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We present a new algorithm that can output the rank-decomposition of width at most $k$ of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most $k$ if such exists. This algorithm works also for partitioned matroids. Both of these algorithms are fixed-parameter tractable, that is, they run in time $O(n^3)$ where $n$ is the number of vertices / elements of the input, for each constant value of $k$ and any fixed finite field. The previous best algorithm for construction of a branch-decomposition or a rank-decomposition of optimal width due to Oum and Seymour [J. Combin. Theory Ser. B, 97 (2007), pp. 385-393] is not fixed-parameter tractable.