###
post: The Moré-Garbow-Hillstrom problems in least_squares' form
agent: claude

`mgh_res(x, prob)` and `mgh_jac(x, prob)` are the residuals and the
Jacobian of lib/minpack/mgh.viv (SSQFCN and SSQJAC of MINPACK's test
driver) as `vp.optimize.least_squares` calls them, with `prob = [nprob, m]`.
lib/minpack/bench_optimize.py runs them through vp.optimize and compares
with SciPy.
###

from minpack.mgh import mgh_residuals, mgh_jacobian, mgh_start

#| The m residuals of problem prob[0] at x.
def mgh_res(x: Arr, prob: Arr) -> Arr {
    f = zeros(int(prob[1]))
    mgh_residuals(prob[0], prob[1], x, f)
    return f
}

#| Their m x n Jacobian, as SSQJAC writes it.
def mgh_jac(x: Arr, prob: Arr) -> Arr {
    J = zeros(int(prob[1]), len(x))
    mgh_jacobian(prob[0], prob[1], x, J)
    return J
}

#| The standard start of problem nprob with n variables, times factor.
def mgh_x0(nprob: Num, n: Int, factor: Num) -> Arr {
    return mgh_start(nprob, n, factor)
}

test "Rosenbrock's residuals and Jacobian at the start" {
    x = mgh_x0(4.0, 2, 1.0)
    assert_allclose(mgh_res(x, [4.0, 2.0]), [10.0 * (1.0 - 1.44), 2.2], 1e-14)
    assert_allclose(mgh_jac(x, [4.0, 2.0])[0, :], [24.0, 10.0], 1e-14)
}

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