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Federico Poloni
Federico Poloni
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There are many solutions --- up to $2^N$ in general. Do you need a specific one? Otherwise, some of the special cases you suggest are simple: for instance, if $C_i=0$ for all $i$ then all zeros is a trivial solution. Or, if all entries are independent from $i$, then you get $N$ copies of the same equation.

In general, as far as I know, a closed-form solution does not exist. Note that by introducing a few auxiliary variables you can turn any polynomial equation in this form, so this form is quite general.

My paper cited in the other answer studies numerical methods for this equation under some particular sign conditions ($A_{ijk} \geq 0$, $C_i \geq 0$, and $B$$-B$ is aan $-M$$M$-matrix.) In most cases the go-to algorithm is Newton's method, so if you were planning to use this to speed up another Newton iteration then probably you are out of luck. :)

There are many solutions --- up to $2^N$ in general. Do you need a specific one? Otherwise, some of the special cases you suggest are simple: for instance, if $C_i=0$ for all $i$ then all zeros is a trivial solution. Or, if all entries are independent from $i$, then you get $N$ copies of the same equation.

In general, as far as I know, a closed-form solution does not exist. Note that by introducing a few auxiliary variables you can turn any polynomial equation in this form, so this form is quite general.

My paper cited in the other answer studies numerical methods for this equation under some particular sign conditions ($A_{ijk} \geq 0$, $C_i \geq 0$, and $B$ is a $-M$-matrix.) In most cases the go-to algorithm is Newton's method, so if you were planning to use this to speed up another Newton iteration then probably you are out of luck. :)

There are many solutions --- up to $2^N$ in general. Do you need a specific one? Otherwise, some of the special cases you suggest are simple: for instance, if $C_i=0$ for all $i$ then all zeros is a trivial solution. Or, if all entries are independent from $i$, then you get $N$ copies of the same equation.

In general, as far as I know, a closed-form solution does not exist. Note that by introducing a few auxiliary variables you can turn any polynomial equation in this form, so this form is quite general.

My paper cited in the other answer studies numerical methods for this equation under some particular sign conditions ($A_{ijk} \geq 0$, $C_i \geq 0$, and $-B$ is an $M$-matrix.) In most cases the go-to algorithm is Newton's method, so if you were planning to use this to speed up another Newton iteration then probably you are out of luck. :)

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Federico Poloni
Federico Poloni
  • 20.2k
  • 2
  • 82
  • 120

There are many solutions --- up to $2^N$ in general. Do you need a specific one? Otherwise, some of the special cases you suggest are simple: for instance, if $C_i=0$ for all $i$ then all zeros is a trivial solution. Or, if all entries are independent from $i$, then you get $N$ copies of the same equation.

In general, as far as I know, a closed-form solution does not exist. Note that by introducing a few auxiliary variables you can turn any polynomial equation in this form, so this form is quite general.

My paper cited in the other answer studies numerical methods for this equation under some particular sign conditions ($A_{ijk} \geq 0$, $C_i \geq 0$, and $B$ is a $-M$-matrix.) In most cases the go-to algorithm is Newton's method, so if you were planning to use this to speed up another Newton iteration then probably you are out of luck. :)