We construct a new numerical method for computing {\it reference} numerical solutions to the self--similar solution to the problem of incompressible laminar flow past a thin flat plate with suction--blowing. The method generates global numerical approximations to the velocity components and their scaled derivatives for arbitrary values of the Reynolds number in the range $[1, \infty)$ on a domain including the boundary layer but excluding a neighbourhood of the leading edge. The method is based on Blasius' approach. Using an experimental error estimate technique it is shown that these numerical approximations are pointwise accurate and that they satisfy pointwise error estimates which are independent of the Reynolds number for the flow. The Reynolds--uniform orders of convergence of the reference numerical solutions, with respect to the number of mesh subintervals used in the solution of Blasius'problem, is at least 0.86 and the error constant is not more than 80. The number of iterations required to solve the nonlinear Blasius problem is independent of the Reynolds number. Therefore the method generates reference numerical solutions with $\varepsilon$--uniform errors of any prescribed accuracy.