The paper presents an algorithm for computing a trust region step in optimization, which is used in trust region methods for minimizing a quadratic function subject to an ellipsoidal constraint. The algorithm is guaranteed to produce a nearly optimal solution in a finite number of iterations. The algorithm is based on solving a subproblem of the form $ \min \{\psi(w): ||w|| \leq \Delta\} $, where $ \psi(w) = g^T w + \frac{1}{2} w^T B w $, with $ B $ being a symmetric matrix. The algorithm is robust and efficient, and it is used in a trust region Newton's method. The paper proves that under reasonable assumptions, the sequence $ \{x_k\} $ generated by Newton's method has a limit point $ x^* $ that satisfies the first and second order necessary conditions for a minimizer of the objective function $ f $. Numerical results for GQTPAR, a Fortran implementation of the algorithm, show that it is quite successful in a trust region method. In the tests, a call to GQTPAR required on average 1.6 iterations. The algorithm is described in detail, including its structure, convergence criteria, and implementation. The paper also discusses the use of the algorithm in trust region methods for unconstrained minimization and proves that the sequence generated by the algorithm converges to a point $ x^* $ with $ \nabla f(x^*) = 0 $ and $ \nabla^2 f(x^*) $ positive semidefinite. The algorithm is shown to have strong convergence properties and is efficient in practice.The paper presents an algorithm for computing a trust region step in optimization, which is used in trust region methods for minimizing a quadratic function subject to an ellipsoidal constraint. The algorithm is guaranteed to produce a nearly optimal solution in a finite number of iterations. The algorithm is based on solving a subproblem of the form $ \min \{\psi(w): ||w|| \leq \Delta\} $, where $ \psi(w) = g^T w + \frac{1}{2} w^T B w $, with $ B $ being a symmetric matrix. The algorithm is robust and efficient, and it is used in a trust region Newton's method. The paper proves that under reasonable assumptions, the sequence $ \{x_k\} $ generated by Newton's method has a limit point $ x^* $ that satisfies the first and second order necessary conditions for a minimizer of the objective function $ f $. Numerical results for GQTPAR, a Fortran implementation of the algorithm, show that it is quite successful in a trust region method. In the tests, a call to GQTPAR required on average 1.6 iterations. The algorithm is described in detail, including its structure, convergence criteria, and implementation. The paper also discusses the use of the algorithm in trust region methods for unconstrained minimization and proves that the sequence generated by the algorithm converges to a point $ x^* $ with $ \nabla f(x^*) = 0 $ and $ \nabla^2 f(x^*) $ positive semidefinite. The algorithm is shown to have strong convergence properties and is efficient in practice.