly 2 or nearly 0, % and v = identity. Hence the 2-norm of U-U2, i.e. the largest entry in s, is about 2. %Part B for k = 1:5; [U,R1] = qr(randn(50)); [V,R1] = qr(randn(50)); S = diag(sort(rand(50,1), 'descend')); A = U*S*V'; [U2,S2,V2] = svd(A); for k1 = 1:50; if norm(U2(:,k1) - U(:,k1)) > 1.95 U2(:,k1) = -U2(:,k1); V2(:,k1) = -V2(:,k1); end end [norm(U-U2) norm(V-V2) norm(A-U2*S2*V2') cond(A)] end % now errors are much smaller. Don't see a clear correlation with cond (A) for k = 1:5; lim1 = 10*rand; [U,R1] = qr(randn(50)); [V,R1] = qr(randn(50)); S = diag(sort(10.^(lim1*rand(50,1)), 'descend')); A = U*S*V'; [U2,S2,V2] = svd(A); for k1 = 1:50; if norm(U2(:,k1) - U(:,k1)) > 1.95 U2(:,k1) = -U2(:,k1); V2(:,k1) = -V2(:,k1); end end [norm(U-U2) norm(V-V2) norm(A-U2*S2*V2') cond(A)] end % Now cond(A) varies over 10 orders of magnitude, % and it seems that the errors are proportional to cond(A) %Part C for k = 1:5; [U,R1] = qr(randn(50)); [V,R1] = qr(randn(50)); S = diag(sort(rand(50,1).^6, 'descend')); A = U*S*V'; [U2,S2,V2] = svd(A); for k1 = 1:50; if norm(U2(:,k1) - U(:,k1)) > 1.9 U2(:,k1) = -U2(:,k1); V2(:,k1) = -V2(:,k1); end end [norm(U-U2) norm(V-V2) norm(A-U2*S2*V2') cond(A)] end % Now when A is ill-conditioned, see much larger errors in U2 and V2 than % in the product U2*S2*V2'. Numerical Linear Algebra: Homework 4 Due on Oct. 28, 2009 Da Kuang da.kuang@cc.gatech.edu 18.1 (a) A∗ A = (A∗ A)−1 = 3 3.0002 3.0002 3.00040002 150020001 −150010000 −150010000 150000000 A+ = (A∗ A)−1 A∗ = 10001 −5000 −5000 −10000 5000 5000 P = AA+ 10 = 0 0.5 0 0.5 0 0.5 0.5 (b) x = A+ b = 1 1 2 y = Ax = 2.0001 2.0001 (c) Da Kuang da.kuang@cc.gatech.edu Homework 4 18.1 (continued) κ(A) = 4.2429 × 104 θ = 39.2306o η=1 (d) b A y 1.2910 5.4775 × 104 x 5.4775 × 104 1.4699 × 109 (e1) For κb→y , it is attained when P (δb) = P δ b . The right singular vector of P with the largest ∗ −14 singular value is v1 = (1, 0, 0) . Set δb = 10 v1 , then computing in MATLAB with exact P yields: P δb Pb δb = 1.2934 b (e2) For κb→x , it is attained when A+ (δb) = A+ δ b . The right singular vector of A+ with the largest singular value is v1 = (−0.8165, 0.4082, 0.4082)∗ . Set δb = 10−14 v1 , then computing in MATLAB with exact A+ yields: δb A+ δb = 5.4239 × 104 A+ b b (e3) ∗ For κA→y , it is most likely to be attained when δA = (δp)vn , where δp is orthogonal to range(A). Compute the full SVD of A: −0.5773 0.8165 0 2.4496 0 −0.7071 −0.7071 A = U ΣV ∗ = −0.5773 −0.4082 −0.7071 0 5.7733 × 10−5 0.7071 −0.7071 −0.5773 −0.4082 0.7071 0 0 The right singular vector of A with the smallest singular value is v2 = (0.7071, −0.7071)∗ . The vector orthogonal to range(A) is u3 = (0, −0.7071, 0.7071)∗ . Then 0 0 ∗ u3 v2 = −0.5 0.5 0.5 −0.5 ∗ Set δA = 10−14 u3 v2 , then computing in MATLAB yields: (A + δA)(A + δA)+ b AA+ b δA = 3.4552 × 104 A Since for κA→y , 5.4775 × 104 is just an upper bound, this ratio of relative change 3.4552 × 104 is acceptable. 18.1 continued on next page. . . Page 2 of 5 Da Kuang da.kuang@cc.gatech.edu Homework 4 18.1 (continued) (e4) For κA→x , it is most likely to be attained when δA = δA1 + δA2 , where δA2 is defined to be δA in (e3). ˜ ˜ ˜ ˜ ˜ ˜ For δA1 , ideally it should satisfy A−1 (δ A)x = A−1 δ A x , where A and δ A are the version of A and δA after change of bases. Since δA2 counts for the dominant part in the expression of upper bound of condition number κA→x , we can simply take δA as δA2 , i.e. as δA in (e3). Computing in MATLAB yields: (A + δA)+ b A+ b δA = 1.4660 × 109 A 20.5 (a) The factorization A = LU takes the form that L is lower-triangular, and U is upper-triangular with all 1s on the diagonal. (b) Suppose A = LU , then AD = L(U D), so the factorization takes the form that L does not change, but the columns of U are rescaled. The solution is D−1 x0 in this case, where x0 is the solution of Ax = b. (c) The final factorization takes the form that A = LDU , where L is lower-triangular with all 1s on the diagonal, D is diagonal, and U is upper-triangular with all 1s on the diagonal. 22.1 We can prove by induction. m = 1: A = [a11 ] = 1 · [u11 ] = LU , where a11 = u11 . In this case, ρ = 1 = 21−1 . m > 1: In the first step of LU factorization, before the factorization we switch the row r with the first row, where ar1 = max1≤i≤m . Let A(1) be the resulting matrix after the first step factorization. Then for any 1 ≤ r, s ≤ m, |a(1) | ≤ |ars | + |a1s | ≤ 2 max |aij | rs i,j Therefore (1) max |aij | ≤ 2 max |aij | i,j i,j After this step, we are left with a submatrix with dimention m − 1 to be factorized. Since ρm−1 ≤ 2m−2 22.1 continued on next page. . . Page 3 of 5 ...
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