High-Speed Sparse Linear Algebra

Mathematica 5 utilizes specialized techniques to make computations involving sparse matrices (matrices in which most of the elements are zero) vastly more efficient for large-scale problems. These improvements make Mathematica highly suitable for large-scale simulations, optimization, or solving of partial or ordinary differential equations--all examples in which sparse matrices are typically involved.

Mathematica 5's implementation of sparse linear algebra is unique because:

• Symbolic preprocessing optimizes the formation of sparse matrices.
• Sparse algorithms are automatically utilized where they would improve performance, without user intervention. (They can be manually evoked where required, too--for example, for comparison with traditional numerical systems.)
• Any dimension (or rank) of array is handled.

The performance in speed and memory utilization of sparse linear algebra operations is on a par with, or better than, those in dedicated numerical systems.

The following graph shows the large difference in size between sparse and dense representations of a matrix with three nonzero elements per row.

Many common matrix operations on sparse arrays are significantly faster than operations on dense arrays. The following graph shows the time it takes to invert a matrix with three nonzero elements per row in Mathematica 5 and 4.2.

		Documentation from The Mathematica Book Manipulating Lists: Sparse Arrays Linear Algebra: Sparse Arrays Matrix Inversion
		Advanced Documentation Linear Algebra
		Documentation from the Reference Guide SparseArray Inverse
		Other Links gigaNumerics (Key Technology)