Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 58

Note that the originalcontents of c were destroyed and replacedbythe multipliers
during the decomposition. The solutionalgorithm for y by forward substitutionis

y(1) = b(1)
fork=2:n
y(k) = b(k) - c(k-1)*y(k-1);

end

The augmented coefficient matrix representing Ux = y is

⎡
⎣

⎤
⎦

d 1 e 1 0 ··· 0 0 y 1

0 d 2 e 2 ··· 0 0 y 2

U

y

00 d 3 ··· 0 0 y 3

=

. . . . . .

000 ··· d n − 1 e n − 1 y n − 1

000 ··· 0 d n y n

Note again that the contents of d were altered from the original values during the
decompositionphase (but e was unchanged). The solution for x isobtainedbyback
substitutionusing the algorithm

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

end

LUdec3

The function LUdec3 contains the codefor the decompositionphase. The original
vectors c and d are destroyed and replacedbythe vectorsofthedecomposedmatrix.

function [c,d,e] = LUdec3(c,d,e)
%LUdecompositionoftridiagonalmatrixA=[c\d\e].
% USAGE: [c,d,e] = LUdec3(c,d,e)

n = length(d);
fork=2:n
lambda = c(k-1)/d(k-1);
d(k) = d(k) - lambda*e(k-1);
c(k-1) = lambda;

end