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Performance Guide

This guide provides tips and best practices for achieving optimal performance with CliffordAlgebras.jl.

Quick performance checklist

  • Prefer sparse operations; avoid unnecessary extend() in hot paths.
  • Keep operands in the same algebra type; avoid converting between algebras.
  • Reuse compiled structures: precompute rotors and reuse them.
  • Aim for type stability; use @inferred in performance-critical code.
  • Avoid excessive prune() calls; use it judiciously after larger expressions.
  • Choose the smallest algebra sufficient for the task (e.g., Cl(2) for 2D rotations).
  • Profile before optimizing; measure allocations and timing.

Understanding the Design

CliffordAlgebras.jl is designed for high performance through several key features:

  1. Sparse Representation: Only non-zero coefficients are stored
  2. Generated Functions: Compile-time specialization for each algebra
  3. Type Stability: Most operations have inferrable return types
  4. Zero-Cost Abstractions: Overhead is eliminated at compile time

Type Stability

Type stability is crucial for Julia performance. Most CliffordAlgebras.jl operations are type-stable:

using CliffordAlgebras

cl3 = CliffordAlgebra(3)
mv1 = cl3.e1 + cl3.e2
mv2 = cl3.e2 + cl3.e3

# These operations are type-stable
@inferred mv1 * mv2          # Geometric product
@inferred mv1 ∧ mv2          # Exterior product  
@inferred ~mv1               # Reverse
@inferred exp(cl3.e1e2)      # Exponential

Non-Type-Stable Operations

Some operations are intentionally not type-stable for flexibility:

# prune() changes the sparse structure
pruned = prune(mv1)  # Return type depends on which coefficients survive

# Use extend() for type stability if needed
extended = extend(mv1)  # Always returns full representation

Memory Efficiency

Sparse vs Dense Representation

CliffordAlgebras.jl automatically uses sparse representation:

cl8 = CliffordAlgebra(8)  # 2^8 = 256 basis elements

# This only stores 2 coefficients, not 256
sparse_mv = cl8.e1 + cl8.e8

# Convert to dense if needed (usually not recommended)
dense_mv = extend(sparse_mv)

Memory Usage Guidelines

  1. Prefer sparse operations: Most functions preserve sparsity
  2. Avoid unnecessary extend(): Only use when type stability is critical
  3. Reuse multivectors: Modify in-place when possible

Compile-Time Optimization

Generated Functions

The package uses @generated functions for optimal performance:

# This generates specialized code for each algebra and multivector type
function (*)(a::MultiVector{CA}, b::MultiVector{CA}) where CA
    # Specialized implementation generated at compile time
end

Specialization Tips

  1. Use concrete types: Avoid MultiVector without type parameters
  2. Consistent algebra types: Don't mix algebras in hot loops
  3. Stable multivector structures: Reuse similar sparse patterns

Benchmarking Examples

Basic Operations

using BenchmarkTools, CliffordAlgebras

cl3 = CliffordAlgebra(3)
mv1 = 1.0 + cl3.e1 + cl3.e2 + cl3.e1e2
mv2 = 2.0 + cl3.e2 + cl3.e3 + cl3.e2e3

# Benchmark geometric product
@benchmark $mv1 * $mv2

# Benchmark exterior product  
@benchmark $mv1 ∧ $mv2

# Benchmark exponential
B = π/4 * cl3.e1e2
@benchmark exp($B)

Large Algebras

# For larger algebras, sparsity becomes crucial
cl6 = CliffordAlgebra(6)  # 64 basis elements

# Sparse multivector (only 3 coefficients)
sparse_mv = cl6.e1 + cl6.e3 + cl6.e6

# Still efficient due to sparsity
@benchmark $sparse_mv * $sparse_mv

Optimization Strategies

1. Precompute Common Operations

# Instead of recomputing rotors
angle = π/6
B = angle * cl3.e1e2
rotor = exp(B)  # Precompute this

# Use the precomputed rotor many times
for vector in vectors
    rotated = rotor ≀ vector
end

2. Use Appropriate Signatures

Choose the minimal signature for your problem:

# For 2D rotations, Cl(2,0,0) is more efficient than Cl(3,0,0)
cl2 = CliffordAlgebra(2)  # 4 basis elements vs 8

# For spacetime, use the exact signature
sta = CliffordAlgebra(1, 3, 0)  # Not Cl(4,0,0)

3. Minimize Allocations

# Good: Reuse multivectors when possible
function rotate_many_vectors!(results, rotor, vectors)
    for (i, v) in enumerate(vectors)
        results[i] = rotor ≀ v
    end
end

# Avoid: Creating new algebras in hot loops
function bad_example()
    for i in 1:1000
        cl = CliffordAlgebra(3)  # Don't do this!
        # ... operations
    end
end

4. Leverage Type Annotations

# Help the compiler with type annotations
function efficient_computation(mv::MultiVector{CA,Float64}) where CA
    result = mv * mv
    return scalar(result)
end

Performance Pitfalls

1. Type Instability

# Bad: Type-unstable function
function unstable_norm(mv)
    if some_condition
        return norm(mv)  # Returns Float64
    else
        return mv        # Returns MultiVector
    end
end

# Good: Type-stable alternatives
function stable_norm(mv)
    return norm(mv)  # Always returns Float64
end

2. Unnecessary Conversions

# Bad: Converting between algebras
cl2_mv = cl2.e1
cl3_mv = MultiVector(cl3, (1.0, 0.0, 0.0))  # Expensive conversion

# Good: Work within one algebra
mv1 = cl3.e1
mv2 = cl3.e2
result = mv1 * mv2

3. Overuse of prune()

# Bad: Excessive pruning
result = mv1 * mv2
result = prune(result)  # Type-unstable and often unnecessary
result = result + mv3
result = prune(result)  # Again!

# Good: Prune only when needed
result = mv1 * mv2 + mv3
# Only prune if you know there are many small coefficients
if need_cleanup
    result = prune(result)
end

Profiling Tools

Memory Allocation

using Profile

function profile_example()
    cl4 = CliffordAlgebra(4)
    mv = cl4.e1 + cl4.e2 + cl4.e3 + cl4.e4
    
    for i in 1:1000
        result = mv * mv
    end
end

# Profile memory allocations
@profile profile_example()
Profile.print()

Type Inference

using Cthulhu

# Inspect generated code
cl3 = CliffordAlgebra(3)
mv1 = cl3.e1
mv2 = cl3.e2

# Descend into the multiplication
@descend mv1 * mv2

Performance Summary

Fast Operations:

  • Geometric product between similar multivectors
  • Exponential of bivectors
  • Grade extraction
  • Reverse and other involutions

Moderate Operations:

  • Operations between very different sparse structures
  • Converting between representations
  • Complex expressions with many terms

Slow Operations:

  • Excessive use of prune()
  • Mixing different algebras
  • Type-unstable code patterns

Best Practices:

  1. Stick to one algebra type per computation
  2. Leverage sparsity naturally
  3. Precompute rotors and other expensive operations
  4. Use @inferred to check type stability
  5. Profile before optimizing
  6. Choose minimal sufficient algebra signatures