/usr/include/madness/mra/convolution1d.h is in libmadness-dev 0.10-3.
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This file is part of MADNESS.
Copyright (C) 2007,2010 Oak Ridge National Laboratory
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
For more information please contact:
Robert J. Harrison
Oak Ridge National Laboratory
One Bethel Valley Road
P.O. Box 2008, MS-6367
email: harrisonrj@ornl.gov
tel: 865-241-3937
fax: 865-572-0680
*/
#ifndef MADNESS_MRA_CONVOLUTION1D_H__INCLUDED
#define MADNESS_MRA_CONVOLUTION1D_H__INCLUDED
#include <madness/world/vector.h>
#include <madness/constants.h>
#include <limits.h>
#include <madness/tensor/tensor.h>
#include <madness/mra/simplecache.h>
#include <madness/mra/adquad.h>
#include <madness/mra/twoscale.h>
#include <madness/tensor/mtxmq.h>
#include <madness/tensor/aligned.h>
#include <madness/tensor/tensor_lapack.h>
#include <algorithm>
/// \file mra/convolution1d.h
/// \brief Compuates most matrix elements over 1D operators (including Gaussians)
/// \ingroup function
namespace madness {
void aligned_add(long n, double* restrict a, const double* restrict b);
void aligned_sub(long n, double* restrict a, const double* restrict b);
void aligned_add(long n, double_complex* restrict a, const double_complex* restrict b);
void aligned_sub(long n, double_complex* restrict a, const double_complex* restrict b);
template <typename T>
static void copy_2d_patch(T* restrict out, long ldout, const T* restrict in, long ldin, long n, long m) {
for (long i=0; i<n; ++i, out+=ldout, in+=ldin) {
for (long j=0; j<m; ++j) {
out[j] = in[j];
}
}
}
/// a(n,m) --> b(m,n) ... optimized for smallish matrices
template <typename T>
inline void fast_transpose(long n, long m, const T* a, T* restrict b) {
// n will always be k or 2k (k=wavelet order) and m will be anywhere
// from 2^(NDIM-1) to (2k)^(NDIM-1).
// for (long i=0; i<n; ++i)
// for (long j=0; j<m; ++j)
// b[j*n+i] = a[i*m+j];
// return;
if (n==1 || m==1) {
long nm=n*m;
for (long i=0; i<nm; ++i) b[i] = a[i];
return;
}
long n4 = (n>>2)<<2;
long m4 = m<<2;
const T* a0 = a;
for (long i=0; i<n4; i+=4, a0+=m4) {
const T* a1 = a0+m;
const T* a2 = a1+m;
const T* a3 = a2+m;
T* restrict bi = b+i;
for (long j=0; j<m; ++j, bi+=n) {
T tmp0 = a0[j];
T tmp1 = a1[j];
T tmp2 = a2[j];
T tmp3 = a3[j];
bi[0] = tmp0;
bi[1] = tmp1;
bi[2] = tmp2;
bi[3] = tmp3;
}
}
for (long i=n4; i<n; ++i)
for (long j=0; j<m; ++j)
b[j*n+i] = a[i*m+j];
}
/// a(i,j) --> b(i,j) for i=0..n-1 and j=0..r-1 noting dimensions are a(n,m) and b(n,r).
/// returns b
template <typename T>
inline T* shrink(long n, long m, long r, const T* a, T* restrict b) {
T* result = b;
if (r == 2) {
for (long i=0; i<n; ++i, a+=m, b+=r) {
b[0] = a[0];
b[1] = a[1];
}
}
else if (r == 4) {
for (long i=0; i<n; ++i, a+=m, b+=r) {
b[0] = a[0];
b[1] = a[1];
b[2] = a[2];
b[3] = a[3];
}
}
else {
MADNESS_ASSERT((r&0x1L)==0);
for (long i=0; i<n; ++i, a+=m, b+=r) {
for (long j=0; j<r; j+=2) {
b[j ] = a[j ];
b[j+1] = a[j+1];
}
}
}
return result;
}
/// actual data for 1 dimension and for 1 term and for 1 displacement for a convolution operator
/// here we keep the transformation matrices
/// !!! Note that if Rnormf is zero then ***ALL*** of the tensors are empty
template <typename Q>
struct ConvolutionData1D {
Tensor<Q> R, T; ///< if NS: R=ns, T=T part of ns; if modified NS: T=\uparrow r^(n-1)
Tensor<Q> RU, RVT, TU, TVT; ///< SVD approximations to R and T
Tensor<typename Tensor<Q>::scalar_type> Rs, Ts; ///< hold relative errors, NOT the singular values..
// norms for NS form
double Rnorm, Tnorm, Rnormf, Tnormf, NSnormf;
// norms for modified NS form
double N_up, N_diff, N_F; ///< the norms according to Beylkin 2008, Eq. (21) ff
/// ctor for NS form
/// make the operator matrices r^n and \uparrow r^(n-1)
/// @param[in] R operator matrix of the requested level; NS: unfilter(r^(n+1)); modified NS: r^n
/// @param[in] T upsampled operator matrix from level n-1; NS: r^n; modified NS: filter( r^(n-1) )
ConvolutionData1D(const Tensor<Q>& R, const Tensor<Q>& T) : R(R), T(T) {
Rnormf = R.normf();
// Making the approximations is expensive ... only do it for
// significant components
if (Rnormf > 1e-20) {
Tnormf = T.normf();
make_approx(T, TU, Ts, TVT, Tnorm);
make_approx(R, RU, Rs, RVT, Rnorm);
int k = T.dim(0);
Tensor<Q> NS = copy(R);
for (int i=0; i<k; ++i)
for (int j=0; j<k; ++j)
NS(i,j) = 0.0;
NSnormf = NS.normf();
}
else {
Rnorm = Tnorm = Rnormf = Tnormf = NSnormf = 0.0;
N_F = N_up = N_diff = 0.0;
}
}
/// ctor for modified NS form
/// make the operator matrices r^n and \uparrow r^(n-1)
/// @param[in] R operator matrix of the requested level; NS: unfilter(r^(n+1)); modified NS: r^n
/// @param[in] T upsampled operator matrix from level n-1; NS: r^n; modified NS: filter( r^(n-1) )
/// @param[in] modified use (un) modified NS form
ConvolutionData1D(const Tensor<Q>& R, const Tensor<Q>& T, const bool modified) : R(R), T(T) {
// note that R can be small, but T still be large
Rnormf = R.normf();
Tnormf = T.normf();
// Making the approximations is expensive ... only do it for
// significant components
if (Rnormf > 1e-20) make_approx(R, RU, Rs, RVT, Rnorm);
if (Tnormf > 1e-20) make_approx(T, TU, Ts, TVT, Tnorm);
// norms for modified NS form: follow Beylkin, 2008, Eq. (21) ff
N_F=Rnormf;
N_up=Tnormf;
N_diff=(R-T).normf();
}
/// approximate the operator matrices using SVD, and abuse Rs to hold the error instead of
/// the singular values (seriously, who named this??)
void make_approx(const Tensor<Q>& R,
Tensor<Q>& RU, Tensor<typename Tensor<Q>::scalar_type>& Rs, Tensor<Q>& RVT, double& norm) {
int n = R.dim(0);
svd(R, RU, Rs, RVT);
for (int i=0; i<n; ++i) {
for (int j=0; j<n; ++j) {
RVT(i,j) *= Rs[i];
}
}
for (int i=n-1; i>1; --i) { // Form cumulative sum of norms
Rs[i-1] += Rs[i];
}
norm = Rs[0];
if (Rs[0]>0.0) { // Turn into relative errors
double rnorm = 1.0/norm;
for (int i=0; i<n; ++i) {
Rs[i] *= rnorm;
}
}
}
};
/// Provides the common functionality/interface of all 1D convolutions
/// interface for 1 term and for 1 dimension;
/// the actual data are kept in ConvolutionData1D
/// Derived classes must implement rnlp, issmall, natural_level
template <typename Q>
class Convolution1D {
public:
typedef Q opT; ///< The apply function uses this to infer resultT=opT*inputT
int k; ///< Wavelet order
int npt; ///< Number of quadrature points (is this used?)
int maxR; ///< Number of lattice translations for sum
Tensor<double> quad_x;
Tensor<double> quad_w;
Tensor<double> c;
Tensor<double> hgT, hg;
Tensor<double> hgT2k;
double arg;
mutable SimpleCache<Tensor<Q>, 1> rnlp_cache;
mutable SimpleCache<Tensor<Q>, 1> rnlij_cache;
mutable SimpleCache<ConvolutionData1D<Q>, 1> ns_cache;
mutable SimpleCache<ConvolutionData1D<Q>, 2> mod_ns_cache;
virtual ~Convolution1D() {};
Convolution1D(int k, int npt, int maxR, double arg = 0.0)
: k(k)
, npt(npt)
, maxR(maxR)
, quad_x(npt)
, quad_w(npt)
, arg(arg)
{
MADNESS_ASSERT(autoc(k,&c));
gauss_legendre(npt,0.0,1.0,quad_x.ptr(),quad_w.ptr());
MADNESS_ASSERT(two_scale_hg(k,&hg));
hgT = transpose(hg);
MADNESS_ASSERT(two_scale_hg(2*k,&hgT2k));
hgT2k = transpose(hgT2k);
// Cannot construct the coefficients here since the
// derived class is not yet constructed so cannot call
// (even indirectly) a virtual method
}
/// Compute the projection of the operator onto the double order polynomials
virtual Tensor<Q> rnlp(Level n, Translation lx) const = 0;
/// Returns true if the block of rnlp is expected to be small
virtual bool issmall(Level n, Translation lx) const = 0;
/// Returns true if the block of rnlp is expected to be small including periodicity
bool get_issmall(Level n, Translation lx) const {
if (maxR == 0) {
return issmall(n, lx);
}
else {
Translation twon = Translation(1)<<n;
for (int R=-maxR; R<=maxR; ++R) {
if (!issmall(n, R*twon+lx)) return false;
}
return true;
}
}
/// Returns the level for projection
//virtual Level natural_level() const {
// return 13;
//}
virtual Level natural_level() const {return 13;}
/// Computes the transition matrix elements for the convolution for n,l
/// Returns the tensor
/// \code
/// r(i,j) = int(K(x-y) phi[n0](x) phi[nl](y), x=0..1, y=0..1)
/// \endcode
/// This is computed from the matrix elements over the correlation
/// function which in turn are computed from the matrix elements
/// over the double order legendre polynomials.
const Tensor<Q>& rnlij(Level n, Translation lx, bool do_transpose=false) const {
const Tensor<Q>* p=rnlij_cache.getptr(n,lx);
if (p) return *p;
// PROFILE_MEMBER_FUNC(Convolution1D); // Too fine grain for routine profiling
long twok = 2*k;
Tensor<Q> R(2*twok);
R(Slice(0,twok-1)) = get_rnlp(n,lx-1);
R(Slice(twok,2*twok-1)) = get_rnlp(n,lx);
R.scale(pow(0.5,0.5*n));
R = inner(c,R);
if (do_transpose) R = transpose(R);
rnlij_cache.set(n,lx,R);
return *rnlij_cache.getptr(n,lx);
};
/// Returns a pointer to the cached modified nonstandard form of the operator
/// @param[in] op_key holds the scale and the source and target translations
/// @return a pointer to the cached modified nonstandard form of the operator
const ConvolutionData1D<Q>* mod_nonstandard(const Key<2>& op_key) const {
const Level& n=op_key.level();
const Translation& sx=op_key.translation()[0]; // source translation
const Translation& tx=op_key.translation()[1]; // target translation
const Translation lx=tx-sx; // displacement
const Translation s_off=sx%2;
const Translation t_off=tx%2;
// we cache translation and source offset
const Key<2> cache_key(n, Vector<Translation,2>{lx, s_off} );
const ConvolutionData1D<Q>* p = mod_ns_cache.getptr(cache_key);
if (p) return p;
// for paranoid me
MADNESS_ASSERT(sx>=0 and tx>=0);
Tensor<Q> R, T, Rm;
// if (!get_issmall(n, lx)) {
// print("no issmall", lx, source, n);
const Translation lx_half = tx/2 - sx/2;
const Slice s0(0,k-1), s1(k,2*k-1);
// print("sx, tx",lx,lx_half,sx, tx,"off",s_off,t_off);
// this is the operator matrix in its actual level
R = rnlij(n,lx);
// this is the upsampled operator matrix
Rm = Tensor<Q>(2*k,2*k);
if (n>0) Rm(s0,s0)=rnlij(n-1,lx_half);
{
// PROFILE_BLOCK(Convolution1D_nstran); // Too fine grain for routine profiling
Rm = transform(Rm,hg);
}
{
// PROFILE_BLOCK(Convolution1D_nscopy); // Too fine grain for routine profiling
T=Tensor<Q>(k,k);
if (t_off==0 and s_off==0) T=copy(Rm(s0,s0));
if (t_off==0 and s_off==1) T=copy(Rm(s0,s1));
if (t_off==1 and s_off==0) T=copy(Rm(s1,s0));
if (t_off==1 and s_off==1) T=copy(Rm(s1,s1));
// if (t_off==0 and s_off==0) T=copy(Rm(s0,s0));
// if (t_off==1 and s_off==0) T=copy(Rm(s0,s1));
// if (t_off==0 and s_off==1) T=copy(Rm(s1,s0));
// if (t_off==1 and s_off==1) T=copy(Rm(s1,s1));
}
{
// PROFILE_BLOCK(Convolution1D_trans); // Too fine grain for routine profiling
Tensor<Q> RT(k,k), TT(k,k);
fast_transpose(k,k,R.ptr(), RT.ptr());
fast_transpose(k,k,T.ptr(), TT.ptr());
R = RT;
T = TT;
}
// }
mod_ns_cache.set(cache_key,ConvolutionData1D<Q>(R,T,true));
return mod_ns_cache.getptr(cache_key);
}
/// Returns a pointer to the cached nonstandard form of the operator
const ConvolutionData1D<Q>* nonstandard(Level n, Translation lx) const {
const ConvolutionData1D<Q>* p = ns_cache.getptr(n,lx);
if (p) return p;
// PROFILE_MEMBER_FUNC(Convolution1D); // Too fine grain for routine profiling
Tensor<Q> R, T;
if (!get_issmall(n, lx)) {
Translation lx2 = lx*2;
#if 0 // UNUSED VARIABLES
Slice s0(0,k-1), s1(k,2*k-1);
#endif
const Tensor<Q> r0 = rnlij(n+1,lx2);
const Tensor<Q> rp = rnlij(n+1,lx2+1);
const Tensor<Q> rm = rnlij(n+1,lx2-1);
R = Tensor<Q>(2*k,2*k);
// R(s0,s0) = r0;
// R(s1,s1) = r0;
// R(s1,s0) = rp;
// R(s0,s1) = rm;
{
// PROFILE_BLOCK(Convolution1D_nscopy); // Too fine grain for routine profiling
copy_2d_patch(R.ptr(), 2*k, r0.ptr(), k, k, k);
copy_2d_patch(R.ptr()+2*k*k + k, 2*k, r0.ptr(), k, k, k);
copy_2d_patch(R.ptr()+2*k*k, 2*k, rp.ptr(), k, k, k);
copy_2d_patch(R.ptr() + k, 2*k, rm.ptr(), k, k, k);
}
//print("R ", n, lx, R.normf(), r0.normf(), rp.normf(), rm.normf());
{
// PROFILE_BLOCK(Convolution1D_nstran); // Too fine grain for routine profiling
R = transform(R,hgT);
}
//print("RX", n, lx, R.normf(), r0.normf(), rp.normf(), rm.normf());
{
// PROFILE_BLOCK(Convolution1D_trans); // Too fine grain for routine profiling
Tensor<Q> RT(2*k,2*k);
fast_transpose(2*k, 2*k, R.ptr(), RT.ptr());
R = RT;
//print("RT", n, lx, R.normf(), r0.normf(), rp.normf(), rm.normf());
//T = copy(R(s0,s0));
T = Tensor<Q>(k,k);
copy_2d_patch(T.ptr(), k, R.ptr(), 2*k, k, k);
}
//print("NS", n, lx, R.normf(), T.normf());
}
ns_cache.set(n,lx,ConvolutionData1D<Q>(R,T));
return ns_cache.getptr(n,lx);
};
Q phase(double R) const {
return 1.0;
}
Q phase(double_complex R) const {
return exp(double_complex(0.0,arg)*R);
}
const Tensor<Q>& get_rnlp(Level n, Translation lx) const {
const Tensor<Q>* p=rnlp_cache.getptr(n,lx);
if (p) return *p;
// PROFILE_MEMBER_FUNC(Convolution1D); // Too fine grain for routine profiling
long twok = 2*k;
Tensor<Q> r;
if (get_issmall(n, lx)) {
r = Tensor<Q>(twok);
}
else if (n < natural_level()) {
Tensor<Q> R(2*twok);
R(Slice(0,twok-1)) = get_rnlp(n+1,2*lx);
R(Slice(twok,2*twok-1)) = get_rnlp(n+1,2*lx+1);
R = transform(R, hgT2k);
r = copy(R(Slice(0,twok-1)));
}
else {
// PROFILE_BLOCK(Convolution1Drnlp); // Too fine grain for routine profiling
if (maxR > 0) {
Translation twon = Translation(1)<<n;
r = Tensor<Q>(2*k);
for (int R=-maxR; R<=maxR; ++R) {
r.gaxpy(1.0, rnlp(n,R*twon+lx), phase(Q(R)));
}
}
else {
r = rnlp(n, lx);
}
}
rnlp_cache.set(n, lx, r);
//print(" SET rnlp", n, lx, r);
return *rnlp_cache.getptr(n,lx);
}
};
/// Array of 1D convolutions (one / dimension)
/// data for 1 term and all dimensions
template <typename Q, int NDIM>
class ConvolutionND {
std::array<std::shared_ptr<Convolution1D<Q> >, NDIM> ops;
Q fac;
public:
ConvolutionND() : fac(1.0) {}
ConvolutionND(const ConvolutionND& other) : fac(other.fac)
{
ops = other.ops;
}
ConvolutionND(std::shared_ptr<Convolution1D<Q> > op, Q fac=1.0) : fac(fac)
{
std::fill(ops.begin(), ops.end(), op);
}
void setop(int dim, const std::shared_ptr<Convolution1D<Q> >& op) {
ops[dim] = op;
}
std::shared_ptr<Convolution1D<Q> > getop(int dim) const {
return ops[dim];
}
void setfac(Q value) {
fac = value;
}
Q getfac() const {
return fac;
}
};
// To test generic convolution by comparing with GaussianConvolution1D
template <typename Q>
class GaussianGenericFunctor {
private:
Q coeff;
double exponent;
int m;
Level natlev;
public:
// coeff * exp(-exponent*x^2) * x^m
GaussianGenericFunctor(Q coeff, double exponent, int m=0)
: coeff(coeff), exponent(exponent), m(m),
natlev(Level(0.5*log(exponent)/log(2.0)+1)) {}
Q operator()(double x) const {
Q ee = coeff*exp(-exponent*x*x);
for (int mm=0; mm<m; ++mm) ee *= x;
return ee;
}
Level natural_level() const {return natlev;}
};
/// Generic 1D convolution using brute force (i.e., slow) adaptive quadrature for rnlp
/// Calls op(x) with x in *simulation coordinates* to evaluate the function.
template <typename Q, typename opT>
class GenericConvolution1D : public Convolution1D<Q> {
private:
opT op;
long maxl; ///< At natural level is l beyond which operator is zero
public:
GenericConvolution1D() {}
GenericConvolution1D(int k, const opT& op, int maxR, double arg = 0.0)
: Convolution1D<Q>(k, 20, maxR, arg), op(op), maxl(LONG_MAX-1) {
// PROFILE_MEMBER_FUNC(GenericConvolution1D); // Too fine grain for routine profiling
// For efficiency carefully compute outwards at the "natural" level
// until several successive boxes are determined to be zero. This
// then defines the future range of the operator and also serves
// to precompute the values used in the rnlp cache.
Level natl = natural_level();
int nzero = 0;
for (Translation lx=0; lx<(1L<<natl); ++lx) {
const Tensor<Q>& rp = this->get_rnlp(natl, lx);
const Tensor<Q>& rm = this->get_rnlp(natl,-lx);
if (rp.normf()<1e-12 && rm.normf()<1e-12) ++nzero;
if (nzero == 3) {
maxl = lx-2;
break;
}
}
}
virtual Level natural_level() const {return op.natural_level();}
struct Shmoo {
typedef Tensor<Q> returnT;
Level n;
Translation lx;
const GenericConvolution1D<Q,opT>& q;
Shmoo(Level n, Translation lx, const GenericConvolution1D<Q,opT>* q)
: n(n), lx(lx), q(*q) {}
returnT operator()(double x) const {
int twok = q.k*2;
double fac = std::pow(0.5,n);
double phix[twok];
legendre_scaling_functions(x-lx,twok,phix);
Q f = q.op(fac*x)*sqrt(fac);
returnT v(twok);
for (long p=0; p<twok; ++p) v(p) += f*phix[p];
return v;
}
};
Tensor<Q> rnlp(Level n, Translation lx) const {
return adq1(lx, lx+1, Shmoo(n, lx, this), 1e-12,
this->npt, this->quad_x.ptr(), this->quad_w.ptr(), 0);
}
bool issmall(Level n, Translation lx) const {
if (lx < 0) lx = 1 - lx;
// Always compute contributions to nearest neighbor coupling
// ... we are two levels below so 0,1 --> 0,1,2,3 --> 0,...,7
if (lx <= 7) return false;
n = natural_level()-n;
if (n >= 0) lx = lx << n;
else lx = lx >> n;
return lx >= maxl;
}
};
/// 1D convolution with (derivative) Gaussian; coeff and expnt given in *simulation* coordinates [0,1]
/// Note that the derivative is computed in *simulation* coordinates so
/// you must be careful to scale the coefficients correctly.
template <typename Q>
class GaussianConvolution1D : public Convolution1D<Q> {
// Returns range of Gaussian for periodic lattice sum in simulation coords
static int maxR(bool periodic, double expnt) {
if (periodic) {
return std::max(1,int(sqrt(16.0*2.3/expnt)+1));
}
else {
return 0;
}
}
public:
const Q coeff; ///< Coefficient
const double expnt; ///< Exponent
const Level natlev; ///< Level to evaluate
const int m; ///< Order of derivative (0, 1, or 2 only)
explicit GaussianConvolution1D(int k, Q coeff, double expnt,
int m, bool periodic, double arg = 0.0)
: Convolution1D<Q>(k,k+11,maxR(periodic,expnt),arg)
, coeff(coeff)
, expnt(expnt)
, natlev(Level(0.5*log(expnt)/log(2.0)+1))
, m(m)
{
MADNESS_ASSERT(m>=0 && m<=2);
// std::cout << "GC expnt=" << expnt << " coeff=" << coeff << " natlev=" << natlev << " maxR=" << maxR(periodic,expnt) << std::endl;
// for (Level n=0; n<5; n++) {
// for (Translation l=0; l<(1<<n); l++) {
// std::cout << "RNLP " << n << " " << l << " " << this->get_rnlp(n,l).normf() << std::endl;
// }
// std::cout << std::endl;
// }
}
virtual ~GaussianConvolution1D() {}
virtual Level natural_level() const {
return natlev;
}
/// Compute the projection of the operator onto the double order polynomials
/// The returned reference is to a cached tensor ... if you want to
/// modify it, take a copy first.
///
/// Return in \c v[p] \c p=0..2*k-1
/// \code
/// r(n,l,p) = 2^(-n) * int(K(2^(-n)*(z+l)) * phi(p,z), z=0..1)
/// \endcode
/// The kernel is coeff*exp(-expnt*z^2)*z^m (with m>0). This is equivalent to
/// \code
/// r(n,l,p) = 2^(-n*(m+1))*coeff * int( ((d/dz)^m exp(-beta*z^2)) * phi(p,z-l), z=l..l+1)
/// \endcode
/// where
/// \code
/// beta = alpha * 2^(-2*n)
/// \endcode
Tensor<Q> rnlp(Level n, Translation lx) const {
int twok = 2*this->k;
Tensor<Q> v(twok); // Can optimize this away by passing in
Translation lkeep = lx;
if (lx<0) lx = -lx-1;
/* Apply high-order Gauss Legendre onto subintervals
coeff*int(exp(-beta(x+l)**2) * z^m * phi[p](x),x=0..1);
The translations internally considered are all +ve, so
signficant pieces will be on the left. Finish after things
become insignificant.
The resulting coefficients are accurate to about 1e-20.
*/
// Rescale expnt & coeff onto level n so integration range
// is [l,l+1]
Q scaledcoeff = coeff*pow(0.5,0.5*n*(2*m+1));
// Subdivide interval into nbox boxes of length h
// ... estimate appropriate size from the exponent. A
// Gaussian with real-part of the exponent beta falls in
// magnitude by a factor of 1/e at x=1/sqrt(beta), and by
// a factor of e^-49 ~ 5e-22 at x=7/sqrt(beta) (and with
// polyn of z^2 it is 1e-20). So, if we use a box of size
// 1/sqrt(beta) we will need at most 7 boxes. Incorporate
// the coefficient into the screening since it may be
// large. We can represent exp(-x^2) over [l,l+1] with a
// polynomial of order 21 to a relative
// precision of better than machine precision for
// l=0,1,2,3 and for l>3 the absolute error is less than
// 1e-23. We want to compute matrix elements with
// polynomials of order 2*k-1+m, so the total order is
// 2*k+20+m, which can be integrated with a quadrature rule
// of npt=k+11+(m+1)/2. npt is set in the constructor.
double fourn = std::pow(4.0,double(n));
double beta = expnt * pow(0.25,double(n));
double h = 1.0/sqrt(beta); // 2.0*sqrt(0.5/beta);
long nbox = long(1.0/h);
if (nbox < 1) nbox = 1;
h = 1.0/nbox;
// Find argmax such that h*scaledcoeff*exp(-argmax)=1e-22 ... if
// beta*xlo*xlo is already greater than argmax we can neglect this
// and subsequent boxes.
// The derivatives add a factor of expnt^m to the size of
// the function at long range.
double sch = std::abs(scaledcoeff*h);
if (m == 1) sch *= expnt;
else if (m == 2) sch *= expnt*expnt;
double argmax = std::abs(log(1e-22/sch)); // perhaps should be -log(1e-22/sch) ?
for (long box=0; box<nbox; ++box) {
double xlo = box*h + lx;
if (beta*xlo*xlo > argmax) break;
for (long i=0; i<this->npt; ++i) {
#ifdef IBMXLC
double phix[80];
#else
double phix[twok];
#endif
double xx = xlo + h*this->quad_x(i);
Q ee = scaledcoeff*exp(-beta*xx*xx)*this->quad_w(i)*h;
// Differentiate as necessary
if (m == 1) {
ee *= -2.0*expnt*xx;
}
else if (m == 2) {
ee *= (4.0*xx*xx*expnt*expnt - 2.0*expnt*fourn);
}
legendre_scaling_functions(xx-lx,twok,phix);
for (long p=0; p<twok; ++p) v(p) += ee*phix[p];
}
}
if (lkeep < 0) {
/* phi[p](1-z) = (-1)^p phi[p](z) */
if (m == 1)
for (long p=0; p<twok; ++p) v(p) = -v(p);
for (long p=1; p<twok; p+=2) v(p) = -v(p);
}
return v;
};
/// Returns true if the block is expected to be small
bool issmall(Level n, Translation lx) const {
double beta = expnt * pow(0.25,double(n));
Translation ll;
if (lx > 0)
ll = lx - 1;
else if (lx < 0)
ll = -1 - lx;
else
ll = 0;
return (beta*ll*ll > 49.0); // 49 -> 5e-22 69 -> 1e-30
};
};
template <typename Q>
struct GaussianConvolution1DCache {
static ConcurrentHashMap<hashT, std::shared_ptr< GaussianConvolution1D<Q> > > map;
typedef typename ConcurrentHashMap<hashT, std::shared_ptr< GaussianConvolution1D<Q> > >::iterator iterator;
typedef typename ConcurrentHashMap<hashT, std::shared_ptr< GaussianConvolution1D<Q> > >::datumT datumT;
static std::shared_ptr< GaussianConvolution1D<Q> > get(int k, double expnt, int m, bool periodic) {
hashT key = hash_value(expnt);
hash_combine(key, k);
hash_combine(key, m);
hash_combine(key, int(periodic));
iterator it = map.find(key);
if (it == map.end()) {
map.insert(datumT(key, std::shared_ptr< GaussianConvolution1D<Q> >(new GaussianConvolution1D<Q>(k,
Q(sqrt(expnt/constants::pi)),
expnt,
m,
periodic
))));
it = map.find(key);
//printf("conv1d: making %d %.8e\n",k,expnt);
}
else {
//printf("conv1d: reusing %d %.8e\n",k,expnt);
}
return it->second;
}
};
}
#endif // MADNESS_MRA_CONVOLUTION1D_H__INCLUDED
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