ABSTRACT
Consider the numerical solution of the linear system Kγ=f, in which K is the positivedefinite and symmetric stiffness matrix. In many problems, it has a large dimension, but is also banded. The matrix can be “triangularized”: K=LLT, in which L is a lower triangular, nonsingular matrix (zeroes in all entries above the diagonal). We can introduce z=LTγ and obtain z by solving Lz=f. Next, γ can be computed by solving LTγ=z. Now Lz=f can be conveniently solved by forward substitution. In particular, Lz=f can be expanded as
(11.1)
Assuming that the diagonal entries are not too small, this equation can be solved, starting from the upper-left entry, using simple arithmetic: z1=f1/l11, z2=[f2−l21z1]/ l22, z3=[f3−l31z1−l32z2]/l33,….
