A set of commonly used algebraic equations defined by C. L. Lawson, and J. J. Dongerra, in a series of papers (see bibliography of LAPACK User's Guide, publication TPD-0003, pp. 112-115, entries [16], [17], [18], [19], and [38].

See Bi-Conjugate Gradient Method.

One of the iterative methods provided through the DITERATIVE package of optimized precondtioned iterative methods.

See Basic Linear Algebra Subprogram.

See Conjugate Gradient Method.

See Bi-Conjugate Gradient Squared Method.

Term used to define LAPACK routines that perform a distinct computational task.

One of the iterative methods provided through the DITERATIVE package of optimized preconditioned iterative methods.

A parallel processing environment in which the NCPUS environment variable is equal to the number of available processors.

Direct solution methods for sparse linear systems transform the matrix A into a product of several other operators so that each of the resulting operators is easy to invert for a given right-hand side b.

Term used to define LAPACK routines used for solving standard types of problems.

The process of scaling a problem before computing its solution.

The mathematical process of resolving a given function, f(x), into its frequency components, which means finding the sequence of constant amplitudes to plug into a Fourier series to reconstruct the original function.

A complex matrix which is equal to the conjugate of its transpose, with either the lower or upper triangle being stored.

Iterative solution methods for sparse linear systems attempt to solve by solving an equivalent system , where M is some approximation to A which is inexpensive to construct and can be easily used to compute z. Unlike direct methods, iterative methods are more special-purpose. There are no general, effective iterative algorithms for an arbitrary sparse linear system.

A public domain library of subroutines for solving dense linear algebra problems, including systems of linear equations, linear least squares problems, eigenvalue problems, and singular value problems. It has been designed for efficiency on high-performance computers.

A set of simultaneous linear algebraic equations.

The process of dividing work done by each available processor into approximately equal amounts.

A parallel processing environment in which users do not know how many processors will be available to a job during run time.

A term that refers to algorithms that combine input and output with computation to solve problems in which the data resides on disk or some other secondary random-access storage device.

A triangular or symmetric matrix in which the full matrix representation is retained while storing only half the matrix elements.

The execution of one instruction per clock period, even those instructions that take several clock periods to complete execution.

A method of execution which allows each step of an operation to pass its result to the next step after only one clock period.

A section of a program that must use a single processor.

A class of problem size in which problems are large enough for vector and parallel processing, but for which parallel processing degrades vector performance.

A linear system which can be described as , where A is an n-by-n matrix, and x and b are n dimensional vectors. A system of this kind is considered sparse if the matrix A has a small percentage of nonzero terms (less than 10%, often less than 1%).

See Symmetric Positive Definite Matrix.

See Small Paralell/Vector Problem .

A recursive algorithm that is slightly faster than the ordinary inner product algorithm. Strassen's algorithm performs the floating-point operations for matrix multiplication in an order differently from the vector method; this can cause round-off problems.

A collection of columns that have the same nonzero pattern.

A method of execution in which the system works on several jobs or processes simultaneously.

A form of parallel processing that uses instruction segmenting and vector registers.

A virtual matrix is similar to a Fortran array, but it cannot be accessed directly from a program. It can only be accessed with calls to specific subroutines. Users do not do any explicit input or output to read from or write to a virtual matrix.

See Vector Problem.

The condition number of a matrix is defined as . A well-conditioned matrix is one for which is small. Although small is relative, if , A can be considered well-conditioned.