User Manual

PolymathPlus is a math solver for students, scientists, and engineers. It numerically solves problems entered in plain text—including linear equations, nonlinear systems, differential equations, and regression analyses—and produces detailed solution reports with statistics, tables, and charts.

Students, engineers, scientists, or anyone with a need for a numerical math problems-solving package will appreciate the simplicity, efficiency, and speed of PolymathPlus.

System of Linear Equations

When solving a simultaneous set of linear equations, you should enter each equation in text format, one per line. The number of equations must be equal to the number of unknown variables for a unique solution to exist.

The following example demonstrates how to input a system of three linear equations to be solved:

In the program, the data would be entered as follows:

				# System of three linear equations
				a + 2*c = 50
				b = -2 + c
				a + 23*c = 12 + b
		

Note: The # symbol marks a comment. Anything after it on the same line is ignored by the solution engine, so you can use it to add notes or explanations.

The Solution Report shows the values of the unknowns at the solution, the problem definition, and any settings applied by the system or hints specified by the user.

Hints

Hints allow you to adjust solution algorithms, error tolerances, and the solution report on a per-program basis (as part of the program syntax). The global defaults—called settings—cannot be changed directly, but you can override them by including hints within a specific program.

System of Nonlinear Equations

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.
Below is an example for solving 2 nonlinear equations with an initial estimate for x, and y at the solution.

Rearranging the equations, and estimating x and y to be 1 at the solution, this would be the data entry to solve this problem:

				f(x) = x^2 + y - 12
				f(y) = 2*x + log(y+2) - 5
				x(0)=1
				y(0)=1
		

you may define any number of auxiliary expressions, to simplify long expressions:

				f(x) = x^2 + y - 12
				f(y) = 2*x + a
				a = log(y+2) - 5
				x(0)=1
				y(0)=1
		

and the resulting solution report gives:

The solver can handle either a single nonlinear equation or a system of simultaneous nonlinear equations. For a single equation, you must provide a range of values that brackets the solution; the program will find all roots within that interval. For a system, you must provide an initial guess for each variable. In this case, each guess is a single value, not a range.

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:

where the expression may contain constants, implicit variables, and explicit variables.
Only real roots (non-complex) can be found. Multiple roots are determined for a single equation where the range of the variable must be provided.

Explicit Algebraic Equations

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.

				# Explicit System of Nonlinear Equations
				a = 1
				b = -12
				c = 35
				disc = b ^2-4*a* c
				x1 = (-b+sqrt(disc))/(2*a)
				x2 = (-b-sqrt(disc))/(2*a)
			

The Solution Report includes the values of the unknowns (x₁ = 7, x₂ = 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.

Single Nonlinear Equation

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:

4x² + x · log(x/2) = 1620

Here's the problem to enter when the solution is expected to be found in the 0 - 100 range.

				# Solve nonlinear equation
				f(x) = 4*x^2 + x*log(x/2) - 1620
				x(min) = 0
				x(max) = 100
		

Note that we can also define an explicit equation for the variable t that can be used to shorten large equations or ones with repeated section.

				# Solve nonlinear equation
				f(x) = 4*x^2 + t - 1620
				t = x*log(x/2)
				x(min) = 0
				x(max) = 100
			

The Solution Report includes the value/s of the unknown (x=20, in this case), the function value/s (f(x)= 1.0e-8), and the values of all the variables at the solution.

Nonlinear Equations

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 # indicate user comments and are being ignored by the program):

				# Solveing two nonlinear equations
				f(x) = 2*x + x*y + 290 - y^2
				f(y) = t + x*y - 105

				t = x*log(y/2)

				x(0) = 2
				y(0) = 10
			

The Solution Report includes the values of the unknowns (x = 5, y = 20), the function values (f(x)= -2.7E-12, f(y)= 1.3E-12 ) and the values of all the additional variable/s at the solution.

Simultaneous Linear and Nonlinear Equations

A mixed system of linear and nonlinear equations can always be solved by treating the entire set as nonlinear. However, if you can isolate some variables and write them explicitly in terms of the remaining nonlinear variables, you reduce the number of variables that require initial guesses and the number of equations that must be written in root form f(x) = 0 and, on top of that, fewer nonlinear equations generally increase the likelihood of a successful solution.

Consider a problem containing three linear and two nonlinear equations in the original form:

These can be solved as a set of 5 nonlinear equations where an initial guess should be provided for each variable.

				# Simultaneous nonlinear equations
				f(c) = 3.8907-ln(6*c)-a
				f(a) = a + b - c
				f(b) = 5*a - 3 - b
				f(d) = 2*c + 4 - 3*b - d
				f(e) = sin(0.1309*d)/0.1 - e

				# Initial guess of the variables 
				# per nonlinear equation
				a(0) = 2
				b(0) = 2
				c(0) = 2
				d(0) = 2
				e(0) = 2
		

The program text above would successfully solve the equations. We could also rewrite the set of equations as two nonlinear equations (root expressions that are set to zero at the solutions), and three explicit expressions. This approach yields:

				# Nonlinear equations & explicit vars
				f(c) = 3.8907-ln(6*c)-a
				f(a) = a + b - c

				# The explicit variables
				b = 5*a - 3
				d = 2*c + 4 - 3*b
				e = sin(0.1309*d)/0.1

				# Initial guess - nonlinear equation vars
				c(0) = 2
				a(0) = 2
			

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:

Constrained Nonlinear Equations

A special solution algorithm is available for problems where some of the variables can have only positive values.
Consider the following set of equations:

In this problem, the variables x₁, x₂ and x₃ are constrained. At the solution, x₁>0 because of physical considerations; x₂ and x₃ must be greater than zero throughout the solution process otherwise a negative value inside the ln function will yield an execution error.
Constraints allowed on the variables of the type x>0 or x<0.
For the solution, the equations for x₂ and x₃ must be rewritten in the implicit form as shown below:

				f(x1) = x1/x2 - 5*ln(0.4*x2/x3) + 4.45977
				f(x2) = x2 + x1 - 1
				f(x3) = x3 + 0.5*x1 - 0.4
		

A solution can be achieved by using a constrained solution algorithm which is done by adding the following hint to the program (at any line location):

#@ NLE_SOLUTION_METHOD_INDEX = 3

The options for constrains of x₁ are 'Any value' that indicates no constraints, 'positive' that means it must be positive at the solution only and 'always positive' that forces the variable to be positive all the way to the solution. For x₁ the 'positive' definition is sufficient. Since the variables x₂ and x₃ must be positive for all calculation leading to the solution, the option 'Always positive' must be marked. The initial guesses for x₂ and x₃ are 0.5 and 0.15 respectively, which are consistent with initial guess of x₁=0.5.

PolymathPlus designates the constrained variables in the initial guess line as follows: Positive value will have >= 0 added to the initial guess value, Always positive values will have > 0 added to the initial guess value.
The entire problem should look like this:

				f(x1) = x1/x2 - 5*ln(0.4*x2/x3) + 4.45977
				f(x2) = x2 + x1 - 1
				f(x3) = x3 + 0.5*x1 - 0.4

				#@ NLE_SOLUTION_METHOD_INDEX = 3

				x1(0)=0.35 >= 0
				x2(0)=0.5 > 0
				x3(0)=0.15 > 0
		

and the solution becomes:

Note that the constrained method may only be used for sets of equations with more than 1 simultaneous nonlinear equation.

Hints

Hints allow you to adjust solution algorithms, error tolerances, and the solution report on a per-program basis (as part of the program syntax). The global defaults—called settings—cannot be changed directly, but you can override them by including hints within a specific program.

Ordinary Differential Equations

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:

				d(x)/d(t) = x + 4*t^2
				d(y)/d(t) = sqrt(y) - x/(5-y)
				x(0)=1
				y(0)=4
				t(0)=0
				t(f)=1
			

or you may define any number of auxiliary expressions, to simplify long and/or repeating sections:

				d(x)/d(t) = x + 4*t^2
				d(y)/d(t) = sqrt(y) - a
				a = x/(5-y)
				x(0)=1
				y(0)=4
				t(0)=0
				t(f)=1
		

And the resulting report would look like:

The problems belonging to this category are comprised of systems of simultaneous first-order ordinary differential equations and explicit algebraic equations. The differential equations must be entered in the following form:

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.

			# Set of consecutive reactions

			d(A)/d(t) = -k1*A
			d(B)/d(t) = k1*A - k2*B
			d(C)/d(t) = k2*B

			k1 = 1
			k2 = 2

			A(0) = 1
			B(0) = 0
			C(0) = 0

			t(0) = 0
			t(f) = 3
		

The Solution Report includes a Table listing the initial, minimal, maximal, and final values of all the variables over the integration range. These are the solver results for the example above:

The graph shows the variation of A, B and C with time.

Note that the same ODE problem can also be entered in a simplified format such as this:

				# Set of consecutive reactions

				A' = -k1*A
				B' = k1*A - k2*B
				C' = k2*B

				k1 = 1
				k2 = 2

				A|1
				B|0
				C|0

				t|0:3
			

Hints

Hints allow you to adjust solution algorithms, error tolerances, and the solution report on a per-program basis (as part of the program syntax). The global defaults—called settings—cannot be changed directly, but you can override them by including hints within a specific program.

Regressions of Data

The known data points must be entered in a strict tabulated format, enclosed within opening and closing square brackets [  ]. Each bracket must be placed on a separate line.
The first line of the table should contain the variable names, separated by spaces. Subsequent lines must contain the corresponding values for the variables, with columns separated by spaces. All rows must have the same number of columns, and blank values are not permitted. For example:

                [
				a    b     xy   
				1    9     1.1  
				2    13    7.87 
				3.1  16    5.67 
				4.2  18.2  8.9  
				5    19    45.6 
				6    20    56.7 
				7    19    66.9 
				]
		

Tip: Use the Edit > Align Table menu item to align the columns nicely.

Polynomial Regression

PolymathPlus can fit a polynomial of degree n with the general form: 

y = P(x) = a₀ + a₁x + a₂x² + . . . + a_nxⁿ

or a linear equation when the equivalent first degree polynomial is specified

y = P(x) = a₀ + a₁x

where a₀, a₁, ..., 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 a₀, a₁, ..., 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.

Correlation of BOD data example

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

				# Straight line fit
				[
				Time  BOD 
				1     0.6 
				2     0.7 
				4     1.5 
				6     1.9 
				8     2.1 
				10    2.6 
				12    2.9 
				14    3.7 
				16    3.5 
				18    3.7 
				20    3.8 
				]

				polyfit Time BOD 1
			

This program yields the the following solution results:

The Solution Report contains a table with the observed (measured) y values, the predicted y values (ycalc) and the residuals (r = y - ycalc) are also displayed.

Fitting the data to a 2nd degree polynomial.
To fit a 2nd degree polynomial, the polyfit statement is revised:

				polyfit Time BOD 2
		

The results obtained in this case are:

To fit a 2nd degree polynomial which passes through the origin, the following polyfit statement is used:

				polyfit Time BOD 2 origin
			

For this polynomial a₀ = 0, by definition, and the rest of the results are:

Below are the plots containing: BOD (the data) vs. P(x) (calculated values), and Delta BOD (residuals) vs. time.

Regression Analysis: Judging by the R² and σ² values, the 2nd degree polynomial represents the data the best. However, the confidence interval value of a₀ shows that this parameter is not significantly different from zero, thus the use of 2nd degree polynomial passing through the origin should be preferred.

4th order Polynomial Regression Example

Consider the following set of Pressure versus Temperature data:

The following operations should be performed on this data:

1. Convert the temperature to K (by adding 273.15 to the value shown)
2. Convert the pressure to kPa (by multiplying the value shown by 0.1333)
3. Fit a fourth degree polynomial to the Pressure (dependent variable) vs. Temperature data.

The program to accomplish these tasks is shown below.

				# Vapor Pressure Correlation

				[
				T_C    P_mmHg 
				-36.7  1      
				-19.6  5      
				-11.5  10     
				-2.6   20     
				7.6    40     
				15.4   60     
				26.1   100    
				42.2   200    
				60.6   400    
				80.1   760    
				]

				T_K = T_C + 273.15
				P_kPa = P_mmHg*0.1333

				polyfit T_K P_kPa 4
		

The Solution Report includes a table listing the parameter values along with their 95% confidence intervals. For this problem, the results are:

The regression plot:
   

The Solution Report includes also the statistical metrics: R², R²_adj, root mean squared deviation and variance (σ²). A table displaying the observed (measured) y values, the predicted y values (y_calc) and the residuals (r = y - y_calc).

To fit a polynomial that passes through the origin the "polyfit" command must also include the parameter "origin". For example:

				polyfit T_K P_kPa 4 origin
			

Multiple Linear Regression

This PolymathPlus option will fit a linear function of the form: 

y(x₁, x₂, ..., x_n) = a₀ + a₁ x₁ + a₂ x₂ + ... + a_n x_n 

where a₀, a₁, ..., a_n are regression parameters fitted to N tabulated values of the independent variables x₁, x₂, ..., x_n and the corresponding dependent variable y. Note that the number of data points must be greater than n+1 (thus N >= n+1). The program calculates the coefficients a₀, a₁, ..., a_n by minimizing the sum of squares of the deviations between the calculated and observed y values.

Use the mlinfit command to perform a multiple linear regression. List the independent variable names first, followed by the dependent variable name. An optional final argument 'origin' forces the fit through the origin (thus a0 = 0).

Simple multiple linear regression example

The following data show bearing wear (y) that may depend on oil viscosity (x₁) and load (x₂).

In this example we compare the linear regression models that includes x₁ and x₂ as independent variables, with a model that includes only x₁.

When the multilinear model is y = a0 + a1x1 + a2x2 we enter the following program:

				# Multiple linear regression
				[
				y    x1    x2   
				293  1.61  851  
				230  15.5  820  
				172  22    1058 
				91   45    1201 
				125  33    1357 
				125  40    1115 
				]

				mlinfit x1 x2 y
		

The results obtained in this case are:

To remove x₂ from the regression model, we set mlinfit x1 y and the revised statement yields the following results:

Judging by the R² and σ² values, the regression model that contains both x₁ and x₂ represents the data better. However, the 95% confidence interval for a₂ includes zero, indicating that the x₂ term is not statistically significant; therefore removing x₂ is justified.

Considering the "vapor pressure" example to fit a regression model of the form:

Define the dependent variable as logP = log(P_kPa) and the independent variables as OneonT = 1/T_K and logT = log(T_K). Then use mlinfit to find the parameter values.

				# VP Correlation by Reduced Riedel Eqs
				[
				T_C     P_mmHg 
				-36.7   1      
				-19.6   5      
				-11.5   10     
				-2.6    20     
				7.6     40     
				15.4    60     
				26.1    100    
				42.2    200    
				60.6    400    
				80.1    760    
				]

				# auxiliary variables derived
				T_K = T_C+273.15
				P_kPa = P_mmHg*0.1333
				logP = log(P_kPa)
				OneonT = 1/T_K
				logT = log(T_K)

				# regression model
				mlinfit OneonT logT logP
			

The Solution Report includes a table listing the parameter values along with their 95% confidence intervals. For this problem, the results are:

Regression Model:

y = a₀ + a₁x₁ + a₂x₂

where y and x_i map to:

logP = a₀ + a₁ OneonT + a₂ logT

Nonlinear Regression

Nonlinear regression involves a general mathematical function (model) of the form:

y = f (x₁, x₂, ..., x_n, a₀, a₁, a₂,...,a_m)

where a₀, a₁, ..., a_n are regression parameters to a set of N tabulated values of x₁, x₂, ..., 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 perform a 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:

				m(ai) = value
		

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 data
Following is the program for finding the optimal parameter values with nlinfit.

				# Nonlinear Regression
				# Antoine Equation parameters

				[
				T       P   
				-36.7   1   
				-19.6   5   
				-11.5   10  
				-2.6    20  
				7.6     40  
				15.4    60  
				26.1    100 
				42.2    200 
				60.6    400 
				80.1    760 
				]

				logP = ln(P)

				# regression model
				nlinfit logP = A-B/(C+T)

				# initial guess for A, B, C
				m(A) = 8
				m(B) = 2000
				m(C) = 200
			

The optimal solution obtained becomes:

with the following statistic results:

Nonlinear Challenges

Nonlinear regression is a powerful method for modeling complex relationships between variables. It enables direct fitting of nonlinear equations to observed data. However, nonlinear regression has some challenges:

• Initial Guesses: Good initial guesses for parameters are required, and poor guesses may lead to failure.
• Convergence: Nonlinear regression algorithms can fail to converge or get stuck in local minima.
• Complexity: Solving nonlinear equations is computationally intensive.

The relationship between time (t) and population (P) follows the model:

where P₀ is the initial population at time t=0, and k is the growth rate constant.
This can be solved using a nonlinear regression model where P is the dependent variable, t is the independent variable, and P₀ and k are the parameters to be determined (model variables), requiring initial estimates.

Nonlinear Regression in PolymathPlus:

		[
		t    P
		0    100
		1    120
		2    150
		3    190
		4    250
		]

		# regression model
		nlinfit P = P0 * exp(k * t)

		# initial guesses
		m(P0) = 90
		m(k) = 0.1
	  

And the solution becomes P0 = 94.943880, and k = 0.23854324

Linearized Model for Exponential Growth

Taking the natural logarithm of both sides:

This transforms the model into a linear equation where ln(P) is the dependent variable, and t is the independent variable. Multiple Linear Regression in PolymathPlus:

			[
			t    P
			0    100
			1    120
			2    150
			3    190
			4    250
			]

			logP = ln(P)

			# regression model
			mlinfit t logP
		

Once solving the (multi)linear model we can map a₀, a₁, a₂, etc. to the model variables we seek while we can be guarenteed to find the best possible solution without needing to provide initial guess.
The solution yields: a₀ = 4.5759337 hence P₀ = exp(4.5759337) = 97.118677, and k = a₁ = 0.22921138

This example involves a single independent variable, resulting in a simple linear regression. In cases with multiple linearly dependent variables, multiple linear regression can be applied.
Complex nonlinear equations can also be linearized by applying transformations such as logarithmic functions, inverse functions, or rearranging terms, simplifying them into linear forms for easier solving. Unlike nonlinear regression, multiple linear regression does not require initial guesses and offers more robust convergence.

Regression Statistics

PolymathPlus provides statistical reports for the regression fit which assess the validity and accuracy of the regression. Each parameter is reported with its 95% confidence interval. If a parameter's 95% confidence interval includes zero, that parameter is not statistically significant at the 5% level.

In addition the following calculations are provided:

The following formulas calculate variables, standard deviation, and chi squared.

The following formulas calculate MAE, MSE, RMSE.

Hints

Hints allow you to adjust solution algorithms, error tolerances, and the solution report on a per-program basis (as part of the program syntax). The global defaults—called settings—cannot be changed directly, but you can override them by including hints within a specific program.

Variables and Expressions

The following objects compose a valid PolymathPlus math expression:

A special if statement is available, with the following syntax:
if (condition) then (expression1) else (expression2)

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 = if (x>0) then (log(x)) else(0)
				b = if (T<minT) then (minT) else (if (T>maxT) then (maxT) else (T))
				Vol_h1 = if (a>5 and c<2) then 1.12 else 7.89
			

Available Functions

PolymathPlus supports a range of standard mathematical functions. Function names must be written in lowercase. For example:

				a = sin(12) + 14
				b = sqrt(a + 25)
				c = exp(ln(a))
			

Settings and Hints

PolymathPlus allows custom tuning of solution algorithms, error tolerance, and basic adjustments to the output solution report. Default settings apply to all solution programs but can be overridden by program-specific hints.

Default Settings

Hints are pre-fixed 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.

		        # demo the use of hints 
				#@ chart_size = 350; 300

				d(x)/d(t) = x + 4*t^2
				d(y)/d(t) = sqrt(y) - x/(5-y)

				t(0)=0
				t(f)=1
				x(0)=1
				y(0)=4