PolymathPlus Manual

PolymathPlus is a generic mathematical problem solver. The problems are entered in a simple text format; and pressing the ► Solve button yields a detailed numerical solution report.

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

The following problem types are supported:

1. Linear Equations (LEQ)
2. Nonlinear Equations (NLE)
3. Ordinary Differential Equations (DEQ)
4. Curve Fitting Regression (REG)

Problem Entry

Problems are entered in plain text.
The # sign indicates a comment; hence any text section following it is ignored by the solution algorithm. When saving problems online, the program name is derived from the first comment line.

System of Linear Equations  👆

The number of equations must be equal to the number of unknowns. All the algebraic expressions must be linear hence only the plus, minus, and multiplication operators are allowed. Expressions like x*x, 1/x, sin(x) are not allowed.

The following example demonstrates the input of a linear system of equations:

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

The Solution Report includes the values of the unknowns at the solution (a = 54, b = -4, and c = -2, for this problem) and the problem definition in both matrix and system of equations formats.

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.

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 will show:


PolymathPlus can numerically solve either a single nonlinear equation or a set of simultaneous nonlinear equations. When solving a single nonlinear equation, the user has to provide a range of values within which the solution exists. The program will find all the solutions within that range. When solving multiple nonlinear equations, the user has to provide a guess for each variable to be found. In this case the guess is a single value and not a range as indicated on the single nonlinear equation.
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.

x = an expression 

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:

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

# 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:

# Solve a system of two nonlinear equations f(x) = 2*x+x*y-y^2 + 290 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  👆

# 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

# Simultaneous nonlinear equations and explicit variables 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 of the nonlinear equation variables 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: a = 1.00011; b = 2.00055; c= 3.00066; d = 3.99967; and e = 4.99964.

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:

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

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.

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

The Solution Report includes a Table listing the initial, minimal, maximal, and final values of all the variables over the integration range. In the example shown, the final values are: A = 0.049787, B = 0.047308 and C = 0.902905. A table containing 50 values of the dependent variables between the initial and final values of the independent variable is also displayed.

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

Regressions of Data  👆

This category of problems involves the fitting of data to various mathematical expressions.  The resulting fitted expressions can be considered by their ability to statistically represent the data. 
PolymathPlus allows the user to fit data sets to

• Polynomials of various degrees
• Linear and multiple linear Regression
• General nonlinear Regression

The known data points should be entered in a strict tabulated form enclosed in square brackets [ ]. Open bracket [ and close bracket ] should get their own dedicated lines.
The 1st line in the table contains the variable names separated by spaces. The rest of the lines contain the values of the variables, columns are seperated by spaces. The number of columns in each line must match, blank values are not allowed.
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 ]

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:

Time BOD days mg/liter ---- -------- 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

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 results a₀ = 0.657797 (±0.374896), (95% confidence intervals shown between parentheses), a₁ = 0.178056 (±0.031674), R² = 0.9472735 and the variance: σ² = 0.0825304.

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

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

polyfit Time BOD 2

The results obtained in this case are a₀ = 0.224387 (±0.353054 ), a₁ = 0.310403 (±0.081692 ), a₂ = -0.006439 (±0.003851), R² = 0.9815572 and σ² = 0.0324762.

To fit a 2^nd 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: a₁ = 0.354779 (±0.044171), a₂ = -0.008236 (±0.002724), R² = 0.9766053 and σ² = 0.0366186.

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.

Consider the following set of Pressure (mm Hg) versus Temperature (deg C) data:

Temperature, T Pressure, P (Deg. C) (mm Hg) --------------- --------------- -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

The following operations need to be carried out with these 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 and confidence intervals on these values.
For this particular problem the following results are shown.

Variable Value Conf Interval --------- ----------- ------------- a0 2121.85 434.312 a1 -33.1504 6.01561 a2 0.196174 0.0310526 a3 -0.000522 7.080E-05 a4 5.28E-07 6.017E-08

The regression plot:
   

The Solution Report includes also the statistical metrics: R2, R2adj, root mean squared deviation and variance. A table displaying the observed (measured) y values, the predicted y values (ycalc) and the residuals (r = y - ycalc).

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, 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 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 the data for y.

Use the mlinfit command to request a multiple linear regression fit. The arguments are the independent variables vector's names, followed by the independent vector name. Additional optional (last) argument is the string 'origin' which indicates that the regression curve should pass through the origin (thus a₀ = 0).

Simple multiple linear regression example

The following data are for the wear of a bearing y that may be dependent upon x₁ = oil viscosity and x₂ = load data.:

y x1 x2 --- ---- ---- 293 1.61 851 230 15.5 820 172 22 1058 91 45 1201 125 33 1357 125 40 1115

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

# 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 a₀ = 360.836 (±118.076), a₁ = -3.75246(±1.774), a₂ = -0.084265(± 0.140313), R² = 0.9835209 and σ² = 159.1951.

To remove x₂ from the regression model, the mlinfit statement is revised:

mlinfit x1 y

After this modification the following results are obtained: a₀ = 292.784(±37.3385 ), a₁ = -4.58727(±1.23994), R² = 0.963462 and σ² = 264.7302 .
Judging by the R² and σ² values, the regression model that contains both x₁ and x₂ represents the data better. However, the confidence interval value of a² shows that this parameter is not significantly different from zero, thus removal of x₂ from the model should be preferred .

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

The dependent variable can be defined by logP=log(P_kPa) and the independent variables defined by OneonT=1/T_K and logT=log(T_K). Then mlinfit can be used 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 and confidence intervals on these values. For this particular problem the following results are shown.

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 request 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's 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 is A = 5.76735, B = 677.094 and C = 153.885.

Regression Statistics  👆

PolymathPlus provides statistical reports for the regression fit which assess the validity and accuracy of the regression. Each fitting variable has a 95% confidence interval. The confidence interval should be smaller than the absolute value of the variable for statistical validity.

In addition the following calculations are provided:

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

The following formulas calculate MAE, MSE, RMSE.

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  👆

A number of standard functions are available for use in the various programs. The arguments of the functions must be enclosed in parentheses. The arguments may be themselves expressions or other functions. The nesting of functions is allowed.

All function names must be given in lower-case letters. The trigonometric functions require that their arguments be given in radians. Conversely, the inverse trigonometric functions give their results in radians.

You should note that the functions require that their arguments be enclosed in parentheses, but that the arguments do not have to be simple numbers. You may have a complicated expression as the argument for a function, and you may even nest the functions, using one function (or an expression including one or more functions) as the argument for another.

Settings and Hints  👆

PolymathPlus allows for custom tuning of the solution algorithms, as well as basic adjustments to the output solution report. Settings are the default parameters and apply to all solution programs. Specific program parameters can be overridden by hints. Settings are generally stored in a dedicated settings file, while hints are program-specific and are stored as part of the program *.pol.

• Settings are default parameters which apply to all programs
• Hints are optional per program tuning which can override the default settings and are saved as part of the program itself

The main settings and hints apply to:

• Solution algorithms and error tolerance tuning
• Report display parameters such as displaying titles, charts, statistics
• Report display tuning such as chart size or significant digits.

List of Global 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:

#@ 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

Linear Equations Programs Hints  👆

Nonlinear Equations Programs Hints  👆

Differential Equations Programs Hints  👆

Regression Programs Hints  👆