cnanlagau

valsi

cnanlagau

type

fu'ivla

creator

krtisfranks

time entered

Sun Apr 1 01:47:34 2018  

English

Definition #70581 - Preferred

 

definition

x₁ is the generalized arithmetic-geometric mean of the elements of the 2-element set x₂ (set; cardinality must be 2) of order x₃ (either single extended-real number xor an unordered pair/2-element set of extended-real numbers).

notes

Elements of x₂ must have the same units/dimensionality; the result has the same units/dimensionality as them. If x₃ = p for a single extended-real number p, then MATH

$x_3=(p,p-1)$

x₃ = (p, p - 1) also. If MATH

$x_3 = (p,q)$

x₃ = (p, q) then the algorithm uses the pth-power mean (cnanfadi) and the qth-power mean; thus MATH

$x_3 = (1,0)$

x₃ = (1, 0) corresponds to the standard arithmetic-geometric mean, MATH

$x_3 = (0, -1)$

x₃ = (0, - 1) corresponds to the geometric-harmonic mean. A poor choice of x₃ will lead to non-convergence of the sequences produced by the algorithm and, thus, leave x₁ undefined (NAN error). x₁ is the value to which each of the sequences produced by the algorithm converge (iff these values are mutually equal). Let M_i denote the unweighted ith-power mean for all i and let MATH

$x_2 = (a_0,g_0),$

x₂ = (a₀, g₀), MATH

$x_3 = (p,q)$

x₃ = (p, q); then the algorithm produces potentially infinite sequences MATH

$a = (a_0, a_1, ...), g = (g_0, g_1, ...)$

a = (a₀, a₁,...), g = (g₀, g₁,...), where MATH

$a_n = M_p(a_<a class="undefined" href="../dict/(n-1)?bg=1;langidarg=2">(n-1)</a>, g_<a class="undefined" href="../dict/(n-1)?bg=1;langidarg=2">(n-1)</a>), g_n = M_q(a_<a class="undefined" href="../dict/(n-1)?bg=1;langidarg=2">(n-1)</a>, g_<a class="undefined" href="../dict/(n-1)?bg=1;langidarg=2">(n-1)</a>)$

a_n = M_p(a_(n-1), g_(n-1)), g_n = M_q(a_(n-1), g_(n-1)) for all positive integers n. Notice that this word is symmetric (x₁ remains constant and none of the sumti change in meaning) under an internal permutation of the elements/entries of x₂ and/or of x₃ (when taken as a pair or set), separately. See also: gau'i'o.

gloss words

• AGM ; arithmetic-geometric mean
• arithmetic-geometric mean

created by

krtisfranks

vote information

1

time

Sun Apr 1 02:55:45 2018

Examples

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