Computation of orthonormal factors for fundamental solution matrices
The paper is concerned with the linear system \[ Y'(t)=A(t)Y(t), \quad Y(0)=Y_0, \quad t\geq 0, \tag{1} \] where \(A:t\in[0,t_f]\to A(t)\in {\mathbb R}^{n\times n}\) is a \(C^{k-1}\) function, \(k\geq 1,\) so that \(Y\in C^k\), \(Y_0\) is a full rank matrix and \(Y(t), Y_0\in {\mathbb R}^{n\times p}, n\geq p.\) The purpose is to compute the \(QR\) factorization of some or all columns of a fundamental solution matrix \(Y(t)\) with the emphasis on the integration of the nonlinear polynomial equation for \(Q.\) The authors consider the situation where \(Q\) can be written as a product of elementary Householder or Givens transformations. In general, these transformations do not vary smoothly on the whole interval. Therefore, in order to obtain a well-defined process the authors use a reimbedding procedure obtaining as a result a piece-wise smooth factorization of \(Q\). In order to obtain, if needed, globally smooth factors, a post-processing based on monitoring signs on the diagonal of \(R\) can be implemented. Numerical results are considered to compare the new methods with the existing techniques.
scientific article; zbMATH DE number 1398755