Download 10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES

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590
CHAPTER 10
NUMERICAL METHODS
A second iteration yields
1
Ax1 2
1
2
1
3
0
2
1
0.60
1.00
0.20 1.00
1.00
2.20
and
1.00
0.45
1
x2 1.00 0.45 .
2.20
2.20
1.00
Continuing this process, you obtain the sequence of approximations shown in Table 10.6.
TABLE 10.6
x0
x1
x2
x3
x4
x5
x6
x7
1.00
1.00
1.00
0.60
0.20
1.00
0.45
0.45
1.00
0.48
0.55
1.00
0.51
0.51
1.00
0.50
0.49
1.00
0.50
0.50
1.00
0.50
0.50
1.00
From Table 10.6 you can approximate a dominant eigenvector of A to be
0.50
x 0.50 .
1.00
Using the Rayleigh quotient, you can approximate the dominant eigenvalue of A to be
3. (For this example you can check that the approximations of x and are exact.)
REMARK:
Note that the scaling factors used to obtain the vectors in Table 10.6,
x1
x2
x3
x4
x5
x6
x7
↓
↓
↓
↓
↓
↓
↓
5.00
2.20
2.82
3.13
3.02
2.99
3.00,
are approaching the dominant eigenvalue 3.
In Example 4 the power method with scaling converges to a dominant eigenvector. The
following theorem states that a sufficient condition for convergence of the power method is
that the matrix A be diagonalizable (and have a dominant eigenvalue).
Theorem 10.3
Convergence of the
Power Method
If A is an n n diagonalizable matrix with a dominant eigenvalue, then there exists a
nonzero vector x0 such that the sequence of vectors given by
Ax0, A2x0, A3x0, A4x0, . . . , Akx0, . . .
approaches a multiple of the dominant eigenvector of A.