Abstract: Various applications are presented of a vector field method of computing one or more eigenvalues and eigenvectors of a symmetric matrix. The vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere. The optimization problems embedded in each iteration of the vector field algorithms admit closed-form solutions making the vector field approach relatively efficient. Among the several vector fields discussed is a family of vector fields called the recursive vector fields. Numerical results are presented that suggest that in some embodiments the recursive vector field method yields implementations that are faster than those based on the QR method. Further, the vector field method preserves, and hence can fully exploit the sparseness on the given matrix to speed up computation even further. Preprocessing that contracts the spectral radius of the given matrix further accelerates the systems.

Abstract: A method of evaluating progress of a cancer comprises calculating a prognostic index by selecting coefficients for each of a plurality of nuclear pleomorphic features using a numerical regression, such that the selected coefficients provide a correlation extremum to times to recurrence. A prognostic index is calculated by measuring nuclear pleomorphic features in a patient, and multiplying the observed values by the corresponding selected coefficients. The resulting prognostic index can provide a superior prediction of the likely time to recurrence in the patient.

Abstract: The present invention is a system and method of a feasible point method, such as a canonical coordinates method, for solving non linear optimization problems. The method goes from a point to another point along a curve of a defined nonlinear surface. An objective function is determined from the plurality of points. Each point is given a value determined from the objective function. The objective function value is maximized to improve computational efficiency of a non linear optimization procedure.

Abstract: The present invention is a system and method of improving computational efficiency of constrained nonlinear problems by utilizing Lie groups and their associated Lie algebras to transform constrained nonlinear problems into equivalent unconstrained problems. A first nonlinear surface including a plurality of points is used to determine a second nonlinear surface that also includes a plurality of points. A reference point is selected from the plurality of points of the second nonlinear surface. An objective function equation is maximized by computing a gradient direction line from the reference point. The reference point is adjusted to the point determined along the gradient direction line having the highest associated value.