---
title: "Explore numerical computing in Swift with MLX"
description: "Use MLX Swift for NumPy-style n-dimensional arrays, lazy GPU execution, convolutions, and automatic differentiation in Swift numerical code."
shoutout: "Unofficial, made with love by Superwall - the best way to monetize your apps."
category: "Swift"
---

# Explore numerical computing in Swift with MLX


[Watch on Apple Developer](https://developer.apple.com/videos/play/wwdc2026/328/)

Use MLX Swift for NumPy-style n-dimensional arrays, lazy GPU execution, convolutions, and automatic differentiation in Swift numerical code.

## TL;DR

- MLX Swift is positioned for mathematical Swift code that operates on n-dimensional arrays, with lazy evaluation enabling automatic GPU execution and function transformations such as `grad`.
- The session compares scalar Swift with MLX Swift for Mandelbrot computation, showing how array operations express the math directly and run across the grid on the GPU.
- Heat distribution is modeled with Jacobi iteration using `conv2d`, then accelerated with Successive Over-Relaxation using an omega parameter and red/black checkerboard masks.
- A curve-fitting example uses automatic differentiation to compute gradients for a mean-squared-error loss and run a gradient descent optimization loop without hand-written derivatives.

## Where MLX Swift fits

MLX Swift is for numerical computing code where the primary goal is expressing mathematical operations clearly while still getting performance. Apple's broader numerical ecosystem still matters: Accelerate provides tuned CPU vector primitives, BNNS provides neural-network building blocks, Metal Performance Shaders provides direct GPU kernels, and Swift Numerics provides types such as `Complex` and generic numeric protocols.

MLX Swift's central abstraction is the n-dimensional array, similar to NumPy. The framework is open source under the MIT license, and the same MLX concepts are available across Swift, Python, C++, and C front ends.

- Use MLX Swift when array-oriented mathematical code is more important than low-level control over kernels or memory bookkeeping.
- Most NumPy-style code patterns translate to MLX Swift with minimal conceptual changes.
- Lazy evaluation builds a compute graph and executes when a value is read or `eval` is called.
- The lazy model also supports automatic GPU execution and automatic differentiation.

## Core array model and lazy evaluation

The power-iteration example demonstrates how matrix-vector algorithms map directly into MLX Swift operations. `MLXArray` operations such as transpose, matrix multiplication, vector normalization, and addition look close to the mathematical notation.

Because operations are lazy, loops should call `eval` when needed to keep the graph from growing without bound. Reading a scalar result also forces computation.

- `.T` gives a transpose.
- `matmul` performs matrix-vector or matrix-matrix multiplication.
- `norm` computes the vector norm used for normalization.
- Use MLX's linear algebra package when full eigensystem routines are needed rather than implementing an iterative method manually.

### Power iteration with MLX Swift arrays

Shows array operations, transposition, matrix multiplication, normalization, and explicit evaluation inside an iterative loop.

```swift
import MLX

let n = 100
let steps = 10
let B = MLXRandom.normal([n, n])
var v = MLXRandom.normal([n])

// Symmetric matrix A = Bᵀ + B
let A = B.T + B

// v ← A v / ‖A v‖
for _ in 0 ..< steps {
    let Av = matmul(A, v)
    v = Av / norm(Av)
    eval(v)
}

// λ = vᵀ A v
let lambda = matmul(matmul(v.T, A), v)
print(lambda)
```

## Array computing with Mandelbrot

The Mandelbrot set is a useful example because the same recurrence, `z = z² + c`, is applied independently across a large grid of complex values. A scalar Swift implementation loops over pixels and iterations manually; the MLX Swift version constructs the whole grid and applies the recurrence to every point at once.

The MLX version removes per-pixel bookkeeping and uses the GPU by default. The exact speedup depends on the algorithm and workload, but the session notes that a 10x improvement is possible for this kind of array-parallel computation.

- Plain Swift is expressive but scalar-at-a-time for this implementation.
- MLX Swift expresses the grid as arrays and applies elementwise complex operations.
- The escape count is accumulated with array comparisons rather than nested pixel management.

### Mandelbrot set in MLX Swift

Builds a complex grid and applies the Mandelbrot recurrence across the full array instead of iterating scalar pixels manually.

```swift
import MLX

let x = linspace(-2.0, 0.5, count: w)
let y = linspace(-1.25, 1.25, count: h).reshaped(h, 1)
let c = x + y.asImaginary()

var z = MLXArray.zeros(like: c)
var counts = MLXArray.zeros(c.shape, dtype: .int16)

for _ in 0 ..< maxIterations {
    z = z * z + c
    counts = counts + (abs(z) .< 2)
}
```

## Stencil computation with convolutions

The heat-distribution example models a room as a 2D temperature grid. Jacobi iteration updates each cell by averaging its four neighbors, which is exactly a stencil operation and can be implemented as a convolution.

Boundary conditions are handled with `which`, an elementwise ternary operation. Heat sources and walls keep fixed values; other cells use the next value computed by `conv2d`.

- Represent the four-neighbor averaging stencil as a 3×3 convolution kernel.
- Use `conv2d(temperature, kernel, padding: 1)` to apply the update across the grid.
- Use a mask with `which` to preserve fixed heat sources or walls.

### Jacobi iteration with conv2d

Implements the four-neighbor heat-update rule as a convolution and applies fixed boundary values with an elementwise conditional.

```swift
let kernel = MLXArray(converting: [
    0, 0.25, 0,
    0.25, 0, 0.25,
    0, 0.25, 0,
]).reshaped(1, 3, 3, 1)

var temperature = heatSources

let next = conv2d(temperature, kernel, padding: 1)
temperature = which(heatMask, heatSources, next)
```

## Faster convergence and automatic differentiation

Jacobi iteration is easy to compute but can require about N² iterations for a grid with side length N. Successive Over-Relaxation uses the same convolution kernel, an omega parameter, and in-place-style updates to converge faster; MLX Swift simulates the in-place effect by updating red and black checkerboard cells in alternating phases.

The curve-fitting example demonstrates MLX Swift's `grad` transformation. A quadratic function and mean-squared-error loss are written normally, then `grad(loss)` produces a function that computes the gradient with respect to the parameter vector. The optimization loop updates parameters with gradient descent and calls `eval` each step.

- SOR uses `ω` to overshoot updates and correct over iterations.
- Red/black checkerboard masks let newly updated neighbors participate in the next half-step.
- `grad` derives gradients automatically, avoiding hand-written derivative code.
- MLX also includes optimizers such as SGD, Adam, and RMSprop for more general optimization workflows.

### Successive Over-Relaxation update

Uses omega and checkerboard masks to approximate in-place SOR updates in an array-oriented MLX computation.

```swift
let ω: Float = 2.0 / (1.0 + sin(Float.pi / Float(max(M, N))))
let redMask = checkerboard(rows: M, cols: N, phase: 0)
let blackMask = checkerboard(rows: M, cols: N, phase: 1)

let sorRed = ω * conv2d(temperature, kernel, padding: 1) + (1 - ω) * temperature
temperature = which(redMask, sorRed, temperature)
temperature = which(heatMask, heatSources, temperature)

let sorBlack = ω * conv2d(temperature, kernel, padding: 1) + (1 - ω) * temperature
temperature = which(blackMask, sorBlack, temperature)
temperature = which(heatMask, heatSources, temperature)
```

### Curve fitting with automatic differentiation

Defines a model and loss, transforms the loss into a gradient function, and runs gradient descent over the parameter array.

```swift
func f(_ θ: MLXArray) -> MLXArray {
    θ[0] + θ[1] * x + θ[2] * x ** 2
}

func loss(_ θ: MLXArray) -> MLXArray {
    mean((f(θ) - y) ** 2)
}

var θ = zeros([numParams])
let gradLoss = grad(loss)

for _ in 0 ..< steps {
    let g = gradLoss(θ)
    θ = θ - learningRate * g
    eval(θ)
}
```

## Toolkit and ecosystem

Beyond the examples, MLX includes linear algebra, FFTs, n-dimensional convolutions, reductions, scans, indexing, random number generation, and more. The session also points to packages built on MLX Swift, including the core `mlx-swift` framework, `mlx-swift-lm` for Swift language-model implementations, and `mlx-swift-examples` for example applications.

Example areas mentioned include LLM integration, stable diffusion, model training and fine-tuning, and the session's numerical examples. MLX's shared concepts across Swift, Python, C++, and C make it practical to prototype in one front end and transfer patterns to another.

- Start with the `mlx-swift` repository for framework documentation and tests.
- Use `mlx-swift-examples` for complete example applications.
- Look at `mlx-swift-lm` for Swift language-model implementations.
- Python-side projects such as `mlx-lm` and `mlx-vlm` show additional ecosystem work.

## Resources

- [MLX Swift LM on GitHub](https://github.com/ml-explore/mlx-swift-lm)
- [MLX Swift Examples](https://github.com/ml-explore/mlx-swift-examples)
- [MLX Examples](https://github.com/ml-explore/mlx-examples)
- [MLX Swift](https://github.com/ml-explore/mlx-swift)
- [MLX LM - Python API](https://github.com/ml-explore/mlx-lm)
- [MLX Explore - Python API](https://github.com/ml-explore/mlx)
- [MLX Framework](https://mlx-framework.org/)
- [MLX](https://ml-explore.github.io/mlx/)

## Related Sessions

- [Explore distributed inference and training with MLX](https://wwdc.ai/2026/233)
- [Run local agentic AI on the Mac using MLX](https://wwdc.ai/2026/232)
- [Explore large language models on Apple silicon with MLX](https://developer.apple.com/videos/play/wwdc2025/298)
- [Get started with MLX for Apple silicon](https://developer.apple.com/videos/play/wwdc2025/315)

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