Syamal Kumar Sen

Solving linear programming problems exactly

A linear programming problem in an inequality form having a bounded solution is solved error-free using an algorithm that sorts the inequalities, removes the redundant ones, and uses the p-adic arithmetic

A concise algorithm to solve over-/under-determined linear systems

An O(mn2) direct algorithm to compute a solution of a system of m linear equations Ax=b with n variables is presented. It is concise and matrix inversion- free. It provides an in-built consistency check and also produces the rank of the matrix A. Further, if necessary, it can prune the redundant rows of A and convert A into a full row rank matrix thus preserving the complete information of the system. In addition, the algo rithm produces the unique projection operator that projects the real (n)-dimensional space orthogonally onto the null space of A and that provides a means of computing a relative error bound for the solution vector as well as a nonnegative solution.

Rank-Augmented LU-Algorithm for Computing Generalized Matrix Inverses

A rank-augmnented LU-algorithm is suggested for computing a generalized inverse of a matrix. Initially suitable diagonal corrections are introduced in (the symmetrized form of) the given matrix to facilitate decomposition; a backward-correction scheme then yields a desired generalized inverse.

Error-free matrix symmetrizers and equivalent symmetric matrices

A symmetrizer of the matrix A is a symmetric solution X that satisfies the matrix equation XA=A′X. An exact matrix symmetrizer is computed by obtaining a general algorithm and superimposing a modified multiple modulus residue arithmetic on this algorithm. A procedure based on computing a symmetrizer to obtain a symmetric matrix, called here an equivalent symmetric matrix, whose eigenvalues are the same as those of a given real nonsymmetric matrix is presented.

Solving linear differential equations as a minimum norm least squares problem with error-bounds

Linear ordinary/partial differential equations (DEs) with linear boundary conditions (BCs) are posed as an error minimization problem. This problem has a linear objective function and a system of linear algebraic (constraint) equations and inequalities derived using both the forward and the backward Taylor series expansion. The DEs along with the BCs are approximated as linear equations/inequalities in terms of the dependent variables and their derivatives so that the total error due to discretization and truncation is minimized. The total error along with the rounding errors render the equations and inequalities inconsistent to an extent or, equivalently, near-consistent, in general. The degree of consistency will be reasonably high provided the errors are not dominant. When this happens and when the equations/inequalities are compatible with the DEs, the minimum value of the total discretization and truncation errors is taken as zero. This is because of the fact that these errors could be negative as well as positive with equal probability due to the use of both the backward and forward series. The inequalities are written as equations since the minimum value of the error (implying error-bound and written/expressed in terms of a nonnegative quantity) in each equation will be zero. The minimum norm least-squares solution (that always exists) of the resulting over-determined system will provide the required solution whenever the system has a reasonably high degree of consistency. A lower error-bound and an upper error-bound of the solution are also included to logically justify the quality/validity of the solution.

On computing an equivalent symmetric matrix for a nonsymmetric matrix

A real or a complex symmetric matrix is defined here as an equivalent symmetric matrix for a real nonsymmetric matrix if both have the same eigenvalues. An equivalent symmetric matrix is useful in computing the eigenvalues of a real nonsymmetric matrix. A procedure to compute equivalent symmetric matrices and its mathematical foundation are presented.

Stabilizing Trench's algorithm to invert symmetric Toeplitz matrices

Presented here is a stable algorithm that uses Zohars formulation of Trenchs algorithm and computes the inverse of a symmetric Toeplitz matrix including those with vanishing or nearvanishing leading minors. The algorithm is based on a diagonal modification of the matrix, and exploits symmetry and persymmetry properties of the inverse matrix.