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# PetscDTPKDEvalJet
Evaluate the jet (function and derivatives) of the Proriol-Koornwinder-Dubiner (PKD) basis for the space of polynomials up to a given degree.  The PKD basis is L2-orthonormal on the biunit simplex (which is used as the reference element for finite elements in PETSc), which makes it a stable basis to use for evaluating polynomials in that domain. 
## Synopsis
```
#include "petscdt.h" 
PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])
```

## Input Parameters

- ***dim -*** the number of variables in the multivariate polynomials
- ***npoints -*** the number of points to evaluate the polynomials at
- ***points -*** [npoints x dim] array of point coordinates
- ***degree -*** the degree (sum of degrees on the variables in a monomial) of the polynomial space to evaluate.  There are ((dim + degree) choose dim) polynomials in this space.
- ***k -*** the maximum order partial derivative to evaluate in the jet.  There are (dim + k choose dim) partial derivatives
in the jet.  Choosing k = 0 means to evaluate just the function and no derivatives



## Output Parameter

- ***p -*** an array containing the evaluations of the PKD polynomials' jets on the points.  The size is ((dim + degree)
choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this
three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet
index; the third (fastest varying) dimension is the index of the evaluation point.





## Notes
The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded
ordering of `PetscDTIndexToGradedOrder()` and `PetscDTGradedOrderToIndex()`.  For example, in 3D, the polynomial with
leading monomial x^2,y^0,z^1, which has degree tuple (2,0,1), which by `PetscDTGradedOrderToIndex()` has index 12 (it is the 13th basis function in the space);
the partial derivative $\partial_x \partial_z$ has order tuple (1,0,1), appears at index 6 in the jet (it is the 7th partial derivative in the jet).

The implementation uses Kirby's singularity-free evaluation algorithm, https://doi.org/10.1145/1644001.1644006.


## See Also
 `PetscDTGradedOrderToIndex()`, `PetscDTIndexToGradedOrder()`, `PetscDTJacobiEvalJet()`

## Level
advanced

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/dm/dt/interface/dt.c.html#PetscDTPKDEvalJet">src/dm/dt/interface/dt.c</A>


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