Math

Macros
#define	INT_COPYSIGN   intCopysignShift
	Defines the implementation of int copysign(int,int); function. More...

Functions
template<int N, class T >
void	tinker::symlusolve (const T *aUpRowMajor, T *b)
	This subroutine uses the LU decomposition method to solve the linear system Ax = b, returning x in b. A is an n by n real symmetric matrix with its upper triangle (including the diagonal) stored by rows. More...
template<class T , class T2 >
T	tinker::maxOf (T a, T2 b)
template<class T , class T2 , class... Ts>
T	tinker::maxOf (T a, T2 b, Ts... cs)
template<class T , class T2 >
T	tinker::minOf (T a, T2 b)
template<class T , class T2 , class... Ts>
T	tinker::minOf (T a, T2 b, Ts... cs)
double	tinker::OUProcess (double t, double x0, double gamma, double a, double b, double R)
	Returns the value of an Ornstein-Uhlenbeck process after time t. More...
constexpr bool	tinker::isPow2 (size_t val)
	Return true if and only if val is a power of 2. More...
constexpr int	tinker::pow2 (int val)
	Integer base 2 power. More...
constexpr long long	tinker::pow2ll (int val)
	Long long integer base 2 power. More...
constexpr int	tinker::floorLog2_constexpr (size_t val)
	The base-2 logarithm of val. More...
constexpr size_t	tinker::pow2Le (size_t val)
	A power of 2 that is less than or equal to val. More...
constexpr size_t	tinker::pow2Ge (size_t val)
	A power of 2 that is greater than or equal to val. More...
int	tinker::floorLog2 (int val)
	Non-constexpr floorLog2(int). More...
int	tinker::floorLog2 (long long val)
	Non-constexpr floorLog2(long long). More...
int	tinker::ceilLog2 (int val)
	Non-constexpr log base 2 ceiling. More...
int	tinker::ceilLog2 (long long val)
	Non-constexpr log base 2 ceiling. More...
template<class T >
T	tinker::random ()
	Returns a random number on [0,1] from a uniform distribution. More...
template<class T >
T	tinker::normal ()
	Returns a random number from a normal Gaussian distribution with a mean of zero and a standard deviation of one. More...
template<class T >
T	tinker::normal (T u, T s)
	Returns a random number from a normal Gaussian distribution with a mean of u and a standard deviation of s. More...
template<class T >
T	tinker::chiSquared (int k)
	Returns a random number as if the square of k independent standard normal N(0,1) random variables were aggregated. More...
void	tinker::randomData (RcOp)
template<class T >
T	tinker::sinhc (T x)
	\( \sinh(x)/x \) More...
template<int DO_DTAPER>
__device__ void	tinker::switchTaper5 (real rik, real cut, real off, real &__restrict__ taper, real &__restrict__ dtaper)
	Smooth function F: [cut,off]->[1,0]. More...
template<class T >
void	tinker::matmul3 (T R[3][3], T A[3][3], T B[3][3])
	Matrix multiplication of two 3 by 3 matrices. \( R = A B \). Matrices are stored in the row-major order (C-style). More...
template<class T >
void	tinker::matmul3 (T R[3][3], T A[3][3])
	Matrix multiplication of two 3 by 3 matrices. \( R = A R \). Matrices are stored in the row-major order (C-style). More...
template<class T >
void	tinker::trimatExp (T ans[3][3], T m[3][3], T t)
	\( \exp(mt) \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style). More...
template<class T >
void	tinker::trimatExpm1c (T ans[3][3], T m[3][3], T t)
	\( (\exp(mt)-I)/(mt) \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style). More...
template<class T >
void	tinker::trimatTExpm1c (T ans[3][3], T m[3][3], T t)
	\( t \frac{\exp(mt)-I}{mt} \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style). More...
void	tinker::zeroOnHost (int &v)
	Zeros variable on host. More...
void	tinker::zeroOnHost (float &v)
	Zeros variable on host. More...
void	tinker::zeroOnHost (double &v)
	Zeros variable on host. More...
void	tinker::zeroOnHost (unsigned long long &v)
	Zeros variable on host. More...
template<class T >
void	tinker::zeroOnHost (T *&ptr)
	Zeros variable on host. More...
template<class T , size_t N>
void	tinker::zeroOnHost (T(&v)[N])
	Zeros variables on host. More...
template<class T , class... Ts>
void	tinker::zeroOnHost (T &v, Ts &... vs)
	Zeros variables on host. More...
template<class T >
void	tinker::zeroOnDevice3Async (int nelem, T *a1, T *a2, T *a3)
	Zeros variables on device. More...
template<class T , int N>
void	tinker::zeroOnDevice3Async (int nelem, T(*a1)[N], T(*a2)[N], T(*a3)[N])
	Zeros variables on device. More...
template<class T >
void	tinker::zeroOnDevice9Async (int nelem, T *a1, T *a2, T *a3, T *a4, T *a5, T *a6, T *a7, T *a8, T *a9)
	Zeros variables on device. More...
__device__ int	tinker::intCopysignShift (int a, int b)
	If b < 0, returns -abs(a), otherwise, returns abs(a). Similar to the 32-bit integer version of Fortran SIGN(A,B). More...
__device__ int	tinker::intCopysignIf (int a, int b)
	If b < 0, returns -abs(a), otherwise, returns abs(a). More...
__device__ int	tinker::ffsnShift (int c1, int n)
	Find the position of the n-th least significant bit set in a 32-bit integer. Returns a value from 0 to 32. More...
__device__ int	tinker::ffsnLoop (int c1, int n)
	An alternative implementation of ffsn using loop. More...

Variables
constexpr real	tinker::pi = M_PI
	\( \pi \) More...
constexpr real	tinker::sqrtpi = 1.77245385090551602730
	\( \sqrt{\pi} \) More...
constexpr real	tinker::radian = 57.2957795130823208768
	\( 180/\pi \) More...
constexpr real	tinker::_1radian = 0.01745329251994329576924
	\( \pi/180 \) More...
using	tinker::real2 = float2
using	tinker::real3 = float3
using	tinker::real4 = float4
__device__ real3	tinker::operator- (real3 a)
__device__ real3	tinker::operator+ (real3 a, real3 b)
__device__ real3	tinker::operator- (real3 a, real3 b)
__device__ real3	tinker::operator* (real a, real3 b)
__device__ real3	tinker::operator* (real3 b, real a)
__device__ void	tinker::operator+= (real3 &a, real3 b)
__device__ void	tinker::operator-= (real3 &a, real3 b)
__device__ void	tinker::operator*= (real3 &a, real b)
__device__ real	tinker::dot3 (real3 a, real3 b)
__device__ real	tinker::dot3 (real3 a, real bx, real by, real bz)
__device__ real	tinker::dot3 (real ax, real ay, real az, real3 b)
__device__ real	tinker::dot3 (real ax, real ay, real az, real bx, real by, real bz)
__device__ real3	tinker::matvec (real xx, real xy, real xz, real yy, real yz, real zz, real3 v)
__device__ real3	tinker::cross (real ax, real ay, real az, real bx, real by, real bz)
__device__ real3	tinker::cross (real ax, real ay, real az, real3 b)
__device__ real3	tinker::cross (real3 a, real bx, real by, real bz)
__device__ real3	tinker::cross (real3 a, real3 b)
template<class T >
__device__ void	tinker::fsinhc1 (T d, T &__restrict__ f1d)
template<class T >
__device__ void	tinker::fsinhc2 (T d, T &__restrict__ f1d, T &__restrict__ f2d)
template<class T >
__device__ void	tinker::fsinhc3 (T d, T &__restrict__ f1d, T &__restrict__ f2d, T &__restrict__ f3d)
template<class T >
__device__ void	tinker::fsinhc4 (T d, T &__restrict__ f1d, T &__restrict__ f2d, T &__restrict__ f3d, T &__restrict__ f4d)
template<class T >
__device__ void	tinker::fsinhc5 (T d, T &__restrict__ f1d, T &__restrict__ f2d, T &__restrict__ f3d, T &__restrict__ f4d, T &__restrict__ f5d)
template<class T >
__device__ void	tinker::fsinhc6 (T d, T &__restrict__ f1d, T &__restrict__ f2d, T &__restrict__ f3d, T &__restrict__ f4d, T &__restrict__ f5d, T &__restrict__ f6d)
template<class T >
__device__ void	tinker::fsinhc7 (T d, T &__restrict__ f1d, T &__restrict__ f2d, T &__restrict__ f3d, T &__restrict__ f4d, T &__restrict__ f5d, T &__restrict__ f6d, T &__restrict__ f7d)

Detailed Description

Macro Definition Documentation

INT_COPYSIGN

#define INT_COPYSIGN   intCopysignShift

Defines the implementation of int copysign(int,int); function.

Typedef Documentation

real2

using tinker::real2 = typedef float2

real3

using tinker::real3 = typedef float3

real4

using tinker::real4 = typedef float4

Function Documentation

ceilLog2() [1/2]

int tinker::ceilLog2 ( int  val )

inline

Non-constexpr log base 2 ceiling.

ceilLog2() [2/2]

int tinker::ceilLog2 ( long long  val )

inline

Non-constexpr log base 2 ceiling.

chiSquared()

template<class T >

T tinker::chiSquared

(

int

k

)

Returns a random number as if the square of k independent standard normal N(0,1) random variables were aggregated.

Note

There is a bug in some (not so) early g++ random/chiSquared_distribution implementations [ref 1 and ref 2], thus gamma distribution is used here via the following equations:

\[ p_{\chi^2}(x|n) = \frac{x^{n/2-1} exp(-x/2)}{2^{n/2} \Gamma(n/2)} \]

\[ p_\Gamma(x|a,b) = \frac{x^{a-1} exp(-x/b)}{b^a \Gamma(a)} \]

\[ p_{\chi^2}(n) = p_\Gamma(n/2,2) \]

cross() [1/4]

__device__ real3 tinker::cross ( real  ax, real  ay, real  az, real  bx, real  by, real  bz  )

inline

cross() [2/4]

__device__ real3 tinker::cross ( real  ax, real  ay, real  az, real3  b  )

inline

cross() [3/4]

__device__ real3 tinker::cross ( real3  a, real  bx, real  by, real  bz  )

inline

cross() [4/4]

__device__ real3 tinker::cross ( real3  a, real3  b  )

inline

dot3() [1/4]

__device__ real tinker::dot3 ( real  ax, real  ay, real  az, real  bx, real  by, real  bz  )

inline

dot3() [2/4]

__device__ real tinker::dot3 ( real  ax, real  ay, real  az, real3  b  )

inline

dot3() [3/4]

__device__ real tinker::dot3 ( real3  a, real  bx, real  by, real  bz  )

inline

dot3() [4/4]

__device__ real tinker::dot3 ( real3  a, real3  b  )

inline

ffsnLoop()

__device__ int tinker::ffsnLoop ( int  c1, int  n  )

inline

An alternative implementation of ffsn using loop.

ffsnShift()

__device__ int tinker::ffsnShift ( int  c1, int  n  )

inline

Find the position of the n-th least significant bit set in a 32-bit integer. Returns a value from 0 to 32.

• If c1 equals 0, always returns 0.
• If n equals 0, always returns 0.
• If n is greater than POPC, which is the number of bits that are set to 1 in c1, returns an undefined value.

Parameters

c1	A 32-bit integer.
n	Ranges from 1 to 32.

floorLog2() [1/2]

int tinker::floorLog2 ( int  val )

inline

Non-constexpr floorLog2(int).

floorLog2() [2/2]

int tinker::floorLog2 ( long long  val )

inline

Non-constexpr floorLog2(long long).

floorLog2_constexpr()

constexpr int tinker::floorLog2_constexpr ( size_t  val )

constexpr

The base-2 logarithm of val.

Parameters

val	Must be greater than 0.

Note

Returns 0 if val is 0.

fsinhc1()

template<class T >

__device__ void tinker::fsinhc1 ( T  d, T &__restrict__  f1d  )

inline

fsinhc2()

template<class T >

__device__ void tinker::fsinhc2 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d  )

inline

fsinhc3()

template<class T >

__device__ void tinker::fsinhc3 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d, T &__restrict__  f3d  )

inline

fsinhc4()

template<class T >

__device__ void tinker::fsinhc4 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d, T &__restrict__  f3d, T &__restrict__  f4d  )

inline

fsinhc5()

template<class T >

__device__ void tinker::fsinhc5 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d, T &__restrict__  f3d, T &__restrict__  f4d, T &__restrict__  f5d  )

inline

fsinhc6()

template<class T >

__device__ void tinker::fsinhc6 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d, T &__restrict__  f3d, T &__restrict__  f4d, T &__restrict__  f5d, T &__restrict__  f6d  )

inline

fsinhc7()

template<class T >

__device__ void tinker::fsinhc7 ( T  d, T &__restrict__  f1d, T &__restrict__  f2d, T &__restrict__  f3d, T &__restrict__  f4d, T &__restrict__  f5d, T &__restrict__  f6d, T &__restrict__  f7d  )

inline

intCopysignIf()

__device__ int tinker::intCopysignIf ( int  a, int  b  )

inline

If b < 0, returns -abs(a), otherwise, returns abs(a).

intCopysignShift()

__device__ int tinker::intCopysignShift ( int  a, int  b  )

inline

If b < 0, returns -abs(a), otherwise, returns abs(a). Similar to the 32-bit integer version of Fortran SIGN(A,B).

Note

Standard C and CUDA libraries only have float and double versions.

isPow2()

constexpr bool tinker::isPow2 ( size_t  val )

constexpr

Return true if and only if val is a power of 2.

matmul3() [1/2]

template<class T >

void tinker::matmul3	(	T	R[3][3],
		T	A[3][3]
	)

Matrix multiplication of two 3 by 3 matrices. \( R = A R \). Matrices are stored in the row-major order (C-style).

Parameters

[in,out]	R	Matrix to be multiplied on the left.
[in]	A	Matrix on the left.

matmul3() [2/2]

template<class T >

void tinker::matmul3	(	T	R[3][3],
		T	A[3][3],
		T	B[3][3]
	)

Matrix multiplication of two 3 by 3 matrices. \( R = A B \). Matrices are stored in the row-major order (C-style).

Parameters

[out]	R	Matrix for the result.
[in]	A	Matrix on the left.
[in]	B	Matrix on the right.

matvec()

__device__ real3 tinker::matvec ( real  xx, real  xy, real  xz, real  yy, real  yz, real  zz, real3  v  )

inline

maxOf() [1/2]

template<class T , class T2 >

T tinker::maxOf	(	T	a,
		T2	b
	)

See also

maxOf(T, T2, Ts...)

maxOf() [2/2]

template<class T , class T2 , class... Ts>

T tinker::maxOf	(	T	a,
		T2	b,
		Ts...	cs
	)

Returns

The maximum value from a variadic list of numbers.

minOf() [1/2]

template<class T , class T2 >

T tinker::minOf	(	T	a,
		T2	b
	)

See also

minOf(T, T2, Ts...)

minOf() [2/2]

template<class T , class T2 , class... Ts>

T tinker::minOf	(	T	a,
		T2	b,
		Ts...	cs
	)

Returns

The minimum value from a variadic list of numbers.

normal() [1/2]

template<class T >

T tinker::normal

(

)

Returns a random number from a normal Gaussian distribution with a mean of zero and a standard deviation of one.

normal() [2/2]

template<class T >

T tinker::normal	(	T	u,
		T	s
	)

Returns a random number from a normal Gaussian distribution with a mean of u and a standard deviation of s.

operator*() [1/2]

__device__ real3 tinker::operator* ( real  a, real3  b  )

inline

operator*() [2/2]

__device__ real3 tinker::operator* ( real3  b, real  a  )

inline

operator*=()

__device__ void tinker::operator*= ( real3 &  a, real  b  )

inline

operator+()

__device__ real3 tinker::operator+ ( real3  a, real3  b  )

inline

operator+=()

__device__ void tinker::operator+= ( real3 &  a, real3  b  )

inline

operator-() [1/2]

__device__ real3 tinker::operator- ( real3  a )

inline

operator-() [2/2]

__device__ real3 tinker::operator- ( real3  a, real3  b  )

inline

operator-=()

__device__ void tinker::operator-= ( real3 &  a, real3  b  )

inline

OUProcess()

double tinker::OUProcess ( double  t, double  x0, double  gamma, double  a, double  b, double  R  )

inline

Returns the value of an Ornstein-Uhlenbeck process after time t.

The Ornstein-Uhlenbeck process is described by \( \frac{dx}{dt} = a - \gamma x + b \frac{dW}{dt} \), and \( x_t = x_0 \exp(-\gamma t) + at \frac{1 - \exp(-\gamma t)}{\gamma t} + b R \sqrt{t} \sqrt{\frac{1 - \exp(-2\gamma t)}{2\gamma t}} \).

Parameters

t	Length of a time interval.
x0	Initial value.
gamma	Friction coefficient.
a	Parameter a.
b	Parameter b.
R	Random number of a standard normal distribution.

pow2()

constexpr int tinker::pow2 ( int  val )

constexpr

Integer base 2 power.

pow2Ge()

constexpr size_t tinker::pow2Ge ( size_t  val )

constexpr

A power of 2 that is greater than or equal to val.

pow2Le()

constexpr size_t tinker::pow2Le ( size_t  val )

constexpr

A power of 2 that is less than or equal to val.

Parameters

val	Must be greater than 0.

pow2ll()

constexpr long long tinker::pow2ll ( int  val )

constexpr

Long long integer base 2 power.

random()

template<class T >

T tinker::random

(

)

Returns a random number on [0,1] from a uniform distribution.

randomData()

void tinker::randomData

(

RcOp

)

sinhc()

template<class T >

T tinker::sinhc

(

T

x

)

\( \sinh(x)/x \)

switchTaper5()

template<int DO_DTAPER>

__device__ void tinker::switchTaper5	(	real	rik,
		real	cut,
		real	off,
		real &__restrict__	taper,
		real &__restrict__	dtaper
	)

Smooth function F: [cut,off]->[1,0].

Derived from the 5th order smoothstep function (-S2(x) + 1):

\[ S_2: [0,1]\rightarrow[0,1] \]

\[ S_2(x) = 6 x^5 - 15 x^4 + 10 x^3 \]

Parameters

[in]	rik	Distance.
[in]	cut	Distance at which switching of the potential begins.
[out]	off	Distance at which the potential energy goes to zero.
[out]	taper	F value.
[out]	dtaper	dF/dx value.

Template Parameters

DO_DTAPER

If false, dtaper will not be calculated and its original value will not be altered.

symlusolve()

template<int N, class T >

void tinker::symlusolve	(	const T *	aUpRowMajor,
		T *	b
	)

This subroutine uses the LU decomposition method to solve the linear system Ax = b, returning x in b. A is an n by n real symmetric matrix with its upper triangle (including the diagonal) stored by rows.

Literature reference:

• W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes (C++), 3rd Ed., Section 2.3, Cambridge University Press (2007).

trimatExp()

template<class T >

void tinker::trimatExp	(	T	ans[3][3],
		T	m[3][3],
		T	t
	)

\( \exp(mt) \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style).

Parameters

[out]	ans	3 by 3 matrix for the result.
[in]	m	3 by 3 upper triangular matrix.
[in]	t	Scalar to scale the matrix m.

Note

If the input matrix is not an upper triangular matrix, the result is undefined.

trimatExpm1c()

template<class T >

void tinker::trimatExpm1c	(	T	ans[3][3],
		T	m[3][3],
		T	t
	)

\( (\exp(mt)-I)/(mt) \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style).

Parameters

[out]	ans	3 by 3 matrix for the result.
[in]	m	3 by 3 upper triangular matrix.
[in]	t	Scalar to scale the matrix m.

Note

If the input matrix is not an upper triangular matrix, the result is undefined.

trimatTExpm1c()

template<class T >

void tinker::trimatTExpm1c	(	T	ans[3][3],
		T	m[3][3],
		T	t
	)

\( t \frac{\exp(mt)-I}{mt} \). Matrix m is 3 by 3 upper triangular. Matrices are stored in the row-major order (C-style).

Parameters

[out]	ans	3 by 3 matrix for the result.
[in]	m	3 by 3 upper triangular matrix.
[in]	t	Scalar to scale the matrix m.

Note

If the input matrix is not an upper triangular matrix, the result is undefined.

zeroOnDevice3Async() [1/2]

template<class T >

void tinker::zeroOnDevice3Async	(	int	nelem,
		T *	a1,
		T *	a2,
		T *	a3
	)

Zeros variables on device.

zeroOnDevice3Async() [2/2]

template<class T , int N>

void tinker::zeroOnDevice3Async	(	int	nelem,
		T(*)	a1[N],
		T(*)	a2[N],
		T(*)	a3[N]
	)

Zeros variables on device.

zeroOnDevice9Async()

template<class T >

void tinker::zeroOnDevice9Async	(	int	nelem,
		T *	a1,
		T *	a2,
		T *	a3,
		T *	a4,
		T *	a5,
		T *	a6,
		T *	a7,
		T *	a8,
		T *	a9
	)

Zeros variables on device.

zeroOnHost() [1/7]

void tinker::zeroOnHost ( double &  v )

inline

Zeros variable on host.

zeroOnHost() [2/7]

void tinker::zeroOnHost ( float &  v )

inline

Zeros variable on host.

zeroOnHost() [3/7]

void tinker::zeroOnHost ( int &  v )

inline

Zeros variable on host.

zeroOnHost() [4/7]

template<class T , class... Ts>

void tinker::zeroOnHost	(	T &	v,
		Ts &...	vs
	)

Zeros variables on host.

zeroOnHost() [5/7]

template<class T >

void tinker::zeroOnHost

(

T *&

ptr

)

Zeros variable on host.

zeroOnHost() [6/7]

template<class T , size_t N>

void tinker::zeroOnHost

(

T(&)

v[N]

)

Zeros variables on host.

zeroOnHost() [7/7]

void tinker::zeroOnHost ( unsigned long long &  v )

inline

Zeros variable on host.

Variable Documentation

_1radian

constexpr real tinker::_1radian = 0.01745329251994329576924

constexpr

\( \pi/180 \)

pi

constexpr real tinker::pi = M_PI

constexpr

\( \pi \)

radian

constexpr real tinker::radian = 57.2957795130823208768

constexpr

\( 180/\pi \)

sqrtpi

constexpr real tinker::sqrtpi = 1.77245385090551602730

constexpr

\( \sqrt{\pi} \)