MATLAB Mathematics - SERC - Index of

6 Sparse Matrices
shows that A has 1887 nonzeros, its complete LU factorization has 16777
nonzeros, its incomplete LU factorization with a drop tolerance of 1e-6 has
10311 nonzeros, and its lu('0') factorization has 1886 nonzeros.
The luinc function has a few other options. See the luinc reference page for
details.
The cholinc function provides drop tolerance and level 0 fill-in Cholesky
factorizations of symmetric, positive definite sparse matrices. See the cholinc
reference page for more information.
Simultaneous Linear Equations
There are two different classes of methods for solving systems of simultaneous
linear equations:
• Direct methods are usually variants of Gaussian elimination. These methods
involve the individual matrix elements directly, through matrix
factorizations such as LU or Cholesky factorization. MATLAB implements
direct methods through the matrix division operators / and \, which you can
use to solve linear systems.
• Iterative methods produce only an approximate solution after a finite number
of steps. These methods involve the coefficient matrix only indirectly,
through a matrix-vector product or an abstract linear operator. Iterative
methods are usually applied only to sparse matrices.
Direct Methods
Direct methods are usually faster and more generally applicable than indirect
methods, if there is enough storage available to carry them out. Iterative
methods are usually applicable to restricted cases of equations and depend
upon properties like diagonal dominance or the existence of an underlying
differential operator. Direct methods are implemented in the core of MATLAB
and are made as efficient as possible for general classes of matrices. Iterative
methods are usually implemented in MATLAB M-files and may make use of
the direct solution of subproblems or preconditioners.
Using a Different Preordering. If A is not diagonal, banded, triangular, or a
permutation of a triangular matrix, backslash (\) reorders the indices of A to
reduce the amount of fill-in — that is, the number of nonzero entries that are
added to the sparse factorization matrices. The new ordering, called a
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