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The solution of a linear quaternionic least squares (QLS) problem can be transformed into that of a linear least squares (LS) problem with JRS-symmetric real coefficient matrix, which is suitable to be solved by developing structured iterative methods when the coefficient matrix is large and sparse. The main aim of this work is to construct a structured preconditioner to accelerate the LSQR convergence. The preconditioner is based on structure-preserving tridiagonalization to the real counterpart of the coefficient matrix of the normal equation, and the incomplete inverse upper–lower factorization related to only one symmetric positive definite tridiagonal matrix rather than four, so it is reliable and has low storage requirements. The performances of the LSQR algorithm with structured preconditioner are demonstrated by numerical experiments.

The total least squares (TLS) is a method of fitting that is appropriate when there are errors in both the observation vector b = b m × 1 and the data matrix A = A m × n . In this paper, we study the quaternion total least squares (QTLS) problem by means of real representations of quaternion matrices, and derive an algebraic technique for finding solutions of the QTLS problem in quaternionic quantum theory.

A split quaternionic least squares problem is one method of solving overdetermined sets of split quaternion linear equations AX ? B that is appropriate when there is error in the matrix B. In this paper, by means of complex representation and real representation of a split quaternion matrix, we study the split quaternionic least squares problem, derive two algebraic methods for finding solutions of the problems in split quaternionic mechanics.

Quaternion least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations $${AX \approx B}$$ and $${AXC \approx B}$$ that are appropriate when there are errors in the matrix B. This paper, by means of complex representation and real representation of a quaternion matrix, studies the QLS problems, and derives two new algebraic techniques for finding solutions of the QLS problem in quaternionic quantum mechanics.

In the study of theory and numerical computations of quantum theory, in order to well understand the perturbation theory, experimental proposals and theoretical discussions underlying the quaternion and complex formulations, one meets problems of approximate solutions of quaternion and complex problems, such as approximate solution of quaternion and complex linear equations A α ~ ? α λ or A α ? ? α λ that is appropriate when there are errors in the vector α and λ , i.e. quaternion and complex Least Squares coneigen-problem in quantum mechanics. This paper, by means of representation of quaternion matrices and complex matrices, studies the problems of quaternion and complex Least Squares coneigen-problem, and give practical algebraic methods of computing approximate coneigenvalues and coneigenvectors for quaternion and complex matrices in quantum mechanics.

In this paper, we introduce a kind of complex representation of quaternion matrices (or quaternion vectors) and quaternion matrix norms, study quaternionic least squares problem with quadratic inequality constraints (LSQI) by means of generalized singular value decomposition of quaternion matrices (GSVD), and derive a practical algorithm for finding solutions of the quaternionic LSQI problem in quaternionic quantum theory.

In the study of theory and numerical computations of quaternionic quantum mechanics and quantum chemistry, one of the most important tasks is to solve the Schrodinger equation ∂∂t|f〉=−A|f〉 with A an anti-self-adjoint real quaternion matrix, and |f〉 an eigenstate to A. The quaternionic Schrodinger equation plays an important role in quaternionic quantum mechanics, and it is known that the study of the quaternionic Schrodinger equation is reduced to the study of quaternionic eigen-equation Aα=αλ with A an anti-self-adjoint real quaternion matrix (time-independent). This paper, by means of complex representation of quaternion matrices, introduces concepts of norms of quaternion matrices, studies the problems of quaternionic Least Squares eigenproblem, and give a practical algebraic technique of computing approximate eigenvalues and eigenvectors of a quaternion matrix in quaternionic quantum mechanics.

By means of complex representation of a quaternion matrix, we study the relationship between the solutions of the quaternion equality constrained least squares problem and that of complex equality constrained least squares problem, and obtain a new technique of finding a solution of the quaternion equality constrained least squares problem.