where α = A, B, C . One possible reversible measure-preserving integration algorithm for the BKNH thermostat can then be derived from the following Trotter factorization: The direct translation technique gives the following pseudo-code: 16 U ( τ ) BKNH = U BKNH B ( τ 4 ) U BKNH C ( τ 2 ) U BKNH B ( τ 4 ) × U BKNH A ( τ ) × U BKNH B ( τ 4 ) U BKNH C ( τ 2 ) U BKNH B ( τ 4 ) . (43) • p → p exp [ − τ 2 p$_{ζ}$ m$_{ζ}$ ] q → q exp − τ 2 p$_{ξ}$ m$_{ξ}$ ζ → ζ + τ 2 p$_{ζ}$ m$_{ζ}$ ξ → ξ + τ 2 p$_{ξ}$ m$_{ξ}$ p$_{η}$ → p$_{η}$ + τ $_{2}$F$_{p}$$_{η}$       : U BKNH C ( τ 2 ) • q → q + τ 4 p m p$_{ζ}$ → p$_{ζ}$ + τ $_{4}$F$_{p}$$_{ζ}$    : U BKNH B ( τ 4 ) • p → p + τF ( q ) p$_{ξ}$ → p$_{ξ}$ + τF$_{p}$$_{ξ}$ η → η + τ p$_{η}$ m$_{η}$ p$_{ζ}$ → p$_{ζ}$ exp [ − τ p$_{η}$ m$_{η}$ ]      : U BKNH A ( τ ) • q → q + τ 4 p m p$_{ζ}$ → p$_{ζ}$ + τ $_{4}$F$_{p}$$_{ζ}$    : U BKNH B ( τ 4 ) • p → p exp [ − τ 2 p$_{ζ}$ m$_{ζ}$ ] q → q exp − τ 2 p$_{ξ}$ m$_{ξ}$ ζ → ζ + τ 2 p$_{ζ}$ m$_{ζ}$ ξ → ξ + τ 2 p$_{ξ}$ m$_{ξ}$ p$_{η}$ → p$_{η}$ + τ $_{2}$F$_{p}$$_{ζ}$      : U BKNH C ( τ 2 ) • q → q + τ 4 p m p$_{ζ}$ → p$_{ζ}$ + τ $_{4}$F$_{p}$$_{ζ}$    : U BKNH B ( τ 4 )