WIAS Preprint No. 1646, (2011)

Shifted linear systems in electromagnetics. Part II: Systems with multiple right-hand sides


  • Schlundt, Rainer
    ORCID: 0000-0002-4424-4301
  • Schmückle, Franz-Josef
  • Heinrich, Wolfgang

2010 Mathematics Subject Classification

  • 35Q60 65F10 65F15 65N22 78M25


  • Microwave device, Maxwell's equations, Scattering matrix, Boundary value problem, PML boundary condition, Eigenvalue problem, Linear algebraic equations, Multiple shifts, Multiple right-hand sides, Krylov subspace method, Polynomial preconditioning, Initial guesses


We consider the solution of multiply shifted linear systems for multiple right-hand sides. The coefficient matrix is symmetric, complex, and indefinite. The matrix is shifted by different multiples of the identity. Such problems arise in a number of applications, including the electromagnetic simulation in the development of microwave and mm-wave circuits and modules.
The properties of microwave circuits can be described in terms of their scattering matrix which is extracted from the orthogonal decomposition of the electric field. We discretize the Maxwell's equations with orthogonal grids using the Finite Integration Technique (FIT).
Some Krylov subspace methods have been used to solve systems with multiple right-hand sides. We use both the block-QMR method and a symmetric band Lanczos process based on coupled recurrences with polynomial preconditioning.
We present a method for providing initial guesses to a linear solver both for systems with multiple shifts and for systems with multiple right-hand sides each with a different shift.

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