HW #3 Solutions: M552 Spring (c)
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1 HW #3 Solutions: M55 Spring 6. (4.-Trefethen & Bau: parts (a), (c) and (e)) Determine the SVDs of the following matrices (by hand calculation): (a) [ 3 (c) (e) [ ANS: In each case we seek A UΣV. The general algorithm is to first find A A (UΣV ) UΣV V Σ ΣV V DV, the spectral decomposition of the hermitian matrix A A. Then take Σ D, noting that Σ may have to be extended to the size of A, i.e. zero rows appended, and solve for U from AV UΣ. However, if A A, then we can use A s spectral decompostion A PDP P D sign(d)p UΣV, where U P, Σ D and V sign(d)p. (a): A A, and in fact A is already in diagonal form. So D A and P I. Then [ [ [ A PDP 3 IDI I D sign(d)i UΣV. [ [ (c): Following the general algorithm, A A I I 4 4, the spectral decompostion of A A. Note that the diagonal entries of D are not decreasing. We can achieve this by interchanging the columns of I and the diagonal entries of D, which simply amounts to reordering the e-pairs of [ [ [ [ [ [ 4 4 A A, giving A A. Then extending, 4 [ Σ and V. Now choose U unitary so that AV UΣ, or [u u u 3. We see that this gives U I. Thus, A [ UΣV. (e): A A, and it is easy to see that [ the e-values of A [ are λ and λ, and corresponding / mutually orthogonal e-vectors p / / and p / (do the computation if you don t see this). Then A PDP [ / / / / [ [ / / / / UΣV.
2 . (4.3-Trefethen & Bau) Write a MATLAB program which, given a real matrix A, plots the right singular vectors v and v in the unit circle and also the left singular vectors[ u and u in the appropriate ellipse, as in Figure 4.. Apply your program to the matrix (3.7), A, and also the matrices of exercise 4.. You may use MATLAB s svd command to generate the singular vectors and values. ANS: Here is the code for a given matrix A: [U,S,Vsvd(A); Sdiag(S); % find SVD th:*pi/56:*pi; dom[cos(th); sin(th); rana*dom; % define points on unit circle and image subplot(,,) plot(dom(,:),dom(,:),. ),axis( image ),grid,hold arrow([,,[v(,),v(,)) arrow([,,[v(,),v(,)) title( Right Singular Vectors ) subplot(,,) plot(ran(,:),ran(,:), r. ),axis( image ),grid,hold arrow([,,s()*[u(,),u(,)) if (abs(s()) > e-) % check that singular value is nonzero arrow([,,s()*[u(,),u(,)) end title( Image & Left Singular Vector(s) ) % arrow.m is a user-contributed M-file available at Results for A [ : Right Singular Vectors Image & Left Singular Vector(s)
3 [ 3 Results for A :.5 Right Singular Vectors Image & Left Singular Vector(s) Results for A [ 3 : 3 Image & Left Singular Vector(s) Right Singular Vectors
4 [ Results for A : Right Singular Vectors Image & Left Singular Vector(s) Results for A [ : Right Singular Vectors Image & Left Singular Vector(s)
5 3. (4.5-Trefethen & Bau) Theorem 4. asserts that every A C m n has an SVD A UΣV. Show that is A is real, then it has a real SVD (U R m m,v R n n ). ANS: If A R m n then A A A T A R n n is symmetric, hence A T A V DV T where D R n n is a real diagonal matrix, with non-negative entries (we proved this in class), and V R n n is a real orthogonal matrix. Hence Σ D R n n is defined. If m > n we can append zero rows to Σ to ensure Σ R m n. What if m < n? Think about it! We can solve for U using AV UΣ, showing that U can be chosen to ensure U R m m since A,Σ and V are all real matrices.
6 [ 4. (5.-Trefethen & Bau) In Example 3. we considered the matrix (3.7), A, and asserted, among other things, that its -norm is approximately.98. Using the SVD, work out (on paper) the exact values of σ min (A) and σ max (A) for this matrix. ANS: To find the singular values, form A A (UΣV ) UΣV V Σ ΣV V DV, the spectral decomposition of the hermitian matrix A A. The e-values are the diagonal entries of D, and Σ D. [ [ [ [ A A A A λi λ 8 λ 9λ + 4, 8 λ whose roots are λ, 9 ± ( 9) 4 4 σ min (A) giving A σ max (A) ± 65. So we have , σ max (A) ,
7 5. (5.4-Trefethen & Bau) Suppose A C m m has an [ SVD A UΣV. Find an eigenvalue decomposition A (5.), XΛX, of the m m hermitian matrix. A ANS: Working on it!
8 6. Find the eigenvalues and a corresponding eigenfunction for the operator d /dx applied to functions with homogeneous boundary conditions on [,, i.e., solutions of y λy, y() y(). Note that there are a infinite number of them, say (λ k,y k (x)) for k,,..., with < λ < λ <... ANS: We have to solve a second order constant coefficient ODE, whose solution must also satisfy the given boundary conditions (BCs). We have y + λy r + λ r, ± λ. The specific form of the solution depends on the sign of λ as follows: λ < : So λ >. The general solution is given by y(x) C e λx + C e λx. Applying the BCs, or in matrix form [ e λ e λ [ C C y() C e λ + C e λ C + C, y() C e λ + C e λ C e λ + C e λ, [ C C since [ e λ e λ so the system has only the trivial solution, thus y(x) which can t be an eigenfunction. e λ e λ, λ : The general solution is given by y(x) C + C x. Applying the BCs, y() C + C C C, y() C C C. So again, we arrive at the trivial solution y(x) which can t be an eigenfunction. λ > : So λ < which gives rise to complex roots r,. In this case the general real solution is given by y(x) C cos λx + C sin λx. Applying the BCs, y() C cos λ + C sin λ C + C C C, y() C sin λ C sin λ. Choosing C gives y(x), the trivial solution. For a nontrivial solution we must have sin λ, which is satisfied by an infinite number of λ >, given by (note λ > ) λk kπ for k,,..., λ k (kπ) for k,,.... These {λ k } k are thus the eigenvalues of the continuous second derivative operator for functions defined on [, with homogeneous BCs. For each k, taking C, a corresponding eigenfunction is given by y k (x) sin λ k x sin kπx. Summarizing, the e-pairs are given by (λ k,y k (x)) ((kπ),sin kπx) for k,,..., and note < λ < λ <... with lim k λ k.
9 7. Consider the discrete analog of the eigenproblem in the previous problem, with d /dx D + D D. To this end, choose N >, let h /N and x i i h for i,,...,n. Note we now have N + grid points with x and x N. So we seek e-pairs which satisfy or in matrix form, ( D u) i u i + u i u i+ h λu i for i,,...,n, u u N, h u u. u N u N λ u u. u N u N (a) Show that (λ k,u k ) is an e-pair for k,,...,n where λ k ( cos kπh)/h and u k is the vector with components (u k ) i sin ikπh. Hint: Use trig identities, and note that sin kπh sin Nkπh. (b) For each N 8 and N 6 plot graphs (using subplot) of the continuous eigenfunctions found in the previous problem, and the discrete eigenvectors from (a) for k,, N/ and N. Thus each plot should have 4 images. Use a linetype for the continuous eigenfunctions and symbols for the discrete eigenvectors. Label and title your graphs. (c) In a single plot display the first 3 eigenvalues vs. there number, i.e., k,,...,3 of the continuous problem along with the N eigenvalues for each N 8,6 and 3. For the continuous case plot the eigenvalues using a linetype with a symbol, and in the discrete case just symbols. Label, title, and place a legend on your plot. Discuss the results. ANS: (a) Noting that sin kπh sin Nkπh. Fix a k, k,...,n. For i,...,n, u k,i + u k,i u k,i+ h sin (i )kπh + sin ikπh sin (i + )kπh h (sin ikπhcos kπh cos ikπhsinkπh)+ sinikπh (sin ikπhcos kπh+cos ikπhsin kπh) h sin ikπh cos kπh + sin ikπh h ( cos kπh) sin ikπh h λ k sin ikπh. λ k u k,i. So we have Au k λ k u k for k,...,n, where A is the matrix representing D. Thus, (λ k,u k ) are e-pairs for k,...,n.
10 (b) Here is the code for N 8: N8; h/n; x:h:; xx:.:; k[ N/ N-; for i:4 subplot(,,i) plot(xx,sin(k(i)*pi*xx),x,sin(k(i)*pi*x), o ) grid,axis( tight ) title([ sin(,numstr(k(i)), \pix): N,numstr(N),, k,numstr(k(i))); end And the graphs for N 8 and N 6: sin(πx): N8, k sin(πx): N8, k sin(4πx): N8, k4 sin(7πx): N8, k sin(πx): N6, k sin(πx): N6, k sin(8πx): N6, k8 sin(5πx): N6, k
11 (c) Here is the code to generate the plot, and the plot itself: N[8 6 3; h./n; NN-; plot(:3,(pi*(:3)).^, -o,... :N(),*(-cos((:N())*pi*h()))/(h()^), *,... :N(),*(-cos((:N())*pi*h()))/(h()^), +,... :N(3),*(-cos((:N(3))*pi*h(3)))/(h(3)^), p ) grid,title( Continuous vs. Discrete spectrum of -d^/dx^ ) ylabel( eigenvalue ),legend( continuous, N8, N6, N3,) continuous N8 N6 N3 Continuous vs. Discrete spectrum of d /dx 8 eigenvalue Discussion: From part (b) we see that the eigenvectors of D are the corresponding eigenfunctions of the continuous problem evaluated at the grid points determined by the specific N. In this sense the discrete problem is clearly a very good approximation to the continuous one. On the other hand, note in the plot above that while for each N the first few discrete eigenvalues are good approximations, this is not the case for the eigenvalues corresponding to the higher modes. Thus one expects these modes in the continuous problems not to be modeled well by the discrete approximation. Note: A Taylor series expansion gives (kπ) λ k (kπ) h( cos kπh) (kπ) (kπh) h( ( + (kπh)4 )) 4 (kπ) ( (kπh) ) h (kπh)4 + (kπ) (kπ) + h ( (kπh) 4 ( ) 4 4 (kπ) 4 h O(h 4 ) O(h ) ) So we do have the expected accuracy of approximation, but remember there is still a term in the error that depends on the eigenvalue being approximated, (kπ) 4 /, which explains, even for larger N, why we do not see rapid convergence of the discrete eigenvalues to the continuous ones as k N. In fact, for a fixed N and k large, we see divergence in the upper spectrum.
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