/
Math.h
1635 lines (1388 loc) · 36.6 KB
/
Math.h
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/*
* 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 3 of the License, or
* (at your option) any later version.
*
* Written (W) 2013 Soumyajit De
* Written (W) 2012 Fernando José Iglesias García
* Written (W) 2011 Siddharth Kherada
* Written (W) 2011 Justin Patera
* Written (W) 2011 Alesis Novik
* Written (W) 2011-2012 Heiko Strathmann
* Written (W) 1999-2009 Soeren Sonnenburg
* Written (W) 1999-2008 Gunnar Raetsch
* Written (W) 2007 Konrad Rieck
* Copyright (C) 1999-2009 Fraunhofer Institute FIRST and Max-Planck-Society
*/
#ifndef __MATHEMATICS_H_
#define __MATHEMATICS_H_
#include <shogun/base/SGObject.h>
#include <shogun/lib/common.h>
#include <shogun/io/SGIO.h>
#include <shogun/base/Parallel.h>
#include <shogun/mathematics/Random.h>
#include <math.h>
#include <stdio.h>
#include <float.h>
#include <pthread.h>
#include <unistd.h>
#include <sys/types.h>
#include <sys/time.h>
#include <time.h>
#ifdef SUNOS
#include <ieeefp.h>
#endif
/// workaround for log2 being a define on cygwin
#ifdef log2
#define cygwin_log2 log2
#undef log2
#endif
/// workaround a bug in std cmath
#ifdef _GLIBCXX_CMATH
#if _GLIBCXX_USE_C99_MATH
#if !_GLIBCXX_USE_C99_FP_MACROS_DYNAMIC
/// Function template definitions [8.16.3].
using std::signbit;
using std::fpclassify;
using std::isfinite;
using std::isinf;
using std::isnan;
using std::isnormal;
using std::isgreater;
using std::isgreaterequal;
using std::isless;
using std::islessequal;
using std::islessgreater;
using std::isunordered;
#endif
#endif
#endif
/// end of workaround a bug in std cmath
#ifdef _WIN32
#ifndef isnan
#define isnan _isnan
#endif
#ifndef isfinite
#define isfinite _isfinite
#endif
#endif //_WIN32
#ifndef NAN
#include <stdlib.h>
#define NAN (strtod("NAN",NULL))
#endif
/* Size of RNG seed */
#define RNG_SEED_SIZE 256
/* Maximum stack size */
#define RADIX_STACK_SIZE 512
/* Stack macros */
#define radix_push(a, n, i) sp->sa = a, sp->sn = n, (sp++)->si = i
#define radix_pop(a, n, i) a = (--sp)->sa, n = sp->sn, i = sp->si
#ifndef DOXYGEN_SHOULD_SKIP_THIS
/** Stack structure */
template <class T> struct radix_stack_t
{
/** Pointer to pile */
T *sa;
/** Number of grams in pile */
size_t sn;
/** Byte in current focus */
uint16_t si;
};
///** pair */
/** thread qsort */
template <class T1, class T2> struct thread_qsort
{
/** output */
T1* output;
/** index */
T2* index;
/** size */
uint32_t size;
/** qsort threads */
int32_t* qsort_threads;
/** sort limit */
int32_t sort_limit;
/** number of threads */
int32_t num_threads;
};
#endif // DOXYGEN_SHOULD_SKIP_THIS
#define COMPLEX64_ERROR_ONEARG(function) \
static inline complex64_t function(complex64_t a) \
{ \
SG_SERROR("CMath::%s():: Not supported for complex64_t\n",\
#function);\
return complex64_t(0.0, 0.0); \
}
#define COMPLEX64_ERROR_TWOARGS(function) \
static inline complex64_t function(complex64_t a, complex64_t b) \
{ \
SG_SERROR("CMath::%s():: Not supported for complex64_t\n",\
#function);\
return complex64_t(0.0, 0.0); \
}
#define COMPLEX64_ERROR_THREEARGS(function) \
static inline complex64_t function(complex64_t a, complex64_t b, complex64_t c) \
{ \
SG_SERROR("CMath::%s():: Not supported for complex64_t\n",\
#function);\
return complex64_t(0.0, 0.0); \
}
#define COMPLEX64_STDMATH(function) \
static inline complex64_t function(complex64_t a) \
{ \
return std::function(a); \
}
#define COMPLEX64_ERROR_SORT(function) \
static void function(complex64_t* output, int32_t b) \
{ \
SG_SERROR("CMath::%s():: Not supported for complex64_t\n",\
#function);\
}
namespace shogun
{
/** random number generator */
extern CRandom* sg_rand;
class CSGObject;
/** @brief Class which collects generic mathematical functions
*/
class CMath : public CSGObject
{
public:
/**@name Constructor/Destructor.
*/
//@{
///Constructor - initializes log-table
CMath();
///Destructor - frees logtable
virtual ~CMath();
//@}
/**@name min/max/abs functions.
*/
//@{
///return the minimum of two integers
//
template <class T>
static inline T min(T a, T b)
{
return (a<=b) ? a : b;
}
/// min not implemented for complex64_t, returns (0.0)+i(0.0) instead
COMPLEX64_ERROR_TWOARGS(min)
///return the maximum of two integers
template <class T>
static inline T max(T a, T b)
{
return (a>=b) ? a : b;
}
/// max not implemented for complex64_t, returns (0.0)+i(0.0) instead
COMPLEX64_ERROR_TWOARGS(max)
///return the value clamped to interval [lb,ub]
template <class T>
static inline T clamp(T value, T lb, T ub)
{
if (value<=lb)
return lb;
else if (value>=ub)
return ub;
else
return value;
}
/// clamp not implemented for complex64_t, returns (0.0)+i(0.0) instead
COMPLEX64_ERROR_THREEARGS(clamp)
///return the absolute value of a number
template <class T>
static inline T abs(T a)
{
// can't be a>=0?(a):(-a), because compiler complains about
// 'comparison always true' when T is unsigned
if (a==0)
return 0;
else if (a>0)
return a;
else
return -a;
}
///return the absolute value of a complex number
static inline float64_t abs(complex64_t a)
{
float64_t a_real=a.real();
float64_t a_imag=a.imag();
return (CMath::sqrt(a_real*a_real+a_imag*a_imag));
}
//@}
/**@name misc functions */
//@{
static inline float64_t round(float64_t d)
{
return ::floor(d+0.5);
}
static inline float64_t floor(float64_t d)
{
return ::floor(d);
}
static inline float64_t ceil(float64_t d)
{
return ::ceil(d);
}
/// signum of type T variable a
template <class T>
static inline T sign(T a)
{
if (a==0)
return 0;
else return (a<0) ? (-1) : (+1);
}
/// signum not implemented for complex64_t, returns (0.0)+i(0.0) instead
COMPLEX64_ERROR_ONEARG(sign)
/// swap e.g. floats a and b
template <class T>
static inline void swap(T &a,T &b)
{
/* register for fast swaps */
register T c=a;
a=b;
b=c;
}
/// x^2
template <class T>
static inline T sq(T x)
{
return x*x;
}
/// x^0.5
static inline float32_t sqrt(float32_t x)
{
return ::sqrtf(x);
}
/// x^0.5
static inline float64_t sqrt(float64_t x)
{
return ::sqrt(x);
}
/// x^0.5
static inline floatmax_t sqrt(floatmax_t x)
{
//fall back to double precision sqrt if sqrtl is not
//available
#ifdef HAVE_SQRTL
return ::sqrtl(x);
#else
return ::sqrt(x);
#endif
}
/// x^0.5, x being a complex64_t
COMPLEX64_STDMATH(sqrt)
/// x^-0.5
static inline float32_t invsqrt(float32_t x)
{
union float_to_int
{
float32_t f;
int32_t i;
};
float_to_int tmp;
tmp.f=x;
float32_t xhalf = 0.5f * x;
// store floating-point bits in integer tmp.i
// and do initial guess for Newton's method
tmp.i = 0x5f3759d5 - (tmp.i >> 1);
x = tmp.f; // convert new bits into float
x = x*(1.5f - xhalf*x*x); // One round of Newton's method
return x;
}
/// x^n
static inline floatmax_t powl(floatmax_t x, floatmax_t n)
{
//fall back to double precision pow if powl is not
//available
#ifdef HAVE_POWL
return ::powl((long double) x, (long double) n);
#else
return ::pow((double) x, (double) n);
#endif
}
static inline int32_t pow(bool x, int32_t n)
{
return 0;
}
static inline int32_t pow(int32_t x, int32_t n)
{
ASSERT(n>=0)
int32_t result=1;
while (n--)
result*=x;
return result;
}
static inline float64_t pow(float64_t x, int32_t n)
{
if (n>=0)
{
float64_t result=1;
while (n--)
result*=x;
return result;
}
else
return ::pow((double)x, (double)n);
}
static inline float64_t pow(float64_t x, float64_t n)
{
return ::pow((double) x, (double) n);
}
/// x^n, x or n being a complex64_t
static inline complex64_t pow(complex64_t x, int32_t n)
{
return std::pow(x, n);
}
static inline complex64_t pow(complex64_t x, complex64_t n)
{
return std::pow(x, n);
}
static inline complex64_t pow(complex64_t x, float64_t n)
{
return std::pow(x, n);
}
static inline complex64_t pow(float64_t x, complex64_t n)
{
return std::pow(x, n);
}
static inline float64_t exp(float64_t x)
{
return ::exp((double) x);
}
/// exp(x), x being a complex64_t
COMPLEX64_STDMATH(exp)
/** @return tangens of input */
static inline float64_t tan(float64_t x)
{
return ::tan((double) x);
}
/// tan(x), x being a complex64_t
COMPLEX64_STDMATH(tan)
/** @return arcus tangens of input */
static inline float64_t atan(float64_t x)
{
return ::atan((double) x);
}
/// atan(x), x being a complex64_t not implemented
COMPLEX64_ERROR_ONEARG(atan)
/** @return arcus tangens of input */
static inline float64_t atan2(float64_t x, float64_t y)
{
return ::atan2((double) x, (double) y);
}
/// atan2(x), x being a complex64_t not implemented
COMPLEX64_ERROR_ONEARG(atan2)
/** @return tangens hyperbolicus of input */
static inline float64_t tanh(float64_t x)
{
return ::tanh((double) x);
}
/// tanh(x), x being a complex64_t
COMPLEX64_STDMATH(tanh)
static inline float64_t log10(float64_t v)
{
return ::log(v)/::log(10.0);
}
/// log10(x), x being a complex64_t
COMPLEX64_STDMATH(log10)
static inline float64_t log2(float64_t v)
{
#ifdef HAVE_LOG2
return ::log2(v);
#else
return ::log(v)/::log(2.0);
#endif //HAVE_LOG2
}
static inline float64_t log(float64_t v)
{
return ::log(v);
}
/// log(x), x being a complex64_t
COMPLEX64_STDMATH(log)
static inline index_t floor_log(index_t n)
{
index_t i;
for (i = 0; n != 0; i++)
n >>= 1;
return i;
}
static inline float64_t sin(float64_t x)
{
return ::sin(x);
}
/// sin(x), x being a complex64_t
COMPLEX64_STDMATH(sin)
static inline float64_t asin(float64_t x)
{
return ::asin(x);
}
/// asin(x), x being a complex64_t not implemented
COMPLEX64_ERROR_ONEARG(asin)
static inline float64_t sinh(float64_t x)
{
return ::asinh(x); //TODO is this correct?
}
/// sinh(x), x being a complex64_t
COMPLEX64_STDMATH(sinh)
static inline float64_t cos(float64_t x)
{
return ::cos(x);
}
/// cos(x), x being a complex64_t
COMPLEX64_STDMATH(cos)
static inline float64_t acos(float64_t x)
{
return ::acos(x);
}
/// acos(x), x being a complex64_t not implemented
COMPLEX64_ERROR_ONEARG(acos)
static inline float64_t cosh(float64_t x)
{
return ::cosh(x);
}
/// cosh(x), x being a complex64_t
COMPLEX64_STDMATH(cosh)
static float64_t area_under_curve(float64_t* xy, int32_t len, bool reversed)
{
ASSERT(len>0 && xy)
float64_t area = 0.0;
if (!reversed)
{
for (int i=1; i<len; i++)
area += 0.5*(xy[2*i]-xy[2*(i-1)])*(xy[2*i+1]+xy[2*(i-1)+1]);
}
else
{
for (int i=1; i<len; i++)
area += 0.5*(xy[2*i+1]-xy[2*(i-1)+1])*(xy[2*i]+xy[2*(i-1)]);
}
return area;
}
static inline int64_t factorial(int32_t n)
{
int64_t res=1;
for (int i=2; i<=n; i++)
res*=i ;
return res ;
}
static void init_random(uint32_t initseed=0)
{
if (initseed==0)
{
struct timeval tv;
gettimeofday(&tv, NULL);
seed=(uint32_t) (4223517*getpid()*tv.tv_sec*tv.tv_usec);
}
else
seed=initseed;
sg_rand->set_seed(seed);
}
static inline uint64_t random()
{
return sg_rand->random_64();
}
static inline uint64_t random(uint64_t min_value, uint64_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline int64_t random(int64_t min_value, int64_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline uint32_t random(uint32_t min_value, uint32_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline int32_t random(int32_t min_value, int32_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline float32_t random(float32_t min_value, float32_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline float64_t random(float64_t min_value, float64_t max_value)
{
return sg_rand->random(min_value, max_value);
}
static inline floatmax_t random(floatmax_t min_value, floatmax_t max_value)
{
return sg_rand->random(min_value, max_value);
}
/// Returns a Gaussian or Normal random number.
/// Using the polar form of the Box-Muller transform.
/// http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform#Polar_form
static inline float32_t normal_random(float32_t mean, float32_t std_dev)
{
// sets up variables & makes sure rand_s.range == (0,1)
float32_t ret;
float32_t rand_u;
float32_t rand_v;
float32_t rand_s;
do
{
rand_u = CMath::random(-1.0, 1.0);
rand_v = CMath::random(-1.0, 1.0);
rand_s = rand_u*rand_u + rand_v*rand_v;
} while ((rand_s == 0) || (rand_s >= 1));
// the meat & potatos, and then the mean & standard deviation shifting...
ret = rand_u*sqrt(-2.0*log(rand_s)/rand_s);
ret = std_dev*ret + mean;
return ret;
}
/// Returns a Gaussian or Normal random number.
/// Using the polar form of the Box-Muller transform.
/// http://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform#Polar_form
static inline float64_t normal_random(float64_t mean, float64_t std_dev)
{
float64_t ret;
float64_t rand_u;
float64_t rand_v;
float64_t rand_s;
do
{
rand_u = CMath::random(-1.0, 1.0);
rand_v = CMath::random(-1.0, 1.0);
rand_s = rand_u*rand_u + rand_v*rand_v;
} while ((rand_s == 0) || (rand_s >= 1));
ret = rand_u*sqrt(-2.0*log(rand_s)/rand_s);
ret = std_dev*ret + mean;
return ret;
}
/// Convenience method for generating Standard Normal random numbers
/// Float: Mean = 0 and Standard Deviation = 1
static inline float32_t randn_float()
{
return normal_random(0.0, 1.0);
}
/// Convenience method for generating Standard Normal random numbers
/// Double: Mean = 0 and Standard Deviation = 1
static inline float64_t randn_double()
{
return normal_random(0.0, 1.0);
}
template <class T>
static int32_t get_num_nonzero(T* vec, int32_t len)
{
int32_t nnz = 0;
for (index_t i=0; i<len; ++i)
nnz += vec[i] != 0;
return nnz;
}
static int32_t get_num_nonzero(complex64_t* vec, int32_t len)
{
int32_t nnz=0;
for (index_t i=0; i<len; ++i)
nnz+=vec[i]!=0.0;
return nnz;
}
static inline int64_t nchoosek(int32_t n, int32_t k)
{
int64_t res=1;
for (int32_t i=n-k+1; i<=n; i++)
res*=i;
return res/factorial(k);
}
/** Builds an array with n linearly spaced elements between start and end.
*
* @param output array with linearly spaced elements within the interval
* @param start beginning of the interval to divide
* @param end upper bound of the interval to divide
* @param n number of elements used to divide the interval
*/
static void linspace(float64_t* output, float64_t start, float64_t end, int32_t n = 100);
/** performs a bubblesort on a given matrix a.
* it is sorted in ascending order from top to bottom
* and left to right */
static void sort(int32_t *a, int32_t cols, int32_t sort_col=0);
static void sort(float64_t *a, int32_t*idx, int32_t N);
/** performs a quicksort on an array output of length size
* it is sorted from in ascending (for type T) */
template <class T>
static void qsort(T* output, int32_t size)
{
if (size<=1)
return;
if (size==2)
{
if (output[0] > output [1])
CMath::swap(output[0],output[1]);
return;
}
//T split=output[random(0,size-1)];
T split=output[size/2];
int32_t left=0;
int32_t right=size-1;
while (left<=right)
{
while (output[left] < split)
left++;
while (output[right] > split)
right--;
if (left<=right)
{
CMath::swap(output[left],output[right]);
left++;
right--;
}
}
if (right+1> 1)
qsort(output,right+1);
if (size-left> 1)
qsort(&output[left],size-left);
}
/// qsort not implemented for comple64_t
COMPLEX64_ERROR_SORT(qsort)
/** performs insertion sort of an array output of length size
* it is sorted from in ascending (for type T) */
template <class T>
static void insertion_sort(T* output, int32_t size)
{
for (int32_t i=0; i<size-1; i++)
{
int32_t j=i-1;
T value=output[i];
while (j >= 0 && output[j] > value)
{
output[j+1] = output[j];
j--;
}
output[j+1]=value;
}
}
/// insertion_sort not implemented for comple64_t
COMPLEX64_ERROR_SORT(insertion_sort)
/** performs a in-place radix sort in ascending order */
template <class T>
inline static void radix_sort(T* array, int32_t size)
{
radix_sort_helper(array,size,0);
}
/// radix_sort not implemented for comple64_t
COMPLEX64_ERROR_SORT(radix_sort)
/*
* Inline function to extract the byte at position p (from left)
* of an 64 bit integer. The function is somewhat identical to
* accessing an array of characters via [].
*/
template <class T>
static inline uint8_t byte(T word, uint16_t p)
{
return (word >> (sizeof(T)-p-1) * 8) & 0xff;
}
/// byte not implemented for complex64_t
static inline uint8_t byte(complex64_t word, uint16_t p)
{
SG_SERROR("CMath::byte():: Not supported for complex64_t\n");
return uint8_t(0);
}
template <class T>
static void radix_sort_helper(T* array, int32_t size, uint16_t i)
{
static size_t count[256], nc, cmin;
T *ak;
uint8_t c=0;
radix_stack_t<T> s[RADIX_STACK_SIZE], *sp, *olds, *bigs;
T *an, *aj, *pile[256];
size_t *cp, cmax;
/* Push initial array to stack */
sp = s;
radix_push(array, size, i);
/* Loop until all digits have been sorted */
while (sp>s) {
radix_pop(array, size, i);
an = array + size;
/* Make character histogram */
if (nc == 0) {
cmin = 0xff;
for (ak = array; ak < an; ak++) {
c = byte(*ak, i);
count[c]++;
if (count[c] == 1) {
/* Determine smallest character */
if (c < cmin)
cmin = c;
nc++;
}
}
/* Sort recursively if stack size too small */
if (sp + nc > s + RADIX_STACK_SIZE) {
radix_sort_helper(array, size, i);
continue;
}
}
/*
* Set pile[]; push incompletely sorted bins onto stack.
* pile[] = pointers to last out-of-place element in bins.
* Before permuting: pile[c-1] + count[c] = pile[c];
* during deal: pile[c] counts down to pile[c-1].
*/
olds = bigs = sp;
cmax = 2;
ak = array;
pile[0xff] = an;
for (cp = count + cmin; nc > 0; cp++) {
/* Find next non-empty pile */
while (*cp == 0)
cp++;
/* Pile with several entries */
if (*cp > 1) {
/* Determine biggest pile */
if (*cp > cmax) {
cmax = *cp;
bigs = sp;
}
if (i < sizeof(T)-1)
radix_push(ak, *cp, (uint16_t) (i + 1));
}
pile[cp - count] = ak += *cp;
nc--;
}
/* Play it safe -- biggest bin last. */
swap(*olds, *bigs);
/*
* Permute misplacements home. Already home: everything
* before aj, and in pile[c], items from pile[c] on.
* Inner loop:
* r = next element to put in place;
* ak = pile[r[i]] = location to put the next element.
* aj = bottom of 1st disordered bin.
* Outer loop:
* Once the 1st disordered bin is done, ie. aj >= ak,
* aj<-aj + count[c] connects the bins in array linked list;
* reset count[c].
*/
aj = array;
while (aj<an)
{
T r;
for (r = *aj; aj < (ak = --pile[c = byte(r, i)]);)
swap(*ak, r);
*aj = r;
aj += count[c];
count[c] = 0;
}
}
}
/// radix_sort_helper not implemented for complex64_t
static void radix_sort_helper(complex64_t* array, int32_t size, uint16_t i)
{
SG_SERROR("CMath::radix_sort_helper():: Not supported for complex64_t\n");
}
/** Performs a quicksort on an array of pointers.
* It is sorted from in ascending (for type T)
*
* Every element is dereferenced once before being compared
*
* @param vector array of pointers to sort
* @param length length of array
*
* */
template <class T>
static void qsort(T** vector, index_t length)
{
if (length==1)
return;
if (length==2)
{
if (*vector[0]>*vector[1])
swap(vector[0],vector[1]);
return;
}
T* split=vector[length/2];
int32_t left=0;
int32_t right=length-1;
while (left<=right)
{
while (*vector[left]<*split)
++left;
while (*vector[right]>*split)
--right;
if (left<=right)
{
swap(vector[left],vector[right]);
++left;
--right;
}
}
if (right+1>1)
qsort(vector,right+1);
if (length-left>1)
qsort(&vector[left],length-left);
}
/// qsort not implemented for complex64_t
static void qsort(complex64_t** vector, index_t length)
{
SG_SERROR("CMath::qsort():: Not supported for complex64_t\n");
}
/// display bits (useful for debugging)
template <class T> static void display_bits(T word, int32_t width=8*sizeof(T))
{
ASSERT(width>=0)
for (int i=0; i<width; i++)
{
T mask = ((T) 1)<<(sizeof(T)*8-1);
while (mask)
{
if (mask & word)
SG_SPRINT("1")
else
SG_SPRINT("0")
mask>>=1;
}
}
}
/// disply_bits not implemented for complex64_t
static void display_bits(complex64_t word,
int32_t width=8*sizeof(complex64_t))
{
SG_SERROR("CMath::display_bits():: Not supported for complex64_t\n");