• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Four-vector wikipedia, lookup

Matrix calculus wikipedia, lookup

System of linear equations wikipedia, lookup

Singular-value decomposition wikipedia, lookup

Eigenvalues and eigenvectors wikipedia, lookup

Gaussian elimination wikipedia, lookup

Matrix multiplication wikipedia, lookup

Cayley–Hamilton theorem wikipedia, lookup

Principal component analysis wikipedia, lookup

Non-negative matrix factorization wikipedia, lookup

Perron–Frobenius theorem wikipedia, lookup

Orthogonal matrix wikipedia, lookup

Jordan normal form wikipedia, lookup

Determinant wikipedia, lookup

Matrix (mathematics) wikipedia, lookup

Transcript
```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
â
â
â
â
â
â
â
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.
```
Related documents