Derivatives in NLSQ

Copyright (C) 2026 Andreas Kloeckner

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import sympy as sp

a1f = sp.Function("a_1")
a2f = sp.Function("a_2")
x, y = sp.symbols("x, y")
g1, g2 = sp.symbols("g_1, g_2")
r1, r2 = sp.symbols("r_1, r_2")

Make a symbolic quantity for r (residual, $\boldsymbol g- \boldsymbol a(\boldsymbol x)$) and phi.

Make sure to use sp.factor on phi:

a1 = a1f(x, y)
a2 = a2f(x, y)
a = sp.Matrix([a1, a2])
xvec = sp.Matrix([x, y])
r = a - sp.Matrix([g1, g2])
phi = sp.Rational(1, 2) * sp.factor(r.T@r)
phi

$\displaystyle \left[\begin{matrix}\frac{\left(- g_{1} + a_{1}{\left(x,y \right)}\right)^{2}}{2} + \frac{\left(- g_{2} + a_{2}{\left(x,y \right)}\right)^{2}}{2}\end{matrix}\right]$

def subst_r(x):
    return x.subs(r[0], r1).subs(r[1], r2)

Find the gradient of $\phi$, substitute in r:

subst_r(sp.Matrix([phi.diff(x), phi.diff(y)]))

$\displaystyle \left[\begin{matrix}r_{1} \frac{\partial}{\partial x} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial}{\partial x} a_{2}{\left(x,y \right)}\\r_{1} \frac{\partial}{\partial y} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial}{\partial y} a_{2}{\left(x,y \right)}\end{matrix}\right]$

Now find the Hessian:

H = subst_r(sp.Matrix([[phi.diff(i).diff(j) for j in [x, y]] for i in [x,y]]))
H

$\displaystyle \left[\begin{matrix}r_{1} \frac{\partial^{2}}{\partial x^{2}} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial x^{2}} a_{2}{\left(x,y \right)} + \left(\frac{\partial}{\partial x} a_{1}{\left(x,y \right)}\right)^{2} + \left(\frac{\partial}{\partial x} a_{2}{\left(x,y \right)}\right)^{2} & r_{1} \frac{\partial^{2}}{\partial y\partial x} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y\partial x} a_{2}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{1}{\left(x,y \right)} \frac{\partial}{\partial y} a_{1}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{2}{\left(x,y \right)} \frac{\partial}{\partial y} a_{2}{\left(x,y \right)}\\r_{1} \frac{\partial^{2}}{\partial y\partial x} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y\partial x} a_{2}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{1}{\left(x,y \right)} \frac{\partial}{\partial y} a_{1}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{2}{\left(x,y \right)} \frac{\partial}{\partial y} a_{2}{\left(x,y \right)} & r_{1} \frac{\partial^{2}}{\partial y^{2}} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y^{2}} a_{2}{\left(x,y \right)} + \left(\frac{\partial}{\partial y} a_{1}{\left(x,y \right)}\right)^{2} + \left(\frac{\partial}{\partial y} a_{2}{\left(x,y \right)}\right)^{2}\end{matrix}\right]$

For comparison, here is $J_{\boldsymbol r}^TJ_{\boldsymbol r}$:

J = subst_r(sp.Matrix([list(r.diff(i)) for i in [x, y]]).T)
J.T @ J

$\displaystyle \left[\begin{matrix}\left(\frac{\partial}{\partial x} a_{1}{\left(x,y \right)}\right)^{2} + \left(\frac{\partial}{\partial x} a_{2}{\left(x,y \right)}\right)^{2} & \frac{\partial}{\partial x} a_{1}{\left(x,y \right)} \frac{\partial}{\partial y} a_{1}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{2}{\left(x,y \right)} \frac{\partial}{\partial y} a_{2}{\left(x,y \right)}\\\frac{\partial}{\partial x} a_{1}{\left(x,y \right)} \frac{\partial}{\partial y} a_{1}{\left(x,y \right)} + \frac{\partial}{\partial x} a_{2}{\left(x,y \right)} \frac{\partial}{\partial y} a_{2}{\left(x,y \right)} & \left(\frac{\partial}{\partial y} a_{1}{\left(x,y \right)}\right)^{2} + \left(\frac{\partial}{\partial y} a_{2}{\left(x,y \right)}\right)^{2}\end{matrix}\right]$

H - J.T @ J

$\displaystyle \left[\begin{matrix}r_{1} \frac{\partial^{2}}{\partial x^{2}} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial x^{2}} a_{2}{\left(x,y \right)} & r_{1} \frac{\partial^{2}}{\partial y\partial x} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y\partial x} a_{2}{\left(x,y \right)}\\r_{1} \frac{\partial^{2}}{\partial y\partial x} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y\partial x} a_{2}{\left(x,y \right)} & r_{1} \frac{\partial^{2}}{\partial y^{2}} a_{1}{\left(x,y \right)} + r_{2} \frac{\partial^{2}}{\partial y^{2}} a_{2}{\left(x,y \right)}\end{matrix}\right]$