Given any fixed constant $0<\varepsilon<1$ and a matrix space
$\mathcal{B}=\langle B_1,\ldots,B_m\rangle\le\mathbb{Q}^{n\times n}$,
we give a deterministic $NC^3$ algorithm that outputs a matrix \(A\in\mathcal{B}\) such that $rank(A)\geq (1-\varepsilon) crk(\mathcal{B})$,
where $crk(\mathcal{B})$ denotes the maximum rank of a matrix in $\mathcal{B}$. This complements the recent breakthrough of Chatterjee, Ghosh, Gurjar, Raj, and Thierauf (ECCC, TR26-100), who gave an $NC$ algorithm for computing the noncommutative rank of symbolic matrices. For commutative rank, Bl\"{a}ser, Jindal, and Pandey previously gave a deterministic polynomial-time approximation scheme (ToC, 2018).

Our algorithm follows a different route from the subspace-design approach of Chatterjee, Ghosh, Gurjar, Raj, and Thierauf. It has two main ingredients. First, using a polynomial-size $4$-wise independent family together with operator scaling
(Gurvits'04, Garg-Gurvits-Oliveira-Wigderson'20), we obtain a scalar matrix whose rank is an absolute constant fraction of $crk(\mathcal{B})$. Second, we boost this constant-factor approximation to a $(1-\varepsilon)$-approximation by analyzing the associated Schur complements through Smith normal form over a discrete valuation ring. This mainly helps in iteratively reducing the rank deficit.