632 CHAP. 11 EIGENVALUES AND EIGENVECTORS. QR Method

Size: px

Start display at page:

Download "632 CHAP. 11 EIGENVALUES AND EIGENVECTORS. QR Method"

Theodora Nash
6 years ago
Views:

1 632 CHAP 11 EIGENVALUES AND EIGENVECTORS QR Method Suppose that A is a real symmetric matrix In the preceding section we saw how Householder s method is used to construct a similar tridiagonal matrix The QR method is used to find all eigenvalues of a tridiagonal matrix Plane rotations similar to those that were introduced in Jacobi s method are used to construct an orthogonal matrix Q 1 = Q and an upper-triangular matrix U 1 = U so that A 1 = A has the factorization (34) A 1 = Q 1 U 1 Then form the product (35) A 2 = U 1 Q 1 Since Q 1 is orthogonal, we can use (34) to see that (36) Q 1 A 1 = Q 1 Q 1U 1 = U 1 Therefore, A 2 can be computed with the formula (37) A 2 = Q 1 A 1 Q 1 Since Q 1 = Q 1 1, it follows that A 2 is similar to A 1 and has the same eigenvalues In general, construct the orthogonal matrix Q k and upper-triangular matrix U k so that (38) A k = Q k U k

2 SEC 114 EIGENVALUES OF SYMMETRIC MATRICES 633 Then define (39) A k+1 = U k Q k = Q k A k Q k Again, we have Q k = Q 1 k, which implies that A k+1 and A k are similar An important consequence is that A k is similar to A and hence has the same structure Specifically, we can conclude that if A is tridiagonal then A k is also tridiagonal for all k Now suppose that A is written as d 1 e 1 e 1 d 2 e 2 e 2 d 3 (40) A = d n 2 e n 2 e n 2 d n 1 e n 1 e n 1 We can find a plane rotation P n 1 that reduces to zero the element of A in location (n, n 1), that is, d 1 e 1 e 1 d 2 e 2 e 2 d 3 (41) P n 1 A = d n 2 q n 2 r n 2 e n 2 p n 1 q n 1 0 p n Continuing in a similar fashion, we can construct a plane rotation P n 2 that will reduce to zero the element of P n 1 A located in position (n 1, n 2) After n 1 steps we arrive at p 1 q 1 r 1 0 p 2 q p 3 rn 4 (42) P 1 P n 1 A = = U qn 3 r n 3 p n 2 q n 2 r n 2 0 p n 1 q n p n Since each plane rotation is represented by an orthogonal matrix, equation (42) implies that (43) Q = P n 1 P n 2 P 1 d n

3 634 CHAP 11 EIGENVALUES AND EIGENVECTORS Direct multiplication of U by Q will produce all zero elements below the lower second diagonal The tridiagonal form of A 2 implies that it also has zeros above the upper second diagonal Investigation will reveal that the terms r j are used only to compute these zero elements Consequently, the numbers {r j } do not need to be stored or used in the computer For each plane rotation P j it is assumed that we store the coefficients c j and s j that define it Then we do not need to compute and store Q explicitly; instead, we can use the sequences {c j } and {s j } together with the correct formulas to unravel the product (44) A 2 = UQ= UP n 1 P n 2 P 1 Acceleration Shifts As outlined above the QR method will work, but convergence is slow even for matrices of small dimension We can add a shifting technique that speeds up the rate of convergence Recall that if λ j is an eigenvalue of A, then λ j s i is an eigenvalue of the matrix B = A s i I This idea is incorporated in the modified step (45) A i s i I = U i Q i ; then form (46) A i+1 = U i Q i for i = 1, 2,, k j, where {s i } is a sequence whose sum is λ j ; that is, λ j = s 1 + s 2 + +s k j At each stage the correct amount of shift is found by using the four elements in the lower-right corner of the matrix Start by finding λ 1 and compute the eigenvalues of the 2 2 matrix [ ] dn 1 e (47) n 1 e n 1 d n They are x 1 and x 2 and are the roots of the quadratic equation (48) x 2 (d n 1 + d n )x + d n 1 d n e n 1 e n 1 = 0 The value s i in equation (45) is chosen to be the root of (48) that is closest to d n Then QR iterating with shifting is repeated until we have e n 1 0 This will produce the first eigenvalue λ 1 = s 1 +s 2 + +s k1 A similar process is repeated with the upper n 1 rows to obtain e n 2 0, and the next eigenvalue is λ 2 Successive iteration is applied to smaller submatrices until we obtain e 2 0 and the eigenvalue λ n 2 Finally, the quadratic formula is used to find the last two eigenvalues The details can be gleaned from Program 115

4 SEC 114 EIGENVALUES OF SYMMETRIC MATRICES 635 Example 119 Find the eigenvalues of the matrix M = In Example 118, a tridiagonal matrix A 1 was constructed that is similar to M Westart our diagonalization process with this matrix: A 1 = The four elements in the lower right corner are d 3 = 14, d 4 = 14, and e 3 = 02 and are used to form the quadratic equation x 2 ( )x + ( 14)(14) ( 02)( 02) = x 2 2 = 0 Calculation produces the roots x 1 = and x 2 = The root closest to d 4 is chosen as the first shift s 1 = , and the first shifted matrix is A 1 s 1 I = Next, the factorization A 1 s 1 I = Q 1 U 1 is computed: Q 1 U 1 = Then the matrix product is computed in the reverse order to obtain A 2 = U 1 Q 1 = The second shift is s 2 = , the second shifted matrix is A 2 s 2 I = Q 2 U 2, and A 3 = U 2 Q 2 =

5 636 CHAP 11 EIGENVALUES AND EIGENVECTORS The third shift is s 3 = , the third shifted matrix is A 3 s 3 I = Q 3 U 3, and A 4 = U 3 Q 3 = The first eigenvalue, rounded to five decimal places is given in the calculation λ 1 = s 1 + s 2 + s 3 = = Next λ 1 is placed in the last diagonal position of A 4 and the process is repeated, but changes are made only in the upper 3 3 corner of the matrix A 4 = In a similar manner, one more shift reduces the entry in the second row and third column to zero (to 10 decimal places): Hence the second eigenvalue is s 4 = , A 4 s 4 I = Q 4 U 4, A 5 = U 4 Q 4 λ 2 = λ 1 + s 4 = = Finally, λ 2 is placed on the diagonal of A 5 in the third row and column to obtain A 5 = The final computation requires finding the eigenvalues of the 2 2 matrix in the upper-left corner of A 5 The characteristic equation is x 2 ( )x + (426081)( ) (265724)(265724) = 0, which reduces to x x = 0 The roots are x 1 = and x 2 = , and the last two eigenvalues are computed with the calculations and λ 3 = λ 2 + x 1 = = λ 4 = λ 2 + x 2 = =

6 SEC 114 EIGENVALUES OF SYMMETRIC MATRICES 637 Program 115 can be used to approximate all the eigenvalues of a symmetric tridiagonal matrix The program follows directly from the previous discussion, but with two notable exceptions First, the MATLAB command eig is used to find the roots of the characteristic equation (48) of each 2 2 submatrix (47) Second, the QR factorization of the matrix A i s i I (45) is executed using the MATLAB command [Q,R]=qr(B), which produces an orthogonal matrix Q and an upper-triangular matrix R, such that B=Q*R (readers will be asked to write their own QRfactorization program)

7 Numerical Methods Using Matlab, 4 th Edition, 2004 John H Mathews and Kurtis K Fink ISBN: Prentice-Hall Inc Upper Saddle River, New Jersey, USA

SEC POWER METHOD Power Method

SEC POWER METHOD Power Method SEC..2 POWER METHOD 599.2 Power Method We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided