Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 136

conditional equations, even when implemented in double quadruple (octuple) pre-
cision (Hida et al. , 2000). Instead, we use the singular value decomposition (SVD)
method. Writing the system (2.187) in matrix notation as

C · G = d , (2.198)

the SVD method factors the complex coe cient matrix C into the triple product

V H C = U · W · , (2.199)

where U and V are unitary matrices and W is a real, diagonal matrix with the
positive singular values of C in descending order down the diagonal. The super-
script H indicates the complex conjugate of the transpose, the Hermitian transpose
or adjoint . A matrix whose Hermitian transpose is equal to the original matrix is
itself said to be Hermitian . A matrix which multiplied by its Hermitian transpose
gives the unit matrix is said to be a unitary matrix, the complex analogue of a real
orthogonal matrix. Similarly, a unitary transformation is the complex analogue of a
similarity transformation with real orthogonal matrices, as both preserve the eigen-
values of a matrix. The inverse of a unitary matrix is equal to its Hermitian trans-
pose. The singular value decomposition of a complex rectangular matrix, as well as
its properties and methods of calculation, are outlined by Golub and Kahan (1965).
Once the SVD of C , (2.199), has been carried out, the solution vector G of the
conditional equations (2.198), giving the DFT, is easily found using the properties
of unitary matrices as

W − 1 U H G = V · · · d . (2.200)

Many of the properties of the coe cient matrix C can be determined from its SVD

(Press et al. (1992, pp. 53-56)). Its condition number is defined as the ratio of the
largest singular value to the smallest. If one or more of the singular values vanish,
no solution exists. The solution (2.200) will be numerically unstable if the recip-
rocal of the condition number approaches machine precision. This suggests setting
the inverses of the smallest singular values to zero in W − 1 , giving W − 1 , to improve
numerical stability, leading to an approximate solution of (2.200) for the DFT,

G = W − 1 U H V · · · d . (2.201)

However, the number of singular values to be eliminated remains unspecified.
A basic advance in the application of the SVD technique to the calculation of
the DFT was made by Palmer (2005). Any acceptable DFT must obey Parseval's
relation (2.197), irrespective of whether the time samples are equally spaced or
unequally spaced. Typically, the mean square in the frequency domain, as repres-
ented by the DFT, is many orders of magnitude larger than the mean square in the