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Math."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Considered herein is a modified Newton method for the numerical solution of nonlinear equations where the Jacobian is approximated using a complex-step derivative approximation. We show that this method converges for sufficiently small complex-step values, which need not be infinitesimal. Notably, when the individual derivatives in the Jacobian matrix are approximated using the complex-step method, the convergence is linear and becomes quadratic as the complex-step approaches zero. However, when the Jacobian matrix is approximated by the nonlinear complex-step derivative approximation, the convergence rate remains quadratic for any appropriately small complex-step value, not just in the limit as it approaches zero. This claim is supported by numerical experiments. 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