# SNESSetPicard
Use `SNES` to solve the system A(x) x = bp(x) + b via a Picard type iteration (Picard linearization) 
## Synopsis
```
#include "petscsnes.h"  
PetscErrorCode SNESSetPicard(SNES snes, Vec r, PetscErrorCode (*bp)(SNES, Vec, Vec, void *), Mat Amat, Mat Pmat, PetscErrorCode (*J)(SNES, Vec, Mat, Mat, void *), void *ctx)
```
Logically Collective


## Input Parameters

- ***snes -*** the `SNES` context
- ***r -*** vector to store function values, may be NULL
- ***bp -*** function evaluation routine, may be NULL
- ***Amat -*** matrix with which A(x) x - bp(x) - b is to be computed
- ***Pmat -*** matrix from which preconditioner is computed (usually the same as Amat)
- ***J  -*** function to compute matrix values, see SNESJacobianFunction() for details on its calling sequence
- ***ctx -*** [optional] user-defined context for private data for the function evaluation routine (may be NULL)



## Notes
It is often better to provide the nonlinear function F() and some approximation to its Jacobian directly and use
an approximate Newton solver. This interface is provided to allow porting/testing a previous Picard based code in PETSc before converting it to approximate Newton.

One can call `SNESSetPicard()` or `SNESSetFunction()` (and possibly `SNESSetJacobian()`) but cannot call both

```none
Solves the equation A(x) x = bp(x) - b via the defect correction algorithm A(x^{n}) (x^{n+1} - x^{n}) = bp(x^{n}) + b - A(x^{n})x^{n}
```
```none
Note that when an exact solver is used this corresponds to the "classic" Picard A(x^{n}) x^{n+1} = bp(x^{n}) + b iteration.
```

Run with -snes_mf_operator to solve the system with Newton's method using A(x^{n}) to construct the preconditioner.

We implement the defect correction form of the Picard iteration because it converges much more generally when inexact linear solvers are used then
the direct Picard iteration A(x^n) x^{n+1} = bp(x^n) + b

There is some controversity over the definition of a Picard iteration for nonlinear systems but almost everyone agrees that it involves a linear solve and some
believe it is the iteration  A(x^{n}) x^{n+1} = b(x^{n}) hence we use the name Picard. If anyone has an authoritative  reference that defines the Picard iteration
different please contact us at petsc-dev@mcs.anl.gov and we'll have an entirely new argument :-).

When used with -snes_mf_operator this will run matrix-free Newton's method where the matrix-vector product is of the true Jacobian of A(x)x - bp(x) -b and
A(x^{n}) is used to build the preconditioner

When used with -snes_fd this will compute the true Jacobian (very slowly one column at at time) and thus represent Newton's method.

When used with -snes_fd_coloring this will compute the Jacobian via coloring and thus represent a faster implementation of Newton's method. But the
the nonzero structure of the Jacobian is, in general larger than that of the Picard matrix A so you must provide in A the needed nonzero structure for the correct
coloring. When using `DMDA` this may mean creating the matrix A with `DMCreateMatrix()` using a wider stencil than strictly needed for A or with a `DMDA_STENCIL_BOX`.
See the commment in src/snes/tutorials/ex15.c.




## See Also
 `SNES`, `SNESGetFunction()`, `SNESSetFunction()`, `SNESComputeFunction()`, `SNESSetJacobian()`, `SNESGetPicard()`, `SNESLineSearchPreCheckPicard()`, `SNESJacobianFunction`

## Level
intermediate

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/snes/interface/snes.c.html#SNESSetPicard">src/snes/interface/snes.c</A>


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[Edit on GitLab](https://gitlab.com/petsc/petsc/-/edit/release/src/snes/interface/snes.c)


[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)  
