where (in the so-called single-step or Seidel method) the assignments are performed sequentially. See V. N. Faddeeva, Computational Methods of Linear Algebra, p.117, Dover Publ., 1959. (Note that many textbooks of linear algebra present however a different, in fact less general, formulation.)
Using this as a hint, we propose to do for block encryption processing of n blocks, x1, x2, ... xn, the follwoing, where the f's are invertible non-linear functions, the r's are pseudo-random numbers and the assignments are performed sequentially (the f's and the r's are (secret) key-dependent and different for different rounds, if more than one rounds are used, computation is mod 2**m for block size of m bits):
Note that we have left out the multiplication with a's, which is deemed a justifiable simplicity since the f's are non-linear and further the r's are pseudo-random. Note also that the effect of block-chaining in the use of the common block ciphers is intrinsically present in our scheme. A viable variant of the scheme is to employ ^r instead of +r.