Numerical Solutions for
Scientific and Engineering Mathematical Problems |

POLYMATH is a proven computational system, which has been specifically created for educational or professional use. This software executes on XP, Vista, Windows 7, 8, 8.1 and 10. ( It operates on 32-bit or 64-bit systems.) The various POLYMATH programs allow the user to apply effective numerical analysis techniques during interactive problem solving on personal computers. Results are presented graphically for easy understanding and for incorporation into papers and reports. Engineers, mathematicians, scientists, students or anyone with a need to solve problems will appreciate the efficiency and speed of problem solution.

Please select from the following options for the Overview:

1)
Click on one of the
Subject Areas Below or Scroll Through this Overview**.**

2) Review the complete Polymath HELP file.

Main Polymath Menu

The POLYMATH task selection window is shown below.

The main options available in this window are the following:

LEQ: **Linear Equations Solver.**
Enter (in matrix form) and solve a new system of simultaneous linear equations.

NLE: **Nonlinear Equations Solver.**
Enter and solve a new system of nonlinear algebraic equations.

DEQ: **Differential Equations Solver.**
Enter and solve a new system of ordinary differential equations.

REG: **Data Table
with Analysis and Regression.**
Enter, analyze, regress, and plot set of data points.

calc: **Calculator.** Enter and
evaluate explicit expressions with a variety of intrinsic functions.

units: **Unit Converter.** Convert
selected units into desired units.

const: **Scientific
Constants.**
Find selected scientific and engineering constants.

setup: **Parameter Settings.**
Modify setup and parameters of numerical solution algorithms.

HELP: Extensive HELP is always available. Also F1 always given content-sensitive HELP for POLYMATH..

- Additional information such as the current filename, problem title, and system time is shown in the status bar.

- The purpose of this program is to allow you to solve systems of linear
algebraic equations where the set of linear equations is inputted in a
matrix-vector form. The elements of the matrix of coefficients and the vector of constants
must be entered.

The data entry screen is shown below. The matrix of coefficients and the vector of coefficients are displayed. Each equation (row) is indicated by a number, and each column by a variable name (x1, x2,..., xn, b).

You can use the cursor to highlight one of the cells. This is the current cell. You can move the highlight by using the arrow keys or the left mouse click. To enter or modify a value in a data cell, move the highlight to the appropriate cell and either start typing in a number or erase and re-type the value already in the cell. Column (variable) names can be changed by selecting a column or a cell and then choosing the "Variable name..." option from the "Edit" drop-down menu or from the menu activated with the right mouse click.

After you finished entering the data the system of equations can be solved by selecting "Solve" from the drop-down "Program" menu or by bringing the cursor to the blue arrow on the screen and pressing the left mouse button. -

Go to Overview Index

- The purpose of this program is to allow you to solve systems of nonlinear algebraic equations. The system may contain nonlinear simultaneous (implicit) and supporting auxiliary (explicit) equations. Only real roots (non-complex) can be found. All equations are checked for correct syntax and other errors upon entry. Multiple roots are given for a single equation.
- The equation input/modification window is shown below.
- The options and information available on this screen are the following:
- Input a new nonlinear (implicit) algebraic equation.
- Solve current problem (pink color when problem correctly entered).
- Select numerical algorithm for solution.

A complete solution report is automatically generated. A portion of this is shown below.

- The purpose of this program is to help you solve systems of first-order ordinary differential equations. The system may contain simultaneous differential and supporting explicit algebraic equations. The equation input/modification window is shown below where the full-screen editor allows experienced users to quickly enter or modify problems..
- The options and information available on this screen are the following:

Input a differential equation with a provide template window.

Input an explicit algebraic equation with a provided template window..

Enter the initial and final values of the independent variable.

Summarize current problem variables.

Export problem to Excel (highlighted when problem is correctly entered).

Solve current problem (pink color when problem correctly entered).

Select numerical integration algorithm.

Show table of integration results. (See representative table.)

Show graph of current problem solution. (See representative graph.)

Present automatic report of problem and problem solution. (See representative report.)

- The data table is used for input, manipulation and storage of numerical data. The data is stored in a column-wise fashion where every column is associated with a name (variable) and can be addressed separately. The stored data can be regressed (link to Regression) (meaning fitting a straight line, various curves and equations to the data using multiple linear, polynomial and nonlinear regression techniques), analyzed (link to data analysis) (meaning interpolated, differentiated, integrated and various statistics are calculated) and plotted. The data table can be accessed directly by selecting the "REG" option when creating a new file. Tabular results of solution of a set of ordinary differential equations will be stored also in the data table.
- Go to Overview Index

- The data entry and manipulation window (Data Table) is shown above. It is much like a spreadsheet. The top row of cells contains the column names (the default names being C01 to C30). The rest are data cells. (The default values, the total number of rows and columns can be changed in the setup dialog box). Column names can be changed. Contents of a column can be defined as function of previously defined columns. Column definitions can be very useful when, for example, transformation functions are used to linearize a nonlinear regression model. The various options are always available to right of the data cells.
**Linear & Polynomial Regression**- This part of the program will fit a polynomial of the form:

y = P(x) = a0 + a1*x + a2*x^2 + . . . + an*x^n

where a0, a1, ..., an are regression parameters to a set of N tabulated values of x (independent variable) versus y (dependent variable). The highest degree allowed for a polynomial is N - 1 (thus n >= N - 1). The program calculates the coefficients a0, a1, ..., an by minimizing the sum of squares of the deviations between the calculated y or P(x) and the data for y.

The window for the Regression/Linear & Polynomial option is shown below. The options available in this window include:

Solve current problem (pink color when problem correctly entered).

Show a report showing the regression model the numerical values and confidence intervals of the parameters and other statistical information. (See typical report.)

Graph the calculated curve (or points) and the data points .

Store Model - Places output model results within Daa Table.

Export problem to Excel (highlighted when problem is correctly entered).

A residual plot is displayed showing the deviation between the data and the calculated values of the dependent variable.

- This part of the program will fit a linear function of the form:

y(x1, x2, ..., xn) = a0 + a1*x1 + a2*x2 + ... + an*xn

where a0, a1, ..., an are regression parameters, to a set of N tabulated values of x1, x2, ..., xn (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 a0, a1, ..., an by minimizing the sum of squares of the deviations between the calculated and the data for y. To carry out a multiple linear regression, select the "Multiple linear" option in the regression solver window.

The window for the Regression/Multiple linear option is shown below. The options available in this window include: - Go to Overview Index
**Nonlinear Regression**- This part of the program will fit a nonlinear function of the form:

y = f (x1, x2, …, xn, a0, a1, a2, …, am)

where a0, a1, …, an are regression parameters to a set of N tabulated values of x1, x2, …, xn (independent variables) versus y (dependent variable). Note that the number of data points must be greater than m + 1 (thus N >= m + 1). The window for the Regression/Multiple linear option is shown below. The options available in this window are discussed in detail in POLYMATH HELP. -

Solve current problem (pink color when problem correctly entered).

Show a report showing the regression model the numerical values and confidence intervals of the parameters and other statistical information. (See typical report.)

Graph the calculated curve (or points) and the data points .

Store Model - Places output model results within Daa Table.

Export problem to Excel (highlighted when problem is correctly entered).

A residual plot is displayed showing the deviation between the data and the calculated values of the dependent variable.

- Go to Overview Index
**Data Analysis**- Tabulated data often cannot be represented satisfactorily by a regression model or it can be difficult to find such a model. The purpose of this part of the program to help you interpolate, differentiate, integrate and plot data for which no regression model is available. The window for selection of the type of the data analysis to be carried out is shown below. The options available in this window are the following.

Solve current problem (pink color when problem correctly entered).

Show a report showing the regression model the numerical values and confidence intervals of the parameters and other statistical information. (See typical report.)

Graph the calculated curve (or points) and the data points .

Store Model - Places output model results within Daa Table.

Export problem to Excel (highlighted when problem is correctly entered).

A residual plot is displayed showing the deviation between the data and the calculated values of the dependent variable.

Solve current problem (pink color when problem correctly entered).

**Interpolation - **
Calculates the value of the dependent variable for a specified
value of the independent variable **.**

**Differentiation - **
Calculates the derivative of the dependent variable for a specified
value of the independent variable.

**Integration - **
Calculates the integral of the dependent variable for a specified
region of the independent variable.

**Dependent variable - **
Select any column from the Data Table as the dependent variable.

**Independent variable - **
Select any column from the Data Table as the independent variable/s.

**Solve with - **
Select the solution method.

**Graph Preparation and Editing**- Graphical presentations can be created from the Data Table or are given as an option from several of the programs. The window for creation of a graph in the Data Table give a selection of the column variable for plotting as shown below.
- The resulting graph shown below has many options. Once the graph has been edited, it can be copied to other applications such as word processors, spreadsheets, or desktop publishing packages.
- The graph editing options can be reached using the icon buttons on the left of the window. Representative options are listed below.

*Max Y-axis** *
- Change the upper bound on the Y (vertical)
axis.

*Min Y-axis** *
- Change the lower bound on the Y (vertical)
axis.

*Max X-axis** *
- Change the upper bound on the X (horizontal)
axis.

*Min X-axis** *
- Change the lower bound on the X (horizontal)
axis.

*Scatter
Sonnected** *- Show the curve connecting the calculated
and/or stored data points.

*Draw points** ***
- **Show the calculated and/or stored data
points.

*Auto scale** *
- Determines whether the program automatically
changes the scale as users remove/add function curves.

*Curves and Functions** *
- Option to remove variables and
to add function plots to the graph.

*Title**-*
Add or change graph title.

*Subtitle** ***
- **Add or change graph subtitle.

*Inverse grid lines** *
- Show grid lines if they are not
shown, remove them if they are shown.

*Legend box** *
- The legend box can be "dragged"
to a different location, if necessary, using the mouse.

In addition to the options presented here, there are options available to change the format of the labels in the X and Y axes, change the width of the various lines and curves and change the size of the points drown. These options can be reached using the various icons.

The "Export to Excel" function of Polymath is found in all program except the Linear Equation Solver. This option becomes available whenever the current problem in Polymath has been entered completely and the Excel icon is active. The desired Excel Workbook must be opened on the computer before the problem is exported to Excel. It is good practice to also solve the problem in Polymath so that the solution in Excel can be compared and verified. Upon export, the problem in Polymath will be completely transferred into Excel to a new Worksheet in the current Excel Workbook. This transfer automatically includes the translation of logical statements and the intrinsic functions from Polymath into equivalent functions within Excel.

This option is shown below when a sample problem is ready for solution within Polymath and the Excel icon is colored green..

A single click on the Excel icon or selection from the Problem menu causes the problem to be automatically exported into Excel as shown below.

The problem within Excel is setup in a very logical format which included the equations as they were in Polymath along with the comments. For this problem, the Excel Add-In called 'Solver Add-In' can then be used to solve the nonlinear equations that are setup for calculation in cells C14 and C15. The 'Solver Add-In' setup is shown below

which provides the following 'Solver Add-In' solution. Note that 'Solver" has minimized the Sum of Squares of the nonlinear equations of this problem as indicated in the boxed cell C16 shown in red in the screen display shown below..

This option is demonstrated for a Polymath program involving three simultaneous ordinary differential equation. The problem that is ready for solution in Polymath is shown below.

A single click on the Excel icon or selection from the Problem menu causes the problem to be automatically exported into Excel on a workbook page. This page is shown below where the entire problem is organized and easily understood by the Polymath equations and the Comments in the original problem notation.

Use of the Polymath ODE_Solver Add-In from the Tools menu given the input window. Note that the 'Reload' button automatically sets up the problem, or the user can do this manually.

The actual ordinary differential equations are present in cells C12, C13, and C14 (indicated in red). There are options in the Polymath ODE_Solver window, and the Adv. button gives a selection of five robust integration algorithms for the numerical integrations including two stiff methods. Intermediate cells can be identified for calcuation during the solution.

The 'Solve' button initiates the numerical solution, and the following solution worksheet is automatically generated. The Cell names from the problem worksheet have been copied into the results that are shown below.

Note that content following the 'Calculated values of DEQ variables' is equivalent to the Polymath 'Report' that summarizes the problem. The differential equation variables and the tabulated variable values are also in this 'Report' worksheet as shown below..

These tabulated variable values can be graphed within Excel to shown the profiles for the problem solution.

This option is shown for a Polymath program that has been completely setup for determining a third degree polynomial as shown below.

Note that this problem is ready for solution as indicated by the pink arrow button and by the green Excel button. A single click on the Excel icon initiates the automatic export to Excel. The resulting worksheet is created in Excel.

The summary table indicating the details of the polynomial fit is found in columns I through M of the spreadsheet.

This option is demonstrated for a Polymath program that is ready for a multiple linear regression.. This problem is shown below.

Since the problem is ready for solution, it can be exported to Excel by clicking on the Excel icon (green). This results in the following display that is created in a new worksheet within an open Excel workbook.

The additional regression output information is shown below.

Note that the statistical output information is automatically displayed in columns K, L, M, and N. These results were automatically generated within Excel with the use of the LINEST built in function.

This option is demonstrated for a Polymath program that is ready for a nonlinear regression.. This problem is shown below.

This problem is ready for solution as indicated by the Pink arrow button. A click on the Excel icon button initiates the automatic export to Excel. The created worksheet is shown below.

The problem in Excel is now ready to solve with the use of 'Solver Add-in' that is provided as an Add-In with Excel. Activation of this Add-In and the setup for this nonlinear regression is shown below.

The solution as achieved by the 'Solver Add-in' is shown below.

The converged values of the nonlinear model are given in cells H4, I4, and J4 (colored pink).

The equations and comments from the Nonlinear Equation Solver and the Differential Equations Solver can be automatically generated in complete MATLAB "M" file format.

**
MATLAB Output
from Nonlinear Equations Solver**

The MATLAB "M" file is placed in the Polymath Nonlinear Equations Report as shown, and this code can be copied directly into the MATLAB editor.

**
MATLAB Output
from Differential Equations Solver**

The MATLAB "M" file is placed in the Polymath Differential Equations Report as shown, and this code can be copied directly into the MATLAB editor.

Polymath has a calculator that can be used from within Polymath or as a separate pop-up utility.

where there are many intrinsic functions and advanced features that can be utilized.

There is a convenient unit conversion capability that is available from within Polymath or as a separate pop-up utility.