The paper under review is focused on the polynomial interpolation problem, i.e. on the computation of a planar interpolation polynomial of \(n\)-th degree passing through a set of \(n+1\) points with no two \(x\)-coordinates equal. A new method -- recursive polynomial interpolation algorithm (RPIA) -- for interpolation polynomial computation is described. RPIA consists in a reformulation of the polynomial interpolation problem and the expression of interpolation polynomials as Schur complements. For the construction of RPIA, the properties of the Schur complements and the Sylvester identity are applied. Lagrange and Newton formulas commonly used to obtain interpolation polynomials are mentioned in the paper, too. The explanation of RPIA construction is completed by several examples where the new algorithm is applied. The suggested method can be applied in various areas of numerical analysis, digital signal processing, computer graphics, etc.