Preconditioned conjugate gradient system

Details Preconditioned conjugate gradient system Preconditioned conjugate gradient system

1990-02-12

1992-08-04

LaRoche, Eugene R.

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364402, G06F 9302

Patent

active

051365383

ABSTRACT:
Apparatus for optimizing the operational state of a system, which follows the Karmarkar method, and which in the course of carrying out this method, obtains a solution to the linear system of equations AD.sup.2 A.sup.T u=p in accordance with a modified conjugate gradient method that incorporates a preconditioning operation. The preconditioning operation includes pre-drop and post-drop procedures that reduce the number of non-zero terms in the matrix to be preconditioned by judiciously discarding values that are smaller than a certain threshold. Use of the preconditioned conjugate gradient method reduces the processing time required for carrying out each iteration in the Karmarkar method. Further improvement in the operating speed of our apparatus is achieved by employing an asymmetric conjugate gradient method, where the preconditioning parameters are adjusted to obviate the need for square root operations in obtaining the above solution; performing matrix product operations in accordance with an outer product procedure and tailoring the code to the specific attributes of the A matrix.

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patent: 4744028 (1988-05-01), Karmarker
Berges, Richard T., "Solving Matrix Problems on Small Computers", Electro-Technology; Sep. 1967; pp. 44, 45.
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Combinatorica, vol. 4, No. 4, 1984, "A New Polynomial-Time Algorithm for Linear Programming", N. Karmarkar, pp. 373-395.

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