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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