# SNESConvergedReason
reason a `SNES` method was determined to have converged or diverged 
## Synopsis
```
typedef enum {                       /* converged */
  SNES_CONVERGED_FNORM_ABS      = 2, /* ||F|| < atol */
  SNES_CONVERGED_FNORM_RELATIVE = 3, /* ||F|| < rtol*||F_initial|| */
  SNES_CONVERGED_SNORM_RELATIVE = 4, /* Newton computed step size small; || delta x || < stol || x ||*/
  SNES_CONVERGED_ITS            = 5, /* maximum iterations reached */
  SNES_BREAKOUT_INNER_ITER      = 6, /* Flag to break out of inner loop after checking custom convergence. */
                                     /* it is used in multi-phase flow when state changes */
  /* diverged */
  SNES_DIVERGED_FUNCTION_DOMAIN      = -1, /* the new x location passed the function is not in the domain of F */
  SNES_DIVERGED_FUNCTION_COUNT       = -2,
  SNES_DIVERGED_LINEAR_SOLVE         = -3, /* the linear solve failed */
  SNES_DIVERGED_FNORM_NAN            = -4,
  SNES_DIVERGED_MAX_IT               = -5,
  SNES_DIVERGED_LINE_SEARCH          = -6,  /* the line search failed */
  SNES_DIVERGED_INNER                = -7,  /* inner solve failed */
  SNES_DIVERGED_LOCAL_MIN            = -8,  /* || J^T b || is small, implies converged to local minimum of F() */
  SNES_DIVERGED_DTOL                 = -9,  /* || F || > divtol*||F_initial|| */
  SNES_DIVERGED_JACOBIAN_DOMAIN      = -10, /* Jacobian calculation does not make sense */
  SNES_DIVERGED_TR_DELTA             = -11,
  SNES_CONVERGED_TR_DELTA_DEPRECATED = -11,

  SNES_CONVERGED_ITERATING = 0
} SNESConvergedReason;
```


The two most common reasons for divergence are
```none
1) an incorrectly coded or computed Jacobian or
```
```none
2) failure or lack of convergence in the linear system (in this case we recommend
```
```none
testing with -pc_type lu to eliminate the linear solver as the cause of the problem).
```


## Diverged Reasons

- ***`SNES_DIVERGED_LOCAL_MIN` -*** this can only occur when using the line-search variant of `SNES`.
The line search wants to minimize Q(alpha) = 1/2 || F(x + alpha s) ||^2_2  this occurs
at Q'(alpha) = s^T F'(x+alpha s)^T F(x+alpha s) = 0. If s is the Newton direction - F'(x)^(-1)F(x) then
you get Q'(alpha) = -F(x)^T F'(x)^(-1)^T F'(x+alpha s)F(x+alpha s); when alpha = 0
Q'(0) = - ||F(x)||^2_2 which is always NEGATIVE if F'(x) is invertible. This means the Newton
direction is a descent direction and the line search should succeed if alpha is small enough.


If F'(x) is NOT invertible AND F'(x)^T F(x) = 0 then Q'(0) = 0 and the Newton direction
is NOT a descent direction so the line search will fail. All one can do at this point
is change the initial guess and try again.

An alternative explanation: Newton's method can be regarded as replacing the function with
its linear approximation and minimizing the 2-norm of that. That is F(x+s) approx F(x) + F'(x)s
so we minimize || F(x) + F'(x) s ||^2_2; do this using Least Squares. If F'(x) is invertible then
s = - F'(x)^(-1)F(x) otherwise F'(x)^T F'(x) s = -F'(x)^T F(x). If F'(x)^T F(x) is NOT zero then there
exists a nontrivial (that is F'(x)s != 0) solution to the equation and this direction is
s = - [F'(x)^T F'(x)]^(-1) F'(x)^T F(x) so Q'(0) = - F(x)^T F'(x) [F'(x)^T F'(x)]^(-T) F'(x)^T F(x)
= - (F'(x)^T F(x)) [F'(x)^T F'(x)]^(-T) (F'(x)^T F(x)). Since we are assuming (F'(x)^T F(x)) != 0
and F'(x)^T F'(x) has no negative eigenvalues Q'(0) < 0 so s is a descent direction and the line
search should succeed for small enough alpha.

Note that this RARELY happens in practice. Far more likely the linear system is not being solved
(well enough?) or the Jacobian is wrong.

`SNES_DIVERGED_MAX_IT` means that the solver reached the maximum number of iterations without satisfying any
convergence criteria. `SNES_CONVERGED_ITS` means that `SNESConvergedSkip()` was chosen as the convergence test;
thus the usual convergence criteria have not been checked and may or may not be satisfied.


## Developer Notes
This must match petsc/finclude/petscsnes.h

The string versions of these are in `SNESConvergedReasons`, if you change any value here you must
also adjust that array.


## See Also
 `SNESSolve()`, `SNESGetConvergedReason()`, `KSPConvergedReason`, `SNESSetConvergenceTest()`

## Level
beginner

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/include/petscsnes.h.html#SNESConvergedReason">include/petscsnes.h</A>

## Examples
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/tutorials/ex30.c.html">src/snes/tutorials/ex30.c.html</A><BR>


---
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[Index of all SNES routines](index.md)  
[Table of Contents for all manual pages](/docs/manualpages/index.md)  
[Index of all manual pages](/docs/manualpages/singleindex.md)  
