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<h1 class="menu-title">ad_trait: Automatic Differentiation for Rust</h1>
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<main>
<h1 id="introduction"><a class="header" href="#introduction">Introduction</a></h1>
<p><code>ad_trait</code> is a powerful, flexible, and easy-to-use Automatic Differentiation (AD) library for Rust.</p>
<h2 id="what-is-automatic-differentiation"><a class="header" href="#what-is-automatic-differentiation">What is Automatic Differentiation?</a></h2>
<p>Automatic Differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. Unlike symbolic differentiation, which produces a mathematical expression for the derivative, or finite differencing, which estimates derivatives from function evaluations, AD computes derivatives exactly (up to machine precision) by applying the chain rule to the program’s elementary operations.</p>
<h2 id="why-ad_trait"><a class="header" href="#why-ad_trait">Why ad_trait?</a></h2>
<ul>
<li><strong>Unified Interface</strong>: Use the same code for forward-mode, reverse-mode, and finite differencing.</li>
<li><strong>Integration</strong>: Seamlessly works with <code>nalgebra</code> and <code>ndarray</code>.</li>
<li><strong>Flexibility</strong>: Define your own differentiable functions using a simple trait.</li>
<li><strong>Performance</strong>: High-performance implementations including multi-tangent forward AD and SIMD acceleration.</li>
<li><strong>Second-Order AD</strong>: Native support for Hessians via recursive dual types.</li>
</ul>
<h2 id="goals"><a class="header" href="#goals">Goals</a></h2>
<p>The primary goal of <code>ad_trait</code> is to make sophisticated automatic differentiation accessible to the Rust ecosystem with a focus on robotics, optimization, and machine learning.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="getting-started"><a class="header" href="#getting-started">Getting Started</a></h1>
<p>Adding <code>ad_trait</code> to your project is straightforward.</p>
<h2 id="installation"><a class="header" href="#installation">Installation</a></h2>
<p>Add the following to your <code>Cargo.toml</code>:</p>
<pre><code class="language-toml">[dependencies]
ad_trait = "0.2.0"
</code></pre>
<h2 id="basic-usage"><a class="header" href="#basic-usage">Basic Usage</a></h2>
<p>The core workflow of <code>ad_trait</code> involves three steps:</p>
<ol>
<li><strong>Implement <code>DifferentiableFunctionTrait</code></strong>: Define your function.</li>
<li><strong>Implement <code>Reparameterize</code></strong>: Allow your function to work with different AD types.</li>
<li><strong>Use <code>FunctionEngine</code></strong>: Wrap your function with a differentiation method.</li>
</ol>
<h3 id="a-simple-example"><a class="header" href="#a-simple-example">A Simple Example</a></h3>
<p>Here’s how to compute the derivative of $f(x) = x^2$:</p>
<pre class="playground"><code class="language-rust">use ad_trait::{AD, DifferentiableFunctionTrait, Reparameterize, FunctionEngine, ForwardAD};
#[derive(Clone)]
struct Square;
impl<T: AD> DifferentiableFunctionTrait<T> for Square {
const NAME: &'static str = "Square";
fn call(&self, inputs: &[T], _freeze: bool) -> Vec<T> {
vec![inputs[0] * inputs[0]]
}
fn num_inputs(&self) -> usize { 1 }
fn num_outputs(&self) -> usize { 1 }
}
impl Reparameterize for Square {
type SelfType<T2: AD> = Square;
}
fn main() {
let func = Square;
let engine = FunctionEngine::new(func.clone(), func, ForwardAD::new());
let x = 3.0;
let (val, grad) = engine.derivative(&[x]);
println!("f(3) = {}", val[0]); // Output: 9
println!("f'(3) = {}", grad[(0, 0)]); // Output: 6
}</code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="core-concepts"><a class="header" href="#core-concepts">Core Concepts</a></h1>
<p><code>ad_trait</code> is built around a few central abstractions that make it highly extensible.</p>
<h2 id="the-ad-type-system"><a class="header" href="#the-ad-type-system">The AD Type System</a></h2>
<p>The cornerstone of the library is the <code>AD</code> trait. Any type that implements <code>AD</code> can be used in a differentiable computation. The library provides several built-in implementations:</p>
<ul>
<li><code>f64</code> and <code>f32</code>: For standard computations without derivative tracking.</li>
<li><code>adfn<N></code>: For forward-mode AD with $N$ tangents.</li>
<li><code>adr</code>: For reverse-mode AD using a global computation graph.</li>
<li><code>f64xn<N></code>: For SIMD-accelerated numerical computations.</li>
</ul>
<h2 id="trait-hierarchy"><a class="header" href="#trait-hierarchy">Trait Hierarchy</a></h2>
<ol>
<li><strong><code>AD</code></strong>: The base numerical trait for differentiation.</li>
<li><strong><code>DifferentiableFunctionTrait<T></code></strong>: Defines how a function is evaluated for a given AD type <code>T</code>.</li>
<li><strong><code>Reparameterize</code></strong>: Bridges the gap between different AD types, allowing a function to be automatically adapted for different differentiation modes.</li>
<li><strong><code>DerivativeMethodTrait</code></strong>: Defines how a derivative is calculated (e.g., Forward, Reverse).</li>
</ol>
<h2 id="the-function-engine"><a class="header" href="#the-function-engine">The Function Engine</a></h2>
<p>The <code>FunctionEngine</code> is the primary interface for users. It wraps a differentiable function and a derivative method, providing a simple way to call the function and get its Jacobian.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="the-ad-trait"><a class="header" href="#the-ad-trait">The AD Trait</a></h1>
<p>The <code>AD</code> trait is the fundamental building block of <code>ad_trait</code>. It defines the arithmetic and mathematical operations required for automatic differentiation.</p>
<h2 id="why-use-a-trait"><a class="header" href="#why-use-a-trait">Why use a Trait?</a></h2>
<p>By using a trait instead of a concrete type, <code>ad_trait</code> allows you to write generic algorithms that can be used for:</p>
<ul>
<li>Standard evaluation (using <code>f64</code>).</li>
<li>First-order derivatives (using <code>adfn<1></code>).</li>
<li>Gradients for many inputs (using <code>adr</code>).</li>
<li>Accelerated vector math (using <code>f64xn</code>).</li>
</ul>
<h2 id="key-methods"><a class="header" href="#key-methods">Key Methods</a></h2>
<ul>
<li><code>constant(f64) -> Self</code>: Creates a new AD value from a constant.</li>
<li><code>to_constant(&self) -> f64</code>: Retrieves the underlying value.</li>
<li><code>ad_num_mode()</code>: Returns the current mode (Float, ForwardAD, etc.).</li>
<li><code>to_other_ad_type<T2: AD>(&self) -> T2</code>: Converts to a different AD type.</li>
</ul>
<h2 id="numerical-operations"><a class="header" href="#numerical-operations">Numerical Operations</a></h2>
<p><code>AD</code> requires many standard numerical traits, including:</p>
<ul>
<li><code>RealField</code> and <code>ComplexField</code> from <code>simba</code>.</li>
<li><code>num_traits::Signed</code>.</li>
<li>Standard operator overloads (<code>Add</code>, <code>Mul</code>, etc.).</li>
</ul>
<p>This ensures that any type implementing <code>AD</code> behaves like a sophisticated number.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="automatic-differentiation-modes"><a class="header" href="#automatic-differentiation-modes">Automatic Differentiation Modes</a></h1>
<p><code>ad_trait</code> supports three primary modes of differentiation, each with its own strengths and weaknesses.</p>
<h2 id="summary-table"><a class="header" href="#summary-table">Summary Table</a></h2>
<div class="table-wrapper">
<table>
<thead>
<tr><th style="text-align: left">Mode</th><th style="text-align: left">Type</th><th style="text-align: left">Speed (Few Inputs)</th><th style="text-align: left">Speed (Many Inputs)</th><th style="text-align: left">Precision</th></tr>
</thead>
<tbody>
<tr><td style="text-align: left"><strong>Forward-Mode</strong></td><td style="text-align: left">Exact</td><td style="text-align: left">Very Fast</td><td style="text-align: left">Slow</td><td style="text-align: left">Exact</td></tr>
<tr><td style="text-align: left"><strong>Reverse-Mode</strong></td><td style="text-align: left">Exact</td><td style="text-align: left">Moderate</td><td style="text-align: left">Very Fast</td><td style="text-align: left">Exact</td></tr>
<tr><td style="text-align: left"><strong>Hessian AD</strong></td><td style="text-align: left">Exact</td><td style="text-align: left">Moderate</td><td style="text-align: left">Slow</td><td style="text-align: left">Exact</td></tr>
<tr><td style="text-align: left"><strong>Finite Differencing</strong></td><td style="text-align: left">Approx</td><td style="text-align: left">Fast</td><td style="text-align: left">Very Slow</td><td style="text-align: left">Low</td></tr>
</tbody>
</table>
</div>
<p>In the following chapters, we will explore each of these modes in detail.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="forward-mode-ad"><a class="header" href="#forward-mode-ad">Forward-Mode AD</a></h1>
<p>Forward-mode automatic differentiation propagates derivatives along with the function evaluation. In <code>ad_trait</code>, this is implemented using the <code>adfn<N></code> type, where <code>N</code> is a <strong>const generic</strong> representing the number of “tangent lanes” (the number of derivatives tracked simultaneously).</p>
<h2 id="how-it-works"><a class="header" href="#how-it-works">How it Works</a></h2>
<p>A forward-mode AD variable can be thought of as a pair $(v, \dot{v})$, where $v$ is the current value and $\dot{v}$ is its tangent (the derivative with respect to some input). Every operation on these variables updates both the value and the tangent using the rules of calculus.</p>
<h2 id="single-tangent-forward-ad"><a class="header" href="#single-tangent-forward-ad">Single-Tangent Forward AD</a></h2>
<p>When using <code>ForwardAD</code>, the library calculates the derivative with respect to one input at a time. To compute a full Jacobian for $M$ inputs, the function is evaluated $M$ times.</p>
<h2 id="multi-tangent-forward-ad"><a class="header" href="#multi-tangent-forward-ad">Multi-Tangent Forward AD</a></h2>
<p>One of the unique features of <code>ad_trait</code> is its support for multiple tangents. By using <code>adfn<N></code>, you can compute up to $N$ columns of the Jacobian in a single forward pass. This is extremely efficient for functions where most work is shared across different input variables.</p>
<h2 id="usage-example"><a class="header" href="#usage-example">Usage Example</a></h2>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>use ad_trait::{ForwardAD, adfn};
// evaluate with ForwardAD
// this will compute derivatives by calling the function once
// per input dimension.
let engine = FunctionEngine::new(func.clone(), func, ForwardAD::new());
<span class="boring">}</span></code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="reverse-mode-ad"><a class="header" href="#reverse-mode-ad">Reverse-Mode AD</a></h1>
<p>Reverse-mode automatic differentiation is the most efficient way to compute gradients of functions with a very large number of inputs (e.g., neural networks). In <code>ad_trait</code>, this is implemented using the <code>adr</code> type.</p>
<h2 id="how-it-works-1"><a class="header" href="#how-it-works-1">How it Works</a></h2>
<p>Unlike forward-mode, reverse-mode AD works in two phases:</p>
<ol>
<li><strong>Forward Pass</strong>: The function is evaluated, and all operations are recorded in a <strong>Global Computation Graph</strong>.</li>
<li><strong>Backward Pass</strong>: The library traverses the graph in reverse, applying the chain rule to compute the gradient with respect to every input variable.</li>
</ol>
<h2 id="the-global-computation-graph"><a class="header" href="#the-global-computation-graph">The Global Computation Graph</a></h2>
<p>Because <code>adr</code> relies on a global graph, certain care must be taken:</p>
<ul>
<li>The graph must be reset between independent differentiation calls (handled automatically by <code>FunctionEngine</code>).</li>
<li>In multi-threaded environments, access to the graph must be synchronized.</li>
</ul>
<h2 id="usage-example-1"><a class="header" href="#usage-example-1">Usage Example</a></h2>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>use ad_trait::ReverseAD;
// evaluate with ReverseAD
// this is best for functions with many inputs and few outputs.
let engine = FunctionEngine::new(func.clone(), func, ReverseAD::new());
<span class="boring">}</span></code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="finite-differencing"><a class="header" href="#finite-differencing">Finite Differencing</a></h1>
<p>Finite Differencing is a classical numerical method for estimating derivatives. It is provided in <code>ad_trait</code> as a baseline and for functions where AD types might be impractical.</p>
<h2 id="how-it-works-2"><a class="header" href="#how-it-works-2">How it Works</a></h2>
<p>The derivative is approximated using the formula:
$$f’(x) \approx \frac{f(x + h) - f(x)}{h}$$
where $h$ is a very small value.</p>
<h2 id="accuracy-vs-precision"><a class="header" href="#accuracy-vs-precision">Accuracy vs. Precision</a></h2>
<p>Finite differencing is an approximation and is subject to both truncation error (making $h$ too large) and round-off error (making $h$ too small). It is generally much less precise than true automatic differentiation.</p>
<h2 id="usage-example-2"><a class="header" href="#usage-example-2">Usage Example</a></h2>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>use ad_trait::FiniteDifferencing;
// evaluate with FiniteDifferencing
let engine = FunctionEngine::new(func.clone(), func, FiniteDifferencing::new());
<span class="boring">}</span></code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="hessian--second-order-ad"><a class="header" href="#hessian--second-order-ad">Hessian / Second-Order AD</a></h1>
<p><code>ad_trait</code> supports computing second-order derivatives (Hessians) by nesting automatic differentiation types. This is achieved through the <code>HyperAD</code> family of types.</p>
<blockquote class="blockquote-tag blockquote-tag-note">
<p class="blockquote-tag-title"><svg viewbox="0 0 16 16" width="18" height="18"><path d="M0 8a8 8 0 1 1 16 0A8 8 0 0 1 0 8Zm8-6.5a6.5 6.5 0 1 0 0 13 6.5 6.5 0 0 0 0-13ZM6.5 7.75A.75.75 0 0 1 7.25 7h1a.75.75 0 0 1 .75.75v2.75h.25a.75.75 0 0 1 0 1.5h-2a.75.75 0 0 1 0-1.5h.25v-2h-.25a.75.75 0 0 1-.75-.75ZM8 6a1 1 0 1 1 0-2 1 1 0 0 1 0 2Z"></path></svg>Note</p>
<p>The bracketed value <code><N></code> (e.g., <code>adfn<N></code>, <code>HessianAD<N></code>) is a <strong>const generic</strong> that specifies the number of <strong>tangent lanes</strong>. For Hessian computation, this typically should match the number of input variables you are differentiating with respect to.</p>
</blockquote>
<div class="table-wrapper">
<table>
<thead>
<tr><th>Mode</th><th>Type</th><th>Best For</th><th>Scaling</th></tr>
</thead>
<tbody>
<tr><td><strong>Forward-over-Forward</strong></td><td><code>HyperAD_ADFN<N></code></td><td>Few inputs ($N < 20$), low memory.</td><td>$O(N^2)$</td></tr>
<tr><td><strong>Forward-over-Reverse</strong></td><td><code>HyperAD_ADR<N></code></td><td>Many inputs, few outputs (e.g., loss functions).</td><td>$O(N)$</td></tr>
</tbody>
</table>
</div>
<h2 id="choosing-the-right-mode"><a class="header" href="#choosing-the-right-mode">Choosing the Right Mode</a></h2>
<p>Selecting the optimal mode depends primarily on the number of input variables ($N$) and the memory constraints of your application.</p>
<h3 id="forward-over-forward-fof"><a class="header" href="#forward-over-forward-fof">Forward-over-Forward (FoF)</a></h3>
<ul>
<li><strong>When to use</strong>: Use this when you have a small number of inputs. It is the most robust mode and has the lowest memory overhead because it does not require building a computation graph.</li>
<li><strong>Efficiency</strong>: The computational cost scales quadratically with the number of inputs ($O(N^2)$). For a function with 10 inputs, it is very fast; for 1000 inputs, it becomes prohibitively slow.</li>
<li><strong>Implementation</strong>: Uses <code>HessianAD<N></code>.</li>
</ul>
<h3 id="forward-over-reverse-for"><a class="header" href="#forward-over-reverse-for">Forward-over-Reverse (FoR)</a></h3>
<ul>
<li><strong>When to use</strong>: Use this for functions with many inputs and a single (or few) outputs, such as a neural network loss function or a complex physics simulation.</li>
<li><strong>Efficiency</strong>: This mode is significantly more efficient for large $N$. Because the inner layer is Reverse-mode AD, a single backpropagation through a tangent value can recover an entire row of the Hessian. This allows the total cost to scale linearly with the number of inputs ($O(N)$) for scalar-valued functions.</li>
<li><strong>Memory</strong>: Higher memory usage than FoF because it must maintain the reverse-mode computation graph.</li>
<li><strong>Implementation</strong>: Uses <code>HessianAD_FOR<N></code>.</li>
</ul>
<h2 id="forward-over-forward-hessian"><a class="header" href="#forward-over-forward-hessian">Forward-over-Forward Hessian</a></h2>
<p>This mode uses <code>HyperAD_ADFN</code>, which is essentially an <code>adfn</code> type where the primary value and tangents are themselves <code>adfn</code> types.</p>
<h3 id="example"><a class="header" href="#example">Example</a></h3>
<pre class="playground"><code class="language-rust">use ad_trait::AD;
use ad_trait::function_engine::FunctionEngine;
use ad_trait::differentiable_function::{DifferentiableFunctionTrait, HessianAD, ToOtherADType};
use ad_trait::hyper_ad::hyper::HyperAD_ADFN;
#[derive(Clone)]
struct MyFunc;
impl<T: AD> DifferentiableFunctionTrait<T> for MyFunc {
const NAME: &'static str = "MyFunc";
fn call(&self, inputs: &[T], _freeze: bool) -> Vec<T> {
let x = inputs[0];
vec![ x * x * x ] // f(x) = x^3
}
fn num_inputs(&self) -> usize { 1 }
fn num_outputs(&self) -> usize { 1 }
}
fn main() {
let inputs = [2.0];
let func = MyFunc;
let engine = FunctionEngine::new(
func.clone(),
func.to_other_ad_type::<HyperAD_ADFN<1>>(),
HessianAD::<1>::new()
);
let (f_res, jacobian_res, hessian_res) = engine.hessian(&inputs);
println!("f(2) = {}", f_res[0]); // 8.0
println!("f'(2) = {}", jacobian_res[(0,0)]); // 12.0
println!("f''(2) = {}", hessian_res[0][(0,0)]); // 12.0
}</code></pre>
<h2 id="forward-over-reverse-hessian"><a class="header" href="#forward-over-reverse-hessian">Forward-over-Reverse Hessian</a></h2>
<p>This mode uses <code>HyperAD_ADR</code>, which uses <code>adr</code> as the inner type. This is useful when you want to combine the benefits of forward and reverse mode.</p>
<h3 id="example-1"><a class="header" href="#example-1">Example</a></h3>
<pre class="playground"><code class="language-rust">use ad_trait::AD;
use ad_trait::function_engine::FunctionEngine;
use ad_trait::differentiable_function::{DifferentiableFunctionTrait, HessianAD_FOR, ToOtherADType};
use ad_trait::hyper_ad::hyper_adr::HyperAD_ADR;
#[derive(Clone)]
struct MyFunc;
impl<T: AD> DifferentiableFunctionTrait<T> for MyFunc {
const NAME: &'static str = "MyFunc";
fn call(&self, inputs: &[T], _freeze: bool) -> Vec<T> {
let x = inputs[0];
vec![ x * x * x ]
}
fn num_inputs(&self) -> usize { 1 }
fn num_outputs(&self) -> usize { 1 }
}
fn main() {
let inputs = [2.0];
let func = MyFunc;
let engine = FunctionEngine::new(
func.clone(),
func.to_other_ad_type::<HyperAD_ADR<1>>(),
HessianAD_FOR::<1>::new()
);
let (f_res, jacobian_res, hessian_res) = engine.hessian(&inputs);
println!("f'(2) = {}", jacobian_res[(0,0)]); // 12.0
println!("f''(2) = {}", hessian_res[0][(0,0)]); // 12.0
}</code></pre>
<h2 id="first-order-derivatives-from-hessian-engines"><a class="header" href="#first-order-derivatives-from-hessian-engines">First-Order Derivatives from Hessian Engines</a></h2>
<p>It is important to note that a <code>FunctionEngine</code> initialized for Hessian computation can still be used for standard first-order derivatives.</p>
<p>Calling <code>engine.derivative(&inputs)</code> on a Hessian-enabled block will return the Jacobian matrix as usual. This is possible because the hyper-dual types used for second-order differentiation internally track the first-order gradients as their primal “tangent” values.</p>
<p>This allows you to maintain a single <code>FunctionEngine</code> instance for both standard gradient-based optimization and second-order methods.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="advanced-topics"><a class="header" href="#advanced-topics">Advanced Topics</a></h1>
<p>Explore the high-performance features and deep technical details of <code>ad_trait</code>.</p>
<ul>
<li><strong>Multi-Tangents</strong>: Scaling forward-mode AD.</li>
<li><strong>SIMD Acceleration</strong>: Leveraging hardware for faster differentiation.</li>
<li><strong>Matrix Operations</strong>: Efficiently handling linear algebra.</li>
</ul>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="multi-tangents"><a class="header" href="#multi-tangents">Multi-Tangents</a></h1>
<p>Multi-tangent forward AD is a powerful optimization technique that allows <code>ad_trait</code> to compute multiple partial derivatives in a single pass of the function.</p>
<h2 id="the-problem"><a class="header" href="#the-problem">The Problem</a></h2>
<p>Standard forward AD computes one column of the Jacobian per pass. If a function has 100 inputs, you need 100 passes. For complex functions, this overhead can be significant.</p>
<h2 id="the-solution-adfnn"><a class="header" href="#the-solution-adfnn">The Solution: <code>adfn<N></code></a></h2>
<p>By setting $N > 1$, the tangent component of the AD type becomes a vector of length $N$. Each multiplication or operation now operates on this vector simultaneously.</p>
<h2 id="when-to-use"><a class="header" href="#when-to-use">When to Use</a></h2>
<ul>
<li>When the input dimension is moderately large (e.g., 2-32).</li>
<li>When the function evaluation is computationally expensive relative to the number of inputs.</li>
<li>When you want to minimize the number of times the function logic is executed.</li>
</ul>
<h2 id="integration-with-simd"><a class="header" href="#integration-with-simd">Integration with SIMD</a></h2>
<p>Multi-tangents pair perfectly with SIMD. By propagating multiple tangents, the underlying hardware can often execute these operations in parallel, providing a “free” speedup.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="simd-acceleration"><a class="header" href="#simd-acceleration">SIMD Acceleration</a></h1>
<p><code>ad_trait</code> provides first-class support for Single Instruction, Multiple Data (SIMD) acceleration through the <code>f64xn<N></code> type.</p>
<h2 id="what-is-simd"><a class="header" href="#what-is-simd">What is SIMD?</a></h2>
<p>SIMD allows a single CPU instruction to operate on multiple pieces of data (usually vectors) at once. This can lead to 4x-8x speedups for arithmetic operations on modern processors.</p>
<h2 id="using-f64xn"><a class="header" href="#using-f64xn">Using <code>f64xn</code></a></h2>
<p>The <code>f64xn<N></code> type implements the <code>AD</code> trait (in <code>SIMDNum</code> mode). It stores $N$ floats and performs element-wise operations using SIMD intrinsics (where available).</p>
<h2 id="requirements"><a class="header" href="#requirements">Requirements</a></h2>
<p>Currently, <code>f64xn</code> requires the <strong>nightly</strong> version of Rust to access the <code>portable_simd</code> feature.</p>
<h2 id="example-2"><a class="header" href="#example-2">Example</a></h2>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>// Requires nightly and #![feature(portable_simd)]
use ad_trait::simd::f64xn;
let a = f64xn::<4>::new([1.0, 2.0, 3.0, 4.0]);
let b = f64xn::<4>::new([5.0, 6.0, 7.0, 8.0]);
let c = a + b; // Computed using a single CPU instruction if possible
<span class="boring">}</span></code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="matrix-operations"><a class="header" href="#matrix-operations">Matrix Operations</a></h1>
<p><code>ad_trait</code> is designed to handle linear algebra efficiently. The <code>AD</code> trait includes requirements for matrix-scalar multiplication, which is the foundation for differentiating through linear systems.</p>
<h2 id="scalar-matrix-multiplication"><a class="header" href="#scalar-matrix-multiplication">Scalar-Matrix Multiplication</a></h2>
<p>Types implementing <code>AD</code> must provide:</p>
<ul>
<li><code>mul_by_nalgebra_matrix</code>: Multiplies an AD scalar by a <code>nalgebra</code> matrix.</li>
<li><code>mul_by_ndarray_matrix_ref</code>: Multiplies an AD scalar by an <code>ndarray</code> array.</li>
</ul>
<h2 id="why-this-is-necessary"><a class="header" href="#why-this-is-necessary">Why this is necessary</a></h2>
<p>By providing these specialized methods at the trait level, <code>ad_trait</code> can ensure that matrix operations are handled correctly for each AD mode. For example, in <code>ForwardAD</code>, the matrix multiplication also propagates the tangent through every element of the matrix.</p>
<h2 id="performance-considerations"><a class="header" href="#performance-considerations">Performance Considerations</a></h2>
<p>When performing large matrix multiplications, it is often more efficient to use a differentiation mode that minimizes the number of passes (like <code>ReverseAD</code> or <code>ForwardADMulti</code>).</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="integration"><a class="header" href="#integration">Integration</a></h1>
<p><code>ad_trait</code> doesn’t exist in a vacuum. It is designed to play well with the rest of the Rust numerical ecosystem.</p>
<ul>
<li><strong>nalgebra</strong>: The standard for linear algebra in Rust.</li>
<li><strong>ndarray</strong>: Flexible multi-dimensional arrays.</li>
</ul>
<p>In this section, we’ll see how to use <code>ad_trait</code> types within these libraries.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="integration-with-nalgebra"><a class="header" href="#integration-with-nalgebra">Integration with nalgebra</a></h1>
<p><code>ad_trait</code> types are fully compatible with <code>nalgebra</code>’s generic matrix types.</p>
<h2 id="example-ad-matrix"><a class="header" href="#example-ad-matrix">Example: AD Matrix</a></h2>
<p>You can create a <code>nalgebra</code> matrix using any <code>AD</code> type:</p>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>use nalgebra::SMatrix;
use ad_trait::adfn;
// A 2x2 matrix of Forward-Mode AD variables
let m = SMatrix::<adfn<1>, 2, 2>::zeros();
<span class="boring">}</span></code></pre>
<h2 id="differentiating-through-nalgebra"><a class="header" href="#differentiating-through-nalgebra">Differentiating through nalgebra</a></h2>
<p>Because <code>adfn</code> and <code>adr</code> implement the relevant traits required by <code>nalgebra</code> (like <code>RealField</code> and <code>ComplexField</code>), you can use standard <code>nalgebra</code> functions (determinant, inverse, multiplication) in your differentiable code.</p>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn my_func<T: AD>(inputs: &[T]) -> Vec<T> {
let m = SMatrix::<T, 2, 2>::from_vec(inputs.to_vec());
let inv = m.try_inverse().unwrap();
vec![inv[(0, 0)]]
}
<span class="boring">}</span></code></pre>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="integration-with-ndarray"><a class="header" href="#integration-with-ndarray">Integration with ndarray</a></h1>
<p><code>ad_trait</code> also supports the <code>ndarray</code> crate, which is commonly used for data science and machine learning.</p>
<h2 id="example-ad-array"><a class="header" href="#example-ad-array">Example: AD Array</a></h2>
<pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>use ndarray::Array2;
use ad_trait::adfn;
let a = Array2::<adfn<1>>::zeros((10, 10));
<span class="boring">}</span></code></pre>
<h2 id="scalar-operations"><a class="header" href="#scalar-operations">Scalar Operations</a></h2>
<p>The <code>AD</code> trait includes <code>mul_by_ndarray_matrix_ref</code>, allowing you to perform scalar-array multiplication efficiently across different AD modes.</p>
<h2 id="generic-algorithms"><a class="header" href="#generic-algorithms">Generic Algorithms</a></h2>
<p>Similar to <code>nalgebra</code>, you can write generic algorithms using <code>ndarray</code> that work with any <code>AD</code> type. This is particularly useful for implementing complex mathematical models that require gradients for optimization.</p>
<div style="break-before: page; page-break-before: always;"></div>
<h1 id="examples"><a class="header" href="#examples">Examples</a></h1>
<p>The <code>ad_trait</code> repository contains several examples that demonstrate the library in action.</p>
<h2 id="built-in-examples"><a class="header" href="#built-in-examples">Built-in Examples</a></h2>
<p>Check the <code>examples/</code> directory in the crate for standalone programs:</p>
<ul>
<li><code>test.rs</code>: A general demonstration of forward and reverse AD.</li>
</ul>
<h2 id="regression-tests"><a class="header" href="#regression-tests">Regression Tests</a></h2>
<p>The <code>tests/regression_tests.rs</code> file contains many examples of differentiating complex functions, including:</p>
<ul>
<li>Multi-variate polynomials.</li>
<li>Matrix-vector multiplication.</li>
<li>Jacobian calculations for multiple outputs.</li>
</ul>
<p>These tests serve as an excellent reference for how to structure your own differentiable functions.</p>
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