Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 149

These equations are satisfied by

( p ( p

q ) 2 q ) 2

r 2 r 2

+ + − − +

σ min = , (2.241)

2

( p ( p

q ) 2 q ) 2 r 2 r 2 + + + − +

σ max = . (2.242)

2

The Wilkinson shift is chosen as the smallest singular value, σ min , of the trailing
matrix Z . Its square is the smallest eigenvalue of the matrix Z T Z .
The initial QR iteration is carried out on the full M M bidiagonal matrix. Thus,

the first row is the leading row of the initial working matrix and the last row, M ,is
the trailing row of the initial working matrix. In a technique called matrix splitting
(Burden and Faires, 1988, pp. 515-519), after each QR iteration on the bidiagonal
matrix B , the upper diagonal element in the leading row is tested for negligibility.
If the upper diagonal element in the leading row is negligibly small compared to
the absolute value of the smallest of its nearest diagonal element neighbours, the
bidiagonal matrix deflates by one dimension. The first row with a non-negligible
upper diagonal element becomes the leading row K of the new working matrix.
The trailing row of the new working matrix becomes the first row N , below the
new leading row, with a negligible upper diagonal element (or, in the case of the
last row, a non-existent upper diagonal element). Then, the new working matrix
begins at row K and ends at row N . Finally, the last iteration is performed on a
2

×

× 2 bidiagonal matrix. For the 5 × 5 matrix we are using as an example, the final

form of B is

upper diagonal of B is,
0.13105D-72

0.92262D-31 -0.25429D-19 -0.36258D-21 0.00000D+00

diagonal of B is,
0.14241D+04

0.28591D+00 0.10321D-01 0.76246D-02 0.15188D-02,

for an assumed machine precision of 10 − 15 . The unitary matrices in the SVD fac-
torisation (2.199) of the example matrix are U ,

real part of matrix is,
0.44717D+00 -0.63165D+00 0.30071D+00 -0.45811D+00 -0.31745D+00
-0.19052D+00 0.14267D+00 -0.24572D+00 -0.22489D+00 -0.46049D+00
-0.28493D+00 -0.31439D-02 -0.82148D-01 0.85482D-01 -0.59629D+00

0.43326D+00 0.30757D+00 0.33106D+00 0.52251D+00 -0.44210D+00

-0.84196D-01 -0.13213D+00 -0.29110D+00 0.12652D+00 -0.31403D+00

imaginary part of matrix is,
-0.21841D-16 -0.20379D-13 0.18805D-11 -0.12338D-11 0.17202D-12
-0.40463D+00 0.28401D+00 0.37157D+00 -0.47903D+00 -0.91814D-01
0.34475D+00 -0.25425D-02 -0.64460D+00 -0.11350D+00 0.43863D-01

0.11092D+00 0.80108D-01 0.29806D+00 0.83846D-01 0.15821D+00

-0.43917D+00 -0.61767D+00 0.75420D-01 0.44029D+00 0.46451D-01,