204
8 A Model for Microstructure Characterization
Upon calling the double integrals on the left G
(x)
m m , G
(y)
n n , respectively, and the
integrals on the right H
(x)
m m and H
(y)
n n , respectively, we have the result
N
m=1
N
n=1
G
(x)
m m G
(y)
n n ψ mn = |λ|
2
N
m=1
N
n=1
H
(x)
m m H
(y)
n n ψ mn , m
, n
= 1, · · · , N ,
(8.17)
or, in vector-matrix notation
G · v = |λ|
2 H · v ,
(8.18)
where G = G (x) ⊗ G (y) , H = H (x) ⊗ H (y) , and v = [ψ 11 , · · · , ψ NN ] T . For the
separable covariances shown in (8.14), we have G (x) = G (y) . This completes the
derivation of the generalized eigenvalue problem. If {f n (x)} is orthogonal, then H
is diagonal, and if {f n (x)} is normalized to unity, then H is the identity matrix, and
the generalized eigenvalue problem reduces to the standard form
G · v = |λ|
2 v .
(8.19)
We have already calculated the matrix elements for the double-exponential
function of (8.14) in (7.20), so we will proceed with the computation of the matrix
elements of the Gaussian function. Using the one-dimensional basis functions that
were defined in (7.17), we have
G
(x)
mm = s
b
−b
b
−b
e
−(x−x ) 2 /L 2 π(x/δ − m)π(x
/δ − m
)dxdx
= s
(m +1)δ
m δ
dx
(m+1)δ
mδ
e
−(x−x ) 2 /L 2 dx
= sL
(m +1)δ
m δ
dx
(m+1)δ/L−x /L
mδ/L−x /L
e
−u 2 du
= s
√
π
2
L
(m +1)δ
m δ
erf
x
/L − mδ/L
− erf
x
/L − (m + 1)δ/L
dx
= s
√
π
2
L
2
(m −m+1)δ/L
(m −m)δ/L
[erf v − erf (v − δ/L)] dv ,
(8.20)
where the error function, erf, and its properties are defined in [81, Chapter 7].
We can get an explicit analytical expression for G
(x)
mm by substituting the power
series representation for erf z,