II-Linear Algebra, and the corresponding FORTRAN subroutine in EISPACK.ĭefinition at line 178 of file EigenMatrix.H. The tridiagonaliseSymmMatrix, symmTridiagonalQL, Hessenberg and realSchur methods are based on the Algol procedures tred2 by Bowdler, Martin, Reinsch, and Wilkinson, Handbook for Auto.Their implementation was based upon EISPACK. The OpenQBMM eigenSolver class derives almost completely from the TNT/JAMA implementation, a public-domain library developed by NIST and MathWorks from 1998 to 2012, available at (Retrieved June 6, 2020).Therefore, no differences between EigenMatrix and eigenSolver (2019) classes should be expected in terms of input-process-output operations. This implementation is an integration of the OpenQBMM eigenSolver class (2019) without any changes to its internal mechanisms.F., Miller, B., Pozo, R., & Remington, K. (No particular order) Hicklin, J., Moler, C., Webb, P.,īoisvert, R. The International Journal of Supercomputer ApplicationsĪnd High Performance Computing, 11(3), 251-263. Template Numerical Toolkit for linear algebra: Handbook for Automatic Computation: Volume II: Linear Algebra. 'Hessenberg' and 'realSchur' (based on ALGOL-procedure:tred2): 'tridiagonaliseSymmMatrix', 'symmTridiagonalQL', Hessian spectrum as in Fig 2c or multiple bulks (two in the case of Fig 2d) with. Passalacqua, A., Heylmun, J., Icardi, M., dominant Hessian eigenvector can be shown, in some cases, to positively. References: OpenFOAM-compatible implementation: In addition, the validity of the equation \(A = v*D*v^T\) depends on the condition number of v, which can be ill-conditioned, or singular for invalidated equations. For the same eigenvalue, the corresponding eigenvector can be real or complex with non-unique entries. Even though eigenvalues of a matrix are unique, eigenvectors of the matrix are not. The columns of v represent eigenvectors corresponding to eigenvalues, satisfying the eigenvalue equation. If A is not symmetric, D becomes a block diagonal matrix wherein the real eigenvalues are present on the diagonal within 1-by-1 blocks, and complex eigenvalues within 2-by-2 blocks, i.e. If A is symmetric, the following relation is satisfied: eigenvalue problem) is:Ī diagonalisable square matrix of dimension m-by-mĪ (non-zero) vector of dimension m (right eigenvector)
Return right eigenvectors in unpacked form. Return real eigenvalues or real part of complex eigenvalues.
eigendecomposition or spectral decomposition) decomposes a diagonalisable nonsymmetric real square matrix into its canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.ĮigenMatrix (const SquareMatrix & A, bool symmetric)Ĭonstruct from a SquareMatrix and symmetry flag.