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x
i
x
i
1
D
1
2
ıt
D
x
i
;i
D
1;:::;m
(8.56)
x
0
.Here
starting from the initial state
i
denotes the time step number.
is never computed by convolving a state vector
x
0
with the discretized kernel (
8.42
), but rather by the discrete-time integration of
the diffusion equation with the explicit numerical scheme
Similarly, the action of exp
.
D
=2/
x
i
x
i
1
D
1
2
ıt
D
x
i
1
;i
D
1;:::;m
(8.57)
such that
I
C
m
x
0
x
m
D
D
=2m
(8.58)
in correspondence with the asymptotic relation (
8.9
) for the Gaussian kernel
B
g
.
Appendix 3
By definition, a Hadamard matrix (HM) is a square matrix whose entries are either
1or1 and whose columns are mutually orthogonal. The simplest way to construct
HMs is the recursive Sylvester algorithm which is based on the obvious property: if
H
N
is an
N
N
Hadamard matrix, then
H
N
H
N
H
N
H
N
H
2N
D
is also an HM. Starting from
H
2
D
Œ1 1
I
1
1
, the HMs with order
N
D
2
n
;n
D
1;2:::
were constructed
“manually” more than a century ago. A more general HM construction algorithm,
which employs the Galois fields theory, was found in 1933. In the present study we
used the MatLab software that only handles the cases when
can be easily constructed. HMs with
N
D
12;20
M=12
,or
M=20
is a
power of 2. Despite this restriction, the available values of
M
were sufficient for
purposes of this chapter.
References
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs and
mathematical tables. Dover Publications, New York
Avramidi IG (1999) Covariant techniques for computation of the heat kernel. Rev Math Phys
11:947-980
Bekas CF, Kokiopoulou E, Saad Y (2007) An estimator for the diagonal of a matrix. Appl Numer
Math 57:1214-1229
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