Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 346

= / | |

and repeat the process.Each iterationwill
increase the dominance of the first term in Eq. (9.29)sothat the process converges to

In subsequentcycles we set v z z

1
λ 1 v 1 x 1 =

1
λ 1 x 1

z =

(atthisstage v 0).

The inverse powermethodalso works with the nonstandard eigenvalue problem

= x 1 ,sothat v 1 = 1, v 2 = v 3 =···=

Ax = λ Bx (9.30)

provided that Eq. (9.28) is replacedby

Az = Bv (9.31)

The alternative is, of course,totransform the problem to standard formbefore apply-
ing the powermethod.

Eigenvalue Shifting

By inspection of Eq. (9.29) we see that the rate of convergence is determinedbythe
strength of the inequality

| λ 1 /λ 2 | < 1 (the second term in the equation). If | λ 2 | is well

| λ 1 |

separated from , the inequalityisstrong and the convergence is rapid. On the other

hand, close proximity of these twoeigenvalues results in very slow convergence.
The rate of convergence can be improvedbyatechniquecalled eigenvalue
shifting . If we let

λ = λ ∗ +

s (9.32)

where s is apredetermined“shift,” the eigenvalue probleminEq. (9.27) istrans-
formed to

λ ∗ + =

Ax ( s ) x

or

A ∗ x = λ ∗ x

(9.33)

where

A ∗ = A − s I (9.34)

λ 1 Solving the transformedprobleminEq. (9.33) by the inverse powermethodyields

λ 1 is the smallest eigenvalueof A ∗ . The corresponding eigenvalueofthe
original problem,

and x 1 , where

λ = λ 1 + s , isthus the eigenvalue closest to s .

Eigenvalueshifting hastwo applications.Anobviousone is the determination of
the eigenvalue closest to a certain value s .For example, if the working speed of a shaft