This paper presents a stochastic approximation (SA) algorithm for finding roots of multivariate gradient equations, which is based on a "simultaneous perturbation" gradient approximation. The algorithm is designed to be more efficient than standard finite difference-based algorithms, particularly in large-dimensional problems. The paper includes theoretical justification for the algorithm, including strong convergence and asymptotic normality results. Numerical experiments demonstrate that the simultaneous perturbation SA (SPSA) algorithm can achieve significantly lower root mean square errors compared to the multivariate Kiefer-Wolfowitz finite difference SA (FDSA) algorithm, especially for large-dimensional problems. The SPSA algorithm requires fewer measurements per iteration and fewer total measurements to achieve the same level of accuracy as the FDSA algorithm. The paper also discusses the relative efficiency of SPSA and FDSA in terms of asymptotic mean square error and the number of measurements required.This paper presents a stochastic approximation (SA) algorithm for finding roots of multivariate gradient equations, which is based on a "simultaneous perturbation" gradient approximation. The algorithm is designed to be more efficient than standard finite difference-based algorithms, particularly in large-dimensional problems. The paper includes theoretical justification for the algorithm, including strong convergence and asymptotic normality results. Numerical experiments demonstrate that the simultaneous perturbation SA (SPSA) algorithm can achieve significantly lower root mean square errors compared to the multivariate Kiefer-Wolfowitz finite difference SA (FDSA) algorithm, especially for large-dimensional problems. The SPSA algorithm requires fewer measurements per iteration and fewer total measurements to achieve the same level of accuracy as the FDSA algorithm. The paper also discusses the relative efficiency of SPSA and FDSA in terms of asymptotic mean square error and the number of measurements required.