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Updating preparing for solvers etc..

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This commit is contained in:
Peter Boyle 2015-05-16 23:35:08 +01:00
parent 9b0aae665f
commit dc6b6bdc96
8 changed files with 1170 additions and 34 deletions
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20
benchmarks/Grid_memory_bandwidth.cc
View File

@ -19,7 +19,7 @@ int main (int argc, char ** argv)
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking fused AXPY bandwidth"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s"<<"\t\t"<<"Gflop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=4;lat<=32;lat+=4){
@ -44,15 +44,16 @@ int main (int argc, char ** argv)
double stop=usecond();
double time = (stop-start)/Nloop*1000;
double flops=lat*lat*lat*lat*Nvec*2;// mul,add
double bytes=3*lat*lat*lat*lat*Nvec*sizeof(Real);
std::cout<<std::setprecision(2) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t"<<flops/time<<std::endl;
}
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking a*x + y bandwidth"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s"<<"\t\t"<<"Gflop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=4;lat<=32;lat+=4){
@ -75,15 +76,16 @@ int main (int argc, char ** argv)
double stop=usecond();
double time = (stop-start)/Nloop*1000;
double flops=lat*lat*lat*lat*Nvec*2;// mul,add
double bytes=3*lat*lat*lat*lat*Nvec*sizeof(Real);
std::cout<<std::setprecision(2) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t"<<flops/time<<std::endl;
}
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking SCALE bandwidth"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s"<<"\t\t"<<"Gflop/s"<<std::endl;
for(int lat=4;lat<=32;lat+=4){
@ -107,14 +109,15 @@ int main (int argc, char ** argv)
double time = (stop-start)/Nloop*1000;
double bytes=2*lat*lat*lat*lat*Nvec*sizeof(Real);
std::cout <<std::setprecision(2) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<std::endl;
double flops=lat*lat*lat*lat*Nvec*1;// mul
std::cout <<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t"<<flops/time<<std::endl;
}
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking READ bandwidth"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s"<<"\t\t"<<"Gflop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=4;lat<=32;lat+=4){
@ -139,7 +142,8 @@ int main (int argc, char ** argv)
double time = (stop-start)/Nloop*1000;
double bytes=lat*lat*lat*lat*Nvec*sizeof(Real);
std::cout<<std::setprecision(2) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<std::endl;
double flops=lat*lat*lat*lat*Nvec*2;// mul,add
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t"<<flops/time<<std::endl;
}

16
benchmarks/Grid_su3.cc
View File

@ -16,7 +16,7 @@ int main (int argc, char ** argv)
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking SU3xSU3 x= x*y"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=2;lat<=24;lat+=2){
@ -40,7 +40,7 @@ int main (int argc, char ** argv)
double bytes=3.0*lat*lat*lat*lat*Nc*Nc*sizeof(Complex);
double footprint=2.0*lat*lat*lat*lat*Nc*Nc*sizeof(Complex);
double flops=Nc*Nc*(6.0+8.0+8.0)*lat*lat*lat*lat;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<footprint<<"\t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<footprint<<" \t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
}
@ -48,7 +48,7 @@ int main (int argc, char ** argv)
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking SU3xSU3 z= x*y"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=2;lat<=24;lat+=2){
@ -71,14 +71,14 @@ int main (int argc, char ** argv)
double bytes=3*lat*lat*lat*lat*Nc*Nc*sizeof(Complex);
double flops=Nc*Nc*(6+8+8)*lat*lat*lat*lat;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
}
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking SU3xSU3 mult(z,x,y)"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=2;lat<=24;lat+=2){
@ -101,14 +101,14 @@ int main (int argc, char ** argv)
double bytes=3*lat*lat*lat*lat*Nc*Nc*sizeof(Complex);
double flops=Nc*Nc*(6+8+8)*lat*lat*lat*lat;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
}
std::cout << "===================================================================================================="<<std::endl;
std::cout << "= Benchmarking SU3xSU3 mac(z,x,y)"<<std::endl;
std::cout << "===================================================================================================="<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << " L "<<"\t\t"<<"bytes"<<"\t\t\t"<<"GB/s\t\t GFlop/s"<<std::endl;
std::cout << "----------------------------------------------------------"<<std::endl;
for(int lat=2;lat<=24;lat+=2){
@ -131,7 +131,7 @@ int main (int argc, char ** argv)
double bytes=3*lat*lat*lat*lat*Nc*Nc*sizeof(Complex);
double flops=Nc*Nc*(8+8+8)*lat*lat*lat*lat;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<"\t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
std::cout<<std::setprecision(3) << lat<<"\t\t"<<bytes<<" \t\t"<<bytes/time<<"\t\t" << flops/time<<std::endl;
}

207
lib/algorithms/LinearOperator.h
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@ -5,14 +5,211 @@
namespace Grid {
// Red black cases?
template<class vtype> class LinearOperatorBase {
/////////////////////////////////////////////////////////////////////////////////////////////
// LinearOperators Take a something and return a something.
/////////////////////////////////////////////////////////////////////////////////////////////
//
// Hopefully linearity is satisfied and the AdjOp is indeed the Hermitian conjugate (transpose if real):
//
// i) F(a x + b y) = aF(x) + b F(y).
// ii) <x|Op|y> = <y|AdjOp|x>^\ast
//
// Would be fun to have a test linearity & Herm Conj function!
/////////////////////////////////////////////////////////////////////////////////////////////
template<class Field> class LinearOperatorBase {
public:
void M(const Lattice<vtype> &in, Lattice<vtype> &out){ assert(0);}
void Mdag(const Lattice<vtype> &in, Lattice<vtype> &out){ assert(0);}
void MdagM(const Lattice<vtype> &in, Lattice<vtype> &out){ assert(0);}
virtual void Op (const Field &in, Field &out) = 0; // Abstract base
virtual void AdjOp (const Field &in, Field &out) = 0; // Abstract base
};
/////////////////////////////////////////////////////////////////////////////////////////////
// Hermitian operators are self adjoint and only require Op to be defined, so refine the base
/////////////////////////////////////////////////////////////////////////////////////////////
template<class Field> class HermitianOperatorBase : public LinearOperatorBase<Field> {
public:
void AdjOp(const Field &in, Field &out) {
return Op(in,out);
};
};
/////////////////////////////////////////////////////////////////////////////////////////////
// Interface defining what I expect of a general sparse matrix
/////////////////////////////////////////////////////////////////////////////////////////////
template<class Field> class SparseMatrixBase {
public:
// Full checkerboar operations
virtual void M (const Field &in, Field &out);
virtual void Mdag (const Field &in, Field &out);
virtual void MdagM(const Field &in, Field &out);
// half checkerboard operaions
virtual void Mpc (const Field &in, Field &out);
virtual void MpcDag (const Field &in, Field &out);
virtual void MpcDagMpc(const Field &in, Field &out);
};
/////////////////////////////////////////////////////////////////////////////////////////////
// Whereas non hermitian takes a generic sparse matrix (e.g. lattice action)
// conforming to sparse matrix interface and builds the full checkerboard non-herm operator
// Op and AdjOp distinct.
// By sharing the class for Sparse Matrix across multiple operator wrappers, we can share code
// between RB and non-RB variants. Sparse matrix is like the fermion action def, and then
// the wrappers implement the specialisation of "Op" and "AdjOp" to the cases minimising
// replication of code.
/////////////////////////////////////////////////////////////////////////////////////////////
template<class SparseMatrix,class Field>
class NonHermitianOperator : public LinearOperatorBase<Field> {
SparseMatrix &_Mat;
public:
NonHermitianOperator(SparseMatrix &Mat): _Mat(Mat){};
void Op (const Field &in, Field &out){
_Mat.M(in,out);
}
void AdjOp (const Field &in, Field &out){
_Mat.Mdag(in,out);
}
};
////////////////////////////////////////////////////////////////////////////////////
// Redblack Non hermitian wrapper
////////////////////////////////////////////////////////////////////////////////////
template<class SparseMatrix,class Field>
class NonHermitianRedBlackOperator : public LinearOperatorBase<Field> {
SparseMatrix &_Mat;
public:
NonHermitianRedBlackOperator(SparseMatrix &Mat): _Mat(Mat){};
void Op (const Field &in, Field &out){
this->Mpc(in,out);
}
void AdjOp (const Field &in, Field &out){ //
this->MpcDag(in,out);
}
};
////////////////////////////////////////////////////////////////////////////////////
// Hermitian wrapper
////////////////////////////////////////////////////////////////////////////////////
template<class SparseMatrix,class Field>
class HermitianOperator : public HermitianOperatorBase<Field> {
SparseMatrix &_Mat;
public:
HermitianOperator(SparseMatrix &Mat): _Mat(Mat) {};
void Op (const Field &in, Field &out){
this->Mpc(in,out);
}
void AdjOp (const Field &in, Field &out){ //
this->MpcDag(in,out);
}
};
////////////////////////////////////////////////////////////////////////////////////
// Hermitian RedBlack wrapper
////////////////////////////////////////////////////////////////////////////////////
template<class SparseMatrix,class Field>
class HermitianRedBlackOperator : public HermitianOperatorBase<Field> {
SparseMatrix &_Mat;
public:
HermitianRedBlackOperator(SparseMatrix &Mat): _Mat(Mat) {};
void Op (const Field &in, Field &out){
this->MpcDagMpc(in,out);
}
};
/////////////////////////////////////////////////////////////
// Base classes for functions of operators
/////////////////////////////////////////////////////////////
template<class Field> class OperatorFunction {
public:
virtual void operator() (LinearOperatorBase<Field> *Linop, const Field &in, Field &out) = 0;
};
/////////////////////////////////////////////////////////////
// Base classes for polynomial functions of operators ? needed?
/////////////////////////////////////////////////////////////
template<class Field> class OperatorPolynomial : public OperatorFunction<Field> {
public:
virtual void operator() (LinearOperatorBase<Field> *Linop,const Field &in, Field &out) = 0;
};
/////////////////////////////////////////////////////////////
// Base classes for iterative processes based on operators
// single input vec, single output vec.
/////////////////////////////////////////////////////////////
template<class Field> class IterativeProcess : public OperatorFunction<Field> {
public:
RealD Tolerance;
Integer MaxIterations;
virtual void operator() (LinearOperatorBase<Field> *Linop,const Field &in, Field &out) = 0;
};
/////////////////////////////////////////////////////////////
// Grand daddy iterative method
/////////////////////////////////////////////////////////////
template<class Field> class ConjugateGradient : public IterativeProcess<Field> {
public:
virtual void operator() (HermitianOperatorBase<Field> *Linop,const Field &in, Field &out) = 0;
};
/////////////////////////////////////////////////////////////
// A little more modern
/////////////////////////////////////////////////////////////
template<class Field> class PreconditionedConjugateGradient : public IterativeProcess<Field> {
public:
void operator() (HermitianOperatorBase<Field> *Linop,
OperatorFunction<Field> *Preconditioner,
const Field &in,
Field &out) = 0;
};
// FIXME : To think about
// Chroma functionality list defining LinearOperator
/*
virtual void operator() (T& chi, const T& psi, enum PlusMinus isign) const = 0;
virtual void operator() (T& chi, const T& psi, enum PlusMinus isign, Real epsilon) const
virtual const Subset& subset() const = 0;
virtual unsigned long nFlops() const { return 0; }
virtual void deriv(P& ds_u, const T& chi, const T& psi, enum PlusMinus isign) const
class UnprecLinearOperator : public DiffLinearOperator<T,P,Q>
const Subset& subset() const {return all;}
};
*/
// Chroma interface defining GaugeAction
/*
template<typename P, typename Q> class GaugeAction
virtual const CreateGaugeState<P,Q>& getCreateState() const = 0;
virtual GaugeState<P,Q>* createState(const Q& q) const
virtual const GaugeBC<P,Q>& getGaugeBC() const
virtual const Set& getSet(void) const = 0;
virtual void deriv(P& result, const Handle< GaugeState<P,Q> >& state) const
virtual Double S(const Handle< GaugeState<P,Q> >& state) const = 0;
class LinearGaugeAction : public GaugeAction< multi1d<LatticeColorMatrix>, multi1d<LatticeColorMatrix> >
typedef multi1d<LatticeColorMatrix> P;
typedef multi1d<LatticeColorMatrix> Q;
virtual void staple(LatticeColorMatrix& result,
const Handle< GaugeState<P,Q> >& state,
int mu, int cb) const = 0;
*/
// Chroma interface defining FermionAction
/*
template<typename T, typename P, typename Q> class FermAct4D : public FermionAction<T,P,Q>
virtual LinearOperator<T>* linOp(Handle< FermState<T,P,Q> > state) const = 0;
virtual LinearOperator<T>* lMdagM(Handle< FermState<T,P,Q> > state) const = 0;
virtual LinOpSystemSolver<T>* invLinOp(Handle< FermState<T,P,Q> > state,
virtual MdagMSystemSolver<T>* invMdagM(Handle< FermState<T,P,Q> > state,
virtual LinOpMultiSystemSolver<T>* mInvLinOp(Handle< FermState<T,P,Q> > state,
virtual MdagMMultiSystemSolver<T>* mInvMdagM(Handle< FermState<T,P,Q> > state,
virtual MdagMMultiSystemSolverAccumulate<T>* mInvMdagMAcc(Handle< FermState<T,P,Q> > state,
virtual SystemSolver<T>* qprop(Handle< FermState<T,P,Q> > state,
class DiffFermAct4D : public FermAct4D<T,P,Q>
virtual DiffLinearOperator<T,Q,P>* linOp(Handle< FermState<T,P,Q> > state) const = 0;
virtual DiffLinearOperator<T,Q,P>* lMdagM(Handle< FermState<T,P,Q> > state) const = 0;
*/
}
#endif

145
lib/algorithms/approx/PolynomialApprox.h Normal file
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@ -0,0 +1,145 @@
#ifndef GRID_POLYNOMIAL_APPROX_H
#define GRID_POLYNOMIAL_APPROX_H
namespace Grid {
////////////////////////////////////////////////////////////////////////////////////////////
// Simple general polynomial with user supplied coefficients
////////////////////////////////////////////////////////////////////////////////////////////
template<class Field>
class Polynomial : public OperatorFunction<Field> {
private:
std::vector<double> _oeffs;
public:
Polynomial(std::vector<double> &_Coeffs) : Coeffs(_Coeffs) {};
// Implement the required interface
void operator() (LinearOperatorBase<Field> &Linop, const Field &in, Field &out) {
Field AtoN = in;
out = AtoN*Coeffs[0];
for(int n=1;n<Coeffs.size();n++){
Field Mtmp=AtoN;
Linop.Op(Mtmp,AtoN);
out=out+AtoN*Coeffs[n];
}
};
};
////////////////////////////////////////////////////////////////////////////////////////////
// Generic Chebyshev approximations
////////////////////////////////////////////////////////////////////////////////////////////
template<class Field>
class Chebyshev : public OperatorFunction<Field> {
private:
std::vector<double> Coeffs;
int order;
double hi;
double lo;
public:
Chebyshev(double _lo,double _hi,int _order, double (* func)(double) ){
lo=_lo;
hi=_hi;
order=_order;
if(order < 2) exit(-1);
Coeffs.resize(order);
for(int j=0;j<order;j++){
double s=0;
for(int k=0;k<order;k++){
double y=cos(M_PI*(k+0.5)/order);
double x=0.5*(y*(hi-lo)+(hi+lo));
double f=func(x);
s=s+f*cos( j*M_PI*(k+0.5)/order );
}
Coeffs[j] = s * 2.0/order;
}
};
double Evaluate(double x) // Convenience for plotting the approximation
{
double Tn;
double Tnm;
double Tnp;
double y=( x-0.5*(hi+lo))/(0.5*(hi-lo));
double T0=1;
double T1=y;
double sum;
sum = 0.5*Coeffs[0]*T0;
sum+= Coeffs[1]*T1;
Tn =T1;
Tnm=T0;
for(int i=2;i<order;i++){
Tnp=2*y*Tn-Tnm;
Tnm=Tn;
Tn =Tnp;
sum+= Tn*Coeffs[i];
}
return sum;
};
// Convenience for plotting the approximation
void PlotApprox(std::ostream &out) {
out<<"Polynomial approx ["<<lo<<","<<hi<<"]"<<std::endl;
for(double x=lo;x<hi;x+=(hi-lo)/50.0){
out <<x<<"\t"<<Evaluate(x)<<std::endl;
}
};
// Implement the required interface; could require Lattice base class
void operator() (LinearOperatorBase<Field> &Linop, const Field &in, Field &out) {
Field T0 = in;
Field T1 = T0; // Field T1(T0._grid); more efficient but hardwires Lattice class
Field T2 = T1;
// use a pointer trick to eliminate copies
Field *Tnm = &T0;
Field *Tn = &T1;
Field *Tnp = &T2;
Field y = in;
double xscale = 2.0/(hi-lo);
double mscale = -(hi+lo)/(hi-lo);
Field *T0=Tnm;
Field *T1=Tn;
// Tn=T1 = (xscale M + mscale)in
Linop.Op(T0,y);
T1=y*xscale+in*mscale;
// sum = .5 c[0] T0 + c[1] T1
out = (0.5*coeffs[0])*T0 + coeffs[1]*T1;
for(int n=2;n<order;n++){
Linop.Op(*Tn,y);
y=xscale*y+mscale*(*Tn);
*Tnp=2.0*y-(*Tnm);
out=out+coeffs[n]* (*Tnp);
// Cycle pointers to avoid copies
Field *swizzle = Tnm;
Tnm =Tn;
Tn =Tnp;
Tnp =swizzle;
}
}
};
}
#endif

710
lib/algorithms/approx/zolotarev.c Executable file
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@ -0,0 +1,710 @@
/* -*- Mode: C; comment-column: 22; fill-column: 79; compile-command: "gcc -o zolotarev zolotarev.c -ansi -pedantic -lm -DTEST"; -*- */
#define VERSION Source Time-stamp: <19-OCT-2004 09:33:22.00 adk@MISSCONTRARY>
/* These C routines evalute the optimal rational approximation to the signum
* function for epsilon < |x| < 1 using Zolotarev's theorem.
*
* To obtain reliable results for high degree approximations (large n) it is
* necessary to compute using sufficiently high precision arithmetic. To this
* end the code has been parameterised to work with the preprocessor names
* INTERNAL_PRECISION and PRECISION set to float, double, or long double as
* appropriate. INTERNAL_PRECISION is used in computing the Zolotarev
* coefficients, which are converted to PRECISION before being returned to the
* caller. Presumably even higher precision could be obtained using GMP or
* similar package, but bear in mind that rounding errors might also be
* significant in evaluating the resulting polynomial. The convergence criteria
* have been written in a precision-independent form. */
#include <math.h>
#include <stdlib.h>
#include <stdio.h>
#define MAX(a,b) ((a) > (b) ? (a) : (b))
#define MIN(a,b) ((a) < (b) ? (a) : (b))
#ifndef INTERNAL_PRECISION
#define INTERNAL_PRECISION double
#endif
#include "zolotarev.h"
#define ZOLOTAREV_INTERNAL
#undef ZOLOTAREV_DATA
#define ZOLOTAREV_DATA izd
#undef ZPRECISION
#define ZPRECISION INTERNAL_PRECISION
#include "zolotarev.h"
#undef ZOLOTAREV_INTERNAL
/* The ANSI standard appears not to know what pi is */
#ifndef M_PI
#define M_PI ((INTERNAL_PRECISION) 3.141592653589793238462643383279502884197\
169399375105820974944592307816406286208998628034825342117068)
#endif
#define ZERO ((INTERNAL_PRECISION) 0)
#define ONE ((INTERNAL_PRECISION) 1)
#define TWO ((INTERNAL_PRECISION) 2)
#define THREE ((INTERNAL_PRECISION) 3)
#define FOUR ((INTERNAL_PRECISION) 4)
#define HALF (ONE/TWO)
/* The following obscenity seems to be the simplest (?) way to coerce the C
* preprocessor to convert the value of a preprocessor token into a string. */
#define PP2(x) #x
#define PP1(a,b,c) a ## b(c)
#define STRINGIFY(name) PP1(PP,2,name)
/* Compute the partial fraction expansion coefficients (alpha) from the
* factored form */
static void construct_partfrac(izd *z) {
int dn = z -> dn, dd = z -> dd, type = z -> type;
int j, k, da = dd + 1 + type;
INTERNAL_PRECISION A = z -> A, *a = z -> a, *ap = z -> ap, *alpha;
alpha = (INTERNAL_PRECISION*) malloc(da * sizeof(INTERNAL_PRECISION));
for (j = 0; j < dd; j++)
for (k = 0, alpha[j] = A; k < dd; k++)
alpha[j] *=
(k < dn ? ap[j] - a[k] : ONE) / (k == j ? ONE : ap[j] - ap[k]);
if(type == 1) /* implicit pole at zero? */
for (k = 0, alpha[dd] = A * (dn > dd ? - a[dd] : ONE); k < dd; k++) {
alpha[dd] *= a[k] / ap[k];
alpha[k] *= (dn > dd ? ap[k] - a[dd] : ONE) / ap[k];
}
alpha[da-1] = dn == da - 1 ? A : ZERO;
z -> alpha = alpha;
z -> da = da;
return;
}
/* Convert factored polynomial into dense polynomial. The input is the overall
* factor A and the roots a[i], such that p = A product(x - a[i], i = 1..d) */
static INTERNAL_PRECISION *poly_factored_to_dense(INTERNAL_PRECISION A,
INTERNAL_PRECISION *a,
int d) {
INTERNAL_PRECISION *p;
int i, j;
p = (INTERNAL_PRECISION *) malloc((d + 2) * sizeof(INTERNAL_PRECISION));
p[0] = A;
for (i = 0; i < d; i++) {
p[i+1] = p[i];
for (j = i; j > 0; j--) p[j] = p[j-1] - a[i]*p[j];
p[0] *= - a[i];
}
return p;
}
/* Convert a rational function of the form R0(x) = x p(x^2)/q(x^2) (type 0) or
* R1(x) = p(x^2)/[x q(x^2)] (type 1) into its continued fraction
* representation. We assume that 0 <= deg(q) - deg(p) <= 1 for type 0 and 0 <=
* deg(p) - deg(q) <= 1 for type 1. On input p and q are in factored form, and
* deg(q) = dq, deg(p) = dp. The output is the continued fraction coefficients
* beta, where R(x) = beta[0] x + 1/(beta[1] x + 1/(...)).
*
* The method used is as follows. There are four cases to consider:
*
* 0.i. Type 0, deg p = deg q
*
* 0.ii. Type 0, deg p = deg q - 1
*
* 1.i. Type 1, deg p = deg q
*
* 1.ii. Type 1, deg p = deg q + 1
*
* and these are connected by two transformations:
*
* A. To obtain a continued fraction expansion of type 1 we use a single-step
* polynomial division we find beta and r(x) such that p(x) = beta x q(x) +
* r(x), with deg(r) = deg(q). This implies that p(x^2) = beta x^2 q(x^2) +
* r(x^2), and thus R1(x) = x beta + r(x^2)/(x q(x^2)) = x beta + 1/R0(x)
* with R0(x) = x q(x^2)/r(x^2).
*
* B. A continued fraction expansion of type 0 is obtained in a similar, but
* not identical, manner. We use the polynomial division algorithm to compute
* the quotient beta and the remainder r that satisfy p(x) = beta q(x) + r(x)
* with deg(r) = deg(q) - 1. We thus have x p(x^2) = x beta q(x^2) + x r(x^2),
* so R0(x) = x beta + x r(x^2)/q(x^2) = x beta + 1/R1(x) with R1(x) = q(x^2) /
* (x r(x^2)).
*
* Note that the deg(r) must be exactly deg(q) for (A) and deg(q) - 1 for (B)
* because p and q have disjoint roots all of multiplicity 1. This means that
* the division algorithm requires only a single polynomial subtraction step.
*
* The transformations between the cases form the following finite state
* automaton:
*
* +------+ +------+ +------+ +------+
* | | | | ---(A)---> | | | |
* | 0.ii | ---(B)---> | 1.ii | | 0.i | <---(A)--- | 1.i |
* | | | | <---(B)--- | | | |
* +------+ +------+ +------+ +------+
*/
static INTERNAL_PRECISION *contfrac_A(INTERNAL_PRECISION *,
INTERNAL_PRECISION *,
INTERNAL_PRECISION *,
INTERNAL_PRECISION *, int, int);
static INTERNAL_PRECISION *contfrac_B(INTERNAL_PRECISION *,
INTERNAL_PRECISION *,
INTERNAL_PRECISION *,
INTERNAL_PRECISION *, int, int);
static void construct_contfrac(izd *z){
INTERNAL_PRECISION *r, A = z -> A, *p = z -> a, *q = z -> ap;
int dp = z -> dn, dq = z -> dd, type = z -> type;
z -> db = 2 * dq + 1 + type;
z -> beta = (INTERNAL_PRECISION *)
malloc(z -> db * sizeof(INTERNAL_PRECISION));
p = poly_factored_to_dense(A, p, dp);
q = poly_factored_to_dense(ONE, q, dq);
r = (INTERNAL_PRECISION *) malloc((MAX(dp,dq) + 1) *
sizeof(INTERNAL_PRECISION));
if (type == 0) (void) contfrac_B(z -> beta, p, q, r, dp, dq);
else (void) contfrac_A(z -> beta, p, q, r, dp, dq);
free(p); free(q); free(r);
return;
}
static INTERNAL_PRECISION *contfrac_A(INTERNAL_PRECISION *beta,
INTERNAL_PRECISION *p,
INTERNAL_PRECISION *q,
INTERNAL_PRECISION *r, int dp, int dq) {
INTERNAL_PRECISION quot, *rb;
int j;
/* p(x) = x beta q(x) + r(x); dp = dq or dp = dq + 1 */
quot = dp == dq ? ZERO : p[dp] / q[dq];
r[0] = p[0];
for (j = 1; j <= dp; j++) r[j] = p[j] - quot * q[j-1];
#ifdef DEBUG
printf("%s: Continued Fraction form: deg p = %2d, deg q = %2d, beta = %g\n",
__FUNCTION__, dp, dq, (float) quot);
for (j = 0; j <= dq + 1; j++)
printf("\tp[%2d] = %14.6g\tq[%2d] = %14.6g\tr[%2d] = %14.6g\n",
j, (float) (j > dp ? ZERO : p[j]),
j, (float) (j == 0 ? ZERO : q[j-1]),
j, (float) (j == dp ? ZERO : r[j]));
#endif /* DEBUG */
*(rb = contfrac_B(beta, q, r, p, dq, dq)) = quot;
return rb + 1;
}
static INTERNAL_PRECISION *contfrac_B(INTERNAL_PRECISION *beta,
INTERNAL_PRECISION *p,
INTERNAL_PRECISION *q,
INTERNAL_PRECISION *r, int dp, int dq) {
INTERNAL_PRECISION quot, *rb;
int j;
/* p(x) = beta q(x) + r(x); dp = dq or dp = dq - 1 */
quot = dp == dq ? p[dp] / q[dq] : ZERO;
for (j = 0; j < dq; j++) r[j] = p[j] - quot * q[j];
#ifdef DEBUG
printf("%s: Continued Fraction form: deg p = %2d, deg q = %2d, beta = %g\n",
__FUNCTION__, dp, dq, (float) quot);
for (j = 0; j <= dq; j++)
printf("\tp[%2d] = %14.6g\tq[%2d] = %14.6g\tr[%2d] = %14.6g\n",
j, (float) (j > dp ? ZERO : p[j]),
j, (float) q[j],
j, (float) (j == dq ? ZERO : r[j]));
#endif /* DEBUG */
*(rb = dq > 0 ? contfrac_A(beta, q, r, p, dq, dq-1) : beta) = quot;
return rb + 1;
}
/* The global variable U is used to hold the argument u throughout the AGM
* recursion. The global variables F and K are set in the innermost
* instantiation of the recursive function AGM to the values of the elliptic
* integrals F(u,k) and K(k) respectively. They must be made thread local to
* make this code thread-safe in a multithreaded environment. */
static INTERNAL_PRECISION U, F, K; /* THREAD LOCAL */
/* Recursive implementation of Gauss' arithmetico-geometric mean, which is the
* kernel of the method used to compute the Jacobian elliptic functions
* sn(u,k), cn(u,k), and dn(u,k) with parameter k (where 0 < k < 1), as well
* as the elliptic integral F(s,k) satisfying F(sn(u,k)) = u and the complete
* elliptic integral K(k).
*
* The algorithm used is a recursive implementation of the Gauss (Landen)
* transformation.
*
* The function returns the value of sn(u,k'), where k' is the dual parameter,
* and also sets the values of the global variables F and K. The latter is
* used to determine the sign of cn(u,k').
*
* The algorithm is deemed to have converged when b ceases to increase. This
* works whatever INTERNAL_PRECISION is specified. */
static INTERNAL_PRECISION AGM(INTERNAL_PRECISION a,
INTERNAL_PRECISION b,
INTERNAL_PRECISION s) {
static INTERNAL_PRECISION pb = -ONE;
INTERNAL_PRECISION c, d, xi;
if (b <= pb) {
pb = -ONE;
F = asin(s) / a; /* Here, a is the AGM */
K = M_PI / (TWO * a);
return sin(U * a);
}
pb = b;
c = a - b;
d = a + b;
xi = AGM(HALF*d, sqrt(a*b), ONE + c*c == ONE ?
HALF*s*d/a : (a - sqrt(a*a - s*s*c*d))/(c*s));
return 2*a*xi / (d + c*xi*xi);
}
/* Computes sn(u,k), cn(u,k), dn(u,k), F(u,k), and K(k). It is essentially a
* wrapper for the routine AGM. The sign of cn(u,k) is defined to be -1 if
* K(k) < u < 3*K(k) and +1 otherwise, and thus sign is computed by some quite
* unnecessarily obfuscated bit manipulations. */
static void sncndnFK(INTERNAL_PRECISION u, INTERNAL_PRECISION k,
INTERNAL_PRECISION* sn, INTERNAL_PRECISION* cn,
INTERNAL_PRECISION* dn, INTERNAL_PRECISION* elF,
INTERNAL_PRECISION* elK) {
int sgn;
U = u;
*sn = AGM(ONE, sqrt(ONE - k*k), u);
sgn = ((int) (fabs(u) / K)) % 4; /* sgn = 0, 1, 2, 3 */
sgn ^= sgn >> 1; /* (sgn & 1) = 0, 1, 1, 0 */
sgn = 1 - ((sgn & 1) << 1); /* sgn = 1, -1, -1, 1 */
*cn = ((INTERNAL_PRECISION) sgn) * sqrt(ONE - *sn * *sn);
*dn = sqrt(ONE - k*k* *sn * *sn);
*elF = F;
*elK = K;
}
/* Compute the coefficients for the optimal rational approximation R(x) to
* sgn(x) of degree n over the interval epsilon < |x| < 1 using Zolotarev's
* formula.
*
* Set type = 0 for the Zolotarev approximation, which is zero at x = 0, and
* type = 1 for the approximation which is infinite at x = 0. */
zolotarev_data* bfm_zolotarev(PRECISION epsilon, int n, int type) {
INTERNAL_PRECISION A, c, cp, kp, ksq, sn, cn, dn, Kp, Kj, z, z0, t, M, F,
l, invlambda, xi, xisq, *tv, s, opl;
int m, czero, ts;
zolotarev_data *zd;
izd *d = (izd*) malloc(sizeof(izd));
d -> type = type;
d -> epsilon = (INTERNAL_PRECISION) epsilon;
d -> n = n;
d -> dd = n / 2;
d -> dn = d -> dd - 1 + n % 2; /* n even: dn = dd - 1, n odd: dn = dd */
d -> deg_denom = 2 * d -> dd;
d -> deg_num = 2 * d -> dn + 1;
d -> a = (INTERNAL_PRECISION*) malloc((1 + d -> dn) *
sizeof(INTERNAL_PRECISION));
d -> ap = (INTERNAL_PRECISION*) malloc(d -> dd *
sizeof(INTERNAL_PRECISION));
ksq = d -> epsilon * d -> epsilon;
kp = sqrt(ONE - ksq);
sncndnFK(ZERO, kp, &sn, &cn, &dn, &F, &Kp); /* compute Kp = K(kp) */
z0 = TWO * Kp / (INTERNAL_PRECISION) n;
M = ONE;
A = ONE / d -> epsilon;
sncndnFK(HALF * z0, kp, &sn, &cn, &dn, &F, &Kj); /* compute xi */
xi = ONE / dn;
xisq = xi * xi;
invlambda = xi;
for (m = 0; m < d -> dd; m++) {
czero = 2 * (m + 1) == n; /* n even and m = dd -1 */
z = z0 * ((INTERNAL_PRECISION) m + ONE);
sncndnFK(z, kp, &sn, &cn, &dn, &F, &Kj);
t = cn / sn;
c = - t*t;
if (!czero) (d -> a)[d -> dn - 1 - m] = ksq / c;
z = z0 * ((INTERNAL_PRECISION) m + HALF);
sncndnFK(z, kp, &sn, &cn, &dn, &F, &Kj);
t = cn / sn;
cp = - t*t;
(d -> ap)[d -> dd - 1 - m] = ksq / cp;
M *= (ONE - c) / (ONE - cp);
A *= (czero ? -ksq : c) * (ONE - cp) / (cp * (ONE - c));
invlambda *= (ONE - c*xisq) / (ONE - cp*xisq);
}
invlambda /= M;
d -> A = TWO / (ONE + invlambda) * A;
d -> Delta = (invlambda - ONE) / (invlambda + ONE);
d -> gamma = (INTERNAL_PRECISION*) malloc((1 + d -> n) *
sizeof(INTERNAL_PRECISION));
l = ONE / invlambda;
opl = ONE + l;
sncndnFK(sqrt( d -> type == 1
? (THREE + l) / (FOUR * opl)
: (ONE + THREE*l) / (opl*opl*opl)
), sqrt(ONE - l*l), &sn, &cn, &dn, &F, &Kj);
s = M * F;
for (m = 0; m < d -> n; m++) {
sncndnFK(s + TWO*Kp*m/n, kp, &sn, &cn, &dn, &F, &Kj);
d -> gamma[m] = d -> epsilon / dn;
}
/* If R(x) is a Zolotarev rational approximation of degree (n,m) with maximum
* error Delta, then (1 - Delta^2) / R(x) is also an optimal Chebyshev
* approximation of degree (m,n) */
if (d -> type == 1) {
d -> A = (ONE - d -> Delta * d -> Delta) / d -> A;
tv = d -> a; d -> a = d -> ap; d -> ap = tv;
ts = d -> dn; d -> dn = d -> dd; d -> dd = ts;
ts = d -> deg_num; d -> deg_num = d -> deg_denom; d -> deg_denom = ts;
}
construct_partfrac(d);
construct_contfrac(d);
/* Converting everything to PRECISION for external use only */
zd = (zolotarev_data*) malloc(sizeof(zolotarev_data));
zd -> A = (PRECISION) d -> A;
zd -> Delta = (PRECISION) d -> Delta;
zd -> epsilon = (PRECISION) d -> epsilon;
zd -> n = d -> n;
zd -> type = d -> type;
zd -> dn = d -> dn;
zd -> dd = d -> dd;
zd -> da = d -> da;
zd -> db = d -> db;
zd -> deg_num = d -> deg_num;
zd -> deg_denom = d -> deg_denom;
zd -> a = (PRECISION*) malloc(zd -> dn * sizeof(PRECISION));
for (m = 0; m < zd -> dn; m++) zd -> a[m] = (PRECISION) d -> a[m];
free(d -> a);
zd -> ap = (PRECISION*) malloc(zd -> dd * sizeof(PRECISION));
for (m = 0; m < zd -> dd; m++) zd -> ap[m] = (PRECISION) d -> ap[m];
free(d -> ap);
zd -> alpha = (PRECISION*) malloc(zd -> da * sizeof(PRECISION));
for (m = 0; m < zd -> da; m++) zd -> alpha[m] = (PRECISION) d -> alpha[m];
free(d -> alpha);
zd -> beta = (PRECISION*) malloc(zd -> db * sizeof(PRECISION));
for (m = 0; m < zd -> db; m++) zd -> beta[m] = (PRECISION) d -> beta[m];
free(d -> beta);
zd -> gamma = (PRECISION*) malloc(zd -> n * sizeof(PRECISION));
for (m = 0; m < zd -> n; m++) zd -> gamma[m] = (PRECISION) d -> gamma[m];
free(d -> gamma);
free(d);
return zd;
}
zolotarev_data* bfm_higham(PRECISION epsilon, int n) {
INTERNAL_PRECISION A, M, c, cp, z, z0, t, epssq;
int m, czero;
zolotarev_data *zd;
izd *d = (izd*) malloc(sizeof(izd));
d -> type = 0;
d -> epsilon = (INTERNAL_PRECISION) epsilon;
d -> n = n;
d -> dd = n / 2;
d -> dn = d -> dd - 1 + n % 2; /* n even: dn = dd - 1, n odd: dn = dd */
d -> deg_denom = 2 * d -> dd;
d -> deg_num = 2 * d -> dn + 1;
d -> a = (INTERNAL_PRECISION*) malloc((1 + d -> dn) *
sizeof(INTERNAL_PRECISION));
d -> ap = (INTERNAL_PRECISION*) malloc(d -> dd *
sizeof(INTERNAL_PRECISION));
A = (INTERNAL_PRECISION) n;
z0 = M_PI / A;
A = n % 2 == 0 ? A : ONE / A;
M = d -> epsilon * A;
epssq = d -> epsilon * d -> epsilon;
for (m = 0; m < d -> dd; m++) {
czero = 2 * (m + 1) == n; /* n even and m = dd - 1*/
if (!czero) {
z = z0 * ((INTERNAL_PRECISION) m + ONE);
t = tan(z);
c = - t*t;
(d -> a)[d -> dn - 1 - m] = c;
M *= epssq - c;
}
z = z0 * ((INTERNAL_PRECISION) m + HALF);
t = tan(z);
cp = - t*t;
(d -> ap)[d -> dd - 1 - m] = cp;
M /= epssq - cp;
}
d -> gamma = (INTERNAL_PRECISION*) malloc((1 + d -> n) *
sizeof(INTERNAL_PRECISION));
for (m = 0; m < d -> n; m++) d -> gamma[m] = ONE;
d -> A = A;
d -> Delta = ONE - M;
construct_partfrac(d);
construct_contfrac(d);
/* Converting everything to PRECISION for external use only */
zd = (zolotarev_data*) malloc(sizeof(zolotarev_data));
zd -> A = (PRECISION) d -> A;
zd -> Delta = (PRECISION) d -> Delta;
zd -> epsilon = (PRECISION) d -> epsilon;
zd -> n = d -> n;
zd -> type = d -> type;
zd -> dn = d -> dn;
zd -> dd = d -> dd;
zd -> da = d -> da;
zd -> db = d -> db;
zd -> deg_num = d -> deg_num;
zd -> deg_denom = d -> deg_denom;
zd -> a = (PRECISION*) malloc(zd -> dn * sizeof(PRECISION));
for (m = 0; m < zd -> dn; m++) zd -> a[m] = (PRECISION) d -> a[m];
free(d -> a);
zd -> ap = (PRECISION*) malloc(zd -> dd * sizeof(PRECISION));
for (m = 0; m < zd -> dd; m++) zd -> ap[m] = (PRECISION) d -> ap[m];
free(d -> ap);
zd -> alpha = (PRECISION*) malloc(zd -> da * sizeof(PRECISION));
for (m = 0; m < zd -> da; m++) zd -> alpha[m] = (PRECISION) d -> alpha[m];
free(d -> alpha);
zd -> beta = (PRECISION*) malloc(zd -> db * sizeof(PRECISION));
for (m = 0; m < zd -> db; m++) zd -> beta[m] = (PRECISION) d -> beta[m];
free(d -> beta);
zd -> gamma = (PRECISION*) malloc(zd -> n * sizeof(PRECISION));
for (m = 0; m < zd -> n; m++) zd -> gamma[m] = (PRECISION) d -> gamma[m];
free(d -> gamma);
free(d);
return zd;
}
#ifdef TEST
#undef ZERO
#define ZERO ((PRECISION) 0)
#undef ONE
#define ONE ((PRECISION) 1)
#undef TWO
#define TWO ((PRECISION) 2)
/* Evaluate the rational approximation R(x) using the factored form */
static PRECISION zolotarev_eval(PRECISION x, zolotarev_data* rdata) {
int m;
PRECISION R;
if (rdata -> type == 0) {
R = rdata -> A * x;
for (m = 0; m < rdata -> deg_denom/2; m++)
R *= (2*(m+1) > rdata -> deg_num ? ONE : x*x - rdata -> a[m]) /
(x*x - rdata -> ap[m]);
} else {
R = rdata -> A / x;
for (m = 0; m < rdata -> deg_num/2; m++)
R *= (x*x - rdata -> a[m]) /
(2*(m+1) > rdata -> deg_denom ? ONE : x*x - rdata -> ap[m]);
}
return R;
}
/* Evaluate the rational approximation R(x) using the partial fraction form */
static PRECISION zolotarev_partfrac_eval(PRECISION x, zolotarev_data* rdata) {
int m;
PRECISION R = rdata -> alpha[rdata -> da - 1];
for (m = 0; m < rdata -> dd; m++)
R += rdata -> alpha[m] / (x * x - rdata -> ap[m]);
if (rdata -> type == 1) R += rdata -> alpha[rdata -> dd] / (x * x);
return R * x;
}
/* Evaluate the rational approximation R(x) using continued fraction form.
*
* If x = 0 and type = 1 then the result should be INF, whereas if x = 0 and
* type = 0 then the result should be 0, but division by zero will occur at
* intermediate stages of the evaluation. For IEEE implementations with
* non-signalling overflow this will work correctly since 1/(1/0) = 1/INF = 0,
* but with signalling overflow you will get an error message. */
static PRECISION zolotarev_contfrac_eval(PRECISION x, zolotarev_data* rdata) {
int m;
PRECISION R = rdata -> beta[0] * x;
for (m = 1; m < rdata -> db; m++) R = rdata -> beta[m] * x + ONE / R;
return R;
}
/* Evaluate the rational approximation R(x) using Cayley form */
static PRECISION zolotarev_cayley_eval(PRECISION x, zolotarev_data* rdata) {
int m;
PRECISION T;
T = rdata -> type == 0 ? ONE : -ONE;
for (m = 0; m < rdata -> n; m++)
T *= (rdata -> gamma[m] - x) / (rdata -> gamma[m] + x);
return (ONE - T) / (ONE + T);
}
/* Test program. Apart from printing out the parameters for R(x) it produces
* the following data files for plotting (unless NPLOT is defined):
*
* zolotarev-fn is a plot of R(x) for |x| < 1.2. This should look like sgn(x).
*
* zolotarev-err is a plot of the error |R(x) - sgn(x)| scaled by 1/Delta. This
* should oscillate deg_num + deg_denom + 2 times between +1 and -1 over the
* domain epsilon <= |x| <= 1.
*
* If ALLPLOTS is defined then zolotarev-partfrac (zolotarev-contfrac) is a
* plot of the difference between the values of R(x) computed using the
* factored form and the partial fraction (continued fraction) form, scaled by
* 1/Delta. It should be zero everywhere. */
int main(int argc, char** argv) {
int m, n, plotpts = 5000, type = 0;
float eps, x, ypferr, ycferr, ycaylerr, maxypferr, maxycferr, maxycaylerr;
zolotarev_data *rdata;
PRECISION y;
FILE *plot_function, *plot_error,
*plot_partfrac, *plot_contfrac, *plot_cayley;
if (argc < 3 || argc > 4) {
fprintf(stderr, "Usage: %s epsilon n [type]\n", *argv);
exit(EXIT_FAILURE);
}
sscanf(argv[1], "%g", &eps); /* First argument is epsilon */
sscanf(argv[2], "%d", &n); /* Second argument is n */
if (argc == 4) sscanf(argv[3], "%d", &type); /* Third argument is type */
if (type < 0 || type > 2) {
fprintf(stderr, "%s: type must be 0 (Zolotarev R(0) = 0),\n"
"\t\t1 (Zolotarev R(0) = Inf, or 2 (Higham)\n", *argv);
exit(EXIT_FAILURE);
}
rdata = type == 2
? higham((PRECISION) eps, n)
: zolotarev((PRECISION) eps, n, type);
printf("Zolotarev Test: R(epsilon = %g, n = %d, type = %d)\n\t"
STRINGIFY(VERSION) "\n\t" STRINGIFY(HVERSION)
"\n\tINTERNAL_PRECISION = " STRINGIFY(INTERNAL_PRECISION)
"\tPRECISION = " STRINGIFY(PRECISION)
"\n\n\tRational approximation of degree (%d,%d), %s at x = 0\n"
"\tDelta = %g (maximum error)\n\n"
"\tA = %g (overall factor)\n",
(float) rdata -> epsilon, rdata -> n, type,
rdata -> deg_num, rdata -> deg_denom,
rdata -> type == 1 ? "infinite" : "zero",
(float) rdata -> Delta, (float) rdata -> A);
for (m = 0; m < MIN(rdata -> dd, rdata -> dn); m++)
printf("\ta[%2d] = %14.8g\t\ta'[%2d] = %14.8g\n",
m + 1, (float) rdata -> a[m], m + 1, (float) rdata -> ap[m]);
if (rdata -> dd > rdata -> dn)
printf("\t\t\t\t\ta'[%2d] = %14.8g\n",
rdata -> dn + 1, (float) rdata -> ap[rdata -> dn]);
if (rdata -> dd < rdata -> dn)
printf("\ta[%2d] = %14.8g\n",
rdata -> dd + 1, (float) rdata -> a[rdata -> dd]);
printf("\n\tPartial fraction coefficients\n");
printf("\talpha[ 0] = %14.8g\n",
(float) rdata -> alpha[rdata -> da - 1]);
for (m = 0; m < rdata -> dd; m++)
printf("\talpha[%2d] = %14.8g\ta'[%2d] = %14.8g\n",
m + 1, (float) rdata -> alpha[m], m + 1, (float) rdata -> ap[m]);
if (rdata -> type == 1)
printf("\talpha[%2d] = %14.8g\ta'[%2d] = %14.8g\n",
rdata -> dd + 1, (float) rdata -> alpha[rdata -> dd],
rdata -> dd + 1, (float) ZERO);
printf("\n\tContinued fraction coefficients\n");
for (m = 0; m < rdata -> db; m++)
printf("\tbeta[%2d] = %14.8g\n", m, (float) rdata -> beta[m]);
printf("\n\tCayley transform coefficients\n");
for (m = 0; m < rdata -> n; m++)
printf("\tgamma[%2d] = %14.8g\n", m, (float) rdata -> gamma[m]);
#ifndef NPLOT
plot_function = fopen("zolotarev-fn.dat", "w");
plot_error = fopen("zolotarev-err.dat", "w");
#ifdef ALLPLOTS
plot_partfrac = fopen("zolotarev-partfrac.dat", "w");
plot_contfrac = fopen("zolotarev-contfrac.dat", "w");
plot_cayley = fopen("zolotarev-cayley.dat", "w");
#endif /* ALLPLOTS */
for (m = 0, maxypferr = maxycferr = maxycaylerr = 0.0; m <= plotpts; m++) {
x = 2.4 * (float) m / plotpts - 1.2;
if (rdata -> type == 0 || fabs(x) * (float) plotpts > 1.0) {
/* skip x = 0 for type 1, as R(0) is singular */
y = zolotarev_eval((PRECISION) x, rdata);
fprintf(plot_function, "%g %g\n", x, (float) y);
fprintf(plot_error, "%g %g\n",
x, (float)((y - ((x > 0.0 ? ONE : -ONE))) / rdata -> Delta));
ypferr = (float)((zolotarev_partfrac_eval((PRECISION) x, rdata) - y)
/ rdata -> Delta);
ycferr = (float)((zolotarev_contfrac_eval((PRECISION) x, rdata) - y)
/ rdata -> Delta);
ycaylerr = (float)((zolotarev_cayley_eval((PRECISION) x, rdata) - y)
/ rdata -> Delta);
if (fabs(x) < 1.0 && fabs(x) > rdata -> epsilon) {
maxypferr = MAX(maxypferr, fabs(ypferr));
maxycferr = MAX(maxycferr, fabs(ycferr));
maxycaylerr = MAX(maxycaylerr, fabs(ycaylerr));
}
#ifdef ALLPLOTS
fprintf(plot_partfrac, "%g %g\n", x, ypferr);
fprintf(plot_contfrac, "%g %g\n", x, ycferr);
fprintf(plot_cayley, "%g %g\n", x, ycaylerr);
#endif /* ALLPLOTS */
}
}
#ifdef ALLPLOTS
fclose(plot_cayley);
fclose(plot_contfrac);
fclose(plot_partfrac);
#endif /* ALLPLOTS */
fclose(plot_error);
fclose(plot_function);
printf("\n\tMaximum PF error = %g (relative to Delta)\n", maxypferr);
printf("\tMaximum CF error = %g (relative to Delta)\n", maxycferr);
printf("\tMaximum Cayley error = %g (relative to Delta)\n", maxycaylerr);
#endif /* NPLOT */
free(rdata -> a);
free(rdata -> ap);
free(rdata -> alpha);
free(rdata -> beta);
free(rdata -> gamma);
free(rdata);
return EXIT_SUCCESS;
}
#endif /* TEST */

85
lib/algorithms/approx/zolotarev.h Executable file
View File

@ -0,0 +1,85 @@
/* -*- Mode: C; comment-column: 22; fill-column: 79; -*- */
#ifdef __cplusplus
extern "C" {
#endif
#define HVERSION Header Time-stamp: <14-OCT-2004 09:26:51.00 adk@MISSCONTRARY>
#ifndef ZOLOTAREV_INTERNAL
#ifndef PRECISION
#define PRECISION double
#endif
#define ZPRECISION PRECISION
#define ZOLOTAREV_DATA zolotarev_data
#endif
/* This struct contains the coefficients which parameterise an optimal rational
* approximation to the signum function.
*
* The parameterisations used are:
*
* Factored form for type 0 (R0(0) = 0)
*
* R0(x) = A * x * prod(x^2 - a[j], j = 0 .. dn-1) / prod(x^2 - ap[j], j = 0
* .. dd-1),
*
* where deg_num = 2*dn + 1 and deg_denom = 2*dd.
*
* Factored form for type 1 (R1(0) = infinity)
*
* R1(x) = (A / x) * prod(x^2 - a[j], j = 0 .. dn-1) / prod(x^2 - ap[j], j = 0
* .. dd-1),
*
* where deg_num = 2*dn and deg_denom = 2*dd + 1.
*
* Partial fraction form
*
* R(x) = alpha[da] * x + sum(alpha[j] * x / (x^2 - ap[j]), j = 0 .. da-1)
*
* where da = dd for type 0 and da = dd + 1 with ap[dd] = 0 for type 1.
*
* Continued fraction form
*
* R(x) = beta[db-1] * x + 1 / (beta[db-2] * x + 1 / (beta[db-3] * x + ...))
*
* with the final coefficient being beta[0], with d' = 2 * dd + 1 for type 0
* and db = 2 * dd + 2 for type 1.
*
* Cayley form (Chiu's domain wall formulation)
*
* R(x) = (1 - T(x)) / (1 + T(x))
*
* where T(x) = prod((x - gamma[j]) / (x + gamma[j]), j = 0 .. n-1)
*/
typedef struct {
ZPRECISION *a, /* zeros of numerator, a[0 .. dn-1] */
*ap, /* poles (zeros of denominator), ap[0 .. dd-1] */
A, /* overall factor */
*alpha, /* coefficients of partial fraction, alpha[0 .. da-1] */
*beta, /* coefficients of continued fraction, beta[0 .. db-1] */
*gamma, /* zeros of numerator of T in Cayley form */
Delta, /* maximum error, |R(x) - sgn(x)| <= Delta */
epsilon; /* minimum x value, epsilon < |x| < 1 */
int n, /* approximation degree */
type, /* 0: R(0) = 0, 1: R(0) = infinity */
dn, dd, da, db, /* number of elements of a, ap, alpha, and beta */
deg_num, /* degree of numerator = deg_denom +/- 1 */
deg_denom; /* degree of denominator */
} ZOLOTAREV_DATA;
#ifndef ZOLOTAREV_INTERNAL
/* zolotarev(epsilon, n, type) returns a pointer to an initialised
* zolotarev_data structure. The arguments must satisfy the constraints that
* epsilon > 0, n > 0, and type = 0 or 1. */
ZOLOTAREV_DATA* bfm_higham(PRECISION epsilon, int n) ;
ZOLOTAREV_DATA* bfm_zolotarev(PRECISION epsilon, int n, int type);
#endif
#ifdef __cplusplus
}
#endif

5
lib/qcd/Grid_qcd_wilson_dop.cc
View File

@ -290,10 +290,7 @@ void WilsonMatrix::MpcDagMpc(const LatticeFermion &in, LatticeFermion &out)
{
return;
}
void WilsonMatrix::MDagM (const LatticeFermion &in, LatticeFermion &out)
{
return;
}
}}

16
lib/qcd/Grid_qcd_wilson_dop.h
View File

@ -9,7 +9,7 @@ namespace Grid {
namespace QCD {
class WilsonMatrix : public LinearOperatorBase<LatticeFermion>
class WilsonMatrix : public SparseMatrixBase<LatticeFermion>
{
//NB r=1;
public:
@ -40,6 +40,11 @@ namespace Grid {
virtual void Mdag (const LatticeFermion &in, LatticeFermion &out);
virtual void MdagM(const LatticeFermion &in, LatticeFermion &out);
// half checkerboard operaions
void Mpc (const LatticeFermion &in, LatticeFermion &out);
void MpcDag (const LatticeFermion &in, LatticeFermion &out);
void MpcDagMpc(const LatticeFermion &in, LatticeFermion &out);
// non-hermitian hopping term; half cb or both
void Dhop(const LatticeFermion &in, LatticeFermion &out);
@ -48,17 +53,10 @@ namespace Grid {
typedef iScalar<iMatrix<vComplex, Nc> > matrix;
// half checkerboard operaions
void MpcDag (const LatticeFermion &in, LatticeFermion &out);
void Mpc (const LatticeFermion &in, LatticeFermion &out);
void MpcDagMpc(const LatticeFermion &in, LatticeFermion &out);
// full checkerboard hermitian
void MDagM (const LatticeFermion &in, LatticeFermion &out);
};
}
}
#endif