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SCIENTIST 3
微分方程曲線套配軟體
Parameter fitting for model equations
軟體代號:304
瀏覽次數:5220
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Features
Micromath Scientist®


Scientist is designed to provide a comprehensive solution to the problem of fitting experimental data on the PC. It includes the capability of solving systems of model equations that can include nonlinear equations, ordinary differential equations and Laplace transforms. Scientist is an application for researchers who "know what's going on" with their data and need to establish solid parameter values to model real-world phenomena. Scientist provides remarkably simple model equation entry, data management, control of initial parameter estimates and constraints on parameter values, as well as complete statistical analysis and publication quality graphical output.


The interactive nature of Scientist leads to a higher likelihood of finding optimal parameter values than if fitting were done in batch mode, without the ability to examine and react to computational results. It also enables users to develop a much greater awareness of the sensitivity of models to parameter values. With Scientist you can develop and fit data to the most complex models involving nonlinear, differential and Laplace transform equations.


Scientist can model phenomena from all scientific and engineering disciplines and is being used in many teaching and research applications including: Physical Chemistry, Organic Chemistry, Pharmaceutical Chemistry, Biophysics, Thermodynamics and Heat Transfer, Kinetics, Genetics, Sociology, Economics, Physics, Mechanical Engineering, Electrical Engineering, Civil Engineering, Applied Mathematics and many others. Models can consist of single functions [defined by several lines of code], multiple functions that can be fit simultaneously, implicit equations or systems of equations, parametric equations (i.e. X and Y both defined in terms of a third variable), differential or integral equations and equations involving Laplace transforms.



Minimization Algorithms


Scientist employs a least squares minimization procedure based on a modification of Powell's algorithm. This algorithm is many times faster than the more common algorithms based on sequential searches involving one parameter at a time. The algorithm is a hybrid that combines the reliability of a steep descent method with the speed of the Gauss-Newton method near convergence. Other minimization methods are also available. These methods may be more effective under certain circumstances. For example, a nonlinear simplex algorithm may be to locate the general location of a minimum (i.e. improving parameter estimates prior to least squares minimization). Steepest descent and Levenberg-Marquardt minimization algorithms are also available.




Scientist provides comprehensive numerical integration of differential equations. This makes Scientist a powerful, easy-to-use model development tool for Scientists, engineers and graduate and undergraduate students in science and engineering. Scientist allows users to focus on science, not software.


The algorithms used in Scientist are adapted from various sources. Scientist implements four standard methods and a method designed to integrate stiff equations (EPISODE):
  • Euler's Method
  • Runge-Kutta Method (Fourth Order)
  • Error Controled Runge-Kutta Method
  • Bulirsch-Stoer Method




The use of Laplace transforms can greatly simplify the solution of models representing very complicated physical systems. The Laplace transform reduces differential equations to algebraic equations in order to solve them. The equations can then be inverted to obtain the solution to the differential equations. Scientist can calculate the numerical inverse of models written as Laplace transforms. This is particularly useful when the inverse transform has no explicit solution.
This technique can be applied to a broad range of scientific, engineering and other technical problems. It also allows the solution of problems that might otherwise be impractical. The inverse Laplace transform may be calculated for a single point, for a curve representing a range of time values, or for a family of curves in situations dependent on both space and time coordinates. Equations involving Laplace transforms can be directly fitted to data, freeing the user from the time consuming iterative parameter refinement process that would otherwise be required.


The algorithms used for the inversion of Laplace transforms are adapted from various sources. Scientist implements both Piessens' method and Weeks' method.




Scientist provides a broad range of statistical output, including parameter estimates, confidence limits, various measures of goodness-to-fit, variance-covariance and correlation information, and analysis of residuals. Confidence limits for parameter estimates are calculated using the customary approach involving a local linearization of the model or a more rigorous approach that locates various points on constant sum of squares contours.