Materialize a bivector exponential

Use bivector_exp when you need coefficients of \(\exp(B)\) for storage, composition, or another Clifford product. The argument is the actual exponent: no minus sign or half-angle is inserted. A compact grade-2 input defaults to all even grades in the result.

import torch
from clifra import make_algebra

algebra = make_algebra(3, dtype=torch.float64)
bivectors = algebra.layout((2,))
even = algebra.layout((0, 2))

exponential = algebra.plan_bivector_exp(input=bivectors, output=even)
theta = torch.tensor(0.4, dtype=torch.float64, requires_grad=True)
B = torch.stack((theta, theta * 0, theta * 0))
value = exponential(B)

expected = torch.stack((theta.cos(), theta.sin(), theta * 0, theta * 0))
derivative, = torch.autograd.grad(value[0], (theta,))

torch.testing.assert_close(value, expected)
torch.testing.assert_close(derivative, -theta.detach().sin())

Here \(e_{12}^2=-1\), giving the circular exponential. A bivector whose square is a positive scalar gives hyperbolic functions; a square-zero bivector gives \(1+B\). General bivectors need not have scalar squares, so applying this two-term formula to their coefficient norm is incorrect.

Use an explicit output layout to select coefficients of the exponential. The output layout selects coefficients from the final result; intermediate powers may use wider layouts. Full-basis input is accepted but projected to grade 2 first; this API is not a general mixed-grade multivector exponential.

Materialization may require larger intermediate coefficient spaces than the requested output. Built-in materialized methods cover dimensions 2 through 12 subject to resource limits. Their numerical routes include finite closures, scaling and squaring, and matrix exponentiation; the Taylor route requires coefficient L1 norm at most 65,536. Large hyperbolic arguments can overflow regardless of the exact algebraic formula.

If only transformed values are needed, use an induced action. Its vector-space representation can avoid materializing the exponentially growing even coefficient space.