Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 59

LUsol3

This is the function for the solutionphase. The vector y overwrites the constant vector
b during the forward substitution. Similarly, the solutionvector x replaces y in the
back substitutionprocess.

functionx=LUsol3(c,d,e,b)
%SolvesA*x=bwhereA=[c\d\e]istheLU
% decomposition of the original tridiagonal A.
%USAGE:x=LUsol3(c,d,e,b)

n = length(d);
fork=2:n %Forwardsubstitution
b(k) = b(k) - c(k-1)*b(k-1);

end
b(n) = b(n)/d(n);

% Back substitution

fork=n-1:-1:1
b(k) = (b(k) -e(k)*b(k+1))/d(k);

end
x=b;

Symmetric Coefficient Matrices

More often than not, coefficient matrices that arise in engineering problems are
symmetric as well as banded. Therefore, it is worthwhile to discover special prop-
erties of such matrices, and learn how to utilize theminthe construction of efficient
algorithms.
If the matrix A issymmetric, then the LU decomposition can be presentedinthe
form

LDL T

A = LU = (2.23)

LL T that where D is a diagonal matrix. An example is Choleski's decomposition A =

was discussedinthe previous article (in thiscase D = I ).ForDoolittle's decomposition

wehave

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

D 1 00 ··· 0 1 L 21 L 31 ··· L n 1

0 D 2 0 ··· 0 0 1 L 32 ··· L n 2

00 D 3 ··· 0 00 1 ··· L n 3

DL T U = =

. . . . . . . . . . . . . .

000 ··· D n 00 0 ··· 1