TY - JOUR
T1 - Polynomials and lemniscates of indefiniteness
AU - Huhtanen, Marko
AU - Nevanlinna, Olavi
PY - 2016/6
Y1 - 2016/6
N2 - For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in C-nxn. A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the L-2-norm algorithms for solving the problem are devised.
AB - For a large indefinite linear system, there exists the option to directly precondition for the normal equations. Matrix nearness problems are formulated to assess the attractiveness of this alternative. Polynomial preconditioning leads to polynomial approximation problems involving lemniscate-like sets, both in the plane and in C-nxn. A natural matrix analytic extension for lemniscates is introduced. Operator theoretically one is concerned with polynomial unitarity and associated factorizations for the inverse. For the speed of convergence and lemniscate asymptotics, the notion of quasilemniscate arises. In the L-2-norm algorithms for solving the problem are devised.
KW - POTENTIAL-THEORY
KW - NORMAL OPERATORS
KW - APPROXIMATION
KW - ITERATIONS
KW - LENGTH
UR - http://link.springer.com/article/10.1007%2Fs00211-015-0745-2
U2 - 10.1007/s00211-015-0745-2
DO - 10.1007/s00211-015-0745-2
M3 - Article
VL - 133
SP - 233
EP - 253
JO - Numerische Mathematik
JF - Numerische Mathematik
SN - 0029-599X
IS - 2
ER -