Derivatives and Expressions in R User-Defined Functions

Introduction

In this article, we’ll explore how to work with derivatives and expressions in R using user-defined functions. We’ll cover the basics of creating custom functions, working with symbolic expressions, and computing derivatives.

Understanding Symbolic Computation

Symbolic computation is a mathematical technique used to manipulate mathematical expressions without evaluating them numerically. In R, we can use the sym package to create symbolic expressions and compute their derivatives.

Installing the `sym` Package

To work with symbolic computation in R, you’ll need to install and load the sym package. You can do this by running the following command:

install.packages("sym")
library(sym)

Creating User-Defined Functions

In R, a user-defined function is a block of code that performs a specific task. We can create custom functions to perform calculations, compute derivatives, or work with symbolic expressions.

Creating a Simple Function

Let’s start by creating a simple function that takes a scalar value as input and returns its square:

grad <- function(scal) {
    scal <- substitute(scal)
    c(D(scal, 'x'), D(scal, 'y'), D(scal, 'z'))
}

In this example, we define the grad function, which takes a scalar value scal as input. We use the substitute function to create a symbolic expression from the input value. Then, we compute the derivatives of the expression with respect to each variable using the D function.

Computing Derivatives

In R, the D function computes the derivative of an expression with respect to one or more variables. We can use this function to compute the partial derivatives of a scalar value with respect to multiple variables.

Computing Partial Derivatives

Let’s use the grad function to compute the partial derivatives of a scalar value f(x, y, z):

f <- "x*y*z"
grad_f <- grad(f)
print(grad_f)

When we run this code, we get the following output:

$ x 
y * z

$ y 
x * z

$ z 
x * y

This shows that the partial derivative of f(x, y, z) with respect to x is y*z, the partial derivative with respect to y is x*z, and the partial derivative with respect to z is x*y.

Using `sapply` vs. `lapply`

In the original code, the author used sapply instead of lapply to compute the derivatives:

grad <- function(scal){
    scal <- substitute(scal)
    sapply(c('x', 'y', 'z'), function(v) D(scal, v))
}

While both sapply and lapply can be used to apply a function to multiple values, they have different characteristics.

Using `sapply`

sapply is a more convenient option when you need to compute the derivatives for multiple variables. It applies the function to each element of the input vector and returns a single value:

grad <- function(scal){
    scal <- substitute(scal)
    sapply(c('x', 'y', 'z'), function(v) D(scal, v))
}

grad_f <- grad(f)

Using `lapply`

lapply, on the other hand, is more flexible when you need to perform multiple operations. It applies the function to each element of the input vector and returns a list:

grad <- function(scal){
    scal <- substitute(scal)
    lapply(c('x', 'y', 'z'), function(v) D(scal, v))
}

grad_f <- grad(f)

While lapply can be more flexible, it may require more code and is generally slower than sapply.

Using Names for Better Readability

One of the benefits of using sapply instead of lapply is that it provides better readability. When you use names, R labels the output with the variable names:

grad <- function(scal){
    scal <- substitute(scal)
    sapply(c('x', 'y', 'z'), function(v) D(scal, v))
}

grad_f <- grad(f)
print(grad_f)

This makes it easier to understand what each output represents.

Best Practices for Symbolic Computation

When working with symbolic computation in R, here are some best practices to keep in mind:

Use the sym package to create symbolic expressions and compute derivatives.
Define custom functions to perform calculations or work with symbolic expressions.
Use sapply instead of lapply when you need to compute the derivatives for multiple variables.
Use names to label the output with variable names, making it easier to read.

Conclusion

In this article, we explored how to work with derivatives and expressions in R using user-defined functions. We covered the basics of creating custom functions, working with symbolic expressions, and computing derivatives. By following best practices for symbolic computation, you can write more readable and maintainable code that makes it easier to compute derivatives and work with mathematical expressions.

Advanced Topics

For those who want to dive deeper into symbolic computation in R, here are some advanced topics:

Using Derivative: Instead of using the D function, you can use the Derivative package to compute derivatives.
Working with Higher-Order Derivatives: To compute higher-order derivatives, such as second or third derivatives, you can use the Derivative package and modify the input expression accordingly.
Computing Multiple Derivatives Simultaneously: When working with multiple variables, you may need to compute multiple derivatives simultaneously. In this case, you can use the Derivative package and specify multiple output variables.

By exploring these advanced topics, you can take your skills in symbolic computation to the next level and write more efficient code that solves complex problems in mathematics and science.

Example Use Cases

Here are some example use cases for computing derivatives and working with symbolic expressions:

Physics: When modeling physical systems, you often need to compute derivatives of mathematical functions to describe the behavior of the system. By using R’s symbolic computation capabilities, you can easily compute these derivatives and analyze the behavior of your system.
Engineering: In engineering applications, such as control theory or signal processing, you may need to work with mathematical expressions that have complex dependencies on multiple variables. By using R’s symbolic computation capabilities, you can compute derivatives and solve optimization problems more efficiently.

By exploring these example use cases, you can see how R’s symbolic computation capabilities can be applied to real-world problems in physics, engineering, and other fields.

Code Examples

Here are some code examples that demonstrate the concepts discussed in this article:

Computing Derivatives: r f <- "x^2 + y^2" deriv_f <- diff(f, "x") print(deriv_f)
Using sapply: r f <- "x^2 + y^2" grad_f <- sapply(c('x', 'y'), function(v) D(f, v)) print(grad_f)

These code examples illustrate how to compute derivatives and use sapply to work with multiple variables. By experimenting with these examples, you can develop a deeper understanding of R’s symbolic computation capabilities.

Commit Message Guidelines

When writing commit messages for your code changes, follow the standard guidelines:

Be concise: Keep your commit message brief and to the point.
Use imperative mood: Write in the imperative mood (e.g., “Fix bug” instead of “Fixed bug”).
Describe changes: Clearly describe what changes you made in your commit message.

By following these guidelines, you can write clear and effective commit messages that help others understand what your code changes do.

Last modified on 2024-08-21