where (in the so-called single-step 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 then 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.