docs: add a README for SciMLJacobianOperators

lib/SciMLJacobianOperators/README.md

@@ -0,0 +1,59 @@
+# SciMLJacobianOperators.jl
+
+SciMLJacobianOperators provides a convenient way to compute Jacobian-Vector Product (JVP)
+and Vector-Jacobian Product (VJP) using
+[SciMLOperators.jl](https://github.com/SciML/SciMLOperators.jl) and
+[DifferentiationInterface.jl](https://github.com/gdalle/DifferentiationInterface.jl).
+
+Currently we have interfaces for:
+
+  - `NonlinearProblem`
+  - `NonlinearLeastSquaresProblem`
+
+and all autodiff backends supported by DifferentiationInterface.jl are supported.
+
+## Example
+
+```julia
+using SciMLJacobianOperators, NonlinearSolve, Enzyme, ForwardDiff
+
+# Define the problem
+f(u, p) = u .* u .- p
+u0 = ones(4)
+p = 2.0
+prob = NonlinearProblem(f, u0, p)
+fu0 = f(u0, p)
+v = ones(4) .* 2
+
+# Construct the operator
+jac_op = JacobianOperator(
+    prob, fu0, u0;
+    jvp_autodiff = AutoForwardDiff(),
+    vjp_autodiff = AutoEnzyme(; mode = Enzyme.Reverse)
+)
+sjac_op = StatefulJacobianOperator(jac_op, u0, p)
+
+sjac_op * v  # Computes the JVP
+# 4-element Vector{Float64}:
+#  4.0
+#  4.0
+#  4.0
+#  4.0
+
+sjac_op' * v  # Computes the VJP
+# 4-element Vector{Float64}:
+#  4.0
+#  4.0
+#  4.0
+#  4.0
+
+# What if we multiply the VJP and JVP?
+snormal_form = sjac_op' * sjac_op
+
+snormal_form * v  # Computes JᵀJ * v
+# 4-element Vector{Float64}:
+#  8.0
+#  8.0
+#  8.0
+#  8.0
+```