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Best approximate solution of matrix equation $$AXB+CYD=E$$. (English) Zbl 1096.15004
The authors consider the matrix equation $$(*)$$ $$AXB+CYD=E$$ where $$A$$, $$B$$, $$C$$, $$D$$ and $$E$$ are given matrices of suitable sizes. It arises in finite element model updating (which emerged in the 1990s in design, construction and maintainance of mechanical systems and civil engineering structures) in connection with the solution of certain partial differential equations. Equation $$(*)$$ is not always consistent. The authors express its best approximate solution (BAS) in the least-squares solution set to a given matrix pair $$(X_f,Y_f)$$.
The work is based on a projection in the finite-dimensional inner product space using generalized singular value decomposition and canonical correlation decomposition. A direct method for computing the BAS is developed. The algorithm for finding the BAS is described in detail, its feasibility and effectiveness are illustrated by an example.

##### MSC:
 15A24 Matrix equations and identities 65F20 Numerical solutions to overdetermined systems, pseudoinverses 65F22 Ill-posedness and regularization problems in numerical linear algebra
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