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---
title : "API Documentation"
author_profile: false
excerpt: "Coordinates"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/coordinates.html
sidebar:
nav : docs
---
The Grid is define on a N-dimensional set of integer coordinates.
The maximum dimension is eight, and indexes in this space make use of the `Coordinate` class.
The coordinate class shares a similar interface to `std::vector<int>`, but contains all data within the
object, and has a fixed maximum length (template parameter).
**Example**:
```c++
const int Nd=4;
Coordinate point(Nd);
for(int i=0;i<Nd;i++)
point[i] = 1;
std::cout<< point <<std::endl;
point.resize(3);
std::cout<< point <<std::endl;
```
This enables the coordinates to be manipulated without heap allocation or thread contention,
and avoids introducing STL functions into GPU code, but does so at the expense of introducing
a maximum dimensionality. This limit is easy to change (`lib/util/Coordinate.h`).

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---
title : "API Documentation"
author_profile: false
excerpt: "Grids"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/grids.html
sidebar:
nav : docs
---
A `Grid` object defines the geometry of a global cartesian array, and through inheritance
provides access to message passing decomposition, the local lattice, and the message passing primitives.
The constructor requires parameters to indicate how the spatial (and temporal) indices
are decomposed across MPI tasks and SIMD lanes of the vector length.
We use a partial vectorisation transformation, must select
which space-time dimensions participate in SIMD vectorisation.
The Lattice containers are defined to have opaque internal layout, hiding this layout transformation.
We define GridCartesian and GridRedBlackCartesian which both inherit from `GridBase`:
```c++
class GridCartesian : public GridBase
class GridRedBlackCartesian: public GridBase
```
The simplest Cartesian Grid constructor distributes across `MPI_COMM_WORLD`:
```c++
/////////////////////////////////////////////////////////////////////////
// Construct from comm world
/////////////////////////////////////////////////////////////////////////
GridCartesian(const Coordinate &dimensions,
const Coordinate &simd_layout,
const Coordinate &processor_grid);
```
A second constructor will create a child communicator from a previously declared Grid.
This allows to subdivide the processor grid, and also to define lattices of differing dimensionalities and sizes,
useful for both Chiral fermions, lower dimensional operations, and multigrid:
```c++
/////////////////////////////////////////////////////////////////////////
// Constructor takes a parent grid and possibly subdivides communicator.
/////////////////////////////////////////////////////////////////////////
GridCartesian(const Coordinate &dimensions,
const Coordinate &simd_layout,
const Coordinate &processor_grid,
const GridCartesian &parent,int &split_rank);
```
The Grid object provides much `internal` functionality to map a lattice site to
a node and lexicographic index. These are not needed by code interfacing
to the data parallel layer.
**Example** (`tests/solver/Test_split_grid.cc`):
```c++
const int Ls=8;
////////////////////////////////////////////
// Grids distributed across full machine
// pick up default command line args
////////////////////////////////////////////
Grid_init(&argc,&argv);
Coordinate latt_size = GridDefaultLatt();
Coordinate simd_layout = GridDefaultSimd(Nd,vComplex::Nsimd());
Coordinate mpi_layout = GridDefaultMpi();
GridCartesian * UGrid = SpaceTimeGrid::makeFourDimGrid(GridDefaultLatt(),
GridDefaultSimd(Nd,vComplex::Nsimd()),GridDefaultMpi());
GridCartesian * FGrid = SpaceTimeGrid::makeFiveDimGrid(Ls,UGrid);
GridRedBlackCartesian * rbGrid = SpaceTimeGrid::makeFourDimRedBlackGrid(UGrid);
GridRedBlackCartesian * FrbGrid = SpaceTimeGrid::makeFiveDimRedBlackGrid(Ls,UGrid);
/////////////////////////////////////////////
// Split into N copies of 1^4 mpi communicators
/////////////////////////////////////////////
Coordinate mpi_split (mpi_layout.size(),1);
GridCartesian * SGrid = new GridCartesian(GridDefaultLatt(),
GridDefaultSimd(Nd,vComplex::Nsimd()),
mpi_split,
*UGrid);
GridCartesian * SFGrid = SpaceTimeGrid::makeFiveDimGrid(Ls,SGrid);
GridRedBlackCartesian * SrbGrid = SpaceTimeGrid::makeFourDimRedBlackGrid(SGrid);
GridRedBlackCartesian * SFrbGrid = SpaceTimeGrid::makeFiveDimRedBlackGrid(Ls,SGrid);
```

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---
title : "API Documentation"
author_profile: false
excerpt: "Data parallel API"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/introduction.html
sidebar:
nav : docs
---
Data parallel array indices are divided into two types.
* Internal indices, such as complex, colour, spin degrees of freedom
* spatial (space-time) indices.
The ranges of all internal degrees are determined by template parameters,
and known at compile time. The ranges of spatial indices are dynamic, run time
values and the Cartesian structure information is contained and accessed via `Grid` objects.
Grid objects are the controlling entity for the decomposition of a distributed `Lattice`
array across MPI tasks, nodes, SIMD lanes, accelerators. Threaded loops are used
as appropriate on host code.
Data parallel operations can only be performed between Lattice objects constructed
from the same Grid pointer. These are called `conformable` operations.
We will focus initially on the internal indices as these are the building blocks assembled
in Lattice container classes. Every Lattice container class constructor requires a Grid object
pointer.

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---
title : "API Documentation"
author_profile: false
excerpt: "Lattice containers"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/lattice_containers.html
sidebar:
nav : docs
---
{% include toc icon="gears" title="Contents" %}
Lattice objects may be constructed to contain the local portion of a distribued array of any tensor type.
For performance reasons the tensor type uses a vector `Real` or `Complex` as the fundamental datum.
Every lattice requires a `GridBase` object pointer to be provided in its constructor. Memory is allocated
at construction time. If a Lattice is passed a RedBlack grid, it allocates
half the storage of the full grid, and may either store the red or black checkerboard. The Lattice object
will automatically track through assignments which checkerboard it refers to.
For example, shifting a Even checkerboard by an odd distance produces an Odd result field.
Struct of array objects are defined, and used in the template parameters to the lattice class.
**Example** (`lib/qcd/QCD.h`):
```c++
template<typename vtype> using iSpinMatrix = iScalar<iMatrix<iScalar<vtype>, Ns> >;
typedef iSpinMatrix<ComplexF> SpinMatrixF; //scalar
typedef iSpinMatrix<vComplexF> vSpinMatrixF;//vectorised
typedef Lattice<vSpinMatrixF> LatticeSpinMatrixF;
```
The full range of QCD relevant lattice objects is given below.
|-----------|------------|----------|-----------|---------------|---------------------------------|--------------------------|
| Lattice | Lorentz | Spin | Colour | scalar_type | Field | Synonym |
|-----------|------------|----------|-----------|---------------|---------------------------------|--------------------------|
|`Vector` | `Scalar` | `Scalar` | `Scalar` | `RealD` | `LatticeRealD` | N/A |
|`Vector` | `Scalar` | `Scalar` | `Scalar` | `ComplexD` | `LatticeComplexD` | N/A |
|`Vector` | `Scalar` | `Scalar` | `Matrix` | `ComplexD` | `LatticeColourMatrixD` | `LatticeGaugeLink` |
|`Vector` | `Vector` | `Scalar` | `Matrix` | `ComplexD` | `LatticeLorentzColourMatrixD` | `LatticeGaugeFieldD` |
|`Vector` | `Scalar` | `Vector` | `Vector` | `ComplexD` | `LatticeSpinColourVectorD` | `LatticeFermionD` |
|`Vector` | `Scalar` | `Vector` | `Vector` | `ComplexD` | `LatticeHalfSpinColourVectorD` | `LatticeHalfFermionD` |
|`Vector` | `Scalar` | `Matrix` | `Matrix` | `ComplexD` | `LatticeSpinColourMatrixD` | `LatticePropagatorD` |
|-----------|------------|----------|-----------|---------------|---------------------------------|--------------------------|
Additional single precison variants are defined with the suffix `F`.
Other lattice objects can be defined using the sort of typedef's shown above if needed.
### Opaque containers
The layout within the container is complicated to enable maximum opportunity for vectorisation, and
is opaque from the point of view of the API definition. The key implementation observation is that
so long as data parallel operations are performed and adjacent SIMD lanes correspond to well separated
lattice sites, then identical operations are performed on all SIMD lanes and enable good vectorisation.
Because the layout is opaque, import and export routines from naturally ordered x,y,z,t arrays
are provided (`lib/lattice/Lattice_transfer.h`):
```c++
unvectorizeToLexOrdArray(std::vector<sobj> &out, const Lattice<vobj> &in);
vectorizeFromLexOrdArray(std::vector<sobj> &in , Lattice<vobj> &out);
```
The Lexicographic order of data in the external vector fields is defined by (`lib/util/Lexicographic.h`):
```c++
Lexicographic::IndexFromCoor(const Coordinate &lcoor, int &lex,Coordinate *local_dims);
```
This ordering is $$x + L_x * y + L_x*L_y*z + L_x*L_y*L_z *t$$
Peek and poke routines are provided to perform single site operations. These operations are
extremely low performance and are not intended for algorithm development or performance critical code.
The following are "collective" operations and involve communication between nodes. All nodes receive the same
result by broadcast from the owning node:
```c++
void peekSite(sobj &s,const Lattice<vobj> &l,const Coordinate &site);
void pokeSite(const sobj &s,Lattice<vobj> &l,const Coordinate &site);
```
The following are executed independently by each node:
```c++
void peekLocalSite(sobj &s,const Lattice<vobj> &l,Coordinate &site);
void pokeLocalSite(const sobj &s,Lattice<vobj> &l,Coordinate &site);
```
Lattices of one tensor type may be transformed into lattices of another tensor type by
peeking and poking specific indices in a data parallel manner:
```c++
template<int Index,class vobj> // Vector data parallel index peek
auto PeekIndex(const Lattice<vobj> &lhs,int i);
template<int Index,class vobj> // Matrix data parallel index peek
auto PeekIndex(const Lattice<vobj> &lhs,int i,int j);
template<int Index,class vobj> // Vector poke
void PokeIndex(Lattice<vobj> &lhs,const Lattice<> & rhs,int i)
template<int Index,class vobj> // Matrix poke
void PokeIndex(Lattice<vobj> &lhs,const Lattice<> & rhs,int i,int j)
```
The inconsistent capitalisation on the letter P is due to an obscure bug in g++ that has not to
our knowledge been fixed in any version. The bug was reported in 2016.
### Global Reduction operations
Reduction operations for any lattice field are provided. The result is identical on each computing node
that is part of the relevant Grid communicator:
```c++
template<class vobj>
RealD norm2(const Lattice<vobj> &arg);
template<class vobj>
ComplexD innerProduct(const Lattice<vobj> &left,const Lattice<vobj> &right);
template<class vobj>
vobj sum(const Lattice<vobj> &arg)
```
### Site local reduction operations
Internal indices may be reduced, site by site, using the following routines:
```c++
template<class vobj>
auto localNorm2 (const Lattice<vobj> &rhs)
template<class vobj>
auto localInnerProduct (const Lattice<vobj> &lhs,const Lattice<vobj> &rhs)
```
### Outer product
A site local outer product is defined:
```c++
template<class ll,class rr>
auto outerProduct (const Lattice<ll> &lhs,const Lattice<rr> &rhs)
```
### Slice operations
Slice operations are defined to operate on one lower dimension than the full lattice. The omitted dimension
is the parameter orthogdim:
```c++
template<class vobj>
void sliceSum(const Lattice<vobj> &Data,
std::vector<typename vobj::scalar_object> &result,
int orthogdim);
template<class vobj>
void sliceInnerProductVector( std::vector<ComplexD> & result,
const Lattice<vobj> &lhs,
const Lattice<vobj> &rhs,
int orthogdim);
template<class vobj>
void sliceNorm (std::vector<RealD> &sn,
const Lattice<vobj> &rhs,
int orthogdim);
```
### Data parallel expression template engine
The standard arithmetic operators and some data parallel library functions are implemented site by site
on lattice types.
Operations may only ever combine lattice objects that have been constructed from the **same** grid pointer.
**Example**:
```c++
LatticeFermionD A(&grid);
LatticeFermionD B(&grid);
LatticeFermionD C(&grid);
A = B - C;
```
Such operations are said to be **conformable** and are the lattice are guaranteed to have the same dimensions
and both MPI and SIMD decomposition because they are based on the same grid object. The conformability check
is lightweight and simply requires the same grid pointers be passed to the lattice objects. The data members
of the grid objects are not compared.
Conformable lattice fields may be combined with appropriate scalar types in expressions. The implemented
rules follow those already documented for the tensor types.
### Unary operators and functions
The following sitewise unary operations are defined:
|-----------------------|---------------------------------------------|
| Operation | Description |
|-----------------------|---------------------------------------------|
|`operator-` | negate |
|`adj` | Hermitian conjugate |
|`conjugate` | complex conjugate |
|`trace` | sitewise trace |
|`transpose` | sitewise transpose |
|`Ta` | take traceles anti Hermitian part |
|`ProjectOnGroup` | reunitarise or orthogonalise |
|`real` | take the real part |
|`imag` | take the imaginary part |
|`toReal` | demote complex to real |
|`toComplex` | promote real to complex |
|`timesI` | elementwise +i mult (0 is not multiplied) |
|`timesMinusI` | elementwise -i mult (0 is not multiplied) |
|`abs` | elementwise absolute value |
|`sqrt` | elementwise square root |
|`rsqrt` | elementwise reciprocal square root |
|`sin` | elementwise sine |
|`cos` | elementwise cosine |
|`asin` | elementwise inverse sine |
|`acos` | elementwise inverse cosine |
|`log` | elementwise logarithm |
|`exp` | elementwise exponentiation |
|`operator!` | Logical negation of integer field |
|`Not` | Logical negation of integer field |
|-----------------------|---------------------------------------------|
The following sitewise applied functions with additional parameters are:
```c++
template<class obj> Lattice<obj> pow(const Lattice<obj> &rhs_i,RealD y);
template<class obj> Lattice<obj> mod(const Lattice<obj> &rhs_i,Integer y);
template<class obj> Lattice<obj> div(const Lattice<obj> &rhs_i,Integer y);
template<class obj> Lattice<obj>
expMat(const Lattice<obj> &rhs_i, RealD alpha, Integer Nexp = DEFAULT_MAT_EXP);
```
### Binary operators
The following binary operators are defined:
```
operator+
operator-
operator*
operator/
```
Logical are defined on LatticeInteger types:
```
operator&
operator|
operator&&
operator||
```
### Ternary operator, logical operatons and where
Within the data parallel level of the API the only way to perform operations
that are differentiated between sites is use predicated execution.
The predicate takes the form of a `LatticeInteger` which is confromable with both
the `iftrue` and `iffalse` argument:
```c++
template<class vobj,class iobj> void where(const Lattice<iobj> &pred,
Lattice<vobj> &iftrue,
Lattice<vobj> &iffalse);
```
This plays the data parallel analogue of the C++ ternary operator:
```c++
a = b ? c : d;
```
In order to create the predicate in a coordinate dependent fashion it is often useful
to use the lattice coordinates.
The `LatticeCoordinate` function:
```c++
template<class iobj> LatticeCoordinate(Lattice<iobj> &coor,int dir);
```
fills an `Integer` field with the coordinate in the N-th dimension.
A usage example is given
**Example**:
```c++
int dir =3;
int block=4;
LatticeInteger coor(FineGrid);
LatticeCoordinate(coor,dir);
result = where(mod(coor,block)==(block-1),x,z);
```
(Other usage cases of LatticeCoordinate include the generation of plane wave momentum phases.)
### Site local fused operations
The biggest limitation of expression template engines is that the optimisation
visibility is a single assignment statement in the original source code.
There is no scope for loop fusion between multiple statements.
Multi-loop fusion gives scope for greater cache locality.
Two primitives for hardware aware parallel loops are provided.
These will operate directly on the site objects which are expanded by a factor
of the vector length (in our struct of array datatypes).
Since the mapping of sites
to data lanes is opaque, these vectorised loops
are *only* appropriate for optimisation of site local operations.
### View objects
Due to an obscure aspect of the way that Nvidia handle device C++11 lambda functions,
it is necessary to disable the indexing of a Lattice object.
Rather, a reference to a lattice object must be first obtained.
The reference is copyable to a GPU, and is able to be indexed on either accelerator code,
or by host code.
In order to prevent people developing code that dereferences Lattice objects in a way that
works on CPU compilation, but fails on GPU compilation, we have decided to remove the ability
to index a lattice object on CPU code.
As a result of Nvidia's constraints, all accesses to lattice objects are required to be made
through a View object.
In the following, the type is `LatticeView<vobj>`, however it is wise to use the C++11 auto keyword
to avoid naming the type. See code examples below.
### thread_loops
The first parallel primitive is the thread_loop
**Example**:
```c++
LatticeField r(grid);
LatticeField x(grid);
LatticeField p(grid);
LatticeField mmp(grid);
auto r_v = r.View();
auto x_v = x.View();
auto p_v = p.View();
auto mmp_v = mmp.View();
thread_loop(s , r_v, {
r_v[s] = r_v[s] - a * mmp_v[s];
x_v[s] = x_v[s] + a*p_v[s];
p_v[s] = p_v[s]*b + r_v[s];
});
```
### accelerator_loops
The second parallel primitive is an accelerated_loop
**Example**:
```c++
LatticeField r(grid);
LatticeField x(grid);
LatticeField p(grid);
LatticeField mmp(grid);
auto r_v = r.View();
auto x_v = x.View();
auto p_v = p.View();
auto mmp_v = mmp.View();
accelerator_loop(s , r_v, {
r_v[s] = r_v[s] - a * mmp_v[s];
x_v[s] = x_v[s] + a*p_v[s];
p_v[s] = p_v[s]*b + r_v[s];
});
```
### Cshift
Site shifting operations are provided using the Cshift function:
```c++
template<class vobj>
Lattice<vobj> Cshift(const Lattice<vobj> &rhs,int dimension,int shift)
```
This shifts the whole vector by any distance shift in the appropriate dimension.
For the avoidance of doubt on direction conventions,a positive shift moves the
lattice site $$x_mu = 1$$ in the rhs to $$x_mu = 0$$ in the result.
**Example** (`benchmarks/Benchmark_wilson.cc`):
```c++
{ // Naive wilson implementation
ref = Zero();
for(int mu=0;mu<Nd;mu++){
tmp = U[mu]*Cshift(src,mu,1);
{
auto ref_v = ref.View();
auto tmp_v = tmp.View();
for(int i=0;i<ref_v.size();i++){
ref_v[i]+= tmp_v[i] - Gamma(Gmu[mu])*tmp_v[i]; ;
}
}
tmp =adj(U[mu])*src;
tmp =Cshift(tmp,mu,-1);
{
auto ref_v = ref.View();
auto tmp_v = tmp.View();
for(int i=0;i<ref_v.size();i++){
ref_v[i]+= tmp_v[i] + Gamma(Gmu[mu])*tmp_v[i]; ;
}
}
}
}
```
### CovariantCshift
Covariant Cshift operations are provided for common cases of boundary condition. These may be further optimised
in future:
```c++
template<class covariant,class gauge>
Lattice<covariant> CovShiftForward(const Lattice<gauge> &Link, int mu,
const Lattice<covariant> &field);
template<class covariant,class gauge>
Lattice<covariant> CovShiftBackward(const Lattice<gauge> &Link, int mu,
const Lattice<covariant> &field);
```
### Boundary conditions
The covariant shift routines occur in namespaces PeriodicBC and ConjugateBC. The correct covariant shift
for the boundary condition is passed into the gauge actions and wilson loops via an
"Impl" template policy class.
The relevant staples, plaquettes, and loops are formed by using the provided method:
```c++
Impl::CovShiftForward
Impl::CovShiftBackward
```
etc... This makes physics code transform appropriately with externally supplied rules about
treating the boundary.
**Example** (`lib/qcd/util/WilsonLoops.h`):
```c++
static void dirPlaquette(GaugeMat &plaq, const std::vector<GaugeMat> &U,
const int mu, const int nu) {
// ___
//| |
//|<__|
plaq = Gimpl::CovShiftForward(U[mu],mu,
Gimpl::CovShiftForward(U[nu],nu,
Gimpl::CovShiftBackward(U[mu],mu,
Gimpl::CovShiftIdentityBackward(U[nu], nu))));
}
```
### Inter-grid transfer operations
Transferring between different checkerboards of the same global lattice:
```c++
template<class vobj> void pickCheckerboard(int cb,Lattice<vobj> &half,const Lattice<vobj> &full);
template<class vobj> void setCheckerboard(Lattice<vobj> &full,const Lattice<vobj> &half);
```
These are used to set up Schur red-black decomposed solvers, for example.
Multi-grid projection between a fine and coarse grid:
```c++
template<class vobj,class CComplex,int nbasis>
void blockProject(Lattice<iVector<CComplex,nbasis > > &coarseData,
const Lattice<vobj> &fineData,
const std::vector<Lattice<vobj> > &Basis);
```
Multi-grid promotion to a finer grid:
```c++
template<class vobj,class CComplex,int nbasis>
void blockPromote(const Lattice<iVector<CComplex,nbasis > > &coarseData,
Lattice<vobj> &fineData,
const std::vector<Lattice<vobj> > &Basis)
```
Support for sub-block Linear algebra:
```c++
template<class vobj,class CComplex>
void blockZAXPY(Lattice<vobj> &fineZ,
const Lattice<CComplex> &coarseA,
const Lattice<vobj> &fineX,
const Lattice<vobj> &fineY)
template<class vobj,class CComplex>
void blockInnerProduct(Lattice<CComplex> &CoarseInner,
const Lattice<vobj> &fineX,
const Lattice<vobj> &fineY)
template<class vobj,class CComplex>
void blockNormalise(Lattice<CComplex> &ip,Lattice<vobj> &fineX)
template<class vobj>
void blockSum(Lattice<vobj> &coarseData,const Lattice<vobj> &fineData)
template<class vobj,class CComplex>
void blockOrthogonalise(Lattice<CComplex> &ip,std::vector<Lattice<vobj> > &Basis)
```
Conversion between different SIMD layouts:
```c++
template<class vobj,class vvobj>
void localConvert(const Lattice<vobj> &in,Lattice<vvobj> &out)
```
Slices between grid of dimension N and grid of dimentions N+1:
```c++
template<class vobj>
void InsertSlice(const Lattice<vobj> &lowDim,Lattice<vobj> & higherDim,int slice, int orthog)
template<class vobj>
void ExtractSlice(Lattice<vobj> &lowDim,const Lattice<vobj> & higherDim,int slice, int orthog)
```
Growing a lattice by a multiple factor, with periodic replication:
```c++
template<class vobj>
void Replicate(Lattice<vobj> &coarse,Lattice<vobj> & fine)
```
That latter is useful to, for example, pre-thermalise a smaller volume and then grow the volume in HMC.
It was written while debugging G-parity boundary conditions.

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---
title : "API Documentation"
author_profile: false
excerpt: "Random number generators"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/random_number_generators.html
sidebar:
nav : docs
---
Grid provides three configure time options for random the number generator engine.
* `sitmo`
* `ranlux48`
* `mt19937`
The selection is controlled by the `--enable-rng=<option>` flag.
Sitmo is the default Grid RNG and is recommended. It is a hash based RNG that is cryptographically secure and has
#. passed the BigCrush tests
#. can Skip forward an arbitrary distance (up to 2^256) in O(1) time
We use Skip functionality to place each site in an independent well separated stream.
The Skip was trivially parallelised, important in a many core node,
and gives very low overhead parallel RNG initialisation.
Our implementation of parallel RNG
* Has passed the BigCrush tests **drawing once from each site RNG** in a round robin fashion.
This test is applied in `tests/testu01/Test_smallcrush.cc`
The interface is as follows::
```c++
class GridSerialRNG {
GridSerialRNG();
void SeedFixedIntegers(const std::vector<int> &seeds);
}
class GridParallelRNG {
GridParallelRNG(GridBase *grid);
void SeedFixedIntegers(const std::vector<int> &seeds);
}
template <class vobj> void random(GridParallelRNG &rng,Lattice<vobj> &l) { rng.fill(l,rng._uniform); }
template <class vobj> void gaussian(GridParallelRNG &rng,Lattice<vobj> &l) { rng.fill(l,rng._gaussian); }
template <class sobj> void random(GridSerialRNG &rng,sobj &l) { rng.fill(l,rng._uniform ); }
template <class sobj> void gaussian(GridSerialRNG &rng,sobj &l) { rng.fill(l,rng._gaussian ); }
```
* Serial RNG's are used to assign scalar fields.
* Parallel RNG's are used to assign lattice fields and must subdivide the field grid (need not be conformable).
It is the API users responsibility to initialise, manage, save and restore these RNG state for their algorithm.
In particular there is no single globally managed RNG state.
Input/Output routines are provided for saving and restoring RNG states.
`lib/parallelIO/BinaryIO.h`:
```c++
////////////////////////////////////////////////////////////////////////////
// Read a RNG; use IOobject and lexico map to an array of state
////////////////////////////////////////////////////////////////////////////
static void readRNG(GridSerialRNG &serial,
GridParallelRNG &parallel,
std::string file,
Integer offset,
uint32_t &nersc_csum,
uint32_t &scidac_csuma,
uint32_t &scidac_csumb)
////////////////////////////////////////////////////////////////////////////
// Write a RNG; lexico map to an array of state and use IOobject
////////////////////////////////////////////////////////////////////////////
static void writeRNG(GridSerialRNG &serial,
GridParallelRNG &parallel,
std::string file,
Integer offset,
uint32_t &nersc_csum,
uint32_t &scidac_csuma,
uint32_t &scidac_csumb)
lib/parallelIO/NerscIO.h::
void writeRNGState(GridSerialRNG &serial,GridParallelRNG &parallel,std::string file);
void readRNG(GridSerialRNG &serial,
GridParallelRNG &parallel,
std::string file,
Integer offset,
uint32_t &nersc_csum,
uint32_t &scidac_csuma,
uint32_t &scidac_csumb);
```
**Example**:
```c++
NerscIO::writeRNGState(sRNG,pRNG,rfile);
```

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---
title : "API Documentation"
author_profile: false
excerpt: "Tensor classes"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/tensor_classes.html
sidebar:
nav : docs
---
The Tensor data structures are built up from fundamental
scalar matrix and vector classes:
```c++
template<class vobj > class iScalar { private: vobj _internal ; }
template<class vobj,int N> class iVector { private: vobj _internal[N] ; }
template<class vobj,int N> class iMatrix { private: vobj _internal[N] ; }
```
These are template classes and can be passed a fundamental scalar or vector type, or
nested to form arbitrarily complicated tensor products of indices. All mathematical expressions
are defined to operate recursively, index by index.
Presently the constants
* `Nc`
* `Nd`
are globally predefined. However, this is planned for changed in future and policy classes
for different theories (e.g. QCD, QED, SU2 etc...) will contain these constants and enable multiple
theories to coexist more naturally.
Arbitrary tensor products of fundamental scalar, vector
and matrix objects may be formed in principle by the basic Grid code.
For Lattice field theory, we define types according to the following tensor
product structure ordering. The suffix "D" indicates either double types, and
replacing with "F" gives the corresponding single precision type.
|Lattice | Lorentz | Spin | Colour | scalar_type | Field |
|---------|----------|-----------|----------|-------------|--------------------------|
|Scalar | Scalar | Scalar | Scalar | RealD | RealD |
|Scalar | Scalar | Scalar | Scalar | ComplexD | ComplexD |
|Scalar | Scalar | Scalar | Matrix | ComplexD | ColourMatrixD |
|Scalar | Vector | Scalar | Matrix | ComplexD | LorentzColourMatrixD |
|Scalar | Scalar | Vector | Vector | ComplexD | SpinColourVectorD |
|Scalar | Scalar | Vector | Vector | ComplexD | HalfSpinColourVectorD |
|Scalar | Scalar | Matrix | Matrix | ComplexD | SpinColourMatrixD |
|---------|----------|-----------|----------|-------------|--------------------------|
The types are implemented via a recursive tensor nesting system.
**Example** we declare:
```c++
template<typename vtype>
using iLorentzColourMatrix = iVector<iScalar<iMatrix<vtype, Nc> >, Nd > ;
typedef iLorentzColourMatrix<ComplexD > LorentzColourMatrixD;
```
**Example** we declare:
```c++
template<typename vtype>
using iLorentzColourMatrix = iVector<iScalar<iMatrix<vtype, Nc> >, Nd > ;
typedef iLorentzColourMatrix<ComplexD > LorentzColourMatrixD;
```
Arbitrarily deep tensor nests may be formed. Grid uses a positional and numerical rule to associate indices for contraction
in the Einstein summation sense.
|----------------|---------|------------|
| Symbolic name | Number | Position |
|----------------|---------|------------|
|`LorentzIndex` | 0 | left |
|`SpinIndex` | 1 | middle |
|`ColourIndex` | 2 | right |
|----------------|---------|------------|
The conventions are that the index ordering left to right are: Lorentz, Spin, Colour. A scalar type (either real
or complex, single or double precision) is be provided to the innermost structure.
### Tensor arithmetic rules (`lib/tensors/Tensor_arith.h`)
Arithmetic rules are defined on these types
The multiplication operator follows the natural multiplication
table for each index, index level by index level.
`Operator *`
|----|----|----|----|
| x | S | V | M |
|----|----|----|----|
| S | S | V | M |
| V | S | S | V |
| M | M | V | M |
|----|----|----|----|
The addition and subtraction rules disallow a scalar to be added to a vector,
and vector to be added to matrix. A scalar adds to a matrix on the diagonal.
`Operator +` and `Operator -`
|----|----|----|----|
| +/-| S | V | M |
|----|----|----|----|
| S | S | - | M |
| V | - | V | - |
| M | M | - | M |
|----|----|----|----|
The rules for a nested objects are recursively inferred level by level from basic rules of multiplication
addition and subtraction for scalar/vector/matrix. Legal expressions can only be formed between objects
with the same number of nested internal indices. All the Grid QCD datatypes have precisely three internal
indices, some of which may be trivial scalar to enable expressions to be formed.
Arithmetic operations are possible where the left or right operand is a scalar type.
**Example**:
```c++
LatticeColourMatrixD U(grid);
LatticeColourMatrixD Udag(grid);
Udag = adj(U);
RealD unitary_err = norm2(U*adj(U) - 1.0);
```
Will provide a measure of how discrepant from unitarity the matrix U is.
### Internal index manipulation (`lib/tensors/Tensor_index.h`)
General code can access any specific index by number with a peek/poke semantic:
```c++
// peek index number "Level" of a vector index
template<int Level,class vtype> auto peekIndex (const vtype &arg,int i);
// peek index number "Level" of a vector index
template<int Level,class vtype> auto peekIndex (const vtype &arg,int i,int j);
// poke index number "Level" of a vector index
template<int Level,class vtype>
void pokeIndex (vtype &pokeme,arg,int i)
// poke index number "Level" of a matrix index
template<int Level,class vtype>
void pokeIndex (vtype &pokeme,arg,int i,int j)
```
**Example**:
```c++
for (int mu = 0; mu < Nd; mu++) {
U[mu] = PeekIndex<LorentzIndex>(Umu, mu);
}
```
Similar to the QDP++ package convenience routines are provided to access specific elements of
vector and matrix internal index types by physics name or meaning aliases for the above routines
with the appropriate index constant.
* `peekColour`
* `peekSpin`
* `peekLorentz`
and
* `pokeColour`
* `pokeSpin`
* `pokeLorentz`
For example, we often store Gauge Fields with a Lorentz index, but can split them into
polarisations in relevant pieces of code.
**Example**:
```c++
for (int mu = 0; mu < Nd; mu++) {
U[mu] = peekLorentz(Umu, mu);
}
```
For convenience, direct access as both an l-value and an r-value is provided by the parenthesis operator () on each of the Scalar, Vector and Matrix classes.
For example one may write
**Example**:
```c++
ColourMatrix A, B;
A()()(i,j) = B()()(j,i);
```
bearing in mind that empty parentheses are need to address a scalar entry in the tensor index nest.
The first (left) empty parentheses move past the (scalar) Lorentz level in the tensor nest, and the second
(middle) empty parantheses move past the (scalar) spin level. The (i,j) index the colour matrix.
Other examples are easy to form for the many cases, and should be obvious to the reader.
This form of addressing is convenient and saves peek, modifying, poke
multiple temporary objects when both spin and colour indices are being accessed.
There are many cases where multiple lines of code are required with a peek/poke semantic which are
easier with direct l-value and r-value addressing.
### Matrix operations
Transposition and tracing specific internal indices are possible using:
```c++
template<int Level,class vtype>
auto traceIndex (const vtype &arg)
template<int Level,class vtype>
auto transposeIndex (const vtype &arg)
```
These may be used as
**Example**:
```c++
LatticeColourMatrixD Link(grid);
ComplexD link_trace = traceIndex<ColourIndex> (Link);
```
Again, convenience aliases for QCD naming schemes are provided via
* `traceColour`
* `traceSpin`
* `transposeColour`
* `transposeSpin`
**Example**:
```c++
ComplexD link_trace = traceColour (Link);
```
The operations only makes sense for matrix and scalar internal indices.
The trace and transpose over all indices is also defined for matrix and scalar types:
```c++
template<class vtype,int N>
auto trace(const iMatrix<vtype,N> &arg) -> iScalar
template<class vtype,int N>
auto transpose(const iMatrix<vtype,N> &arg ) -> iMatrix
```
Similar functions are:
* `conjugate`
* `adjoint`
The traceless anti-Hermitian part is taken with:
```c++
template<class vtype,int N> iMatrix<vtype,N>
Ta(const iMatrix<vtype,N> &arg)
```
Reunitarisation (or reorthogonalisation) is enabled by:
```c++
template<class vtype,int N> iMatrix<vtype,N>
ProjectOnGroup(const iMatrix<vtype,N> &arg)
```
**Example**:
```c++
LatticeColourMatrixD Mom(grid);
LatticeColourMatrixD TaMom(grid);
TaMom = Ta(Mom);
```
### Querying internal index structure
Templated code may find it useful to use query functions on the Grid datatypes they are provided.
For example general Serialisation and I/O code can inspect the nature of a type a routine has been
asked to read from disk, or even generate descriptive type strings:
```c++
////////////////////////////////////////////////////
// Support type queries on template params:
////////////////////////////////////////////////////
// int _ColourScalar = isScalar<ColourIndex,vobj>();
// int _ColourVector = isVector<ColourIndex,vobj>();
// int _ColourMatrix = isMatrix<ColourIndex,vobj>();
template<int Level,class vtype> int isScalar(void)
template<int Level,class vtype> int isVector(void)
template<int Level,class vtype> int isMatrix(void)
```
**Example** (`lib/parallelIO/IldgIO.h`):
```c++
template<class vobj> std::string ScidacRecordTypeString(int &colors, int &spins, int & typesize,int &datacount) {
/////////////////////////////////////////
// Encode a generic tensor as a string
/////////////////////////////////////////
typedef typename getPrecision<vobj>::real_scalar_type stype;
int _ColourN = indexRank<ColourIndex,vobj>();
int _ColourScalar = isScalar<ColourIndex,vobj>();
int _ColourVector = isVector<ColourIndex,vobj>();
int _ColourMatrix = isMatrix<ColourIndex,vobj>();
int _SpinN = indexRank<SpinIndex,vobj>();
int _SpinScalar = isScalar<SpinIndex,vobj>();
int _SpinVector = isVector<SpinIndex,vobj>();
int _SpinMatrix = isMatrix<SpinIndex,vobj>();
int _LorentzN = indexRank<LorentzIndex,vobj>();
int _LorentzScalar = isScalar<LorentzIndex,vobj>();
int _LorentzVector = isVector<LorentzIndex,vobj>();
int _LorentzMatrix = isMatrix<LorentzIndex,vobj>();
std::stringstream stream;
stream << "GRID_";
stream << ScidacWordMnemonic<stype>();
if ( _LorentzVector ) stream << "_LorentzVector"<<_LorentzN;
if ( _LorentzMatrix ) stream << "_LorentzMatrix"<<_LorentzN;
if ( _SpinVector ) stream << "_SpinVector"<<_SpinN;
if ( _SpinMatrix ) stream << "_SpinMatrix"<<_SpinN;
if ( _ColourVector ) stream << "_ColourVector"<<_ColourN;
if ( _ColourMatrix ) stream << "_ColourMatrix"<<_ColourN;
if ( _ColourScalar && _LorentzScalar && _SpinScalar ) stream << "_Complex";
typesize = sizeof(typename vobj::scalar_type);
if ( _ColourMatrix ) typesize*= _ColourN*_ColourN;
else typesize*= _ColourN;
if ( _SpinMatrix ) typesize*= _SpinN*_SpinN;
else typesize*= _SpinN;
};
```
### Inner and outer products
We recursively define (`tensors/Tensor_inner.h`), ultimately returning scalar in all indices:
```c++
/////////////////////////////////////////////////////////////////////////
// innerProduct Scalar x Scalar -> Scalar
// innerProduct Vector x Vector -> Scalar
// innerProduct Matrix x Matrix -> Scalar
/////////////////////////////////////////////////////////////////////////
template<class l,class r>
auto innerProductD (const iScalar<l>& lhs,const iScalar<r>& rhs)
template<class l,class r,int N>
auto innerProductD (const iVector<l,N>& lhs,const iVector<r,N>& rhs)
template<class l,class r,int N>
auto innerProductD (const iMatrix<l,N>& lhs,const iMatrix<r,N>& rhs)
template<class l,class r>
auto innerProduct (const iScalar<l>& lhs,const iScalar<r>& rhs)
template<class l,class r,int N>
auto innerProduct (const iVector<l,N>& lhs,const iVector<r,N>& rhs)
template<class l,class r,int N>
auto innerProduct (const iMatrix<l,N>& lhs,const iMatrix<r,N>& rhs)
```
The sum is always performed in double precision for the `innerProductD` variant.
We recursively define (`tensors/Tensor_outer.h`):
```c++
/////////////////////////////////////////////////////////////////////////
// outerProduct Scalar x Scalar -> Scalar
// Vector x Vector -> Matrix
/////////////////////////////////////////////////////////////////////////
template<class l,class r>
auto outerProduct (const iScalar<l>& lhs,const iScalar<r>& rhs)
template<class l,class r,int N>
auto outerProduct (const iVector<l,N>& lhs,const iVector<r,N>& rhs)
```
### Functions of Tensor
The following unary functions are defined, which operate element by element on a tensor
data structure:
```c++
sqrt();
rsqrt();
sin();
cos();
asin();
acos();
log();
exp();
abs();
Not();
toReal();
toComplex();
```
Element wise functions are defined for::
```c++
div(tensor,Integer);
mod(tensor,Integer);
pow(tensor,RealD);
```
Matrix exponentiation (as opposed to element wise exponentiation is implemented via power series in::
```c++
Exponentiate(const Tensor &r ,RealD alpha, Integer Nexp = DEFAULT_MAT_EXP)
```
the exponentiation is distributive across vector indices (i.e. proceeds component by component for a `LorentzColourMatrix`).
Determinant is similar::
```c++
iScalar Determinant(const Tensor &r )
```

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---
title : "API Documentation"
author_profile: false
excerpt: "Vectorisation"
header:
overlay_color: "#5DADE2"
permalink: /docs/API/vectorisation.html
sidebar:
nav : docs
---
Internally, Grid defines a portable abstraction SIMD vectorisation, via the following types:
* `vRealF`
* `vRealD`
* `vComplexF`
* `vComplexD`
These have the usual range of arithmetic operators and functions acting upon them. They do not form
part of the API, but are mentioned to (partially) explain the need for controlling the
layout transformation in lattice objects.