Given a list of d (d>1) real numbers {α_1, α_2, ..., α_d}, the Jacobi-Perron algorithm calculates a multidimensional continued fraction that simultaneously approximates the given real numbers. Start setting: α_i^(0) = α_i for 1<=i<=d. Define a_i^(n) = floor(α_i^(n)) for 1<=i<=d and n>=0. Recursively define α_d^(n) = 1/(α_1^(n - 1) - a_1^(n - 1)) α_i^(n) = α_d^(n) (α_(i + 1)^(n - 1) - a_(i + 1)^(n - 1)) for 1<=i<=d - 1. Then the simultaneous approximations α_i≈p_i^(n)/q^(n) for 1<=i<=d can be obtained from A_n = I_(d + 1)·B_1·B_2·...·B_(n - 1), where B_n = (0 | 0 | ... | 0 | 1 1 | 0 | ... | 0 | a_1^(n) 0 | 1 | ... | 0 | a_2^(n) ⋮ | ⋮ | ⋱ | ⋮ | ⋮ 0 | 0 | ... | 1 | a_d^(n)) and A_n = (q^(n - d) | q^(n - d + 1) | ... | q^(n - 1) | q^(n) p_1^(n - d) | p_1^(n - d + 1) | ... | p_1^(n - 1) | p_1^(n) p_2^(n - d) | p_2^(n - d + 1) | ... | p_2^(n - 1) | p_2^(n) ⋮ | ⋮ | ⋱ | ⋮ | ⋮ p_d^(n - d) | p_d^(n - d + 1) | ... | p_d^(n - 1) | p_d^(n)). If α_i^(n) element Z^+ for some n and i the algorithm is interrupted and continued with the remaining α_i^(n).
continued fraction
ContinuedFraction | FromContinuedFraction | ContinuedFractionK | Convergents
Fritz Schweiger. Multidimensional Continued Fractions. 2000.