☰ PolymathPlus
PolymathPlus

Math-Solver for Students, Scientists, and Engineers
⬤ Enter a problem in plain text  ▶ Press the Solve button  ⬤ Get a numeric solution

Linear Equations Nonlinear Equations
Differential Equations Regression & Curve Fit

Embracing our vision, we offer a vibrant, free version of our user-friendly Math Solver.

New users can use the solver for free, either as guests or by registering and logging in. This allows them to explore sample programs and test the solver with their own. Registered users unlock more capabilities and can solve larger problems than guests. For even greater access, affordable paid licenses are available.

         

Our internal algorithms have been widely used for decades by engineering faculties at universities around the world. Students and professionals appreciate the application's ease of use and its detailed solution reports.

Solver Modules

Linear Equations Solver

Here's an example of 3 linear equations to be solved simultaneously:

Below are the equations to be entered into the PolymathPlus to solve the problem above:

a + 2*c = 50 b = -2 + c a + 23*c = 12 + b

Nonlinear Equations Solver

When solving simultaneous nonlinear equations, the equations have to be written as root expressions (expressions that are equal to zero in the solution). In addition, an initial guess should be provided for each nonlinear variable.
Below is a sample problem and data entry for solving 2 nonlinear equations:

nle.png

When entering the problem into PolymathPlus, we should enter the root expressions, each associated to a nonlinear variable function. We should also provide an initial estimate for each nonlinear variable. In the example below, we have 2 nonlinear variables x and y, hence 2 root functions, and we provide an initial guess of 1 for both x, and y.

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

Note that we may also define any number of auxiliary expressions, to simplify long expressions, in this example we define a as an auxiliary expression:

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

To solve a single nonlinear equation (including any number of optional auxiliary expressions), you must provide minimum and maximum estimates for the nonlinear variable. Check this example solving for V:

f(V) = (P+a/(V^2))*(V-b)-R*T P=56 R=0.08206 T=450 Tc=405.5 Pc=111.3 Pr=P/Pc a=27*(R^2*Tc^2/Pc)/64 b=R*Tc/(8*Pc) Z=P*V/(R*T) V(min)=0.4 V(max)=1

Differential Equations Solver

Consider this set of ordinary differential quations to be solved:

ode1.png

Given the initial conditions x0=1, y0=4, at t0=0, and having the integration finish at tf=1,
the text below should be entered into PolymathPlus:

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

The following alternative compatible syntax can also be used for the same 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

The output solution displays a clear table of results and an integration chart.

ode4.png

Curve Fitting

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.
Check the Sample Library for a comprehensive set of examples.

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

lin_reg_0.png

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.

# 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 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

The Strength and Challenges with Nonlinear Regression

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.
To address these challenges, linearization can be used. Linearization transforms a nonlinear model into a linear one, which can be solved using multiple linear regression (MLR). Linear regression guarantees convergence and provides a simpler, robust solution.
Below, we demonstrate both methods.

Solving Exponential Growth with Nonlinear Model

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

P(t)=P0ekt

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

Solving Exponential Growth with Linearized Model

Taking the natural logarithm of both sides:

ln(P) = ln(P0) + k t

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.

A wider range of examples for all program types can be found in the Sample Library.

Program Settings and Hints

PolymathPlus enables users to customize the format of solution reports and the selection of algorithms using hints. These hints are stored within the PolymathPlus program file and are prefixed with #@.

The sample program below includes several hint lines, used to modify the number of digits displayed in the report, adjust the size and content of the chart, and display a table of intermediate data points in the solution report.

# ODE program with custom hints # for the solution report format #@Report_fix_digits = 12 #@Chart_Size = 400 250 #@Chart_Y_Curves = A B | A2 #@DEQ_Intermediate_data_points = 20 #@Report_show_Settings = yes #@Report_Show_header = yes A' = -k1*A B' = k2*A k1 = 1.2 k2 = 1.5 A2 = 32*A^2.8/(B+1.6) A|1 B|0.2 t|0:5
Site License Admin Access

This section provides information for organizations purchasing bulk licenses. Bulk licenses enable the distribution of online access to multiple users and optionally the installation of a Windows desktop solver into their public labs or virtual environment computers.

IT Access Form

If you have already registered your site, you may access the site admin panel using the form below. The site admin panel allows you to view, purchase, or renew the site license; and provides instructions on distributing the software.

  

New Site Registration

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

 DetailExample
1.Site nameArizona State University
2.IT contact emailjohn123@asu.edu
3.Backup emailabra2759@asu.edu
4.Email extension/s@asu.edu, @student.asu.edu
5.Site AddressABC College
123 University Avenue
College Town, CA 98765
United States

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 vari us payment methods, including credit/debit cards, digital wallets, and Apple/Google Pay.
For costs and details on license types, please check the licenses section.

Sample Library

Below you will find sample problems that have been solved using PolymathPlus. Click on any of them to explore the simple syntax and clear solution in more detail.

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PolymathPlus Sample

Simon

The boss of all bosses

Linear and Explicit Equations

Nonlinear Equations

Differential Equations

Linear/Polynomial Regression

Multilinear Regression

Nonlinear Regression

Licenses

Our free license is powerful and available at no cost. Get started with it with simple registration and login. For advanced users, affordable paid licenses with expanded capabilities are also available, starting at just $15 annually.

Individual Licenses

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

Capacity & Features Free Std Pro
Online storage capacity 5 100 ∞
Nonlinear equations 5 10 100
Differential equations 5 10 2000
Supporting auxiliary variables 10 50 2000
Regression data points 20 100 ∞
Regression model variables 5 10 ∞
Intermediate data points table 50 300 1000
Solver requests per month 100 ∞ ∞
Online solver ✅ ✅ ✅
Windows desktop application   ✅ ✅
Dynamic charts   ✅ ✅
Advanced tools     ✅
Annual Cost excluding sales tax (USD) Free $15 🛒 $50 🛒
   ✱ Login and Registration is required for Std/Pro licenses.
   ✱ Some advanced features are available only in the Windows desktop version for Std/Pro licenses.
   ∞ Maximized within reasonable resource limits.
Access a curated set of features encompassing a wide range of functionalities on our online solver. We are happy to provide an accessible version of our software for everyone.
Designed specifically for students, this license offers a comprehensive set of features at an affordable price. With the student license, users gain access to advanced mathematical tools and functions, making it an ideal choice for educational purposes.
Tailored to meet the needs of professionals and researchers, the professional license offers the complete range of features and capabilities of our solver software.

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 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
$370 per 50 users
$450 per 100 users
$1250 per 50 users
$1500 per 100 users

Institutions with site licenses may install the desktop version on their public computers. Additional details are available on the Site Admin page.

Online and Desktop Programs

PolymathPlus is available online and via a Windows desktop application. The desktop version can be downloaded from the user’s profile page by those who have purchased student or professional licenses. It can also be installed on public computers at sites with a site license, managed by the site IT administrator.

The online web-based application offers the following advantages:

  • Accessibility:Users can access the software from any device with internet connectivity, eliminating the need for software installations and updates.
  • Cross-Platform Compatibility:The online version is compatible with various operating systems and devices, including Windows, macOS, Linux, Android, and iOS, providing a seamless experience across platforms.
  • Collaboration and Sharing:Users can easily collaborate with peers, share calculations, and work on projects collectively, fostering a sense of community and encouraging knowledge exchange.
Questions & Answers

Click any question below for an answer:

➕ 1. What is PolymathPlus?
➕ 2. Is PolymathPlus free?
➕ 3. Can I use PolymathPlus from my Windows/Mac laptop?
➕ 4. How can I try the online solver?
➕ 5. What type of licenses does PolymathPlus offer?
➕ 6. Which mathematical algorithms does PolymathPlus use?
➕ 7. What is the history of PolymathPlus?
➕ 8. Does PolymathPlus offer a desktop version package?
➕ 9. What is the user profile page?
➕ 10. What is the 'Site Admin' page, and who can use it?
➕ 11. I am interested in procuring a site license for my organization. What steps should I take?
➕ 12. How can I link my existing account to a site license my organization purchased?
➕ 13. Troubleshooting: Desktop application fails with WebView2 controllers compatibility
➕ 14. Troubleshooting: Too many steps in routine or step size underflow
➕ 15. Troubleshooting: LNSRCH: Roundoff problem in lnsrch
➕ 16. How can I support PolymathPlus?
About Us

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.

admin@polymathplus.org
sales@polymathplus.org
+1 530 405 9303
Pre-scheduled calls: 08:00-10:00 AM ET
Maintenance Status

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 world's premier, user-friendly, yet advanced math solver, accessible and affordable for users worldwide, empowering learners, professionals, and enthusiasts to confidently conquer mathematical challenges.
Terms and Conditions

PolymathPlus does not warrant that the functions contained in the program will meet your requirements or that the operation of the program will be uninterrupted or error-free. In no event will PolymathPlus or its distributors or their dealers be liable to you for any damages, including any lost profit, lost savings, lost patience, or other incidental or consequential damage.
Please carefully read the terms of service indicated in the link below.

Terms of Service (TOS)

 🔸  EULA  🔸  Accessibility  🔸  BCP
 🔸  CMP  🔸  InfoSec  🔸  HECVAT

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