# 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


## Input Parameters

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



## Output Parameter

- ***FB -*** The Smoothed Fischer-Burmeister function vector



## Notes
The Smoothed Fischer-Burmeister function is defined as
```none
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]





## See Also
 `Vec`, `VecFischer()`, `MatDFischer()`, `MatDSFischer()`

## Level
developer

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/tao/util/tao_util.c.html#VecSFischer">src/tao/util/tao_util.c</A>


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[Edit on GitLab](https://gitlab.com/petsc/petsc/-/edit/release/src/tao/util/tao_util.c)


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