chisl_meth / Лаб 6 QR-алгоритм, метод итераций / QR-A

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QR METHOD

The basis of the method for calculating the eigenvalues of is the fact that an real matrix can be written as

factorization of

where is orthogonal and is upper triangular. The method is efficient for the calculation of all eigenvalues of a matrix.

are constructed so that is upper

triangular. These matrices can be chosen as orthogonal matrices and are called householder matrices. Since the 's are orthogonal, the stability of the eigenvalue problem will not be worsened (this is proved in numerical analysis texts). If we let

then we have and

We discuss the construction of the 's presently. First, we state how the factorization of is used to find eigenvalues of . We define

by this process:

Step 1.

Set , and .

Step 2.

First set ; then factor as ( factorization of ).

Step 3.

First, set ; then factor as ( factorization of ).

Step 4.

Set ; then factor as ( factorization of ).

At the th step, a matrix is found, first, by using and from the previous step; second, is factored into . Thus a factorization takes place at each step. Matrix will tend toward a triangular or nearly triangular form. Thus the eigenvalues of will be

easy to calculate. The importance is that if the eigenvalues can be ordered as , then the following is true:

As increases the eigenvalues of approach the eigenvalues of .

The proof of this fact is well beyond the scope of this book. Before applying the algorithm to some examples, we discuss the

factorization of a matrix .

The idea in factorization is to first find which, when multiplied on the left of , will produce zeros below . That is, we want

After this is done, we find which will produce

The process is continued until we have

The problem is to find the matrices. It turns out that the matrices

can be chosen as orthogonal matrices. In fact, the construction proceeds as follows.

To construct :

1. Pull column out of the matrix (just if ):

2. Normalize this column vector, and call the new vector

3.Set (choose + if ).

4.Set . Also set

and

for

5. Write

Note that . 6. For the matrix

matrices, because of their form, are called householder matrices. It can be shown that householder matrices are orthogonal.

Definition 6.3.1 A householder matrix is any matrix of the form , where .

Theorem 6.3.1 Householder matrices are orthogonal.

Proof. We show that . By definition, then, would be orthogonal. First we note that is symmetric:

Now

Example 1 Find a factorization of

keeping four digits to the right of the decimal point.

Solution The first column normalized is

The ``diagonal'' element is 0.8165. We want zeros below it. First we calculate :

The minus sign was chosen since . Now we set

Thus

Multiply by ; we obtain

To construct , we look at column 2 of :

Normalized, this is

Actually we obtain 0.0008. Because of rounding, we called this zero.

Therefore,

and we have

Finally,

Therefore,

From the example just calculated, we see that finding the factorization for a matrix is tedious by hand. A computer, of course, is necessary to find factorizations and, therefore, to use the method for finding eigenvalues.

For a matrix, only one householder matrix must be found, so we consider the factorization for a general matrix

The first column, normalized, is

Actually 0.0001. Because of rounding we call this zero.

Now , so we can write sign , where sign if and sign if .

sign

where sign if and sign if . Since , we have sign . For we have

Therefore,

snf

Since is symmetric, .

Example 2 Find a factorization of

Solution Using the formulas as above, we find that

Using the formulas for the case, we now calculate the eigenvalues of

by the method.

Example 3 Use the method to calculate the eigenvalues of

(The true eigenvalues are 4 and 9.)

Solution We use the formulas for the case each time we need a

factorization. The calculated matrices are listed below (rounded); after step 3, only is listed.

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

Step 12:

Approximate eigenvalues are on the diagonal.

In Example 3, appeared to be converging to a diagonal matrix; of

course, the diagonal elements are the approximate eigenvalues. This illustrates the following important result.

Theorem 6.3.2 Let be a real matrix with eigenvalues satisfying

Then matrices in the method will converge to an upper triangular matrix with diagonal entries , . If is symmetric, matrices converge to a diagonal matrix with the eigenvalues on the diagonal.

If the hypotheses of Theorem 6.3.2 are not satisfied by , the

method may fail. If the difference in the magnitudes of the eigenvalues is small, convergence of the method can be slow.

Example 4 Applying the method to attempt calculation of eigenvalues of