Geology Reference
In-Depth Information
The (
N
−
2)-length vector of second derivatives,
f
2
,
f
3
,...,
f
N
−
2
,
f
N
−
1
T
f
N
−
2
=
,
(1.561)
is related to the
N
-length vector of function values,
(
f
1
,
f
2
,...,
f
N
−
1
,
f
N
)
T
f
=
,
(1.562)
by the matrix equation
A
f
N
−
2
=
B
f
,
(1.563)
where
A
is the (
N
−
2)
×
(
N
−
2) tridiagonal matrix
⎝
⎠
a
11
a
12
1
a
22
a
23
.
.
1
A
=
,
(1.564)
a
N
−
3,
N
−
3
a
N
−
3,
N
−
2
a
N
−
2,
N
−
3
a
N
−
2,
N
−
2
and
B
is the (
N
−
2)
×
N
matrix
⎝
⎠
b
11
b
12
b
13
b
22
b
23
b
24
.
.
.
b
N
−
3,
N
−
3
b
N
−
3,
N
−
2
b
N
−
3,
N
−
1
b
N
−
2,
N
−
2
b
N
−
2,
N
−
1
b
N
−
2,
N
B
=
.
(1.565)
Matrix
A
can be augmented by the identity matrix and inverted entirely by
row operations alone, as in the Gauss-Jordan technique. A forward pass elimin-
ates elements on the sub-diagonal, while a backward pass eliminates elements on
the super-diagonal. Then,
f
N
−
2
=
(
A
−
1
B
)
f
. The matrix
A
−
1
B
has dimensions
−
×
(
N
N
. A new first row can be added by a combination of its first and second
rows, as indicated by (1.554), then a new last row can be added as the combination
of its last and second last rows, as indicated by (1.555). The full vector of second
derivatives,
f
=
2)
(
f
1
,
f
2
,...,
f
N
−
1
,
f
N
)
T
, is then related to the vector of function
values by
f
=
C
f
,
(1.566)
with
C
afull
N
N
square matrix. The subroutine SPMAT finds the matrix
C
connecting the second derivatives to the function values at the nodes:
×
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