petsc-master 2020-01-26
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# VecSFischer

Evaluates the Smoothed Fischer-Burmeister function for complementarity problems.

### Synopsis

```#include "petsctao.h"
PetscErrorCode VecSFischer(Vec X, Vec F, Vec L, Vec U, PetscReal mu, Vec FB)
```
Logically Collective on vectors

### Input Parameters

 X - current point F - function evaluated at x L - lower bounds U - upper bounds mu - smoothing parameter

### Output Parameters

FB -The Smoothed Fischer-Burmeister function vector

### Notes

The Smoothed Fischer-Burmeister function is defined as
```       phi(a,b) := sqrt(a*a + b*b + 2*mu*mu) - a - b
```
and is used reformulate a complementarity problem as a semismooth system of equations.

### The result of this function is done by cases

 l[i] == - infinity, u[i] == infinity -- fb[i] = -f[i] - 2*mu*x[i] l[i] == - infinity, u[i] finite -- fb[i] = phi(u[i]-x[i], -f[i], mu) l[i] finite, u[i] == infinity - - fb[i] = phi(x[i]-l[i], f[i], mu) l[i] finite < u[i] finite - - fb[i] = phi(x[i]-l[i], phi(u[i]-x[i], -f[u], mu), mu) otherwise l[i] == u[i] - - fb[i] = l[i] - x[i]