Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 161

In the recursions from (4.27), the vectors q (s n and e (s n are obtained by inter
changeably forming their quotients and differences. This is the basis for the
name 'quotientdifference' for the Rutishauser algorithm. The QD algorithm
is one of the most extensively employed computational tools throughout nu
merical analysis. Transparently, the vectors q (s n and e (s n can be stored in a
twodimensional table as a double array of a lozenge form via

q (0)
1

e (1)
0

e (0)
1

q (1)
1

q (0)
2

e (2)
0

e (1)
1

e (0)
2

. . .

q (2)
1

q (1)
2

(4.28)

e (3)
0

e (2)
1

e (1)
2

. q (3)
1

. . .

q (2)
2

. e (3)
1

e (2)
2

. . . . . .

where the first column is filled with zeros e (0 0 = 0. It follows from the ta
ble (4.28) that the subscript (n) and superscript (s) denote a column and a
counterdiagonal, respectively. Moreover, the columns of the arrays q (s) and

n

e (s) are interleaved. Here, the starting values are e (s)

0

= 0 (s = 1, 2,...) and

n

q (s)

1 = c s+1 /c s (s = 0, 1, 2,...). Subsequent arrays q (s n and e (s n are obtained
by two intertwined recursions of quantities that are located at the vertices
(corners) of the lozenge in the table (4.28). The column containing only the
vectors e (s)

are generated through the differences e (s) = q (s+1) −q (s) + e (s+1)

n−1

n n n n

(n = 1, 2,... and s = 0, 1, 2,...). Similarly, the columns with the arrays

q (s) are constructed by means of the quotients q (s) = q (s+1)

n−1 e (s+1) n−1 /e (s) n−1 (n =

2, 3,... and s = 0, 1, 2,...). This is the complete procedure by which the QD
algorithm creates one column at a time by alternatively generating the quo
tients and differences of the q and equantities through the recursion relations
from (4.27).

n n

4.7 The Gordon product-difference recursive algorithm
The wellknown Lanczos algorithm [5, 148] has numerical di culties includ
ing the loss of orthogonality of the elements of the Lanczos basis set{ψ n }.
This often seriously deteriorates the accuracy of the coupling parameters that
are computed during the construction of the state vectors{ψ n }. Because the