A Fast Method for Solving a Class of Tridiagonal Linear Systems

The solution of linear systems having real, symmetric,
diagonally dominant,tridiagonal coefficient 
matrices with constant diagonals is considered.  It is
proved that the diagonals of the LU decomposition 
converges when floating-point precision.  It is also
proved that the computed LU decomposition converges 
when floating-point arithmetic is used and that the limits
of the LU diagonals using floating point are 
roughly within machine precision of the limits using
real arithmetic.  This fact is exploited to reduce 
the number of floating-point operations required to
solve a linear system from 8n-7 to 5n+2k-3, where 
k is much less than n, the order of the matrix.  If the
elements of the subdiagonals and superdiagonals 
are 1, then only 4n+2k-3 operations are needed.  The
entire LU decomposition takes k words of storage, 
and considerable savings in array subscripting are achieved.
 Upper and lower bounds on k are obtained 
in terms of the ratio of the coefficient matrix diagonal
constants and parameters of the floating-point 
number system.  Various generalizations of these results are discussed.

CACM January, 1974

Malcolm, M. A.
Palmer, J.

numerical linear algebra, linear systems,
Toeplitz matrices, tridiagonal matrices

5 5.1 5.11 5.14 5.17

CA740102 JB January 18, 1978  2:50 PM

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