Monday, February 1, 2010

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 that a good initial approximation is known. Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision. The inverse power method is then invoked to refine the numerical values and gain full precision. To discuss the situation, we will need the following definitions.

If [Graphics:Images/PowerMethodMod_gr_1.gif] is an eigenvalue of A that is larger in absolute value than any other eigenvalue, it is called the dominant eigenvalue. An eigenvector [Graphics:Images/PowerMethodMod_gr_2.gif] corresponding to [Graphics:Images/PowerMethodMod_gr_3.gif] is called a dominant eigenvector.

An eigenvector V is said to be normalized if the coordinate of largest magnitude is equal to unity (i.e., the largest coordinate in the vector V is the number 1).

Remark:

It is easy to normalize an eigenvector [Graphics:Images/PowerMethodMod_gr_4.gif] by forming a new vector [Graphics:Images/PowerMethodMod_gr_5.gif] where [Graphics:Images/PowerMethodMod_gr_6.gif] and [Graphics:Images/PowerMethodMod_gr_7.gif].

Theorem (Power Method):

Assume that the n×n matrix A has n distinct eigenvalues [Graphics:Images/PowerMethodMod_gr_8.gif] and that they are ordered in decreasing magnitude; that is, [Graphics:Images/PowerMethodMod_gr_9.gif]. If [Graphics:Images/PowerMethodMod_gr_10.gif] is chosen appropriately, then the sequences [Graphics:Images/PowerMethodMod_gr_11.gif] and [Graphics:Images/PowerMethodMod_gr_12.gif] generated recursively by

[Graphics:Images/PowerMethodMod_gr_13.gif]
and
[Graphics:Images/PowerMethodMod_gr_14.gif]

where
[Graphics:Images/PowerMethodMod_gr_15.gif] and [Graphics:Images/PowerMethodMod_gr_16.gif], will converge to the dominant eigenvector [Graphics:Images/PowerMethodMod_gr_17.gif] and eigenvalue [Graphics:Images/PowerMethodMod_gr_18.gif], respectively. That is,

[Graphics:Images/PowerMethodMod_gr_19.gif] and [Graphics:Images/PowerMethodMod_gr_20.gif].

Remark:

If [Graphics:Images/PowerMethodMod_gr_21.gif] is an eigenvector and [Graphics:Images/PowerMethodMod_gr_22.gif], then some other starting vector must be chosen.

Speed of Convergence

In the iteration in the theorem uses the equation

[Graphics:Images/PowerMethodMod_gr_23.gif],

and the coefficient of
[Graphics:Images/PowerMethodMod_gr_24.gif] that is used to form [Graphics:Images/PowerMethodMod_gr_25.gif] goes to zero in proportion to [Graphics:Images/PowerMethodMod_gr_26.gif]. Hence, the speed of convergence of [Graphics:Images/PowerMethodMod_gr_27.gif] to [Graphics:Images/PowerMethodMod_gr_28.gif] is governed by the terms [Graphics:Images/PowerMethodMod_gr_29.gif]. Consequently, the rate of convergence is linear. Similarly, the convergence of the sequence of constants [Graphics:Images/PowerMethodMod_gr_30.gif] to [Graphics:Images/PowerMethodMod_gr_31.gif] is linear. The Aitken [Graphics:Images/PowerMethodMod_gr_32.gif] method can be used for any linearly convergent sequence [Graphics:Images/PowerMethodMod_gr_33.gif] to form a new sequence,

[Graphics:Images/PowerMethodMod_gr_34.gif],

that converges faster. The Aitken
[Graphics:Images/PowerMethodMod_gr_35.gif] can be adapted to speed up the convergence of the power method.

Shifted-Inverse Power Method

We will now discuss the shifted inverse power method. It requires a good starting approximation for an eigenvalue, and then iteration is used to obtain a precise solution. Other procedures such as the
QM and Givens’ method are used first to obtain the starting approximations. Cases involving complex eigenvalues, multiple eigenvalues, or the presence of two eigenvalues with the same magnitude or approximately the same
magnitude will cause computational difficulties and require more advanced methods. Our illustrations will focus on the case where the eigenvalues are distinct. The shifted inverse power method is based on the following three results (the proofs are left as exercises).

Theorem (Shifting Eigenvalues):

Suppose that [Graphics:Images/PowerMethodMod_gr_36.gif],V is an eigenpair of A. If [Graphics:Images/PowerMethodMod_gr_37.gif] is any constant, then [Graphics:Images/PowerMethodMod_gr_38.gif],V is an eigenpair of the matrix [Graphics:Images/PowerMethodMod_gr_39.gif].

Theorem (Inverse Eigenvalues):

Suppose that [Graphics:Images/PowerMethodMod_gr_40.gif],V is an eigenpair of A. If [Graphics:Images/PowerMethodMod_gr_41.gif], then [Graphics:Images/PowerMethodMod_gr_42.gif],V is an eigenpair of the matrix [Graphics:Images/PowerMethodMod_gr_43.gif].

Theorem (Shifted-Inverse Eigenvalues):

Suppose that [Graphics:Images/PowerMethodMod_gr_44.gif],V is an eigenpair of A. If [Graphics:Images/PowerMethodMod_gr_45.gif], then [Graphics:Images/PowerMethodMod_gr_46.gif],V is an eigenpair of the matrix [Graphics:Images/PowerMethodMod_gr_47.gif].

Theorem (Shifted-Inverse Power Method):

Assume that the n×n matrix A has distinct eigenvalues [Graphics:Images/PowerMethodMod_gr_48.gif] and consider the eigenvalue [Graphics:Images/PowerMethodMod_gr_49.gif]. Then a constant [Graphics:Images/PowerMethodMod_gr_50.gif] can be chosen so that [Graphics:Images/PowerMethodMod_gr_51.gif] is the dominant eigenvalue of [Graphics:Images/PowerMethodMod_gr_52.gif]. Furthermore, if [Graphics:Images/PowerMethodMod_gr_53.gif] is chosen appropriately, then the sequences [Graphics:Images/PowerMethodMod_gr_54.gif] and [Graphics:Images/PowerMethodMod_gr_55.gif] generated recursively by

[Graphics:Images/PowerMethodMod_gr_56.gif]
and
[Graphics:Images/PowerMethodMod_gr_57.gif]

where
[Graphics:Images/PowerMethodMod_gr_58.gif] and [Graphics:Images/PowerMethodMod_gr_59.gif], will converge to the dominant eigenpair [Graphics:Images/PowerMethodMod_gr_60.gif],[Graphics:Images/PowerMethodMod_gr_61.gif] of the matrix [Graphics:Images/PowerMethodMod_gr_62.gif]. Finally, the corresponding eigenvalue for the matrix A is given by the calculation

[Graphics:Images/PowerMethodMod_gr_63.gif]

Remark. For practical implementations of this Theorem, a linear system solver is used to compute [Graphics:Images/PowerMethodMod_gr_64.gif] in each step by solving the linear system [Graphics:Images/PowerMethodMod_gr_65.gif].

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Introductory Methods of Numerical Analysis