1. An apparatus for outputting a sequence of vectors comprising: (a) a first storage which stores vector x of dimension at least 1; (b) a first calculator which calculates vector x′&equals;f(x) which utilizes a first rational vector map f to which the vector x stored in said first storage is input; (c) a second storage which stores vector y of dimension at least 1; (d) a second calculator which calculates vector y′&equals;g(x, y) which utilizes a second rational vector map g to which the vector x stored in said first storage and the vector y of dimension at least 1 stored in said second storage are input; (e) an output which outputs vector z′ which is association of the vector x′ resulting from said first calculator and the vector y′ resulting from said second calculator; (f) a first update which updates said first storage by storing the vector x′ resulting from said first calculator; and (g) a second update which updates said second storage by storing the vector y′ resulting from said second calculator, where the rational vector map is a map which converts vector of dimension at least 1 having rational number components into another vector of dimension at least 1 having rational number components.