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# Matrix Calibration for Domino Operation in APL

IP.com Disclosure Number: IPCOM000087000D
Original Publication Date: 1976-Nov-01
Included in the Prior Art Database: 2005-Mar-03
Document File: 2 page(s) / 22K

IBM

## Related People

Ho, CW: AUTHOR [+1]

## Abstract

The APL system is a widely used program product. One of the standard set of operators in APL is the domino operator, which inverts a matrix and solves a set of linear simultaneous equations. However, when the given matrix is stiff, i.e., the magnitudes of the largest and smallest eigen values of the matrix are widely separated, the domino operator fails to work even though the matrix is nonsingular. The stiff matrix appears very extensively in dealing with practical engineering problems on a computer, such as electronic circuit simulation, electronic device analysis and the like.

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Matrix Calibration for Domino Operation in APL

The APL system is a widely used program product. One of the standard set of operators in APL is the domino operator, which inverts a matrix and solves a set of linear simultaneous equations. However, when the given matrix is stiff, i.e., the magnitudes of the largest and smallest eigen values of the matrix are widely separated, the domino operator fails to work even though the matrix is nonsingular. The stiff matrix appears very extensively in dealing with practical engineering problems on a computer, such as electronic circuit simulation, electronic device analysis and the like.

A simple method is set forth below to check the stiffness of the matrix to be solved by the domino operator in the APL system. If the stiffness of the matrix, as measured by a condition number, exceeds a certain threshold, the matrix is first calibrated to reduce the condition number and then solved by the domino operator. The method of calibration is straightforward to implement and the operation count is proportional to only n where n is the dimension of the matrix. It is therefore small compared to the total time spent by the domino operator which is proportional to n/3/.

The condition number of the matrix is defined as:

(Image Omitted)

Where aij is the (i, j) -th element in matrix
A. If the condition number exceeds 10/9/, the matrix is calibrated by dividing every row of the matrix by its norm dii where

(Image Omitted)

and the matrix A i...