Related Research Articles
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics. However, the reverse operation is, in general, much more challenging.
In mathematics and computing, the Levenberg–Marquardt algorithm, also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and reasonable starting parameters, the LMA tends to be slower than the GNA. LMA can also be viewed as Gauss–Newton using a trust region approach.
LAPACK is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008). The routines handle both real and complex matrices in both single and double precision. LAPACK relies on an underlying BLAS implementation to provide efficient and portable computational building blocks for its routines.
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication. They are the de facto standard low-level routines for linear algebra libraries; the routines have bindings for both C and Fortran. Although the BLAS specification is general, BLAS implementations are often optimized for speed on a particular machine, so using them can bring substantial performance benefits. BLAS implementations will take advantage of special floating point hardware such as vector registers or SIMD instructions.
The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function. Since a sum of squares must be nonnegative, the algorithm can be viewed as using Newton's method to iteratively approximate zeroes of the components of the sum, and thus minimizing the sum. In this sense, the algorithm is also an effective method for solving overdetermined systems of equations. It has the advantage that second derivatives, which can be challenging to compute, are not required.
EISPACK is a software library for numerical computation of eigenvalues and eigenvectors of matrices, written in FORTRAN. It contains subroutines for calculating the eigenvalues of nine classes of matrices: complex general, complex Hermitian, real general, real symmetric, real symmetric banded, real symmetric tridiagonal, special real tridiagonal, generalized real, and generalized real symmetric matrices. In addition it includes subroutines to perform a singular value decomposition.
John Emory Dennis, Jr. is an American mathematician who has made major contributions in mathematical optimization. Dennis is currently a Noah Harding professor emeritus and research professor in the department of computational and applied mathematics at Rice University in Houston, Texas. His research interests include optimization in engineering design. He is the founder and editor-in-chief of the SIAM Journal on Optimization. In 2010, he was elected a Fellow of the Society for Industrial and Applied Mathematics.
CUTEr is an open source testing environment for optimization and linear algebra solvers. CUTEr provides a collection of test problems along with a set of tools to help developers design, compare, and improve new and existing test problem solvers.
Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. The "full" Newton's method requires the Jacobian in order to search for zeros, or the Hessian for finding extrema. Some iterative methods that reduce to Newton's method, such as SLSQP, may be considered quasi-Newtonian.
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations,
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965.
In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from different viewpoints. Its name refers to the geometrical bundles of light rays originating from each 3D feature and converging on each camera's optical center, which are adjusted optimally according to an optimality criterion involving the corresponding image projections of all points.
Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box–Cox transformed regressors ().
MINOS is a Fortran software package for solving linear and nonlinear mathematical optimization problems. MINOS may be used for linear programming, quadratic programming, and more general objective functions and constraints, and for finding a feasible point for a set of linear or nonlinear equalities and inequalities.
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.
INTLAB (INTerval LABoratory) is an interval arithmetic library using MATLAB and GNU Octave, available in Windows and Linux, macOS. It was developed by S.M. Rump from Hamburg University of Technology. INTLAB was used to develop other MATLAB-based libraries such as VERSOFT and INTSOLVER, and it was used to solve some problems in the Hundred-dollar, Hundred-digit Challenge problems.
References
- J. J. Moré, B. S. Garbow, and K. E. Hillstrom, User Guide for MINPACK-1, Argonne National Laboratory Report ANL-80-74, Argonne, Ill., 1980.
- J. J. Moré, D. C. Sorensen, K. E. Hillstrom, and B. S. Garbow, The MINPACK Project, in Sources and Development of Mathematical Software, W. J. Cowell, ed., Prentice-Hall, pages 88–111, 1984.
- ↑ Dennis, J. E.; Schnabel, Robert B. (1996). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics. pp. 228, 233, 265. ISBN 9781611971200.