Setup

See Also: Troubleshooting Nonlinear Equations  Troubleshooting Differential Equations

Several general program parameters and parameters of numerical solution algorithms can be adjusted by the user. The adjustable parameters can be reached by pressing “setup”. After changing a parameter its value is kept until it is changed again by the user or its default value is "restored". There is also the "Reset All" option to restore all the parameter values to their default value. Note that reloading POLYMATH does not change the stored parameter values. 

Following is a brief description of the adjustable parameters, their default values and their significance in numerical computations.

Data table

1. Initial number of columns: The total number of columns in the table - Default value = 30

2. Initial number of rows: The total number of rows in the table - Default value = 300

General

1. Show Polymath logo upon startup -Default value = True

2. Show recent files startup dialog - Default value = False

3. Show status bar - Default value = True

4. Maximize sub windows - Default value = True

5. Undo buffer size - Default value = 20

Graph

1. Use vertical gridlines - Default value = False.
2. Use horizontal gridlines - Default value = False.
3. Default marker size - Default value = 70
4. Default title = Graph Title
5. Default subtitle - blank
6. Query for save on close - Default value = False

Linear equations (LIN)

1. Initial number of equations - Default value = 5
2. Show equations in report  - Default value = False

Nonlinear equations (NLE)

1. Reporting digits - Default value = 9
2. Calculate initial guess values - Default value = False
3. Font name - Default value = Arial
4. Font size - Default value = 9.75
5. Auto color syntax - Default value = True

6. Auto unpaint on Byte size - Default value = 70
7. Show comments in report - Default value = True

8. Show MATLAB formatted problem in report - Default value = False

The parameter names and their default values in the various algorithms are shown in the table below:

In the iterative solution of the nonlinear algebraic equations the algorithms fastnewt, safenewt and fastbroyd use the absolute function values and the relative x corrections for convergence test. If the maximal, absolute function value is smaller than “Tolerance f” or the maximum, relative x correction is smaller than “Tolerance x” the iterations are stopped and the solution is accepted as a valid solution. Safenewt and safebroyd include function minimization along the Newton or Broyden directions. If a normalized gradient value along this direction is smaller than “Tolerance min” it is assumed that the convergence is to a local minimum, the iterations are stopped and the solution is not accepted. In “constrained” the norm of the relative x correction values is checked against the “Relative error tolerance”. If the norm is smaller than the tolerance the iterations are stopped and the solution is accepted.

In case of no convergence, possible remedial actions include tolerance, and maximum number of iteration changes. For suggestions in this respect see the “Troubleshooting” section.

Ordinary differential equations (DEQ)

1. Reporting digits - Default value = 9
2. Intermediate data points - Default value = 100

• Intermediate data points: The number of equally spaced points along the solution where values are being stored. Default value = 100. If the solution curves appear to be non-smooth, broken lines the number of data points must be increased (maximum 500 points). Note however that increasing the number of data points may lead to memory shortage problems especially if there are many equations in the system.

3. Show intermediate data points on report - Default value = False
4. Font name - Default value = Arial
5. Font size - Default value = 9.75

6. Auto color syntax - Default value = True

7. Auto unpaint on Byte size - Default value = 70

8. Show comments in report - default value = True

9. Show MATLAB formatted problem in report - Default value = False

The additional parameter names and their default values in the various algorithms are shown in the table below:

In the numerical solution of ordinary differential equations, all algorithms in the program adjust the integration step-size in order to reduce the estimate of the local truncation error below a specified error tolerance. “Initial step size” is the step size with which the integration starts, the step size is being increased or decreased so that to keep the maximal truncation error estimate (scaled with the respective dependent variable values) below the “Scaled truncation error tolerance”. If the specified tolerance requires reducing the step-size below the “Minimum allowed step-size” the integration stops and the error message “Step-size is too small” is displayed. Possible remedial actions that can be taken in such case are discussed in the “Troubleshooting” section.

Regression, Nonlinear

mrqmin

1. Maximum number of iterations -  Default value = 64

2.Tolerance: Tolerance for the sum of squares of the differences between the calculated and measured values of the dependent variable. Default value =0.0001

 

LM (Levenberg-Marquardt)

The convergence rate of the Levenberg-Marquardt algorithm, used for solution, can often be slow. Furthermore, the default error tolerance may not be sufficient to obtain accurate results. It is a good practice to run a problem for which solution has been found, starting with the parameter values that were found as solution with smaller error tolerance. If the converged solution yields parameter values close to the previous ones the solution is accurate enough. Otherwise the tolerance should be further reduced.