Backward error analysis of specified eigenpairs for sparse matrix polynomials

This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$ ‐symmetric, T$$ T $$ ‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$ ‐even, T$$ T $$ ‐odd, H$$ H $$ ‐even, H$$ H $$ ‐odd, T$$ T $$ ‐palindromic, T$$ T $$ ‐anti‐palindromic, H$$ H $$ ‐palindromic, and H$$ H $$ ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.