Yu. O. Vorontsov

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + XTB = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.

Conditions for unique solvability of the matrix equation AX + X*B = C

Conditions for the unique solvability of the matrix equation AX + X*B = C are formulated in terms of the eigenvalues and the Kronecker structure of the matrix pencil A + λB* associated with this equation.

Numerical algorithm for solving the matrix equation AX + X* B = C

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X*B = C is suggested. By applying the QZ-algorithm to the original equation, it is transformed into an equation of the same type with triangular matrix coefficients A and B. The resulting matrix equation is equivalent to the sequence of a system of linear equations with a smaller order of the coefficients of the desired solution. Using numerical examples, the authors simulate a situation where the conditions of a unique solution are “almost” violated. Deterioration of the calculated solutions is in this case followed.

Numerical solution of matrix equations of the form X + AXTB = C

A review of numerical methods for solving matrix equations of the form X + AXTB = C is given. The methods under consideration were implemented in the Matlab environment. The performances of these methods are compared, including the case where the conditions for unique solvability are “almost” violated.

Modifying a numerical algorithm for solving the matrix equation X + AXTB = C

Certain modifications are proposed for a numerical algorithm solving the matrix equation X + AXTB = C. By keeping the intermediate results in storage and repeatedly using them, it is possible to reduce the total complexity of the algorithm from O(n4) to O(n3) arithmetic operations.

Numerical solution of Sylvester matrix equations in the self-adjoint case

Algorithms of the Bartels-Stewart type for numerically solving Sylvester matrix equations of modest size are modified for the case where the linear operators associated with these equations are self-adjoint. The superiority of the modified algorithms over the original ones is illustrated by numerical results.

Numerical solution of the matrix equations AX + X T B = C and AX + X*B = C in the self-adjoint case

The numerical algorithms for solving equations of the type AX + XTB = C or AX + X*B = C that were earlier proposed by the authors are now modified for the situations where these equations can be regarded as self-adjoint ones. The economy in computational time and work achieved through these modifications is illustrated by numerical results.

Numerical solution of matrix equations of the Stein type in the self-adjoint case

The algorithms for solving the equations X − AXTB = C and X − AX*B = C proposed by the authors in earlier publications are now modified for the case where these equations can be regarded as self-adjoint ones. The economy in the computational time and work achieved through these modifications is illustrated by numerical results.

Numerical algorithm for solving sesquilinear matrix equations of a certain class

AbstractA relationship is found between the solutions to the sesquilinear matrix equation X*DX + AX + X*B + C = 0, where all the matrix coefficients are n × n matrices, and the neutral subspaces of the 2n × 2n matrix