abacus-develop/source/source_base/math_integral.cpp at develop · AroundPeking/abacus-develop · GitHub

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#include "math_integral.h"

#include <stddef.h> // use size_t

#include <cassert>

#include <algorithm>

#include <functional>

#include <cmath>

#include "constants.h"

namespace ModuleBase

{

Integral::Integral(){}

Integral::~Integral(){}

// Peize Lin accelerate 2017-10-02

void Integral::Simpson_Integral

(

const int mesh,

const double *func,

const double *rab,

double &asum

)

{

// simpson's rule integration. On input:

// mesh = mhe number of grid points (should be odd)

// func(i)= function to be integrated

// rab(i) = r(i) * dr(i)/di * di

// For the logarithmic grid not including r=0 :

// r(i) = r_0*exp((i-1)*dx) ==> rab(i)=r(i)*dx

// For the logarithmic grid including r=0 :

// r(i) = a(exp((i-1)*dx)-1) ==> rab(i)=(r(i)+a)*dx

// Output in asum = \sum_i c_i f(i)*rab(i) = \int_0^\infty f(r) dr

// where c_i are alternativaly 2/3, 4/3 except c_1 = c_mesh = 1/3

// simpson's rule integrator for function stored on the

// radial logarithmic mesh

// routine assumes that mesh is an odd number so run check

if (mesh % 2 == 0)

{

std::cout << "\n error in subroutine simpson ";

std::cout << "\n routine assumes mesh is odd but mesh = "

<< mesh << std::endl;

return;

}

asum = 0.00;

const double r12 = 1.00 / 12.00;

double f3 = func [0] * rab [0] * r12;

for (int i = 1;i < mesh;i += 2)

{

const double f1 = f3;

const double f2 = func [i] * rab [i] * r12;

f3 = func [i + 1] * rab [i + 1] * r12;

asum += 4.00 * f1 + 16.00 * f2 + 4.00 * f3;

}

return;

}// end subroutine simpson

// Peize Lin accelerate 2017-10-02

void Integral::Simpson_Integral

(

const int mesh,

const double * const func,

const double * const rab,

double &asum

)

{

/* simpson's rule integration. On input:

! mesh = mhe number of grid points (should be odd)

! func(i)= function to be integrated

! rab(i) = r(i) * dr(i)/di * di

! For the logarithmic grid not including r=0 :

! r(i) = r_0*exp((i-1)*dx) ==> rab(i)=r(i)*dx

! For the logarithmic grid including r=0 :

! r(i) = a(exp((i-1)*dx)-1) ==> rab(i)=(r(i)+a)*dx

! Output in asum = \sum_i c_i f(i)*rab(i) = \int_0^\infty f(r) dr

! where c_i are alternativaly 2/3, 4/3 except c_1 = c_mesh = 1/3

// simpson's rule integrator for function stored on the

// radial logarithmic mesh

// routine assumes that mesh is an odd number so run check

assert(mesh&1);

asum = 0.00;

const size_t end = mesh-2;

for( size_t i=1; i!=end; i+=2 )

{

const double f1 = func[i]*rab[i];

asum += f1 + f1 + func[i+1]*rab[i+1];

}

const double f1 = func[mesh-2]*rab[mesh-2];

asum += f1+f1;

asum += asum;

asum += func[0]*rab[0] + func[mesh-1]*rab[mesh-1];

asum /= 3.0;

return;

}// end subroutine simpson

// Peize Lin accelerate 2017-10-02

void Integral::Simpson_Integral

(

const int mesh,

const double * const func,

const double dr,

double &asum

)

{

/* simpson's rule integration. On input:

! mesh = mhe number of grid points (should be odd)

! func(i)= function to be integrated

! rab(i) = r(i) * dr(i)/di * di

! For the logarithmic grid not including r=0 :

! r(i) = r_0*exp((i-1)*dx) ==> rab(i)=r(i)*dx

! For the logarithmic grid including r=0 :

! r(i) = a(exp((i-1)*dx)-1) ==> rab(i)=(r(i)+a)*dx

! Output in asum = \sum_i c_i f(i)*rab(i) = \int_0^\infty f(r) dr

! where c_i are alternativaly 2/3, 4/3 except c_1 = c_mesh = 1/3

// simpson's rule integrator for function stored on the

// radial logarithmic mesh

// routine assumes that mesh is an odd number so run check

assert(mesh&1);

asum = 0.00;

const size_t end = mesh-2;

for(size_t i=1; i!=end; i+=2 )

{

const double f1 = func[i];

asum += f1 + f1 + func[i+1];

}

const double f1 = func[mesh-2];

asum += f1+f1;

asum += asum;

asum += func[0] + func[mesh-1];

asum *= dr/3.0;

return;

}// end subroutine simpson

// Peize Lin add 2016-02-14

void Integral::Simpson_Integral_0toall

(

const int mesh,

const double * const func,

const double * const rab,

double * const asum

)

{

// asum(r) = \int_{r'=0}^{r} dr' f(r')

const double r2=1.00/2.00, r3=1.00/3.00;

asum[0] = 0.00;

double f3 = func [0] * rab [0];

for( int i=1; i<mesh; i+=2)

{

const double f1 = f3;

const double f2 = func[i] * rab[i] ;

f3 = func[i+1] * rab[i+1] ;

asum[i] = asum[i-1] + r2*( f1 + f2);

if(i+1<mesh)

{

asum[i+1] = asum[i-1] + r3*( f1 + 4.00*f2 + f3 );

}

return;

}

// Peize Lin add 2016-02-14

// faster but still have bug

/*void Integral::Simpson_Integral_alltoinf

(

const int mesh,

const double *func,

const double *rab,

double *asum

)

{

// asum(r) = \int_{r'=r}^{+\infty} dr' f(r')

// = \inf_{r'=r}^{mesh} dr' f(r')

const double r2=1.00/2.00, r3=1.00/3.00;

asum[mesh-1] = 0.00;

const int odd_mesh = (mesh-1)^~1;

double f1 = func[odd_mesh] * rab[odd_mesh];

for( size_t i=(mesh-3)|1; i>0; i-=2)

{

const double f3 = f1;

if( i+3==mesh )

{

const double f4 = func[mesh-1] * rab[mesh-1];

asum[mesh-2] = r2*(f3 + f4);

}

const double f2 = func[i] * rab[i] ;

f1 = func[i-1] * rab[i-1] ;

asum[i-1] = asum[i+1] + r3*( f1 + 4.00*f2 + f3 );

asum[i] = asum[i-1] - r2*( f1 + f2);

}

return;

}*/

// Peize Lin add 2016-06-11

// a little lower

void Integral::Simpson_Integral_alltoinf

(

const int mesh,

const double * const func,

const double * const rab,

double * const asum

)

{

Integral::Simpson_Integral_0toall( mesh, func, rab, asum );

const double asum_all = asum[mesh-1];

for (int i = 0;i < mesh; ++i)

{

asum[i] = asum_all - asum[i];

}

return;

}

double Integral::simpson(const int n, const double* const f, const double dx)

{

assert(n >= 2);

if (n == 4)

{ // Simpson's 3/8 rule

return 3.0 * dx / 8 * (f[0] + 3.0 * f[1] + 3.0 * f[2] + f[3]);

}

if (n == 2)

{

return 0.5 * dx * (f[0] + f[1]);

}

if (n % 2 == 1)

{ // composite Simpson's 1/3 rule

double sum = 0.0;

for (int i = 1; i != n-2; i += 2)

{

sum += 2.0 * f[i] + f[i+1];

}

sum += 2.0 * f[n-2];

sum *= 2.0;

sum += f[0] + f[n-1];

return sum * dx / 3.0;

}

else

{ // composite Simpson's 1/3 rule for the first n-4 intervals plus Simpson's 3/8 rule for the last 3 intervals

return simpson(n-3, f, dx) + simpson(4, &f[n-4], dx);

}

double Integral::simpson(const int n, const double* const f, const double* const h)

{

// Simpson's rule for irregularly-spaced grid

// The treatment for even number of grid points is the same as that of the regularly-spaced grid case.

assert( n >= 2 );

assert( std::all_of(h, h+(n-1), [](double x){return x > 0.0;}) );

if (n == 4)

{

double w = h[0] + h[1] + h[2];

return w / 12.0 * ( 2.0 + ((h[1]+h[2])/h[0]-1.0) * (h[2]/(h[0]+h[1])-1.0) ) * f[0]

+ std::pow(w,3) / 12.0 * (h[0]+h[1]-h[2]) / (h[0]*h[1]*(h[1]+h[2])) * f[1]

+ std::pow(w,3) / 12.0 * (h[2]+h[1]-h[0]) / (h[2]*h[1]*(h[1]+h[0])) * f[2]

+ w / 12.0 * ( 2.0 + ((h[1]+h[0])/h[2]-1.0) * (h[0]/(h[2]+h[1])-1.0) ) * f[3];

}

if (n == 2)

{

return 0.5 * h[0] * (f[0] + f[1]);

}

if (n % 2 == 1)

{

double sum = 0.0;

for (int i = 0; i < n/2; ++i)

{

double hrp = h[2*i+1] / h[2*i];

double hrm = h[2*i] / h[2*i+1];

sum += (h[2*i+1] + h[2*i]) / 6.0 * ( (2.0-hrp)*f[2*i] + (2.0+hrp+hrm)*f[2*i+1] + (2.0-hrm) * f[2*i+2]);

}

return sum;

}

else

{

return simpson(n-3, f, h) + simpson(4, &f[n-4], &h[n-4]);

}

void Integral::Gauss_Legendre_grid_and_weight(const int n, double *x, double *weights)

{

assert( n >= 1 );

double z = 0.0;

double z1 = 0.0;

double p1 = 1.0;

double p2 = 0.0;

double p3 = 0.0;

double pp = 0.0;

int half_grid_num = static_cast<int>((n+1)/2);

for(int i = 1; i <= half_grid_num; i++)

{

z = cos(ModuleBase::PI * (i - 0.25) / (n + 0.5));

while(true)

{

p1 = 1.0;

p2 = 0.0;

for(int j = 1; j <= n; j++)

{

p3 = p2;

p2 = p1;

p1 = ((2.0 * j - 1.0) * z * p2 - (j - 1.0) * p3) / j;

}

pp = n * (p2 - z * p1) / (1.0 - z*z);

z1 = z;

z = z1 - p1 / pp;

if (std::abs(z - z1) < 1e-13) break;

}

x[i-1] = -z;

x[n-i] = z;

weights[i-1] = 2.0 / ((1.0 - z * z) * pp * pp);

weights[n-i] = weights[i-1];

}

void Integral::Gauss_Legendre_grid_and_weight(const double xmin, const double xmax, const int n, double *x, double *weights)

{

assert( n >= 1 );

double xl = (xmax - xmin) * 0.5;

double xmean = (xmax + xmin) * 0.5;

double z = 0.0;

double z1 = 0.0;

double p1 = 1.0;

double p2 = 0.0;

double p3 = 0.0;

double pp = 0.0;

int half_grid_num = static_cast<int>((n+1)/2);

for(int i = 1; i <= half_grid_num; i++)

{

z = cos(ModuleBase::PI * (i - 0.25) / (n + 0.5));

while(true)

{

p1 = 1.0;

p2 = 0.0;

for(int j = 1; j <= n; j++)

{

p3 = p2;

p2 = p1;

p1 = ((2.0 * j - 1.0) * z * p2 - (j - 1.0) * p3) / j;

}

pp = n * (p2 - z * p1) / (1.0 - z*z);

z1 = z;

z = z1 - p1 / pp;

if (std::abs(z - z1) < 1e-13) break;

}

x[i-1] = xmean - xl * z;

x[n-i] = xmean + xl * z;

weights[i-1] = 2.0 * xl / ((1.0 - z * z) * pp * pp);

weights[n-i] = weights[i-1];

}

const double Integral::Lebedev_Laikov_grid110_x[110] = {

1.000000000000000, -1.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.577350269189626, -0.577350269189626,

0.577350269189626, -0.577350269189626, 0.577350269189626, -0.577350269189626, 0.577350269189626, -0.577350269189626, 0.185115635344736, -0.185115635344736,

0.185115635344736, -0.185115635344736, 0.185115635344736, -0.185115635344736, 0.185115635344736, -0.185115635344736, 0.185115635344736, -0.185115635344736,

0.185115635344736, -0.185115635344736, 0.185115635344736, -0.185115635344736, 0.185115635344736, -0.185115635344736, 0.965124035086594, -0.965124035086594,

0.965124035086594, -0.965124035086594, 0.965124035086594, -0.965124035086594, 0.965124035086594, -0.965124035086594, 0.690421048382292, -0.690421048382292,

0.690421048382292, -0.690421048382292, 0.690421048382292, -0.690421048382292, 0.690421048382292, -0.690421048382292, 0.690421048382292, -0.690421048382292,

0.690421048382292, -0.690421048382292, 0.690421048382292, -0.690421048382292, 0.690421048382292, -0.690421048382292, 0.215957291845848, -0.215957291845848,

0.215957291845848, -0.215957291845848, 0.215957291845848, -0.215957291845848, 0.215957291845848, -0.215957291845848, 0.395689473055942, -0.395689473055942,

0.395689473055942, -0.395689473055942, 0.395689473055942, -0.395689473055942, 0.395689473055942, -0.395689473055942, 0.395689473055942, -0.395689473055942,

0.395689473055942, -0.395689473055942, 0.395689473055942, -0.395689473055942, 0.395689473055942, -0.395689473055942, 0.828769981252592, -0.828769981252592,

0.828769981252592, -0.828769981252592, 0.828769981252592, -0.828769981252592, 0.828769981252592, -0.828769981252592, 0.478369028812150, -0.478369028812150,

0.478369028812150, -0.478369028812150, 0.878158910604066, -0.878158910604066, 0.878158910604066, -0.878158910604066, 0.478369028812150, -0.478369028812150,

0.478369028812150, -0.478369028812150, 0.878158910604066, -0.878158910604066, 0.878158910604066, -0.878158910604066, 0.000000000000000, 0.000000000000000,

0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000

};

const double Integral::Lebedev_Laikov_grid110_y[110] = {

0.000000000000000, 0.000000000000000, 1.000000000000000, -1.000000000000000, 0.000000000000000, 0.000000000000000, 0.577350269189626, 0.577350269189626,

-0.577350269189626, -0.577350269189626, 0.577350269189626, 0.577350269189626, -0.577350269189626, -0.577350269189626, 0.185115635344736, 0.185115635344736,

-0.185115635344736, -0.185115635344736, 0.185115635344736, 0.185115635344736, -0.185115635344736, -0.185115635344736, 0.965124035086594, 0.965124035086594,

-0.965124035086594, -0.965124035086594, 0.965124035086594, 0.965124035086594, -0.965124035086594, -0.965124035086594, 0.185115635344736, 0.185115635344736,

-0.185115635344736, -0.185115635344736, 0.185115635344736, 0.185115635344736, -0.185115635344736, -0.185115635344736, 0.690421048382292, 0.690421048382292,

-0.690421048382292, -0.690421048382292, 0.690421048382292, 0.690421048382292, -0.690421048382292, -0.690421048382292, 0.215957291845848, 0.215957291845848,

-0.215957291845848, -0.215957291845848, 0.215957291845848, 0.215957291845848, -0.215957291845848, -0.215957291845848, 0.690421048382292, 0.690421048382292,

-0.690421048382292, -0.690421048382292, 0.690421048382292, 0.690421048382292, -0.690421048382292, -0.690421048382292, 0.395689473055942, 0.395689473055942,

-0.395689473055942, -0.395689473055942, 0.395689473055942, 0.395689473055942, -0.395689473055942, -0.395689473055942, 0.828769981252592, 0.828769981252592,

-0.828769981252592, -0.828769981252592, 0.828769981252592, 0.828769981252592, -0.828769981252592, -0.828769981252592, 0.395689473055942, 0.395689473055942,

-0.395689473055942, -0.395689473055942, 0.395689473055942, 0.395689473055942, -0.395689473055942, -0.395689473055942, 0.878158910604066, 0.878158910604066,

-0.878158910604066, -0.878158910604066, 0.478369028812150, 0.478369028812150, -0.478369028812150, -0.478369028812150, 0.000000000000000, 0.000000000000000,

0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.478369028812150, -0.478369028812150,

0.478369028812150, -0.478369028812150, 0.878158910604066, -0.878158910604066, 0.878158910604066, -0.878158910604066

};

const double Integral::Lebedev_Laikov_grid110_z[110] = {

0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 1.000000000000000, -1.000000000000000, 0.577350269189626, 0.577350269189626,

0.577350269189626, 0.577350269189626, -0.577350269189626, -0.577350269189626, -0.577350269189626, -0.577350269189626, 0.965124035086594, 0.965124035086594,

0.965124035086594, 0.965124035086594, -0.965124035086594, -0.965124035086594, -0.965124035086594, -0.965124035086594, 0.185115635344736, 0.185115635344736,

0.185115635344736, 0.185115635344736, -0.185115635344736, -0.185115635344736, -0.185115635344736, -0.185115635344736, 0.185115635344736, 0.185115635344736,

0.185115635344736, 0.185115635344736, -0.185115635344736, -0.185115635344736, -0.185115635344736, -0.185115635344736, 0.215957291845848, 0.215957291845848,

0.215957291845848, 0.215957291845848, -0.215957291845848, -0.215957291845848, -0.215957291845848, -0.215957291845848, 0.690421048382292, 0.690421048382292,

0.690421048382292, 0.690421048382292, -0.690421048382292, -0.690421048382292, -0.690421048382292, -0.690421048382292, 0.690421048382292, 0.690421048382292,

0.690421048382292, 0.690421048382292, -0.690421048382292, -0.690421048382292, -0.690421048382292, -0.690421048382292, 0.828769981252592, 0.828769981252592,

0.828769981252592, 0.828769981252592, -0.828769981252592, -0.828769981252592, -0.828769981252592, -0.828769981252592, 0.395689473055942, 0.395689473055942,

0.395689473055942, 0.395689473055942, -0.395689473055942, -0.395689473055942, -0.395689473055942, -0.395689473055942, 0.395689473055942, 0.395689473055942,

0.395689473055942, 0.395689473055942, -0.395689473055942, -0.395689473055942, -0.395689473055942, -0.395689473055942, 0.000000000000000, 0.000000000000000,

0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.878158910604066, 0.878158910604066,

-0.878158910604066, -0.878158910604066, 0.478369028812150, 0.478369028812150, -0.478369028812150, -0.478369028812150, 0.878158910604066, 0.878158910604066,

-0.878158910604066, -0.878158910604066, 0.478369028812150, 0.478369028812150, -0.478369028812150, -0.478369028812150

};

const double Integral::Lebedev_Laikov_grid110_w[110] = {

0.048107465851397, 0.048107465851397, 0.048107465851397, 0.048107465851397, 0.048107465851397, 0.048107465851397, 0.123071735281670, 0.123071735281670,

0.123071735281670, 0.123071735281670, 0.123071735281670, 0.123071735281670, 0.123071735281670, 0.123071735281670, 0.103191734088330, 0.103191734088330,

0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330,

0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.103191734088330, 0.124945096872513, 0.124945096872513,

0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513,

0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.124945096872513, 0.120580249028528, 0.120580249028528,

0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528,

0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.120580249028528, 0.121830917385521, 0.121830917385521,

0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521,

0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521, 0.121830917385521

};

}

abacus-develop/source/source_base/math_integral.cpp at develop · AroundPeking/abacus-develop · GitHub

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