ABSTRACT
We consider numerical solution of the linear system Kg¼ f, in which K is the positive definite and symmetric stiffness matrix. In many problems it has a large dimension, but is also banded. The matrix may be ‘‘triangularized’’ to yield the form K¼LLT, in which L is a lower triangular nonsingular matrix (zeroes in all entries above the diagonal). We may introduce z¼LTg and obtain z by solving Lz¼ f. Next g can be computed by solving LTg¼ z. We now see that Lz¼ f can be conveniently solved by forward substitution. Lz¼ f may be expanded as
l11 0 : : : 0
l21 l22 : : : :
l31 l32 l33 : : :
: : : : : :
: : : : : 0
ln1 ln2 : : : lnn
2 6666666664
3 7777777775
z1
z2
z3
:
:
zn
0 BBBBBBBBB@
1 CCCCCCCCCA ¼
f1
f2
f3
:
:
fn
0 BBBBBBBBB@
1 CCCCCCCCCA
(9:1)
Assuming that the diagonal entries are not too small, this equation can be solved, starting from the upper left entry, using simple arithmetic: z1¼ f1=l11, z2¼ [ f2 l21z1]=l22, z3¼ [ f3 l31z1 l32z2]=l33, . . . .
