434 CHAPTER 15. NUMERICAL METHODS, EIGENVALUES

Applying the QR algorithm to this matrix yields the following sequence of matrices.

A1 =

 1.2353 1.9412 4.3657−.39215 1.5425 5.3886×10−2

−.16169 −.18864 .22222

...

A12 =

 9.1772×10−2 .63089 −2.0398−2.8556 1.9082 −3.1043

1.0786×10−2 3.4614×10−4 1.0

At this point the bottom two terms on the left part of the bottom row are both very small

so it appears the real eigenvalue is near 1.0. The complex eigenvalues are obtained fromsolving

det

(λ

(1 00 1

)−

(9.1772×10−2 .63089−2.8556 1.9082

))= 0

This yieldsλ = 1.0− .98828i, 1.0+ .98828i

Example 15.3.9 The equation x4+x3+4x2+x−2 = 0 has exactly two real solutions. Youcan see this by graphing it. However, the rational root theorem from algebra shows neitherof these solutions are rational. Also, graphing it does not yield any information about thecomplex solutions. Lets use the QR algorithm to approximate all the solutions, real andcomplex.

A matrix whose characteristic polynomial is the given polynomial is−1 −4 −1 21 0 0 00 1 0 00 0 1 0

Using the QR algorithm yields the following sequence of iterates for Ak

A1 =

.99999 −2.5927 −1.7588 −1.29782.1213 −1.7778 −1.6042 −.99415

0 .34246 −.32749 −.917990 0 −.44659 .10526

...

A9 =

−.83412 −4.1682 −1.939 −.7783

1.05 .14514 .2171 2.5474×10−2

0 4.0264×10−4 −.85029 −.616080 0 −1.8263×10−2 .53939

