PolymathPlus Manual 
PolymathPlus is a generic mathematical problem solver. The problems are entered in a simple text format; and pressing the ► Solve button yields a detailed numerical solution report.
Students, engineers, scientists, or anyone with a need for a numerical math problemssolving package will appreciate the simplicity, efficiency, and speed of PolymathPlus.
The following problem types are supported:
Problems are entered in plain text.
The
The number of equations must be equal to the number of unknowns.
All the algebraic expressions must be linear hence only the plus, minus, and multiplication operators are allowed. Expressions like x*x
, 1/x
, sin(x)
are not allowed.
The following example demonstrates the input of a linear system of equations:
Given the following set of 3 linear equations to be solved:
The Solution Report includes the values of the unknowns at the solution (a = 54, b = 4, and c = 2, for this problem) and the problem definition in both matrix and system of equations formats.
When solving simultaneous nonlinear equations, the equations have to be arranged as root expressions (expressions that are equal to zero at the solution).
In addition, an initial guess should be provided for each nonlinear variable.
Rearranging the equations, and estimating x
and y
to be 1 at the solution, this would be the data entry to solve this problem:
you may define any number of auxiliary expressions, to simplify long expressions:
and the resulting solution report will show:
PolymathPlus can numerically solve either a single nonlinear equation or a set of simultaneous nonlinear equations.
When solving a single nonlinear equation, the user has to provide a range of values within which the solution exists.
The program will find all the solutions within that range.
When solving multiple nonlinear equations, the user has to provide a guess for each variable to be found. In this case the guess is a single value and not a range as indicated on the single nonlinear equation.
The nonlinear equations have to be arranged such that all terms are on one side and the other side should be set to zero, essentially building the root expression. The algorithm will search the independent variables which cause the root expression to become zero, or very close to zero.
Additional auxiliary variables can be defined to simplify expressions for repetitive terms.
The problems belonging to this category contain implicit nonlinear algebraic equations that require an iterative solution and explicit algebraic equations where individual variables can be expressed as explicit functions of other variables and constants. An implicit nonlinear equation must be entered in the form:
f(x) = an expression
where x
is the chosen variable name, and f(x)
is an expression that will have the value of zero at the solution.
An explicit equation should be rewritten in the form:
x = an expression
where the expression may contain constants, implicit variables, and explicit
variables.
Only real roots (noncomplex) can be found. Multiple roots are determined
for a single equation where the range of the variable must be provided.
This type of problem contains a set of explicit algebraic equations. Each equation must start with a variable name followed by the equal sign and a linear or nonlinear algebraic expression. The equations may be entered in any order. During the solution, the equations are automatically ordered to achieve a solution. The following system of explicit equations calculates the two roots of a quadratic equation. Note that both problem setups have the same solution.
The Solution Report includes the values of the unknowns (x_{1} = 7, x_{2} = 5) and all the variables at the solution.
Note that the equations do not need to be ordered for solution. Keep this option in mind for your general explicit calculational needs.
This option involves the solution of a single implicit nonlinear
equation with any number of explicitly defined variables. The implicit
nonlinear equation must be arranged to an expression which is equal to zero
at the solution.
For the variable associated with the implicit equation,
both minimal and maximal initial estimates must be provided. Multiple roots
are reported if the problem has several solutions (roots) between the
minimal and maximal initial estimates.
The equations and initial estimates
can be entered in any order. PolymathPlus always reorders the equations
during the problem solution.
Solve a nonlinear equation to find the value of x:
The Solution Report includes the value/s of the unknown (x=20, in this case), the function value/s (f(x)= 1.0e8), and the values of all the variables at the solution.
This option involves the solution of a system comprising of several, implicit, nonlinear algebraic equations with any number of explicitly defined variables. All the implicit equations must be rewritten in the form:
f(x) = an expression
where x is a variable that does not appear on the left hand side of any of the explicit equations and the "expression" must be equal to 0 at the solution. Initial estimates must be provided for all the implicit variables using the syntax: x(0) = a value. The equations may be written in any order.
Example, Solving a set of 2 nonlinear equations:
The equations can be entered in the following manner (note that lines starting with
The Solution Report includes the values of the unknowns (x = 5, y = 20), the function values (f(x)= 2.7E12, f(y)= 1.3E12 ) and the values of all the additional variable/s at the solution.
These can be solved as a set of 5 nonlinear equations where an initial guess should be provided for each variable.
Note that both programs give the same solution, but the second program requires initial estimates for variables a and c only. The solution is calculated as: a = 1.00011; b = 2.00055; c= 3.00066; d = 3.99967; and e = 4.99964.
A special solution algorithm is available for problems where some of the variables can have only positive values.
Consider the following set of equations:
having
#@ NLE_SOLUTION_METHOD_INDEX = 3
and the solution becomes:
Note that the constrained method may only be used for sets of equations with more than 1 simultaneous nonlinear equation.
The boundary conditions are: x(0)=1, y(0)=4, t(0)=0, t(f)=1
.
This is the data entry to solve the problem:
or you may define any number of auxiliary expressions, to simplify long and/or repeating sections:
d(x)/d(t) = an expression
Here t is the independent variable and x is a dependent variable. Note that other independent and dependent variable names are possible. Auxiliary algebraic equations must be entered in the form:
x = an expression
where the same naming rules apply.
An expression may contain any combination of constants and variables. However, the auxiliary equations must be explicit (not simultaneous). PolymathPlus will sort and order the equations and will indicate if a simultaneous situation exists (by warning of 'circular dependency').
Initial conditions must be specified for all the variables defined by differential equations, as well as the independent variable. A final value must also be specified for the independent variable.
The Solution Report includes a Table listing the initial, minimal, maximal, and final values of all the variables over the integration range. In the example shown, the final values are: A = 0.049787, B = 0.047308 and C = 0.902905. A table containing 50 values of the dependent variables between the initial and final values of the independent variable is also displayed.
Note that the same ODE problem can also be entered in a simplified format such as this:
This category of problems involves the fitting of data to various mathematical expressions. The
resulting fitted expressions can be considered by their ability to statistically
represent the data.
PolymathPlus allows the user to fit data sets to
The known data points should be entered in a strict tabulated form enclosed in square brackets [ ]
. Open bracket [
and close bracket ]
should get their own dedicated lines.
The 1st line in the table contains the variable names separated by spaces. The rest of the lines contain the values of the variables, columns are seperated by spaces. The number of columns in each line must match, blank values are not allowed.
For example:
PolymathPlus can fit a polynomial of degree n with the general form:
or a linear equation when the equivalent first degree polynomial is specified
where a_{0}, a_{1}, ..., a_{n} are regression parameters to a set of N tabulated values of x (a single independent variable) versus y (a single dependent variable). The highest degree allowed for a polynomial is N  1 (thus n >= N  1). The program calculates the coefficients a_{0}, a_{1}, ..., a_{n} by minimizing the sum of squares of the deviations between the calculated y or P(x) and the data for y.
A polynomial regression is specified by the "polyfit" command that must be followed by the name of the independent variable vector, name of the dependent variable vector, a number representing the degree of the polynomial, and an optional string 'origin' which indicates that the polynomial regression curve must go through the origin. These names must be separated by commas.
Consider the following BOD (biological oxygen demand) data versus Time:
Fitting a straight line (linear regression) to the BOD vs. Time data. BOD = a0 + a1*Time
This program yields the results a_{0} = 0.657797 (±0.374896), (95% confidence intervals shown between parentheses), a_{1} = 0.178056 (±0.031674), R^{2} = 0.9472735 and the variance: σ^{2} = 0.0825304.
In the Solution Report a table containing the observed (measured) y values, the predicted y values (ycalc) and the residuals (r = y ycalc) are also displayed.
Fitting the data to a 2^{nd} degree polynomial.
To fit a 2^{nd} degree polynomial, the polyfit statement is revised:
The results obtained in this case are a_{0} = 0.224387 (±0.353054 ), a_{1} = 0.310403 (±0.081692 ), a_{2} = 0.006439 (±0.003851), R^{2} = 0.9815572 and σ^{2} = 0.0324762.
To fit a 2^{nd} degree polynomial which passes through the origin, the following polyfit statement is used:
For this polynomial a_{0} = 0, by definition, and the rest of the results are: a_{1} = 0.354779 (±0.044171), a_{2} = 0.008236 (±0.002724), R^{2} = 0.9766053 and σ^{2} = 0.0366186.
Below are the plots containing: BOD (the data) vs. P(x) (calculated values), and Delta BOD (residuals) vs. time.
Regression Analysis: Judging by the R^{2 }and σ^{2} values, the 2nd degree polynomial represents the data the best. However, the confidence interval value of a_{0} shows that this parameter is not significantly different from zero, thus the use of 2nd degree polynomial passing through the origin should be preferred.
Consider the following set of Pressure (mm Hg) versus Temperature (deg C) data:
The following operations need to be carried out with these data:
The program to accomplish these tasks is shown below.
The Solution Report includes a Table listing the parameter values
and confidence intervals on these values.
For this particular problem the following results are shown.
The regression plot:
The Solution Report includes also the statistical metrics: R2, R2adj, root mean squared deviation and variance. A table displaying the observed (measured) y values, the predicted y values (ycalc) and the residuals (r = y  ycalc).
To fit a polynomial that passes through the origin the "polyfit"
command must also include the parameter "origin".
For example:
This PolymathPlus option will fit a linear function of the form:
where a_{0}, a_{1}, ..., a_{n} are regression parameters, to a set of N tabulated values of x_{1}, x_{2}, ..., x_{n} (independent variables) versus y (dependent variable). Note that the number of data points must be greater than n+1 (thus N >= n+1). The program calculates the coefficients a_{0}, a_{1}, ..., a_{n} by minimizing the sum of squares of the deviations between the calculated and the data for y.
Use the mlinfit command to request a multiple linear regression fit. The arguments are the independent variables vector's names, followed by the independent vector name. Additional optional (last) argument is the string 'origin' which indicates that the regression curve should pass through the origin (thus a_{0} = 0).
The following data are for the wear of a bearing y that may be dependent upon x_{1} = oil viscosity and x_{2} = load data.:
In this example we compare the linear regression models that includes x_{1} and x_{2} as independent variables, with a model that includes only x_{1}.
y = a0 + a1*x1 + a2*x2
The results obtained in this case are a_{0} = 360.836 (±118.076), a_{1} = 3.75246(±1.774), a_{2} = 0.084265(± 0.140313), R^{2} = 0.9835209 and σ^{2} = 159.1951.
To remove x_{2} from the regression model, the mlinfit statement is revised:
After this modification the following results are obtained: a_{0}
= 292.784(±37.3385 ), a_{1} = 4.58727(±1.23994), R^{2} =
0.963462 and σ^{2} = 264.7302 .
Judging by the
R^{2}
and
σ^{2}
values, the
regression model that contains both x_{1 }and x_{2}
represents the data better. However, the confidence interval value of a^{2}
shows that this parameter is not significantly different from zero, thus
removal of x_{2} from the model should be preferred
.
Considering the "vapor pressure" example to fit a regression model of the form:
The dependent variable can be defined by logP=log(P_kPa)
and the independent variables defined by OneonT=1/T_K
and logT=log(T_K)
.
Then mlinfit
can be used to find the parameter values.
The Solution Report includes a Table listing the parameter values and confidence intervals on these values. For this particular problem the following results are shown.
Nonlinear regression involves a general mathematical function (model) of the form:
where a_{0}, a_{1}, ..., a_{n} are regression parameters
to a set of N tabulated values of x_{1}, x_{2}, ..., x_{n}
(independent variables) versus y (dependent variable). Note that the number of
data points must be greater than m + 1 (thus N >= m + 1).
Use the nlinfit command to request nonlinear regression fit. The argument is the nonlinear regression model: the dependent variable vector name on the left side of the "=" sign, the independent variable vectors names and the parameters on the right side. Initial estimates must be provided for all the parameter values using expressions of the form:
Fitting a nonlinear equation (Antoine) to the "vapor pressure example". The regression model is of the following form:
where P is pressure (mmHg) , T is temperature (deg C) and A, B and C
are parameters, need to be fitted to the "vapor pressure" example's
data.
Following is the program for finding the optimal parameter values
with nlinfit.
The optimal solution obtained is A = 5.76735, B = 677.094 and C = 153.885.
PolymathPlus provides statistical reports for the regression fit which assess the validity and accuracy of the regression. Each fitting variable has a 95% confidence interval. The confidence interval should be smaller than the absolute value of the variable for statistical validity.
In addition the following calculations are provided:
The following formulas calculate variables, standard deviation, and chi squared.
The following formulas calculate MAE, MSE, RMSE.
The following objects compose a valid PolymathPlus math expression:
Expression Objects  Description 
Number 
A number can be represented in decimal notation or engineering notation.
Engineering notation can be used to indicate powers of 10 using 'e' or 'E' For example, 0.123e5 is the same as 0.123*10^(5). Note that only the period "." is used as the decimal delimiter symbol and comma nor spaces must not be used to indicate digits grouping.
Sample valid
PolymathPlus
numbers:
Invalid PolymathPlus numbers: 
Variable Name 
Variable names must begin with an alphabetical character and can contain
alphabetic and numeric characters or underscore "_". Variable names are casesensitive, i.e. 't' and 'T' are two different variables. Special characters such as $, &, #, @ etc. are not allowed.
Samples: 
Arithmetic Operator 
Returns the arithmetic operations of the binary components. For example '2+3' returns 5 The '^' symbol is used for exponentiation. The +, , *, / symbols are used for addition, subtraction, multiplication, and division respectively. The precedence of operators is ^, (* or /), (+ or ) which is invoked working from left to right in an expression.
Samples: 
Function Name 
Various mathematical function names are available. Examples: sin, cos,
exp, ... The full list of available PolymathPlus functions is presented below.
Samples: 
Condition Operator 
These operators return "1" if the condition met, and "0" if the
condition does not meet. Examples: The expression '3>6' returns 0. The expression '4>=4' returns 1.
Valid conditional operators: 
Boolean Operator 
The Boolean operators return "1" if the condition is met, and "0" if the
condition is not met. Example: The expression '(2>3) or (1>0)' returns 1. The expression '1 and 0' returns 0.
Valid Boolean operators: 
IfThenElse 
There are three caseinsensitive keywords which compose the ifthenelse
expression structure. The three keywords are: If, Then, Else.
Samples: 
A special if
statement is available, with the following syntax:
The parentheses are optional.
The condition may include the following operators: and, or (Boolean
operators), > (greater than), < (less than), >= (greater than or equal), <=
(less than or equal), == (equals).
The expressions may be any formula, including another if
statement (nested statements).
For example:
A number of standard functions are available for use in the various programs. The arguments of the functions must be enclosed in parentheses. The arguments may be themselves expressions or other functions. The nesting of functions is allowed.
Function Name  Description 
abs ( )  absolute value 
arccos ( )  trigonometric inverse cosine with result in radians 
arccosec ( )  trigonometric inverse cosecant with result in radians 
arccosech ( )  inverse hyperbolic cosecant 
arcsech ()  inverse hyperbolic secant 
arccosh ( )  inverse hyperbolic cosine 
arccotan ( )  trigonometric inverse cotangent with result in radians 
arccotanh ( )  inverse hyperbolic cotangent 
arcsec ( )  trigonometric inverse secant with result in radians 
arcsin ( )  trigonometric inverse sine with result in radians 
arcsinh ( )  inverse hyperbolic sine 
arctan ( )  trigonometric inverse tangent with result in radians 
arctanh ( )  inverse hyperbolic tangent 
cbrt ( )  cubic root 
cos ( )  trigonometric cosine with argument in radians 
cosec ( )  trigonometric cosecant with argument in radians 
cosech ( )  hyperbolic cosecant 
cosh ( )  hyperbolic cosine 
cotan ( )  trigonometric cotangent with argument in radians 
coth ( )  hyperbolic cotangent 
exp ( )  exponential (e^x ) 
erf ( )  error function 
exp10 ( )  exponential of 10 (10^x ) 
exp2 ( )  exponential of 2 (2^x ) 
fact ( N )  factorial of integer part of number N (this only operates on a number) 
frac ( )  fractional part 
int ( )  integer part 
ln ( )  natural logarithm to the base e 
log ( )  logarithm to the base 10 
psi ( )  psi function 
rand ( ) 
Returns a random number between 01. A parameter such as 1 or 2 should be provided to this function. 
round ( )  rounded value 
sec ( )  trigonometric secant with argument in radians 
sech ( )  hyperbolic secant 
sign ( )  returns + 1 or 0 or 1 
sin ( )  trigonometric sine with argument in radians 
sinh ( )  hyperbolic sine 
sqrt ( )  square root 
tan ( )  trigonometric tangent with argument in radians 
tanh ( )  hyperbolic tangent 
All function names must be given in lowercase letters. The trigonometric functions require that their arguments be given in radians. Conversely, the inverse trigonometric functions give their results in radians.
You should note that the functions require that their arguments be enclosed in parentheses, but that the arguments do not have to be simple numbers. You may have a complicated expression as the argument for a function, and you may even nest the functions, using one function (or an expression including one or more functions) as the argument for another.
PolymathPlus allows for custom tuning of the solution algorithms, as well as basic adjustments to the output solution report. Settings are the default parameters and apply to all solution programs.
Specific program parameters can be overridden by hints. Settings are generally stored in a dedicated settings file, while hints are programspecific and are stored as part of the program *.pol
.
The main settings and hints apply to:
#  Setting Name  Default  Type 

1  Report_show_header  Yes  Boolean 
2  Report_fix_digits  10  Integer 
3  Report_verbose  No  Boolean 
4  Report_show_Source  Yes  Boolean 
5  Report_show_Equations  No  Boolean 
6  Report_show_charts  Yes  Boolean 
7  Report_chart_height  350  Integer 
8  Report_chart_width  450  Integer 
9  Report_formatted_equations  Yes  Boolean 
10  Report_formatted_equations_color  No  Boolean 
11  Report_formatted_equations_scale  1.8  Number 
12  NLE_SOLUTION_METHOD_INDEX  1  Integer 
13  NLE_tol_x  1E07  Number 
14  NLE_tol_f  1E07  Number 
15  NLE_min_tol  1E07  Number 
16  NLE_eps  1E07  Number 
17  NLE_max_iter  150  Integer 
18  DEQ_SOLUTION_METHOD_INDEX  0  Integer 
19  DEQ_Intermediate_data_points  50  Integer 
20  DEQ_RKF_H  1E06  Number 
21  DEQ_RKF_TETOL  1E06  Number 
22  DEQ_EPS  1E05  Number 
23  DEQ_H1  0.001  Number 
24  DEQ_HMIN  1E08  Number 
25  NLR_SOLUTION_METHOD_INDEX  1  Integer 
26  NLR_maxiter  64  Integer 
27  NLR_tol  0.0001  Number 
28  APP_WINDOW_WIDTH  1200  Integer 
29  APP_WINDOW_HEIGHT  800  Integer 
30  APP_WINDOW_MAXIMIZED  No  Boolean 
Hints are prefixed with #@
in the program file.
The following sample DEQ problem has a hint line to propose the size of the chart on the solution report:
#  Hint syntax sample & Description 

1  #@Report_fix_digits = 8 
Significant digits to display when presenting numbers  
2  #@Report_show_header = no 
Indicates whether a header text is displayed in the solution report Valid values are yes or no 

3  #@Report_show_Settings = no 
Valid values are yes or no  
4  #@Report_show_Matlab = no 
Indicates whether a Matlab formatted problem should be displayed in the solution report Valid values are yes or no 
#  Hint syntax sample & Description 

1  #@Report_fix_digits = 8 
Significant digits to display when presenting numbers  
2  #@Report_show_header = no 
Indicates whether a header text is displayed in the solution report  
3  #@Report_show_charts = no 
Indicates whether charts should be displayed in the solution report Valid values are yes or no 

4  #@Report_show_Matlab = no 
Indicates whether a Matlab formatted problem should be displayed in the solution report Valid values are yes or no 

5  #@Report_show_root_function_values = no 
Valid values are yes or no  
6  #@Report_show_Settings = no 
Valid values are yes or no  
7  #@Chart_Size = 400; 450 
Indicates the size of the chart to be drawn in the report Values are Width; Height 

8  #@NLE_SOLUTION_METHOD_INDEX = 1 
Algorithm used to solve the NLE system. Valid method indexes are: 0=FastNewton, 1=SafeNewton, 2=SafeBroydn, 3=Constrained 

9  #@NLE_tol_x = 1E07 
10  #@NLE_tol_f = 1E07 
11  #@NLE_min_tol = 1E07 
12  #@NLE_eps = 1E07 
13  #@NLE_max_iter = 150 
#  Hint syntax sample & Description 

1  #@Report_fix_digits = 8 
Significant digits to display when presenting numbers  
2  #@Report_show_header = no 
Indicates whether a header text is displayed in the solution report  
3  #@Report_show_charts = no 
Indicates whether charts should be displayed in the solution report Valid values are yes or no 

4  #@Report_show_Matlab = no 
Indicates whether a Matlab formatted problem should be displayed in the solution report Valid values are yes or no 

5  #@Report_show_Settings = no 
Valid values are yes or no  
6  #@Report_Show_Intermediate_data_points = no 
Valid values are yes or no  
7  #@DEQ_Intermediate_data_points = 50 
8  #@DEQ_RKF_H = 1E06 
9  #@DEQ_RKF_TETOL = 1E06 
10  #@DEQ_EPS = 1E05 
11  #@DEQ_H1 = 0.001 
12  #@DEQ_HMIN = 1E08 
13  #@DEQ_SOLUTION_METHOD_INDEX = 3 
DEQ solution method algorithm. Valid indexes are: 0=RKF45, 1=RKF56, 2=BS, 3=Stiff, 4=StiffBS 

14  #@Chart_Size = 400; 450 
Indicates the size of the chart to be drawn in the report Values are Width; Height 

15  #@Chart_Y_Curves = A; B; k3; k4; k5 
Indicates the variables names, separated by semicolon, to be drawn on the solution graph. Invalid names will be ignored. You should replace the default entries. 

16  #@Chart_GridLines = 1; 1; 1; 1 
Indicates the gridlines width. Width 0 is essentially no gridlines. Values are: MajorX; MinorX; MajorY; MinorY 

17  #@Chart_X_Scale = 0; 11; 1; 0.2 
Indicates the X axis scale. Values are (steps and padding values are optional): Minimum value; Maximum value; Major step; Minor step; Minimum padding; Maximum padding 

18  #@Chart_Y_Scale = 30; 380; 50; 10; 0.3; 0.3 
Indicates the X axis scale. Values are (steps and padding values are optional): Minimum value; Maximum value; Major step; Minor step; Minimum padding; Maximum padding 

19  #@Chart_Title = 
20  #@Chart_SubTitle = 
21  #@Chart_X_Axis_Title = x (min) 
22  #@Chart_Grayscale = no 
Valid values are yes or no  
23  #@Chart_inner_background_html_color = #FFFFFF 
24  #@Chart_outer_background_html_color = #FFFFFF 
25  #@Chart_text_font_size = 12 
#  Hint syntax sample & Description 

1  #@Report_fix_digits = 8 
Significant digits to display when presenting numbers  
2  #@Report_show_header = no 
Indicates whether a header text is displayed in the solution report  
3  #@Report_show_data_points = no 
Display calculated datapoints table Valid values are yes or no 

4  #@Report_show_settings = no 
Show program settings Valid values are yes or no 

5  #@Report_show_charts = no 
Indicates whether charts should be displayed in the solution report Valid values are yes or no 

6  #@Report_show_Matlab = no 
Indicates whether a Matlab formatted problem should be displayed in the solution report Valid values are yes or no 

7  #@Chart_Size = 400; 450 
Indicates the size of the chart to be drawn in the report Values are Width; Height 

8  #@Chart_GridLines = 1; 1; 1; 1 
Indicates the gridlines width. Width 0 is essentially no gridlines. Values are: MajorX; MinorX; MajorY; MinorY 

9  #@Chart_X_Scale = 0; 11; 1; 0.2 
Indicates the X axis scale. Values are (steps and padding values are optional): Minimum value; Maximum value; Major step; Minor step; Minimum padding; Maximum padding 

10  #@Chart_Y_Scale = 30; 380; 50; 10; 0.3; 0.3 
Indicates the X axis scale. Values are (steps and padding values are optional): Minimum value; Maximum value; Major step; Minor step; Minimum padding; Maximum padding 

11  #@Chart_Title = 
12  #@Chart_SubTitle = 
13  #@Chart_X_Axis_Title = x 
14  #@Chart_inner_background_html_color = #FFFFFF 
15  #@Chart_outer_background_html_color = #F5F5F5 
16  #@Chart_text_font_size = 12 
17  #@Chart2_Size = 200; 150 
18  #@Chart2_X_GridLines = yes 
Valid values are yes or no  
19  #@Chart2_Y_GridLines = yes 
Valid values are yes or no  
20  #@Chart2_X_Scale = 0; 11; 1 
21  #@Chart2_Y_Scale = 1; 1; 0.2 
22  #@Chart2_Title = 
23  #@Chart2_SubTitle = 
24  #@Chart2_X_Axis_Title = x 
25  #@Chart2_inner_background_html_color = #FFFFFF 
26  #@Chart2_outer_background_html_color = #F5F5F5 
27  #@Chart2_text_font_size = 12 
28  #@NLR_SOLUTION_METHOD_INDEX = 1 
Algorithm used to solve the NLR model. Valid method indexes are: 0=MRQMIN, 1=LM 

29  #@NLR_maxiter = 64 
30  #@NLR_tol = 0.0001 