implement RationalFunction class

[crowlogic]

Unified Polynomial Representation Including Remainders and Divisors for Rational Functions

Objective: Refine the expression parser to include a unified representation that integrates remainders and divisors into the polynomial structure, with a specific emphasis on handling rational functions, particularly Lommel polynomials.

Details:

• Unified Representation: Enhance the polynomial data structure to incorporate remainders and divisors directly. This integration simplifies mathematical operations by keeping all necessary data within a single framework.
• Boolean Flag isIntegral: Use a boolean to indicate whether a result is integral or has a remainder. This flag aids in the quick determination of the polynomial's state and streamlines subsequent operations.
• Lommel Polynomials: Although not technically polynomials but rational functions (or "polynomials in $1/z$"), Lommel polynomials can be seamlessly managed with the proposed system. This representation is crucial for handling functions where standard polynomial structures are insufficient.

Example: Considering this:

• Start with polynomial $1$.
• Divide by $x^2$ to produce a result of $0$ with a remainder of $x$.
• This state is represented as: {polynomial: $0$, remainder: $x$, divisor: $x^2$}.
• Multiplying by $x$ updates the representation to:

$${\text{polynomial: } 0, \text{remainder: } x^2, \text{divisor: } x^2}$$

Which reduces to

$${\text{polynomial: } 0, \text{remainder: } 1, \text{divisor: } x}$$

Now multiply by x

$${\text{polynomial: } 0, \text{remainder: } x, \text{divisor: } x}$$

Which reduces to

$${\text{polynomial: } 1, \text{remainder: } 0, \text{divisor: } 1}$$

[crowlogic]

wait , wtf, thats just a rational function. so, just finish https://github.com/crowlogic/arb4j/issues/417

[crowlogic]

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \title{Development of a Generic Rational Function Class} \author{Your Name} \date{\today}

\begin{document}

\maketitle

\section{Introduction} This document outlines the development of the \texttt{RationalFunction} class, designed to manage and compute rational functions by extending the \texttt{RealPolynomial} class. This approach aims to integrate this functionality into an existing expression compiler, enhancing its capabilities to handle more complex mathematical expressions.

\section{Structural Design} The \texttt{RationalFunction} class includes three primary components: \begin{itemize} \item \textbf{Value} ($V(x)$): A polynomial representing the non-fractional part of the function. \item \textbf{Remainder} ($R(x)$): When combined with the \textbf{Divisor}, represents the fractional part of the function. \item \textbf{Divisor} ($D(x)$): Denotes the denominator in the fractional part, where $D(x) \neq 0$. \end{itemize} Each component is managed distinctly to maintain the integrity and accuracy of calculations within the class.

\section{Operational Definitions} Operations within the \texttt{RationalFunction} are defined as follows:

\subsection{Addition} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$, the result is: [ \frac{R_1(x)D_2(x) + R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) + V_2(x) ]

\subsection{Subtraction} Subtraction follows a similar structure: [ \frac{R_1(x)D_2(x) - R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) - V_2(x) ]

\subsection{Multiplication} For multiplication: [ \left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right) \cdot \left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right) ] [ = \frac{R_1(x)R_2(x)}{D_1(x)D_2(x)} + \frac{R_1(x)V_2(x)}{D_1(x)} + \frac{V_1(x)R_2(x)}{D_2(x)} + V_1(x)V_2(x) ]

\subsection{Division} And for division: [ \frac{\left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right)}{\left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right)} ] [ = \frac{R_1(x)D_2(x) + V_1(x)D_1(x)D_2(x)}{R_2(x)D_1(x) + V_2(x)D_1(x)D_2(x)} ]

\section{Conclusion} The \texttt{RationalFunction} class significantly enhances the capabilities of the existing expression compiler, enabling it to handle a wider range of mathematical expressions efficiently. This integration ensures that both polynomial and rational functions are processed accurately and robustly, adhering to algebraic identities and operational integrity.

\end{document}

[crowlogic]

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \title{Development of a Generic Rational Function Class} \author{Your Name} \date{\today}

\begin{document}

\maketitle

\section{Introduction} This document outlines the development of the \texttt{RationalFunction} class, designed to manage and compute rational functions by extending the \texttt{RealPolynomial} class. This approach aims to integrate this functionality into an existing expression compiler, enhancing its capabilities to handle more complex mathematical expressions.

\section{Structural Design} The \texttt{RationalFunction} class includes three primary components: \begin{itemize} \item \textbf{Value} ($V(x)$): A polynomial representing the non-fractional part of the function. \item \textbf{Remainder} ($R(x)$): When combined with the \textbf{Divisor}, represents the fractional part of the function. \item \textbf{Divisor} ($D(x)$): Denotes the denominator in the fractional part, where $D(x) \neq 0$. \end{itemize} Each component is managed distinctly to maintain the integrity and accuracy of calculations within the class.

\section{Operational Definitions} Operations within the \texttt{RationalFunction} are defined as follows:

\subsection{Addition} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$, the result is: [ \frac{R_1(x)D_2(x) + R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) + V_2(x) ]

\subsection{Subtraction} Subtraction follows a similar structure: [ \frac{R_1(x)D_2(x) - R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) - V_2(x) ]

\subsection{Multiplication} For multiplication: [ \left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right) \cdot \left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right) ] [ = \frac{R_1(x)R_2(x)}{D_1(x)D_2(x)} + \frac{R_1(x)V_2(x)}{D_1(x)} + \frac{V_1(x)R_2(x)}{D_2(x)} + V_1(x)V_2(x) ]

\subsection{Division} And for division: [ \frac{\left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right)}{\left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right)} ] [ = \frac{R_1(x)D_2(x) + V_1(x)D_1(x)D_2(x)}{R_2(x)D_1(x) + V_2(x)D_1(x)D_2(x)} ]

\section{Handling of Remainders in Operations} In the context of polynomial arithmetic, here's how operations generally behave with regards to remainders: \begin{itemize} \item \textbf{Addition and Subtraction}: These operations do not inherently produce new remainders. Instead, existing remainders are algebraically combined or subtracted. \item \textbf{Multiplication}: Incorporates and manages the remainders without necessarily generating new remainders outside of the resulting expressions. \item \textbf{Division}: Remainders are a fundamental part of the operation if complete divisibility is not achieved. \end{itemize}

\section{Conclusion} The \texttt{RationalFunction} class significantly enhances the capabilities of the existing expression compiler, enabling it to handle a wider range of mathematical expressions efficiently. This integration ensures that both polynomial and rational functions are processed accurately and robustly, adhering to algebraic identities and operational integrity.

\end{document}

[crowlogic]

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \title{Development of a Generic Rational Function Class} \author{Your Name} \date{\today}

\begin{document}

\maketitle

\section{Introduction} This document outlines the development of the \texttt{RationalFunction} class, designed to manage and compute rational functions by extending the \texttt{RealPolynomial} class. This approach aims to integrate this functionality into an existing expression compiler, enhancing its capabilities to handle more complex mathematical expressions.

\section{Structural Design} The \texttt{RationalFunction} class includes three primary components: \begin{itemize} \item \textbf{Value} ($V(x)$): A polynomial representing the non-fractional part of the function. \item \textbf{Remainder} ($R(x)$): When combined with the \textbf{Divisor}, represents the fractional part of the function. \item \textbf{Divisor} ($D(x)$): Denotes the denominator in the fractional part, where $D(x) \neq 0$. \end{itemize} Each component is managed distinctly to maintain the integrity and accuracy of calculations within the class.

\section{Operational Definitions} Operations within the \texttt{RationalFunction} are defined as follows:

\subsection{Addition} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$, the result is: [ \frac{R_1(x)D_2(x) + R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) + V_2(x) ]

\subsection{Subtraction} Subtraction follows a similar structure: [ \frac{R_1(x)D_2(x) - R_2(x)D_1(x)}{D_1(x)D_2(x)} + V_1(x) - V_2(x) ]

\subsection{Multiplication} For multiplication: [ \left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right) \cdot \left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right) ] [ = \frac{R_1(x)R_2(x)}{D_1(x)D_2(x)} + \frac{R_1(x)V_2(x)}{D_1(x)} + \frac{V_1(x)R_2(x)}{D_2(x)} + V_1(x)V_2(x) ]

\subsection{Division} And for division: [ \frac{\left(\frac{R_1(x)}{D_1(x)} + V_1(x)\right)}{\left(\frac{R_2(x)}{D_2(x)} + V_2(x)\right)} ] [ = \frac{R_1(x)D_2(x) + V_1(x)D_1(x)D_2(x)}{R_2(x)D_1(x) + V_2(x)D_1(x)D_2(x)} ]

\section{Handling of Remainders in Operations} In the context of polynomial arithmetic, here's how operations generally behave with regards to remainders: \begin{itemize} \item \textbf{Addition and Subtraction}: These operations do not inherently produce new remainders. Instead, existing remainders are algebraically combined or subtracted. \item \textbf{Multiplication}: Incorporates and manages the remainders without necessarily generating new remainders outside of the resulting expressions. \item \textbf{Division}: Remainders are a fundamental part of the operation if complete divisibility is not achieved. \end{itemize}

\section{Conclusion} The \texttt{RationalFunction} class significantly enhances the capabilities of the existing expression compiler, enabling it to handle a wider range of mathematical expressions efficiently. This integration ensures that both polynomial and rational functions are processed accurately and robustly, adhering to algebraic identities and operational integrity.

\end{document}

[crowlogic]

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \title{Development of a Generic Rational Function Class} \author{Your Name} \date{\today}

\begin{document}

\maketitle

\section{Introduction} This document outlines the development of the \texttt{RationalFunction} class, designed to manage and compute rational functions by extending the \texttt{RealPolynomial} class. This approach aims to integrate this functionality into an existing expression compiler, enhancing its capabilities to handle more complex mathematical expressions.

\section{Structural Design} The \texttt{RationalFunction} class includes three primary components: \begin{itemize} \item \textbf{Value} ($V(x)$): A polynomial representing the non-fractional part of the function. \item \textbf{Remainder} ($R(x)$): When combined with the \textbf{Divisor}, represents the fractional part of the function. \item \textbf{Divisor} ($D(x)$): Denotes the denominator in the fractional part, where $D(x) \neq 0$. \end{itemize} Each component is managed distinctly to maintain the integrity and accuracy of calculations within the class.

\section{Operational Definitions} Operations within the \texttt{RationalFunction} are defined as follows:

\subsection{Addition} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x) + V_2(x) ] [ \text{Resulting Remainder} = R_1(x)D_2(x) + R_2(x)D_1(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Subtraction} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x) - V_2(x) ] [ \text{Resulting Remainder} = R_1(x)D_2(x) - R_2(x)D_1(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Multiplication} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x)V_2(x) + \frac{V_1(x)R_2(x)}{D_2(x)} + \frac{R_1(x)V_2(x)}{D_1(x)} ] [ \text{Resulting Remainder} = R_1(x)R_2(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Division} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = \frac{V_1(x)D_2(x) + R_1(x)}{R_2(x) + V_2(x)D_2(x)} ] [ \text{Resulting Remainder} = R_1(x)D_2(x) + V_1(x)D_1(x)D_2(x) ] [ \text{Resulting Divisor} = R_2(x)D_1(x) + V_2(x)D_1(x)D_2(x) ]

\section{Handling of Remainders in Operations} In the context of polynomial arithmetic, here's how operations generally behave with regards to remainders:

\begin{itemize} \item \textbf{Addition}: When adding polynomials that include remainders, the resulting remainder is the algebraic combination of the input remainders. This can result in a new remainder if the combination is not fully reducible. \item \textbf{Subtraction}: When subtracting polynomials that include remainders, the resulting remainder is the algebraic subtraction of the input remainders. \item \textbf{Multiplication}: Multiplication of polynomials with remainders combines these remainders, which may result in a non-zero remainder if the product of the polynomials does not perfectly reduce. \item \textbf{Division}: Division inherently results in a remainder if the numerator is not completely divisible by the denominator. \end{itemize}

\section{Conclusion} The \texttt{RationalFunction} class significantly enhances the capabilities of the existing expression compiler, enabling it to handle a wider range of mathematical expressions efficiently. This integration ensures that both polynomial and rational functions are processed accurately and robustly, adhering to algebraic identities and operational integrity.

\end{document}

[crowlogic]

\documentclass{article} \usepackage[utf8]{inputenc} \usepackage{amsmath} \title{Development of a Generic Rational Function Class} \author{Your Name} \date{\today}

\begin{document}

\maketitle

\section{Introduction} This document outlines the development of the \texttt{RationalFunction} class, designed to manage and compute rational functions by extending the \texttt{RealPolynomial} class. This approach aims to integrate this functionality into an existing expression compiler, enhancing its capabilities to handle more complex mathematical expressions.

\section{Structural Design} The \texttt{RationalFunction} class includes three primary components: \begin{itemize} \item \textbf{Value} ($V(x)$): A polynomial representing the non-fractional part of the function. \item \textbf{Remainder} ($R(x)$): When combined with the \textbf{Divisor}, represents the fractional part of the function. \item \textbf{Divisor} ($D(x)$): Denotes the denominator in the fractional part, where $D(x) \neq 0$. \end{itemize} Each component is managed distinctly to maintain the integrity and accuracy of calculations within the class.

\section{Operational Definitions} Operations within the \texttt{RationalFunction} are defined as follows:

\subsection{Addition} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x) + V_2(x) ] [ \text{Resulting Remainder} = R_1(x)D_2(x) + R_2(x)D_1(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Subtraction} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x) - V_2(x) ] [ \text{Resulting Remainder} = R_1(x)D_2(x) - R_2(x)D_1(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Multiplication} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = V_1(x)V_2(x) + \frac{V_1(x)R_2(x)}{D_2(x)} + \frac{R_1(x)V_2(x)}{D_1(x)} ] [ \text{Resulting Remainder} = R_1(x)R_2(x) ] [ \text{Resulting Divisor} = D_1(x)D_2(x) ]

\subsection{Division} Given two rational functions $\frac{R_1(x)}{D_1(x)} + V_1(x)$ and $\frac{R_2(x)}{D_2(x)} + V_2(x)$: [ \text{Resulting Value} = \left(\frac{V_1(x)D_2(x) + R_1(x)}{D_1(x)}\right) \cdot \left(\frac{D_1(x)}{V_2(x)D_1(x) + R_2(x)}\right) ] [ \text{Resulting Remainder} = R_1(x)D_2(x) + V_1(x)D_1(x)D_2(x) ] [ \text{Resulting Divisor} = R_2(x)D_1(x) + V_2(x)D_1(x)D_2(x) ]

\section{Handling of Remainders in Operations} In the context of polynomial arithmetic, here's how operations generally behave with regards to remainders:

\begin{itemize} \item \textbf{Addition}: When adding polynomials that include remainders, the resulting remainder is the algebraic combination of the input remainders. This can result in a new remainder if the combination is not fully reducible. \item \textbf{Subtraction}: When subtracting polynomials that include remainders, the resulting remainder is the algebraic subtraction of the input remainders. \item \textbf{Multiplication}: Multiplication of polynomials with remainders combines these remainders, which may result in a non-zero remainder if the product of the polynomials does not perfectly reduce. \item \textbf{Division}: Division inherently results in a remainder if the numerator is not completely divisible by the denominator. \end{itemize}

\section{Conclusion} The \texttt{RationalFunction} class significantly enhances the capabilities of the existing expression compiler, enabling it to handle a wider range of mathematical expressions efficiently. This integration ensures that both polynomial and rational functions are processed accurately and robustly, adhering to algebraic identities and operational integrity.

\end{document}

[crowlogic]

@Override public RealRationalFunction div(RealRationalFunction x, int prec, RealRationalFunction result) { if (this.value != null && x.value.divisor != null) { this.value.mul(x.value.divisor, prec, new RealPolynomial()); //V1(x)*D2(x) }

if (x.value != null && this.value.divisor != null) {
    x.value.mul(this.value.divisor, prec, new RealPolynomial()); //V2(x)*D1(x)
}

if (this.value.remainder != null && x.value.divisor != null) {
    this.value.remainder.mul(x.value.divisor, prec, new RealPolynomial()); //R1(x)*D2(x)
}

if (x.value.remainder != null && this.value != null) {
    x.value.remainder.mul(this.value, prec, new RealPolynomial()); //R2(x)*V1(x)
}

V1D2.div(V2D1, prec, result.value); // Resulting Value = V1(x)D2(x)/V2(x)D1(x)

R1D2.sub(R2V1, prec, new RealPolynomial()).div(V2D1, prec, result.value.remainder); // Calculate remainder

if (this.value.divisor != null) {
    this.value.divisor.mul(x.value.divisor, prec, result.value.divisor); // Calculate divisor
}

// Direct Reduction Using ARBLIB's 'reduce'
result.reduce(); // Simplify the result

return result;

}

[crowlogic]

You're absolutely right! Leveraging your existing expression compiler and parser is a much more efficient and elegant approach for implementing these formulas. It eliminates the need to manually code polynomial operations and ensures high performance due to the compiled nature of the expressions. Here's a suggested approach using your expression compiler:

• Define Expressions: Translate the formulas from the screenshot into your expression compiler's syntax. For example: Resulting Value: V1 D2 / (V2 D1) Resulting Remainder: (R1 D2 - R2 V1) / (V2 D1) Resulting Divisor: D1 D2

• Compile Expressions: Use your compiler to generate JVM bytecode for these expressions. This bytecode will effectively implement the polynomial operations.

• Division Method:

  • Arguments: Take two instances of your real function class as input.
  • Extract Components: Extract the polynomials V1, R1, D1, V2, R2, and D2.
  • Evaluate Expressions: Pass the extracted polynomials as variables to the compiled expressions. Execute the compiled bytecode to obtain the resulting value, remainder, and divisor polynomials.
  • Create Output: Construct a new real function instance and populate it with the results.
  • Return: Return the newly created instance. Advantages:

• Conciseness: The implementation becomes much shorter and easier to read.

• Performance: Compiled expressions are typically faster than interpreted code.

• Flexibility: You can easily modify or extend the formulas by changing the expressions.

• Maintainability: The code becomes less error-prone as you're not manually implementing complex polynomial operations. Implementation Notes:

• Ensure that your expression compiler supports the necessary polynomial operations (addition, subtraction, multiplication, division).

• Handle potential division-by-zero errors in your expressions or within your RealFunction class.

• Test the implementation thoroughly to ensure the results match the expected behavior. Feel free to ask if you have any specific questions about using your expression compiler or need further guidance on implementing this approach.