| Numerical Solutions for Scientific and Engineering Mathematical Problems |
POLYMATH is a proven computational system, which has been specifically created for educational or professional use. 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..
Input a new nonlinear (implicit) algebraic equation.
Input a new explicit algebraic equation.
Input minimum and maximum
values for single nonlinear equation.
View the currently defined
problem variables.
Export
current problem to Excel.
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.
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.)
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).
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).
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.
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.
