Add alpha/beta scaling factors for einsum operations

[GiggleLiu]

Description

Julia's OMEinsum supports scaling factors in einsum operations via einsum!:

einsum!(rule, ix, iy, x, y, α, β)  # y = α * einsum(x) + β * y

This allows in-place accumulation with scaling, which is important for:

• BLAS-style operations (GEMM uses α and β)
• Gradient accumulation in training loops
• Memory-efficient iterative algorithms

Current Behavior

Rust omeinsum-rs only supports the basic form:

let result = einsum::<Standard<f64>, _, _>(&[&a, &b], &[&[0, 1], &[1, 2]], &[0, 2]);

Proposed API

// Option 1: Separate function
einsum_scaled::<Standard<f64>, _, _>(
    &[&a, &b], 
    &[&[0, 1], &[1, 2]], 
    &[0, 2],
    alpha,      // scale factor for result
    beta,       // scale factor for existing output
    &mut output // in-place output
);

// Option 2: Builder pattern
ein("ij,jk->ik")
    .alpha(2.0)
    .beta(0.5)
    .execute_into(&[&a, &b], &mut output);

Use Cases

1. Gradient accumulation: grad += learning_rate * new_grad
2. BLAS compatibility: Match cuBLAS/MKL GEMM interface
3. Iterative solvers: x_new = α * A @ x + β * b

Implementation Notes

• cuTENSOR already supports alpha/beta (see src/backend/cuda/cutensor/)
• CPU backend would need to add scaling in contraction loops
• Should work with all algebra types (Standard, Tropical)

Related

• Julia: OMEinsum.jl/src/einsum.jl - unary_einsum!, binary_einsum!
• cuTENSOR: Already has alpha/beta in contract API