Solve engineering math problems without writing code.

Our free license is powerful and available at no cost. Get started with it by simply registering and logging in. For advanced users, affordable paid licenses with expanded capabilities are also available, starting at just $15 annually.

Why upgrade?

Individual Licenses

The following table lists some of the program key capacities and features per license.

Users can purchase individual Student or Pro licenses through their online profile page.

Site Licenses

Site licenses are designed for institutions and organizations that require multi-user access to our software. With a site license, you have the flexibility to purchase a bulk quantity of either student or professional licenses, such as 100 student licenses for a site. Designated site administrators will have online management capabilities through the IT Site Admin page. They will be provided with a unique activation key for both the online program and the desktop application.

Site licenses will be sold in bulk quantities of 50 licenses at a 50% discount or 100 licenses at a 70% discount (compared to the individual license costs).

Site License	Annual Cost
Std	$370	per 50 users
	$450	per 100 users
Pro	$1250	per 50 users
	$1500	per 100 users

Institutions with site licenses may install the desktop version on their public computers.

New Site Registration

To initiate the registration of a new site, please reach out to us and provide the following details:

Additionally, please let us know if you are a tax-exempt site (attach a certificate if applicable). After you provide these details, we will set up your site and provide an access key that you can use to obtain invoices and purchase licenses. We accept various payment methods, including credit/debit cards, digital wallets, and Apple/Google Pay.

Our support team typically replies to account, licensing, and common troubleshooting requests within one business day. For technical questions, enhancement requests, or bug reports, email admin@polymathplus.org with a brief description and any screenshots; for licensing or purchase inquiries, please include the account email and any relevant site details.

Below you'll find common support questions and answers:

1 What is PolymathPlus?

PolymathPlus is math-solving software designed for students, scientists, and engineers, built to handle various numerical problems such as nonlinear equations, ordinary differential equations, linear/polynomial/nonlinear curve-fitting regression, and more.

At its core, it is a deterministic computational engine that applies well-established, rigorously validated numerical algorithms to solve mathematical and engineering models. For the same model and inputs, it produces reproducible results, which is essential for verification, traceability, and design confidence. This is fundamentally different from heuristic AI systems, which can generate variable outputs and may not always provide the consistency or numerical reliability required for engineering problem-solving.

Our goal is to deliver a user-friendly yet powerful math-solving tool that remains accessible and affordable for users around the world. With PolymathPlus, you can enter a math problem in plain text and instantly receive a numeric solution accompanied by a comprehensive report—all with a single click.

2 Is PolymathPlus free?

PolymathPlus offers a free, feature-rich online package.

PolymathPlus also offers affordable premium packages designed to handle more advanced tasks. These plans can solve larger and more complex problems, including cases involving more than five simultaneous nonlinear equations or ordinary differential equations

You can try the solver immediately as a guest — no account needed. A free registration is required to access all solver packages without guest limits.

The free package is designed for individual users. Organizations are asked to register their site and purchase a site license for their users.

3 Does PolymathPlus also work on mobile devices?

Yes — PolymathPlus works on virtually any modern device.

The online version works through all major browsers—including Chrome, Safari, Opera, Firefox, and Edge—on both Windows and macOS laptops and desktops. You can also access PolymathPlus on Android and iOS smartphones or tablets using the supported web browsers.

       

For users with paid licenses, we also offer a dedicated Windows desktop version.

4 I have an old Polymath 6.x license that no longer works. What are my options?

The legacy Polymath 6.x software is no longer supported, as the organization behind it has been dissolved. PolymathPlus was developed by a new, independent organization, maintaining full backward compatibility with the syntax of Polymath 6.1. At the same time, it delivers major advancements in algorithm optimization, capabilities, and underlying technology.

PolymathPlus is available online as a web application in both free and paid versions. The paid desktop version is offered with affordable Std and Pro licenses for individual users.

5 How can I try the online solver?

You can open the solver as a guest for a quick trial. Guest sessions have a brief wait between solves and do not support saving work. Register for a free account to save programs and reduce the wait time.

Click Login to register. After free sign-up, your password will be emailed to you, giving you access to the Free-tier solver.

6 What types of licenses does PolymathPlus offer?

PolymathPlus provides three types of individual user licenses: Free Std and Pro.
The Std and Pro licenses are priced at 15 USD and 50 USD per year respectively.

PolymathPlus also offers discounted site licenses for organizations.
Click here for more details about the capabilities of each license and the available site-license options.

7 Which mathematical algorithms does PolymathPlus use?

PolymathPlus uses well-established, finely tuned numerical solver algorithms.

These algorithms have been rigorously tested over more than 20 years of solving real-world engineering problems.

8 What is the history of PolymathPlus?

The core development team of PolymathPlus also created the earlier Polymath application, which was widely utilized as a numerical solver package within the global community of Chemical Engineers.
The development was also encouraged by:

	CACHE	The Computer Aids for Chemical Engineering Education Corporation, as part of the
	AIChE	The American Institute of Chemical Engineers

After the dissolution of the Polymath team/organization, the principal developer took the initiative to rekindle the project. Drawing from more than 25 years of software development and extensive research in numerical packages, a new package was crafted.

9 Does PolymathPlus offer a desktop version package?

Yes — a Windows desktop version is available for Std and Pro individual licenses, which can be downloaded from the user profile page. However, the free license does not include the desktop package. Sites with an active site license can install the desktop version on their public lab computers, but users covered by the site license can only use the desktop version on those public computers.

At present, our focus remains on the online and Windows desktop platforms, and there aren't immediate plans for a desktop version compatible with Mac or Linux. Nonetheless, this direction could be influenced by user demand and available resources.

The online version works on all common platforms, including Windows, Mac, Linux, iOS, and Android.

10 What is the user profile page?

The user profile page shows your preferences and current license status. It also lets you upgrade your license by purchasing an individual or site license.

To claim a site license, you will need an access key, which is provided exclusively to members of organizations that have purchased a bulk site license.

11 What is the 'Site Admin' page, and who can use it?

The Site Admin page is designed for IT administrators who manage a site license. Access requires a dedicated access key.

The page provides guidance on configuring the site license and tracking license distribution and usage.

12 How do I get a site license for my organization?

To register a new site, please ✉ contact us with the following information:

1. Site name (i.e. Arizona State University)
2. IT contact email and backup email
3. Email extension(s) for the organization (i.e. @asu.edu, @student.asu.edu)
4. Site address

We will then provide an access key. The Site Admin IT contact can use this key to access the Site Admin page, where they can set up invoices, manage licenses, and follow deployment instructions.

This enables granting STD or PRO licenses to site users for online access or on-site desktop use.

13 How can I link my existing account to a site license my organization purchased?

First, go to the PolymathPlus website and log in (or register if you haven’t already).

Then:

1. Open your Profile page (via the link in the menu bar).
2. In the Licenses section at the bottom, check - 'Claim Site License'

3. Enter the site-license key provided by your IT administrator.
4. Click the button to Save the profile.
5. Log out and then log back in.
6. You should now see the license type and your organization’s name in the title/status bar of the Solver page.

Note: The site-license key is intended only for authorized members of your organization.

14 Installing the desktop application on Windows

The desktop installer MSI package can be downloaded from the profile page once on STD or PRO licenses.
Please note these important points when installing the desktop application:

1. If an older version is installed, uninstall it via Control Panel (Add/Remove Programs).
2. It is recommended to confirm the installer's file integrity.
   You can do so by comparing the SHA256 value on the website with this PowerShell command output.
   Get-FileHash "SetupPolymathPlus_7057.msi" -Algorithm SHA256
3. When launching the MSI, if you see a "Windows protected your PC" message, follow these:
   
4. The message "User Account Control: Do you want to allow this app to make changes to your device?" is a Windows security feature that appears before an application makes administrative changes. To proceed with the installation, click "Yes".

15 Troubleshooting: Getting Object reference error when trying to log-in on desktop app

When you use Tools > Login in the desktop application, you may encounter the error: "Object reference not set to an instance of an object", and the application crashes.

This is a known bug in the older desktop release version SetupPolymathPlus_7051.msi.
To resolve this issue, please download and install the latest version from your profile page.

If you still wish to use an older version, please follow these steps to bypass the issue and complete the registration:

1. Open the application and go to Tools > Login.
2. Make sure the "Remember Me" checkbox is selected when entering your email and password, then click OK.
3. The first time you do this, you may still see the "Object reference" error and the program may crash.
4. Relaunch the application and repeat the login process via Tools > Login.
5. Registration should now complete successfully and be confirmed in the lower-right corner of the status bar.

16 Troubleshooting: Desktop application fails with WebView2 controllers compatibility

The Windows desktop application relies on the Microsoft Edge WebView2 Runtime. If Edge is not already installed on your computer, you will need to install the WebView2 Runtime separately.

If the application errors with 'The system cannot find the file specified (0x80070002)', download and install the WebView2 Runtime from the Microsoft link below:
WebView2 Evergreen Bootstrapper

17 Troubleshooting: Too many steps in routine or step size underflow

This error occurs when the solver's adaptive step-size control reduces the step size excessively, preventing the equation set from converging. This results in the solver taking too many steps without reaching a solution.

To resolve this, try the following steps:

1. First ensure the equations are entered correctly as intended. For example: unintentionally using multiplier instead of division: a/b*c vs. a/(b*c)
2. Verify that all variables are scaled appropriately, ensuring uniformity in magnitude.
3. Verify unit conversions
4. Consider reducing the integration range between t(0) and t(f)
5. You may consider switching to another solving algorithm. You can do these steps by using program hints.
   ODE solution method algorithm indexes are: 0=RKF45, 1=RKF56, 2=BS, 3=Stiff, 4=StiffBS.
   The default solution is 1=RKF56. Here's an example for hint to change to Stiff.

   #@DEQ_SOLUTION_METHOD_INDEX = 3
6. Finally, in rare cases, you may consider adjusting the step size parameters for better efficiency.
   You can do these steps by using program hints.
   Sample hints to control the step size and the error tolerance:

   #@DEQ_RKF_H = 1E-06
   #@DEQ_RKF_TETOL = 1E-06
   #@DEQ_EPS = 1E-05
   #@DEQ_H1 = 0.001

18 Troubleshooting: LNSRCH: Roundoff problem in lnsrch

This error occurs when arithmetic operations produce numbers that are too large or too small, resulting in Infinity or NaN. Check your initial and final values for scale issues, and pay particular attention to power operations, which can produce extremely large numbers.

19 How can I support PolymathPlus?

By purchasing the software, you are supporting its sustainability and ongoing development.

We are also actively seeking sponsors. PolymathPlus benefits universities, organizations, and their communities — and sponsors receive recognition for supporting a widely-used tool.

Organizations interested in sponsoring our activities can ✉ contact us directly.

PolymathPlus is designed to solve systems of simultaneous initial value problems (IVPs) involving first-order ordinary differential equations (ODEs).

Higher-order differential equations can be transformed into systems of first-order equations using standard mathematical techniques. Likewise, partial differential equations (PDEs) can be converted into ODEs for numerical solution using methods such as the method of lines.

Each equation in PolymathPlus must be written in first-order form, where the derivative of a dependent variable is isolated on the left-hand side and expressed as an algebraic function of independent and dependent variables. This structure ensures compatibility with the program's numerical solvers.

Consider this set of ordinary differential equations to be solved:

$$\{\begin{array}{c}x\text{'}=x+4t^2\\ y\text{'}=\sqrt{y}-\frac{x}{5-y}\end{array}$$

Given these initial and boundary conditions:

$$x_0=1,y_0=4,\text{at}{t}_0=0,\text{with integration ending at}{t}_f=1$$

The equations can be entered in PolymathPlus using the following syntax:

x' = x + 4*t^2 y' = sqrt(y) - x/(5-y) x|1 y|4 t|0:1

You can also use either of the following two compatible syntax formats for the same problem:

The output shows a table of results along with an integration chart.

In addition to differential equations, PolymathPlus allows users to define explicit variables—algebraic expressions that depend on differential variables, the independent variable, and/or other explicit variables, provided no circular dependencies exist. These variables are computed directly from known values and can be used to simplify equations, define intermediate results, or support the overall structure of the model.

The example below illustrates a typical input format for solving a system of ordinary differential equations (ODEs) in PolymathPlus, including the use of explicit variables.

$$\{\begin{array}{c}T\text{'}=\frac{0.8(500-T)+r_A(-40000)}{40·5}\\ x\text{'}=-\frac{r_A}{}/5\\ y\text{'}=-0.015(1-0.5x)\frac{T}{450}\frac{1}{2y}\end{array}$$
$$\begin{array}{c}k=0.5·\exp \left(5032\right(\frac{1}{450}-\frac{1}{T}\left)\right)\\ C_A=0.271(1-x)\frac{450}{T}\frac{1}{1-0.5x}y\\ C_C=0.271·0.5x\frac{450}{T}\frac{1}{1-0.5x}y\\ K_c=25000\exp \left(\frac{-40000}{8.314}\right(\frac{1}{450}-\frac{1}{T}\left)\right)\\ r_A=-k(C_A^2-\frac{C_C}{K_c})\end{array}$$
$$\begin{array}{c}{T}_0=450\\ x_0=0\\ y_0=1\\ W_0=0W_{\mathrm{f}}=20\end{array}$$

To enter this system of equations into PolymathPlus, you can use the following syntax:

# differential equations (ODE) T' = (0.8*(500-T)+rA*(-40000))/(40*5) x' = -rA/5 y' = -0.015*(1-0.5*x)*(T/450)/(2*y) # explicit variables / equations k = 0.5*exp(5032*(1/450-1/T)) CA = 0.271*(1-x)*(450/T)/(1-0.5*x)*y CC = 0.271*.5*x*(450/T)/(1-0.5*x)*y Kc = 25000*exp((-40000)/8.314*(1/450-1/T)) rA = -k*(CA^2-CC/Kc) # initial values for the dependent # differential variables T | 450 x | 0 y | 1 # initial and final values for the # independent variable W | 0 : 20

When using implicit multipliers syntax, we can simplify the expressions to:

# differential equations (ODE) T' = (0.8(500-T) + rA(-40000))/(40*5) x' = -rA/5 y' = -0.015(1 - 0.5x)(T/450)/(2y) # explicit variables / equations k = 0.5 exp(5032(1/450-1/T)) CA = 0.271(1-x)(450/T)/(1-0.5x)*y CC = 0.271*0.5*x(450/T)/(1-0.5x)*y Kc = 25000 exp((-40000)/8.314*(1/450-1/T)) rA = -k(CA^2-CC/Kc) # initial values for the dependent # differential variables T | 450 x | 0 y | 1 # initial and final values for the # independent variable W | 0 : 20

PolymathPlus supports curve fitting for linear, polynomial, multi-linear, and nonlinear regression. The report evaluates model variables, generates a regression chart and residual plots, and provides statistics on model accuracy.

Below is an example of the data entry required to solve a linear regression model for a given set of data points.
This example demonstrates a polynomial regression model of order 1 (i.e., linear regression), which finds the best-fitting linear equation for the given data points.

# Linear regression example # Straight line fit BOD vs Time [ t 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 ] Time = t + 10 polyfit Time BOD 1

Nonlinear Regression

Below is an example of a data entry for solving nonlinear regression model for a given set of data points. The model variables to be found are a and b, for which we should also provide an initial guess.

The model eqaution to fit is as follows:

$$y=\frac{ax}{b+x}$$

The data entry for this nonlinear regression model is as follows:

# Example: Nonlinear Regression [ x y g 0.5 1.255 1.2 0.387 1.25 1.3 0.24 1.189 1.4 0.136 1.124 2 0.04 0.783 2.1 0.011 0.402 2.2 ] # Nonlinear regression model nlinfit y = a * x / (b + x) # Initial guess of the # regression model variables a,b m(a) = 2 m(b) = 1

and the solution becomes:

Discrete Integration

PolymathPlus can calculate a definite integral from known discrete data points. The solver interpolates the supplied x-y data, builds a continuous curve from those points, and calculates the area under the curve over the requested interval. This is useful when the function is known through measurements, tabulated data, or simulation output rather than through a closed-form equation.

The default settings are designed to work well in almost all practical cases. Advanced users can still override the defaults by choosing the interpolation method and the number of sub-sections used for the numerical integration. Increasing the number of sub-sections can improve accuracy for rapidly changing curves, while simpler interpolation may be preferred for piecewise-linear data.

[ x t 1 1.7 2 2.9 3 4.9 4 8.4 5 14.2 6 24.1 7 41.0 8 69.8 9 118.6 ] y = t+25 # Integration model integrate y(x) 3 8.6

In this example, PolymathPlus integrates y(x) from x = 3 to x = 8.6. By default, it uses Akima interpolation with Simpson integration and 100 sub-sections.

Optional forms provide direct control when needed: use integrate y(x) 2 3 90 akm for Akima interpolation over 2-3 with 90 sub-sections, integrate y(x) 1 2 50 ccs for CCS interpolation over 1-2 with 50 sub-sections, or integrate y(x) 5 9 80 lin for linear interpolation over 5-9 with 80 sub-sections.

⚠ Experimental This NLP solver is a new feature, still in pilot and 🧪 validation mode.

PolymathPlus can find the minimum of a mathematical model expression while respecting user-defined constraints. This type of task is known as constrained nonlinear programming (NLP). The solver determines the variable values that best minimize the objective function, and the resulting report includes charts and statistics that show how well the model fits.

To set up an NLP problem in PolymathPlus, start by specifying the objective function you want to minimize using the min: keyword. Constraints are added as expressions that must equal zero for equality constraints (use ... = 0), or be greater than zero for inequality constraints (use ... > 0). Write each constraint as a mathematical expression followed by the appropriate comparison.
Please Note: Be sure to rewrite each constraint so that all terms appear on one side of the equation or inequality, leaving zero on the other side.

A key requirement for the NLP solver is to provide bounds for each variable. For every variable, you must enter an initial guess, a lower bound, and an upper bound in the following format: variable | guess lower_bound upper_bound
This helps the solver search only within the meaningful range for your model.

Below is an example of how to enter data for a constrained optimization problem. In this example, the goal is to minimize a squared distance function, subject to nonlinear constraints that define the allowed region.

Mathematically, the problem is expressed as:

The corresponding data entry and the solution report are as follows:

# objective function to minimize min: (x - 2)^2 + (y - 3)^2 # region constraints x*y - 2 > 0 25 - x^2 - y^2 > 0 x > 0 y > 0 # Initial guess and bounds x|1.6 0 10 y|1.3 0 10

Since the global minimum of the function is located at $(2,3)$ and this point satisfies all constraints (where $2\times 3=6\ge 2$ and $2^2+3^2=13\le 25$), the solver converges to the optimal value of min F* = 0.

LSQ Optimization

Least Squares (LSQ) optimization is a specific type of nonlinear programming (NLP) used to fit mathematical models to observed data. The main goal is to minimize the sum of squared differences (residuals) between the actual data points and the values predicted by the model. This approach is widely used in scientific modeling, as it provides a quantitative measure of how well the model matches the data under various equality and inequality constraints.

To perform LSQ optimization in PolymathPlus, you provide a set of observed data points and define a mathematical model that describes the relationship between variables. The solver then adjusts the model parameters to find the best fit, minimizing the total squared error.

Below is an example of how to enter data for a least squares optimization problem. In this example, the model uses a saturation kinetics form (which is commonly applied in enzyme kinetics and other scientific fields). The objective is to find the parameter values that make the model curve fit the data points as closely as possible.

The saturation kinetics form model:

The equation above represents a common model used to describe how a response variable y changes with respect to an independent variable x, where - we seek to find the best fitting model variables V_max (represented by a) and K_m (represented by b), by minimizing the residuals of calculated y_calc values compared to the observed y data points.

In the PolymathPlus program, we should define the data-table of the known variable. We should also define the model expression to optimize using the fit: keyword followed by the math expression.

fit: y = a*x/(b + x)

The program will attempt to find the best suited model variables a and b, by minimizing the sum of squared residuals:

Here is the PolymathPlus program to solve the LSQ optimization problem described above:

[ x y 0.05 0.1818 0.10 0.3333 0.20 0.5714 0.50 1.0000 1.00 1.3333 2.00 1.6000 ] # LSQ minimization fit fit: y = a*x/(b + x) # Initial guess and lower/upper bounds a|1.5 0 5 b|0.3 0 5

and the solution becomes:

The verified solution for this dataset yields a ≈ 2.0 (representing V_max) and b ≈ 0.5 (representing K_m). By evaluating the residual plots provided in the report, users can confirm that the error is randomly distributed, indicating that the nonlinear model chosen is an appropriate fit for the data.

The program solution report includes charts to visualize the accuracy of the fit:

PolymathPlus began as a Windows desktop application and expanded its offerings to include an online web application. Our passion and dedication drive us to continually innovate and refine our solutions. We are committed to providing a top-notch numerical solver package to tackle mathematical challenges with ease and confidence.

The development of our software had been encouraged by CACHE - The Computer Aids for Chemical Engineering Education Corporation, as part of the American Institute of Chemical Engineers.

Drawing from more than 25 years of software development and extensive research in numerical packages, we have crafted the new PolymathPlus package. We maintain ties with many US-based universities and colleges, as well as institutions across other regions worldwide.

Our Vision

Deliver the premier user-friendly, advanced math solver—accessible and affordable worldwide—the first choice for learners, professionals, and enthusiasts.

Our research is dedicated to advancing numerical methods and optimization techniques to strengthen the capabilities of PolymathPlus. The following key research areas highlight their potential impact across disciplines commonly applied in science and engineering, particularly in design, modeling, and control.

Optimization

PDE Solvers

Simplex

ODE Optimizers

Chebyshev and Fourier

These research efforts represent our ongoing commitment to improving computational mathematics. While we work to incorporate these advancements into PolymathPlus, we aim to provide users with reliable and efficient tools for solving complex mathematical problems.

Below are sample reports generated by solving PolymathPlus programs. The input for each program is shown at the top of every report. Each group features example problems and their corresponding solution reports.

Solution report

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