Solve a system of two equations in which the existence of l n ( α α + β ) function makes some limitations in iterations of the Newton-Raphson method. { l n ( α α + β ) − c s 1 + ∑ i ∈ A c i ( α − i − ( α + β ) − i ) l n ( β α + β ) − c s 1 + ∑ i ∈ A c i ( β − i − ( α + β ) − i )

Question

Solve a system of two equations in which the existence of   l  n      (          α              α        +        β              )   function makes some limitations in iterations of the Newton-Raphson method.                              {                                                    l                n                                  (                                      α                                          α                      +                      β                                                        )                                −                                  c                                      s                    1                                                  +                                  ∑                                      i                    ∈                    A                                                                                        c                    i                                                        (                                          α                                              −                        i                                                              −                    (                    α                    +                    β                                          )                                              −                        i                                                              )                                                                                                      l                n                                  (                                      β                                          α                      +                      β                                                        )                                −                                  c                                      s                    1                                                  +                                  ∑                                      i                    ∈                    A                                                                                        c                    i                                                        (                                          β                                              −                        i                                                              −                    (                    α                    +                    β                                          )                                              −                        i                                                              )

hitturn35 · Accepted Answer

Put:   X  =  log  ⁡      (          α              α        +        β              )    =  log  ⁡      (          1              1        +                  β          α                      )  Then working in terms of X will ensure that the logarithm is always well defined. You then need to define another independent variable Y such that a linarization of Y in terms of small changes in   α and   β does not become almost linearly dependent on the way the change in X depends on small changes in   α and   β. Since X depends on the ratio of   α and   β, you can choose Y to be a function of the product of   α and   β, so:  Y  =  α  βmight work well, at least you&#039;ll have elminated two potential problems with Newton-Raphson.

Aron Heath · Accepted Answer

We can simply substitute   α by       a    2   and   β by       b    2  .

Solve a system of two equations in which the existence of ln(α/(α+β)) function makes some limitations in iterations of the Newton-Raphson method.

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