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SECTION 10.3
POWER METHOD FOR APPROXIMATING EIGENVALUES
587
The Power Method
Like the Jacobi and Gauss-Seidel methods, the power method for approximating eigenvalues is iterative. First assume that the matrix A has a dominant eigenvalue with corresponding dominant eigenvectors. Then choose an initial approximation x0 of one of the dominant
eigenvectors of A. This initial approximation must be a nonzero vector in Rn. Finally, form
the sequence given by
x1 Ax0
x2 Ax1 A(Ax0) A2x0
x3 Ax2 A(A2x0) A3x0
.
.
.
xk Axk1 A(Ak1x0) Akx0.
For large powers of k, and by properly scaling this sequence, you will see that you obtain
a good approximation of the dominant eigenvector of A. This procedure is illustrated in
Example 2.
EXAMPLE 2
Approximating a Dominant Eigenvector by the Power Method
Complete six iterations of the power method to approximate a dominant eigenvector of
A
Solution
2 12
.
5
1
Begin with an initial nonzero approximation of
x0 1.
1
Then obtain the following approximations.
Iteration
2 12
5
2 12
1 5
2 12
1 5
2 12
1 5
2 12
1 5
2 12
1 5
“Scaled” Approximation
10
x1 Ax0 1
1 4
x2 Ax1 4 10
x3 Ax2 x4 Ax3 x5 Ax4 x6 Ax5 1
10
28
64
10 22
28
64
22 46
136
280
46 94
136
280
94 190
568
1.00
4
2.50
1.00
10
2.80
1.00
22
2.91
1.00
46
2.96
1.00
94
2.98
1.00
190
2.99