A frontal solver, conceived by Bruce Irons, is an approach to solving sparse linear systems which is used extensively in

finite element analysis The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...

. It is a variant of

Gauss elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

that automatically avoids a large number of operations involving zero terms. A frontal solver builds a LU or

Cholesky decomposition In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...

of a sparse matrix given as the assembly of element matrices by assembling the matrix and eliminating equations only on a subset of elements at a time. This subset is called the front and it is essentially the transition region between the part of the system already finished and the part not touched yet. The whole sparse matrix is never created explicitly. Only parts of the matrix are assembled as they enter the front. Processing the front involves

dense matrix In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse ...

operations, which use the CPU efficiently. In a typical implementation, only the front is in

memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered, ...

, while the factors in the decomposition are written into files. The element matrices are read from files or created as needed and discarded. A multifrontal solver of

Duff Duff may refer to: People * Duff (surname) * Duff (given name) * Duff (nickname) * Karen Duffy, an actress, model, and former MTV VJ once known as "Duff" * Duff Roman, on-air name of Canadian radio personality and executive David Mostoway (bo ...

and

Reid Reid is a surname of Scottish origin. It means "red". People with the surname * Alan Reid (disambiguation) * Alex Reid (disambiguation), includes Alexander Reid * Amanda Reid, Australian Paralympic athlete * Amanda Reid (taxonomist), Australia ...

is an improvement of the frontal solver that uses several independent fronts at the same time. The fronts can be worked on by different

processors A central processing unit (CPU), also called a central processor, main processor or just processor, is the electronic circuitry that executes instructions comprising a computer program. The CPU performs basic arithmetic, logic, controlling, and ...

, which enables

parallel computing Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different fo ...

. SeeIain S Duff , Albert M Erisman , John K Reid, Direct methods for sparse matrices, Oxford University Press, Inc., New York, NY, 1986 for a monograph exposition.

References

Numerical linear algebra Numerical software {{mathapplied-stub

See also

References