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random.cpp
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/**
* @file random.c
* @brief Random utility procedures for BayeSys3.
*
// History: Random.c 17 Nov 1994 - 13 Sep 2003
//
// Acknowledgments:
// "Numerical Recipes", Press et al, for ideas
// "Handbook of Mathematical Functions", Abramowitz and Stegun, for formulas
// Peter J Acklam website, for inverse normal code
//-----------------------------------------------------------------------------
/*
Copyright (c) 1994-2003 Maximum Entropy Data Consultants Ltd,
114c Milton Road, Cambridge CB4 1XE, England
This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
This library 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
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
#include "license.txt"
*/
#include <stddef.h>
#include <time.h>
#include <math.h>
#include <float.h>
#include "random.h"
static const double SHIFT32 = 1.0 / 4294967296.0; // 2^-32
static const double HalfPi = 1.57079632679489661922; // pi/2
static const double TwoxPi = 6.28318530717958647688; // 2*pi
static const double SqrPi2 = 1.25331413731550025120; // sqrt(pi/2)
static const double Sqr2Pi = 2.50662827463100050240; // sqrt(2*pi)
static const double LogSqr2Pi = 0.91893853320467274177; // log(sqrt(2*pi))
// Internal prototypes
static void Positive2 (Rand_t, double, double, double, double*, double*);
static void Positive2A (double, double, double, double,
double*, double*, double*);
static void Positive2B (double, double, double, double,
double*, double*, double*);
static void Positive2C (double, double, double, double,
double*, double*, double*);
static void Positive2D (double, double, double*, double*, double*);
static void Posneg2 (Rand_t, double, double, double, double, double,
double*, double*);
static void Posneg2A (double, double, double, double, double, double,
double*, double*, double*);
static void Posneg2B (double, double, double, double, double, double,
double*, double*, double*);
static void Posneg2C (double, double, double, double, double, double,
double*, double*, double*);
static void Posneg2D (double, double, double, double, double, double,
double*, double*, double*);
static double eigenerf2 (double, double, double, double, double);
static double erfint2 (double, double);
static double wedge (double, double, double);
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: RanInit
//
// Purpose: Initialise generator of 4 unsigneds for other Random procedures
//
// Format: Rand[0] + Rand[1] * 2^32 = # calls to Ranint since initialisation
// Rand[2] = generator offset, re-randomised every 2^32 calls
// Rand[3] = extra random integer available after each call
//
// History: John Skilling 15 Jan 2002, 31 Oct 2002
//-----------------------------------------------------------------------------
//
int RanInit( // O Seed, either from input or time
Rand_t Rand, // O Random generator state [4]
int seed) // I Seed: +ve = value, -ve = time seed
{
unsigned j, k;
k = 1;
for ( j = 0; k; ++j )
k += k;
if ( j != 32 ) // Check 32-bit arithmetic
return E_RAN_ARITH;
if ( seed < 0 )
{
seed = (int)time(NULL);
if ( seed < 0 )
seed = ~seed; // still OK after A.D.2030
}
Rand[0] = Rand[1] = 0; // 64-bit counter
Rand[2] = 1013904223 + 1664525 * seed; // sticky offset
Rand[3] = 1013904223 + 1664525 * Rand[2]; // extra random integer
return seed;
}
#if 0
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative initialiser #1, crude congruential.
//-----------------------------------------------------------------------------
void RanInit1(
Rand_t Rand, // O Random generator state [1]
int seed) // I Non-negative seed
{
Rand[0] = seed;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative initialiser #2, double congruential with shuffling.
//-----------------------------------------------------------------------------
void RanInit2(
Rand_t Rand, // O Random generator state [35]
int seed) // I Non-negative seed
{
int i, j;
i = (seed > 2147483397) ? seed - 2147483397 : seed + 1; // [1...2147483398]
Rand[33] = i;
for ( j = 39; j >= 0; j-- )
{
i = 40014 * i - 2147483563 * (i / 53668);
if ( i < 0 )
i += 2147483563; // [1...2147483562]
if ( j < 32 )
Rand[j] = i;
}
Rand[34] = Rand[0];
Rand[32] = i;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative initialiser #3, Knuth's subtractive.
//-----------------------------------------------------------------------------
void RanInit3(
Rand_t Rand, // O Random generator state [112]
int seed) // I Non-negative seed
{
unsigned i, j, k, m;
m = (unsigned)(161803398 - seed);
Rand[54] = m;
Rand[109] = 0;
k = 1;
for ( i = 1; i < 55; ++i )
{
j = (21 * i - 1) % 55;
Rand[j] = k;
k = m - k;
m = Rand[j];
Rand[54+i] = i;
}
for ( k = 0; k < 4; ++k )
for ( i = 0; i < 55; ++i )
Rand[i] -= Rand[(i + 31) % 55];
Rand[110] = 0;
Rand[111] = 31;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative initialiser #4, 64-bit hashing.
//-----------------------------------------------------------------------------
void RanInit4(
Rand_t Rand, // O Random generator state [3]
int seed) // I Non-negative seed
{
Rand[0] = Rand[1] = 0;
Rand[2] = seed;
}
#endif
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ranint
//
// Purpose: Random integer sample, in [-2^31,2^31).
//
// Method: 64-bit hashing is on #calls + offset to produce 32-bit candidate
// with algorithm from ran4 in "Numerical Recipes".
// After each 2^32 calls, the 64-bit offset is incremented by
// congruential generators on its 32-bit halves.
//
// Notes: (1) Period is just less than 2^95, extremely large.
// (2) All random calls are directed through this procedure.
// (3) After a call, Rand[3] is available as another random integer.
//
// History: John Skilling 28 Jan 2002, 31 Oct 2002, 17 Dec 2002
//-----------------------------------------------------------------------------
//
int Ranint( // O 32-bit integer
Rand_t Rand) // I O Random generator state [4]
{
unsigned m, n; // 64-bit register
unsigned u, v, w;
int i;
// 64-bit counter, for hashing
if ( !(Rand[0] ++) )
{
Rand[1] ++;
// occasional update offset in Rand[2]
i = Rand[2] >> 1; // [0...2147483647]
i -= i / 24683721; // [0...2147483561]
i++; // [1...2147483562]
i = 40014 * i - 2147483563 * (i / 53668);
if ( i < 0 ) // (40014 * i) % 2147483563
i += 2147483563; // [1...2147483562]
i--; // [0...2147483561]
i += i / 24683720; // [0...2147483647] (with holes)
Rand[2] = i << 1; // even
if ( 1013904223 + 1664525 * i < 0 ) // chance
Rand[2]++; // even or odd
}
// Two double steps of 64-bit hash
n = Rand[0] + Rand[2];
m = Rand[1] + Rand[2];
w = n ^ 0xbaa96887;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
m ^= (((u >> 16) | (u << 16)) ^ 0xb4f0c4a7) + w * v;
w = m ^ 0x1e17d32c;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
n ^= (((u >> 16) | (u << 16)) ^ 0x178b0f3c) + w * v;
w = n ^ 0x03bcdc3c;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
m ^= (((u >> 16) | (u << 16)) ^ 0x96aa3a59) + w * v;
w = m ^ 0x0f33d1b2;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
n ^= (((u >> 16) | (u << 16)) ^ 0xaa5835b9) + w * v;
Rand[3] = m;
return n;
}
#if 0
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative generator #1, crude congruential.
//-----------------------------------------------------------------------------
int Ranint1( // O 32-bit integer
Rand_t Rand) // I O Random generator state [1]
{
Rand[0] = Rand[0] * 1664525 + 1013904223;
return Rand[0];
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative generator #2, double congruential with shuffling.
//-----------------------------------------------------------------------------
int Ranint2( // O Odd integer with 85 holes: ONLY (2^31 - 85) VALUES
Rand_t Rand) // I O Random generator state [35]
{
int i, j, k;
// First congruential sequence
i = Rand[32];
i = 40014 * i - 2147483563 * (i / 53668);
if ( i < 0 )
i += 2147483563;
Rand[32] = i; // [1...2147483562]
// Second congruential sequence
k = Rand[33];
k = 40692 * k - 2147483399 * (k / 52774);
if ( k < 0 )
k += 2147483399;
Rand[33] = k; // [1...2147483398]
// Subtract from shuffled table
j = Rand[34] / 67108862; // [0...31]
k = Rand[j] - k; // [-2147483397...2147483561]
if ( k < 0 )
k += 2147483563; // (2^31 - 85) values
Rand[34] = k; // [0...2147483562]
Rand[j] = i;
// Expand result k
k += k / 24970740; // [0...2^31) with holes at 24970740*[1...85]
return (k << 1) | 1;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative generator #3, Knuth's subtractive.
//-----------------------------------------------------------------------------
int Ranint3( // O 32-bit integer
Rand_t Rand) // I O Random generator state [112]
{
unsigned j, k;
j = Rand[110];
Rand[110] = Rand[55+j];
k = Rand[111];
Rand[111] = Rand[55+k];
return (Rand[j] -= Rand[k]);
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Kernel of alternative generator #4, 64-bit hashing.
//-----------------------------------------------------------------------------
int Ranint4( // O 32-bit integer
Rand_t Rand) // I O Random generator state [3]
{
unsigned u, v, w, m, n;
// 64-bit counter, for hashing
if ( !(Rand[0] ++) )
Rand[1] ++;
// 64-bit hash
n = Rand[0] + Rand[2];
m = Rand[1] + Rand[2];
w = n ^ 0xbaa96887;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
m ^= (((u >> 16) | (u << 16)) ^ 0xb4f0c4a7) + w * v;
w = m ^ 0x1e17d32c;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
n ^= (((u >> 16) | (u << 16)) ^ 0x178b0f3c) + w * v;
w = n ^ 0x03bcdc3c;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
m ^= (((u >> 16) | (u << 16)) ^ 0x96aa3a59) + w * v;
w = m ^ 0x0f33d1b2;
v = w >> 16;
w &= 0xffff;
u = (v - w) * (v + w);
n ^= (((u >> 16) | (u << 16)) ^ 0xaa5835b9) + w * v;
return n;
}
#endif
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Randouble
//
// Purpose: Random single-precision doubleing-point sample.
// The 2^23 allowed values are odd multiples of 2^-24,
// symmetrically placed in strict interior of (0,1).
//
// Notes: (1) Tuned to 23-bit mantissa in "double" format.
// (2) Uses Ranint.
//
// History: John Skilling 20 Oct 2002
//-----------------------------------------------------------------------------
//
float Ranfloat( // O Value
Rand_t Rand) // I O Random generator state
{
unsigned u = ((unsigned)Ranint(Rand)
& 0xfffffe00) ^ 0x00000100; // switch lowest (2^-24) bit ON
return (float)u * (float)SHIFT32;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Randouble
//
// Purpose: Random double-precision doubleing-point sample.
// The 2^52 allowed values are odd multiples of 2^-53,
// symmetrically placed in strict interior of (0,1).
//
// Notes: (1) Tuned to 52-bit mantissa in "double" format.
// (2) Uses one call to Ranint to get 64 random bits, with extra
// random integer available in Rand[3].
// (3) All doubleing-point random calls are directed through this code,
// except Rangauss which uses the extra random integer in Rand[3].
//
// History: John Skilling 6 May 1995, 3 Dec 1995, 24 Aug 1996
// 20 Oct 2002, 17 Dec 2002
//-----------------------------------------------------------------------------
//
double Randouble( // O Value within (0,1)
Rand_t Rand) // I O Random generator state
{
unsigned hi, lo;
hi = (unsigned)Ranint(Rand); // top 32 bits
lo = (Rand[3] // low bits
& 0xfffff000) ^ 0x00000800; // switch lowest (2^-53) bit ON
return ((double)hi + (double)lo * SHIFT32) * SHIFT32;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Rangauss
//
// Purpose: Sample from Gaussian N(0,1)
//
// Notes: (1) Cannot overflow; |value| < 10 standard deviations
// (2) 2nd sample y*sqrt(...) could be available, best with y = ...+ 0.5
//
// History: JS 3 Oct 2002 Half of Box-Muller
// 17 Dec 2002 Use Rand[3] for extra 40% speedup
//-----------------------------------------------------------------------------
//
double Rangauss( // O Value
Rand_t Rand) // I O Random generator state
{
static const double RR = 4611686018427387904.0; // 2^62
double x, y, r;
do
{
x = Ranint(Rand) + 0.5; // (-2^31,2^31), not 0
y = (int)Rand[3]; // [-2^31,2^31)
r = x * x + y * y;
}
while ( r >= RR ); // disc, radius 2^31
return x * sqrt(2.0 * log(RR / r) / r); // Waste 2nd Box-Muller sample
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Rancauchy
//
// Purpose: Draw sample from
// prob(x) = 1 / (1 + x * x)
//
// Notes: Cannot overflow; |value| <= 2^53/pi
//
// History: JS 3 Oct 2002
//-----------------------------------------------------------------------------
//
double Rancauchy( // O Value
Rand_t Rand) // I O Random generator state
{
double t = tan(HalfPi * (Randouble(Rand) - 0.5));
return (1.0 - t * t) / (2.0 * t);
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Rangamma
//
// Purpose: Draw sample from
// -1+c -x
// prob(x) = x e / Gamma(c) for x in [0, inf)
//
// Notes: x=0 can be returned, especially if c is small, but x=inf cannot.
//
// History: JS 24 Jan 1994, 19 Oct 1995, 24 Aug 1996, 3 Oct 2002
//-----------------------------------------------------------------------------
//
double Rangamma( // O Value
Rand_t Rand, // I O Random generator state
double c) // I Exponent
{
double a, q, r, g, e;
g = c - 1.0;
if (g > 0.0)
{
e = sqrt(g + c) / g;
do
{
do
{
r = Rancauchy(Rand);
q = e * r + 1.0;
}
while (q <= 0.0);
}
while ( log(Randouble(Rand) / (1.0 + r * r))
> g * (log(q) - q + 1.0) ); // Accept?
q *= g;
}
else
{
r = 1.0 / (1.0 + c * exp(-1.0)); // prob(bounding p < 1)
do
{
if ( Randouble(Rand) >= r ) // Bounding x > 1
{
q = 1.0 - log(Randouble(Rand)); // exp(-q) in (1,inf)
a = g * log(q); // log(Accept ratio)
}
else // Bounding x <= 1
{
q = exp(log(Randouble(Rand)) / c); // q^(-1+c) in (0,1)
a = -q; // log(Accept ratio)
}
}
while ( log(Randouble(Rand)) > a );
}
return q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ranpoiss
//
// Purpose: Draw sample from Poisson
//
// -c j
// prob(j) = e c / j! ( 0 <= c <~ 4000000000 )
//
// Notes: Distribution is truncated at 2^32-1,
// so input parameter c should not approach 2^32.
//
// History: JS 15 May 1998, 24 Mar 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
unsigned Ranpoiss( // O Value
Rand_t Rand, // I O Random generator state
double c) // I Mean
{
unsigned j, k;
double a, b, r, t, y;
if ( c <= 0.0 )
return 0;
if ( c < 70.0 ) // Direct is faster for small c
{
j = k = (unsigned)c;
t = exp(j * log(c) - c - logfactorial(j));
for ( ; ; )
{
r = Randouble(Rand);
if ( t >= r )
return j;
a = b = t;
do
{
if ( a > b )
{
if ( (t += a *= j-- / c) >= r )
return j;
}
else
{
if ( (t += b *= c / ++k) >= r )
return k;
}
}
while ( b > 0.0 ); // Trap failure from rounding error
}
}
else
{
b = sqrt(2.0 * c);
do
{
do
{
y = Rancauchy(Rand);
r = c + b * y;
}
while ( r < 0.0 );
j = (unsigned)r;
t = j * log(c) - logfactorial(j) - c;
}
while ( log(Randouble(Rand) / (1.731 * (1.0 + y * y) * b)) > t );
return j;
}
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ranbinom
//
// Purpose: Draw sample from binomial
//
// n! j n-j
// prob(j) = -------- p (1-p) 0 <= j <= n
// j!(n-j)!
//
// History: JS 24 Mar 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
unsigned Ranbinom( // O Value
Rand_t Rand, // I O Random generator state
unsigned n, // I Range
double p) // I Mean/Range
{
unsigned j;
double a, b, g, q, r, t, u, w, y;
if ( n == 0 )
return 0;
if ( p <= 0.0 )
return 0;
if ( p >= 1.0 )
return n;
q = 1.0 - p;
if ( n < 100 )
{
u = w = j = (unsigned)(p * n + p);
t = exp(j * log(p) + (n - j) * log(q)
+ logfactorial(n) - logfactorial(j) - logfactorial(n - j));
for ( ; ; )
{
r = Randouble(Rand);
if ( t >= r )
return j;
a = b = t;
do
{
if ( a > b )
{
a *= u / p;
u -= 1.0;
a *= q / (n - u);
t += a;
if ( t >= r )
return (unsigned)u;
}
else
{
b *= (n - w) / q;
w += 1.0;
b *= p / w;
t += b;
if ( t >= r )
return (unsigned)w;
}
}
while ( a + b > 0.0 ); // Trap failure from rounding error
}
}
else
{
b = sqrt(2.0 * n * p * q + 0.25);
g = logfactorial(n);
do
{
do
{
y = Rancauchy(Rand);
r = n * p + b * y + 0.5;
}
while ( r < 0.0 || r >= n + 1.0 );
j = (unsigned)r;
t = g - logfactorial(j) - logfactorial(n - j)
+ j * log(p) + (n - j) * log(q);
}
while ( log(Randouble(Rand) / (0.8190 * (1.0 + y * y) * b)) > t );
return j;
}
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ranbeta
//
// Purpose: Draw sample from beta distribution
// r n-r
// prob(x) = x (1 - x) / Z , Z = (n+1)! / r! (n-r)!
//
// Method: Rejection beneath Cauchy of phenomenologically adequate width.
//
// History: John Skilling 28 Nov 2000
//-----------------------------------------------------------------------------
//
double Ranbeta( // O Value
Rand_t Rand, // I O Random generator state
int r, // I Number of successes >= 0
int n) // I Total number >= r
{
double p, q, x, y, c;
x = (double)n;
if ( r <= 0 )
x = 1.0 - exp(log(Randouble(Rand)) / (x + 1.0));
else if ( r >= n )
x = exp(log(Randouble(Rand)) / (x + 1.0));
else
{
p = (double)r / x;
q = 1.0 - p;
c = x / (3.0 * p * q); // less than curvature at top
do
{
do
{
y = Rancauchy(Rand) / sqrt(c);
x = p + y; // x from Cauchy upper bound
}
while ( x <= 0.0 || x >= 1.0 ); // x within range
y = r * log(x / p) + (n - r) * log((1.0 - x) / q) // log(true)
+ log(1.0 + c * y * y); // -log(Cauchy)
}
while ( log(Randouble(Rand)) > y ); // accept?
}
return x;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Rangeom
//
// Purpose: Draw integer from truncated geometric distribution
// j
// prob(j) proportional a for j in [0,N-1]
//
// where a = alpha/(alpha+1), alpha > 0.0
//
// Special case N = 0 means N = 2^32 for which
//
// <j> = alpha , var(j) = alpha(alpha+1)
//
// History: John Skilling 3 Apr 2001 Linearly truncated version
// 17 Dec 2002 Hard truncation
//-----------------------------------------------------------------------------
//
unsigned Rangeom( // O Value
Rand_t Rand, // I O Random generator state
double alpha, // I Parameter
unsigned N) // I Supremum (0 interpreted as 2^32)
{
double a, r;
if ( N == 1 || alpha <= 0.0 )
return 0;
if ( alpha * DBL_EPSILON >= 0.5 )
return Rangrid(Rand, N);
a = alpha / (alpha + 1.0);
do
{
r = Randouble(Rand);
if ( N )
r *= 1.0 - pow(a, (double)N);
r = log(1.0 - r) / log(a);
}
while ( r >= N ); // safety
return (int)r;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Rangrid
//
// Purpose: Random integer from [0,bound-1].
//
// Method: Development of Ranint.
//
// History: John Skilling 6 May 1995 - 15 Jan 2002
//-----------------------------------------------------------------------------
//
unsigned Rangrid( // O Value
Rand_t Rand, // I O Random generator state
unsigned bound) // I Supremum (0 interpreted as 2^32)
{
unsigned i, M, N;
if ( bound == 0 )
return (unsigned)Ranint(Rand);
N = (unsigned)(-1) / bound;
M = N * bound;
do i = (unsigned)Ranint(Rand);
while ( i >= M );
return i / N;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ranperm
//
// Purpose: Random permutation of {0,1,2,...,n-1}.
//
// History: John Skilling 23 May 1998, 24 Jan 1999, 9 Nov 2000
//-----------------------------------------------------------------------------
//
void Ranperm(
Rand_t Rand, // I O Random generator state
int n, // I Dimension
int* perm) // O Output permutation
{
int i, m;
for ( m = 0; m < n; ++m )
{
i = Rangrid(Rand, (unsigned)(m + 1));
if ( i < m )
perm[m] = perm[i];
perm[i] = m;
}
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran2gauss
//
// Purpose: Draw sample from normalised form of
//
// exp( - g.x - x.A.x / 2 ) ( A non-negative )
//
// History: JS 31 Dec 2001
//-----------------------------------------------------------------------------
//
void Ran2gauss(
Rand_t Rand, // I O Random generator state
double g1, // I x gradient at origin
double g2, // I y gradient at origin
double A11, // I xx curvature >= 0
double A12, // I xy curvature, A12*A12 <= A11*A22
double A22, // I yy curvature >= 0
double* x, // O x sample position
double* y) // O y sample position
{
double d = A11 * A22 - A12 * A12;
double xbar = (g2 * A12 - g1 * A22) / d;
double ybar = (g1 * A12 - g2 * A11) / d;
*x = Rangauss(Rand) * sqrt(A22 / d);
*y = Rangauss(Rand) / sqrt(A22) - *x * A12 / A22;
*x += xbar;
*y += ybar;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran1pos
//
// Purpose: Draw positive sample (x +ve) from normalised form of
//
// exp( - g x - A x^2 / 2 )
//
// History: JS 30 Nov 1997
//-----------------------------------------------------------------------------
//
double Ran1pos( // O sample value
Rand_t Rand, // I O random generator state
double g, // I coeff of x
double A) // I coeff of x^2, >= 0
{
double a, x;
a = sqrt(A);
if ( 0.3 * a > g ) // Reject from completed normal curve
{
do x = Rangauss(Rand) / a - g / A;
while ( x < 0.0 );
}
else // Reject from bounding exponential
{
do x = -log(Randouble(Rand)) / g;
while ( -log(Randouble(Rand)) < A * x * x / 2.0 );
}
return x;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran2pos
//
// Purpose: Draw sample in positive quadrant from normalised form of
//
// exp( - g1 x - g2 y - (A11 x^2 + 2 A12 x y + A22 y^2) / 2 )
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
void Ran2pos(
Rand_t Rand, // I O random generator state
double g1, // I coeff of |x|, > 0
double g2, // I coeff of |y|, > 0
double A11, // I coeff of x^2, >= 0
double A12, // I coeff of xy, |A12| < sqrt(A11*A22)
double A22, // I coeff of y^2, >= 0
double* x, // O sample
double* y) // O sample
{
double a;
if ( A12 * A12 <= DBL_EPSILON * A11 * A22 )
{
*x = Ran1pos(Rand, g1, A11);
*y = Ran1pos(Rand, g2, A22);
return;
}
g1 /= sqrt(A11);
g2 /= sqrt(A22);
a = A12 / sqrt(A11 * A22);
if ( a > 1.0 )
a = 1.0;
if ( a < -1.0 )
a = -1.0;
if ( g2 <= g1 )
Positive2(Rand, g1, g2, -a, x, y);
else
Positive2(Rand, g2, g1, -a, y, x);
*x /= sqrt(A11);
*y /= sqrt(A22);
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Positive2
//
// Purpose: Draw sample in positive quadrant from normalised form of
//
// exp( - u x - v y - (x^2 - 2 a x y + y^2) / 2 )
//
// with u >= v as well as |a| < 1
//
// Method:
// Apart from the simple special case u > 0.3 or so, v > 0.3 or so:
//
// Sample x from | exp -x^2/2 - u x + (a x - v)^2/2 , ax > v
// | exp -x^2/2 - u x , ax < v
//
// then
// Sample y from | exp -(y - a x + v)^2 / 2 , ax > v
// | exp - y^2 / 2 , ax < v
// with
// Acceptance | 1 , ax > v
// | exp (a x - v)y , ax < v
//
// Provided u >= v, these algorithms always sample with O(1) efficiency.
//
// General sampler for ANY logconcave function of x (such as the above) is
// exp( height + ycap + q * (x - xcap) ), x > xcap (q < 0)
// exp( height + ycap + p * (x - xcap) ), x < xcap (p > 0)
// where (xcap,ycap) is apex of required curve over all x, height = O(1) +ve,
// and p,q define left and right exponentials tangent to the required curve
// A x^2 + B x + C (with A < 0), according to
// p = Lslope(x,y, A,B,C) = 2*A*x + B + 2 * sqrt(-A *(y - A*x*x - B*x - C))
// q = Rslope(x,y, A,B,C) = 2*A*x + B - 2 * sqrt(-A *(y - A*x*x - B*x - C))
// This method always samples with O(1) efficiency.
//
// Subsidiary codes crafted to evaluate all sqrt arguments definitely +ve,
// left slope p definitely +ve, and right slope q definitely -ve.
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Positive2(
Rand_t Rand, // I O random generator state
double u, // I coeff of x
double v, // I coeff of y
double a, // I coeff of xy, 0 < |a| <= 1
double* x0, // O sample
double* y0) // O sample
{
double height = 0.5000;
double x, y, accept, xcap, p, q, r, s, dx;
if ( v > 0.3000 )
// Special case: sample around (0,0) from exp - u x - v y
{
do
{
x = -log(Randouble(Rand)) / u;
y = -log(Randouble(Rand)) / v;
accept = a * x * y - 0.5 * (x * x + y * y);
}
while ( accept < log(Randouble(Rand)) );
}
else
// General sampler ....
{
// Bi-exponential for x
for ( ; ; )
{
if ( a > 0.0 )
{
if ( v < 0.0 )
Positive2A(u, v, a, height, &xcap, &p, &q);
else
Positive2B(u, v, a, height, &xcap, &p, &q);
}
else
{
if ( v < 0.0 )
Positive2C(u, v, a, height, &xcap, &p, &q);
else
Positive2D(u, height, &xcap, &p, &q);
}
// Sample x
for ( ; ; )
{
s = (xcap == 0.0 || Randouble(Rand) < p / (p - q)) ? q : p;
dx = log(Randouble(Rand)) / s;
x = dx + xcap;
if ( x <= 0.0 )
continue;
accept = -dx * (u + xcap + 0.5 * dx);
r = a * x - v;
if ( r > 0.0 )
accept += a * dx * (0.5 * a * (x + xcap) - v);
accept -= height + s * dx;
if ( log(Randouble(Rand)) < accept )
break;
}
// Sample y|x
if ( - v + a * x > 0.0 )
{
y = - v + a * x + Rangauss(Rand);
if ( y >= 0.0 )
break;
}
else
{
y = Rangauss(Rand);
if ( y >= 0.0 && log(Randouble(Rand)) < (- v + a * x) * y )
break;
}
}
}
// Exit
*x0 = x;
*y0 = y;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Positive2A
//
// Purpose: Set bi-exponential sampler for Positive2 in case a > 0, v <= 0
// x = inf |
// | exp - x^2 / 2 - u x + (a x - v)^2 / 2
// x = 0 |
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Positive2A(
double u, // I >= v
double v, // I <= 0
double a, // I > 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double y0, m0, xcap, ycap;
double d, p, q;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries (0,y0)...
// and slopes m0
y0 = 0.5 * v * v;
m0 = -(u + a * v);
// Find apex from slope=0
if ( m0 <= 0.0 ) // apex at 0
{
xcap = 0.0;
ycap = height + y0;
p = 0.0; // arbitrary
q = m0 - sqrt(2.0 * d * height); // Rslope(xcap, ycap, -d/2, m0, v*v/2)
}
else // apex in (0,inf)
{
xcap = m0 / d;
ycap = height + 0.5 * (m0 * xcap + v * v);
if ( ycap - y0 <= m0 * xcap )
p = sqrt(2.0 * d * height); // Lslope(xcap, ycap, -d/2, m0, v*v/2)
else
p = (ycap - y0) / xcap;
q = -sqrt(2.0 * d * height); // Rslope(xcap, ycap, -d/2, m0, v*v/2)
}
*xc = xcap;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Positive2B
//
// Purpose: Set bi-exponential sampler for Positive2 in case a > 0, v >= 0
//
// | exp - x^2 / 2 - u x + (a x - v)^2 / 2
// x2 = v/a |
// | exp - x^2 / 2 - u x
// x = 0 |
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Positive2B(
double u, // I >= v
double v, // I >= 0
double a, // I > 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x2, y2, m2;
double d, q, r;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries (0,y0)...(x2,y2)...
// and slopes +ve,-ve m2
x2 = v / a;
y2 = x2 * (- u - 0.5 * x2);
m2 = - u - x2;
// Apex necessarily at 0
r = (height - y2) + m2 * x2;
if ( r <= 0.0 ) // Rslope(0, height, -1/2., -u, 0)
q = -u - sqrt(2.0 * height);
else // Rslope(0, height, -d/2, -u-a*v, v*v/2)
q = -(u + a * v) - sqrt(2.0 * d * (d * height + a * a * r));
*xc = 0.0;
*p0 = 0.0; // arbitrary
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Positive2C
//
// Purpose: Set bi-exponential sampler for Positive2 in case a < 0, v <= 0
//
// | exp - x^2 / 2 - u x
// x1 = -v/-a |
// | exp - x^2 / 2 - u x + (a x - v)^2 / 2
// x = 0 |
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Positive2C(
double u, // I >= v
double v, // I <= 0
double a, // I < 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x1, y0, y1, m0, m1, xcap, ycap;
double d, p, q, r;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries (0,y0)...(x1,y1)...
// and slopes m0 m1
x1 = v / a;
y1 = x1 * (- u - 0.5 * x1);
m1 = - u - x1; // -ve
y0 = 0.5 * v * v;
m0 = -(u + a * v);
// Find apex from slope=0
if ( m0 <= 0.0 ) // apex at 0
{
xcap = 0.0;
ycap = height + 0.5 * v * v;
p = 0.0; // arbitrary
r = (ycap - y1) + m1 * x1;
if ( r <= 0.0 ) // Rslope(0, ycap, -d/2, -u-a*v, v*v/2)
q = m0 - sqrt(2.0 * d * height);
else // Rslope(0, ycap, -1/2., -u, 0)
q = (v - u) + 2.0 * height / (v - sqrt(2.0 * ycap));
}
else // apex in (0,x1)
{
xcap = m0 / d;
ycap = height + 0.5 * (d * xcap * xcap + v * v);
r = (ycap - y0) - m0 * xcap;
if ( r <= 0.0 ) // Lslope(xcap, ycap, -d/2, -u-a*v, v*v/2)
p = sqrt(2.0 * d * height);
else
p = (ycap - y0) / xcap;
r = (ycap - y1) - m1 * (xcap - x1);
if ( r <= 0.0 ) // Rslope(xcap, ycap, -d/2, -u-a*v, v*v/2)
q = -sqrt(2.0 * d * height);
else // Rslope(xcap, ycap, -1/2., -u, 0)
q = (a * (u - v) + (1.0 + a) * v) * a / d
- sqrt(2.0 * r + (xcap - x1) * (xcap - x1));
}
*xc = xcap;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Positive2D
//
// Purpose: Set bi-exponential sampler for Positive2 in case a < 0, v >= 0
// x = inf |
// | exp - x^2 / 2 - u x
// x = 0 |
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Positive2D(
double u, // I >= v >= 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
// Apex necessarily at 0
*xc = 0.0;
*p0 = 0.0; // arbitrary
*q0 = -u - sqrt(2.0 * height); // Rslope(0, height, -1/2., -u, 0)
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran1posneg
//
// Purpose: Draw sample from normalised form of
//
// exp( - u|x| - f x - A x^2 / 2 )
//
// Method:
// General sampler for ANY logconcave function of x (such as the above) is
// exp( height + ycap + q * (x - xcap) ), x > xcap (q < 0)
// exp( height + ycap + p * (x - xcap) ), x < xcap (p > 0)
// where (xcap,ycap) is apex of required curve over all x, height = O(1) +ve,
// and p,q define left and right exponentials tangent to the required curve
// a x^2 + b x + c (with A < 0), according to
// p = Lslope(x,y, a,b,c) = 2*a*x + b + 2 * sqrt(-a *(y - a*x*x - b*x - c))
// q = Rslope(x,y, a,b,c) = 2*a*x + b - 2 * sqrt(-a *(y - a*x*x - b*x - c))
// This method always samples with O(1) efficiency.
//
// History: JS 31 Dec 2001, 3 Oct 2000
//-----------------------------------------------------------------------------
//
double Ran1posneg( // O sample value
Rand_t Rand, // I O random generator state
double f, // I coeff of x
double u, // I coeff of |x|, > 0
double A) // I coeff of x^2, >= 0
{
static const double height = 0.5000;
double x, xcap, ycap;
double accept, p, q, r, m9, m0;
// Special case of no curvature
if ( A <= 0.0 )
{
if ( Randouble(Rand) < 0.5 * (1.0 + f / u) )
return log(Randouble(Rand)) / (u - f);
else
return -log(Randouble(Rand)) / (u + f);
}
// Scale general case to unit curvature
f /= sqrt(A);
u /= sqrt(A);
m9 = u - f;
m0 = -u - f;
// Set dominating bi-exponential sampler
if ( m9 < 0.0 ) // Apex in x -ve
{
xcap = m9;
ycap = height + 0.5 * xcap * xcap;
p = sqrt(2.0 * height); // Lslope(xcap, ycap, -1/2., m9, 0)
if ( ycap <= m9 * xcap )
q = -sqrt(2.0 * height); // Rslope(xcap, ycap, -1/2., m9, 0)
else
{
r = ycap - m0 * xcap;
if ( r <= 0.0 )
q = ycap / xcap;
else // Rslope(xcap, ycap, -1/2., m0, 0)
q = -2.0 * u - sqrt(2.0 * r + m9 * m9);
}
}
else if ( m0 <= 0.0 ) // Apex at x = 0
{
xcap = 0.0;
ycap = height;
p = m9 + sqrt(2.0 * height); // Lslope(xcap, ycap, -1/2., m9, 0)
q = m0 - sqrt(2.0 * height); // Rslope(xcap, ycap, -1/2., m0, 0)
}
else // Apex in x +ve
{
xcap = m0;
ycap = height + 0.5 * xcap * xcap;
if ( ycap <= m0 * xcap )
p = sqrt(2.0 * height); // Lslope(xcap, ycap, -1/2., m0, 0)
else
{
r = ycap - m9 * xcap;
if ( r <= 0.0 )
p = ycap / xcap;
else // Lslope(xcap, ycap, -1/2., m9, 0)
p = 2.0 * u + sqrt(2.0 * r + m0 * m0);
}
q = -sqrt(2.0 * height); // Rslope(xcap, ycap, -1/2., m0, 0)
}
// Sample x
do
{
// accept = -log(sampler)
if ( Randouble(Rand) < p / (p - q) )
{
x = xcap + log(Randouble(Rand)) / q;
accept = -(ycap + q * (x - xcap));
}
else
{
x = xcap + log(Randouble(Rand)) / p;
accept = -(ycap + p * (x - xcap));
}
// accept += log(true)
if ( x > 0.0 )
accept += x * (- u - f - 0.5 * x);
else
accept += x * (u - f - 0.5 * x);
}
while ( log(Randouble(Rand)) > accept );
// Exit
return x / sqrt(A);
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran2posneg
//
// Purpose: Draw sample from normalised form of
//
// exp( - u|x| - v|y| - f x - g y - (A11 x^2 + 2 A12 x y + A22 y^2) / 2 )
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
void Ran2posneg(
Rand_t Rand, // I O random generator state
double f, // I coeff of x
double g, // I coeff of y
double u, // I coeff of |x|, > 0
double v, // I coeff of |y|, > 0
double A11, // I coeff of x^2, >= 0
double A12, // I coeff of xy, |A12| < sqrt(A11*A22)
double A22, // I coeff of y^2, >= 0
double* x, // O sample
double* y) // O sample
{
double a;
if ( A12 * A12 <= DBL_EPSILON * A11 * A22 )
{
*x = Ran1posneg(Rand, f, u, A11);
*y = Ran1posneg(Rand, g, v, A22);
return;
}
f /= sqrt(A11);
u /= sqrt(A11);
g /= sqrt(A22);
v /= sqrt(A22);
a = A12 / sqrt(A11 * A22);
if ( a > 1.0 )
a = 1.0;
if ( a < -1.0 )
a = -1.0;
if ( f <= 0.0 )
{
if ( g <= 0.0 )
{
if ( -g - v >= -f - u )
Posneg2(Rand, -f, -g, u, v, -a, x, y);
else
Posneg2(Rand, -g, -f, v, u, -a, y, x);
}
else
{
if ( g - v >= -f - u )
Posneg2(Rand, -f, g, u, v, a, x, y);
else
Posneg2(Rand, g, -f, v, u, a, y, x);
*y = - *y;
}
}
else
{
if ( g <= 0.0 )
{
if ( -g - v >= f - u )
Posneg2(Rand, f, -g, u, v, a, x, y);
else
Posneg2(Rand, -g, f, v, u, a, y, x);
}
else
{
if ( g - v >= f - u )
Posneg2(Rand, f, g, u, v, -a, x, y);
else
Posneg2(Rand, g, f, v, u, -a, y, x);
*y = - *y;
}
*x = - *x;
}
*x /= sqrt(A11);
*y /= sqrt(A22);
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Posneg2
//
// Purpose: Draw sample from normalised form of
//
// exp( - u|x| - v|y| + f x + g y - (x^2 - 2 a x y + y^2) / 2 )
//
// with f >= 0, g >= 0, f - u <= g - v
// as well as u > 0, v > 0, 0 < |a| <= 1
//
// Method:
// Apart from the simple special case u - f > 0.3 or so, v - g > 0.3 or so:
//
// | exp -x^2/2 + f x - u|x| + (g - v + a x)^2/2 , g-v+ax > 0
// Sample x from | exp -x^2/2 + f x - u|x| , |g+ax| < v
// | exp -x^2/2 + f x - u|x| + (g + v + a x)^2/2 , g+v+ax < 0
// then
// | exp -(y - g + v - a x)^2 / 2 , g-v+ax > 0
// Sample y from | exp - y^2 / 2 , |g+ax| < v
// | exp -(y - g - v - a x)^2 / 2 , g+v+ax < 0
// with
// | exp - v (|y| - y) , g-v+ax > 0
// Acceptance | exp - v |y| + (g + a x) y , |g+ax| < v
// | exp - v (|y| + y) , g+v+ax < 0
//
// Provided f-u <= g-v, these algorithms always sample with O(1) efficiency.
//
// General sampler for ANY logconcave function of x (such as the above) is
// exp( height + ycap + q * (x - xcap) ), x > xcap (q < 0)
// exp( height + ycap + p * (x - xcap) ), x < xcap (p > 0)
// where (xcap,ycap) is apex of required curve over all x, height = O(1) +ve,
// and p,q define left and right exponentials tangent to the required curve
// A x^2 + B x + C (with A < 0), according to
// p = Lslope(x,y, A,B,C) = 2*A*x + B + 2 * sqrt(-A *(y - A*x*x - B*x - C))
// q = Rslope(x,y, A,B,C) = 2*A*x + B - 2 * sqrt(-A *(y - A*x*x - B*x - C))
// This method always samples with O(1) efficiency.
//
// Subsidiary codes crafted to evaluate all sqrt arguments definitely +ve,
// left slope p definitely +ve, and right slope q definitely -ve.
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Posneg2(
Rand_t Rand, // I O random generator state
double f, // I coeff of x, >= 0
double g, // I coeff of y, >= 0
double u, // I coeff of |x|, > 0
double v, // I coeff of |y|, > 0
double a, // I coeff of xy, -1 < a < 1
double* x0, // O sample
double* y0) // O sample
{
double height = 0.5000;
double x, y, accept, xcap, p, q, r, s, dx;
if ( v - g > 0.3000 )
// Special case: sample around (0,0) from exp f x + g y - u|x| - v|y|
{
do
{
x = (Randouble(Rand) > 0.5 * (1.0 - f / u))
? log(Randouble(Rand)) / (f - u)
: log(Randouble(Rand)) / (f + u);
y = (Randouble(Rand) > 0.5 * (1.0 - g / v))
? log(Randouble(Rand)) / (g - v)
: log(Randouble(Rand)) / (g + v);
accept = a * x * y - 0.5 * (x * x + y * y);
}
while ( accept < log(Randouble(Rand)) );
*x0 = x;
*y0 = y;
return;
}
// General bi-exponential for x
do
{
if ( a > 0.0 )
{
if ( g > v )
Posneg2A(f + u, f - u, g + v, g - v, a, height, &xcap, &p, &q);
else
Posneg2B(f + u, f - u, g + v, g - v, a, height, &xcap, &p, &q);
}
else
{
if ( g > v )
Posneg2C(f + u, f - u, g + v, g - v, a, height, &xcap, &p, &q);
else
Posneg2D(f + u, f - u, g + v, g - v, a, height, &xcap, &p, &q);
}
// Sample x
do
{
s = (Randouble(Rand) < p / (p - q)) ? q : p;
dx = log(Randouble(Rand)) / s;
x = dx + xcap;
if ( x > 0.0 )
accept = dx * (f - u - 0.5 * (xcap + x));
else
accept = dx * (f + u - 0.5 * (xcap + x));
r = g - v + a * x;
if ( r > 0.0 )
accept += a * dx * (r - 0.5 * a * dx);
r = g + v + a * x;
if ( r < 0.0 )
accept += a * dx * (r - 0.5 * a * dx);
accept -= height + s * dx;
}
while ( log(Randouble(Rand)) > accept );
// Sample y|x
if ( g - v + a * x > 0.0 )
{
y = g - v + a * x + Rangauss(Rand);
accept = (y >= 0.0) ? 0.0 : 2.0 * v * y;
}
else if ( g + v + a * x < 0.0 )
{
y = g + v + a * x + Rangauss(Rand);
accept = (y > 0.0) ? -2.0 * v * y : 0.0;
}
else
{
y = Rangauss(Rand);
accept = (g + a * x) * y - v * fabs(y);
}
}
while ( accept < 0.0 && accept < log(Randouble(Rand)) );
// Exit
*x0 = x;
*y0 = y;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Posneg2A
//
// Purpose: Set bi-exponential sampler for Posneg2 in case a > 0, g >= v
//
// | exp - x^2 / 2 + f x - u x + (g - v + a x)^2 / 2
// x = 0 |
// | exp - x^2 / 2 + f x + u x + (g - v + a x)^2 / 2
// x2 = -(g-v)/a |
// | exp - x^2 / 2 + f x + u x
// x1 = -(g+v)/a |
// | exp - x^2 / 2 + f x + u x + (g + v + a x)^2 / 2
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Posneg2A(
double fpu, // I f + u > 0
double fmu, // I f - u <= gmv
double gpv, // I g + v > 0
double gmv, // I g - v >= 0
double a, // I > 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x1, x2, y0, y1, y2, m0, m1, m2, m9, xcap, ycap;
double d, p, q, r, s, t, u;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries ...(x1,y1)...(x2,y2)...(0,y0)...
// and slopes m1 m2 m9,m0
x1 = -gpv / a;
y1 = x1 * (fpu - 0.5 * x1);
m1 = fpu - x1;
x2 = -gmv / a;
y2 = x2 * (fpu - 0.5 * x2);
m2 = fpu - x2;
y0 = 0.5 * gmv * gmv;
m9 = fpu + a * gmv; // +ve
m0 = fmu + a * gmv;
// Find apex from slope=0
if ( m0 <= 0.0 ) // apex at 0
{
xcap = 0.0;
ycap = height + y0;
if (ycap <= y2 - m2*x2) // Lslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
p = m9 + sqrt(2.0 * d * height);
else
{
r = (ycap - y1) + m1 * x1;
if ( r <= 0.0 ) // Lslope(xcap, ycap, -1/2., fpu, 0)
p = fpu + sqrt(2.0 * ycap);
else // Lslope(xcap,ycap,-d/2,fpu+a*gpv,gpv*gpv/2)
p = (fpu + a * gpv) + sqrt(2.0 * d * (d * ycap + a * a * r));
} // Rslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
q = m0 - sqrt(2.0 * d * height);
}
else // apex in (0,inf)
{
xcap = m0 / d;
ycap = height + 0.5 * (d * xcap * xcap + gmv * gmv);
if (ycap - y0 <= m0*xcap) // Lslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
p = sqrt(2.0 * d * height);
else
{
r = (ycap - y0) - m9 * xcap;
if ( r <= 0.0 )
p = (ycap - y0) / xcap;
else
{
s = (ycap - y2) - m2 * (xcap - x2);
if ( s <= 0.0 ) // Lslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
p = (fpu - fmu) + sqrt(d * (2.0 * r + d * xcap * xcap));
else
{
t = (ycap - y1) - m1 * (xcap - x1);
if ( t <= 0.0 )
{ // Lslope(xcap,ycap,-1/2.,fpu,0)
u = 2.0 * s - 2.0 * xcap * x2 + x2 * x2;
p = fpu + u / (xcap + sqrt(u + xcap * xcap));
}
else
{ // Lslope(xcap,ycap,-d/2,fpu+a*gpv,gpv*gpv/2)
u = fmu + a * gmv + d * gpv / a;
p = ((fpu - fmu) + a * (gpv - gmv))
+ sqrt(2.0 * d * t + u * u);
}
}
}
} // Rslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
q = -sqrt(2.0 * d * height);
}
*xc = xcap;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Posneg2B
//
// Purpose: Set bi-exponential sampler for Posneg2 in case a > 0, g <= v
//
// | exp - x^2 / 2 + f x - u x + (g - v + a x)^2 / 2
// x2 = (v-g)/a |
// | exp - x^2 / 2 + f x - u x
// x = 0 |
// | exp - x^2 / 2 + f x + u x
// x1 = -(v+g)/a |
// | exp - x^2 / 2 + f x + u x + (g + v + a x)^2 / 2
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Posneg2B(
double fpu, // I f + u > 0
double fmu, // I f - u <= gmv
double gpv, // I g + v > 0
double gmv, // I g - v <= 0
double a, // I > 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x1, x2, y1, y2, m1, m2;
double d, p, q, t;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries ...(x1,y1)...(0,y0)...(x2,y2)...
// and slopes m1 +ve,-ve m2
x1 = -gpv / a;
y1 = x1 * (fpu - 0.5 * x1);
m1 = fpu - x1;
x2 = -gmv / a;
y2 = x2 * (fmu - 0.5 * x2);
m2 = fmu - x2;
// Apex necessarily at 0
t = m1 * x1 - (y1 - height);
if ( t <= 0.0 ) // Lslope(0,height,-1/2.,fpu,0)
p = fpu + sqrt(2.0 * height);
else // Lslope(0,height,-d/2,fpu+a*gpv,gpv*gpv/2)
p = (fpu + a * gpv) + sqrt(d * (2.0 * t + d * x1 * x1));
t = m2 * x2 - (y2 - height);
if ( t <= 0.0 ) // Rslope(0,height,-1/2.,fmu,0)
q = fmu - sqrt(2.0 * height);
else // Rslope(0,height,-d/2,fmu+a*gmv,gmv*gmv/2)
q = (fmu + a * gmv) - sqrt(d * (2.0 * t + d * x2 * x2));
*xc = 0.0;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Posneg2C
//
// Purpose: Set bi-exponential sampler for Posneg2 in case a < 0, g >= v
//
// | exp - x^2 / 2 + f x - u x + (g + v + a x)^2 / 2
// x2 = (g+v)/-a |
// | exp - x^2 / 2 + f x - u x
// x1 = (g-v)/-a |
// | exp - x^2 / 2 + f x - u x + (g - v + a x)^2 / 2
// x = 0 |
// | exp - x^2 / 2 + f x + u x + (g - v + a x)^2 / 2
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Posneg2C(
double fpu, // I f + u > 0
double fmu, // I f - u <= gmv
double gpv, // I g + v > 0
double gmv, // I g - v >= 0
double a, // I < 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x1, x2, y0, y1, y2, m0, m1, m2, m9, xcap, ycap;
double d, p, q, r, s, t, u, v, w;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries ...(0,y0)...(x1,y1)...(x2,y2)...
// and slopes m9,m0 m1 m2
x1 = -gmv / a;
y1 = x1 * (fmu - 0.5 * x1);
m1 = fmu - x1; // -ve
x2 = -gpv / a;
y2 = x2 * (fmu - 0.5 * x2);
m2 = fmu - x2;
y0 = 0.5 * gmv * gmv;
m9 = fpu + a * gmv;
m0 = fmu + a * gmv;
// Find apex from slope=0
if ( m9 < 0.0 ) // apex in (-inf,0)
{
xcap = m9 / d;
ycap = height + 0.5 * (d * xcap * xcap + gmv * gmv);
p = sqrt(2. * d * height);// Lslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
r = (ycap - y0) - m9 * xcap;
if ( r <= 0.0 ) // Rslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
q = -sqrt(2.0 * d * height);
else
{
s = ycap - y0 - m0 * xcap;
if ( s <= 0.0 )
q = (ycap - y0) / xcap;
else
{
t = (ycap - y1) - m1 * (xcap - x1);
if ( t <= 0.0 ) // Rslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
q = (fmu - fpu) - sqrt(2.0 * d * s + m9 * m9);
else
{
u = (ycap - y2) - m2 * (xcap - x2);
if ( u <= 0.0 )
{ // Rslope(xcap,ycap,-1/2.,fmu,0)
v = 2.0 * t + (x1 - xcap) * (x1 - xcap);
w = (x1 * (1.0 - a) - 2.0 * xcap) * x1 * (1.0 + a);
q = (fmu - gmv)
- (2.0 * t + w) / (gmv - xcap + sqrt(v));
}
else
{ // Rslope(xcap,ycap,-d/2,fmu+a*gpv,gpv*gpv/2)
v = d * (2.0 * u + d * (x2 - xcap) * (x2 - xcap));
w = (fmu - fpu) - a * (gmv - gpv);
q = w - sqrt(v);
}
}
}
}
}
else if ( m0 <= 0.0 ) // apex at 0
{
xcap = 0.0;
ycap = height + 0.5 * gmv * gmv;
// Lslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
p = m9 + sqrt(2.0 * d * height);
r = ycap - y1 + m1 * x1;
if ( r <= 0.0 ) // Rslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
q = m0 - sqrt(2.0 * d * height);
else
{
s = ycap - y2 + m2 * x2;
if ( s <= 0.0 ) // Rslope(xcap,ycap,-1/2.,fmu,0)
q = (fmu - gmv) - 2.0 * height / (gmv + sqrt(2.0 * ycap));
else // Rslope(xcap,ycap,-d/2,fmu+a*gpv,gpv*gpv/2)
q = (a * (gpv - gmv) + m0) - sqrt(d * (2.0 * s + d * x2 * x2));
}
}
else // apex in (0,x1)
{
xcap = m0 / d;
ycap = height + 0.5 * (d * xcap * xcap + gmv * gmv);
r = ycap - y0 - m0 * xcap;
if ( r <= 0.0 ) // Lslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
p = sqrt(2.0 * d * height);
else
{
s = ycap - y0 - m9 * xcap;
if ( s <= 0.0 )
p = (ycap - y0) / xcap;
else // Lslope(xcap,ycap,-d/2,fpu+a*gmv,gmv*gmv/2)
p = (fpu - fmu) + sqrt(d * (2.0 * s + d * xcap * xcap));
}
r = ycap - y1 - m1 * (xcap - x1);
if ( r <= 0.0 ) // Rslope(xcap,ycap,-d/2,fmu+a*gmv,gmv*gmv/2)
q = -sqrt(2.0 * d * height);
else
{
s = ycap - y2 - m2 * (xcap - x2);
if ( s <= 0.0 )
{ // Rslope(xcap,ycap,-1/2.,fmu,0)
v = 2.0 * r + (fmu - x1) * (fmu - x1) * (1.0 + a * a) / d;
w = (a * (gmv - fmu) - (1.0 + a) * gmv) * a / d;
q = -v / (w + sqrt(2.0 * r + (xcap - x1) * (xcap - x1)));
}
else // Rslope(xcap,ycap,-d/2,fmu+a*gpv,gpv*gpv/2)
q = a * (gpv - gmv)
- sqrt(d * (2.0 * s + d * (xcap - x2) * (xcap - x2)));
}
}
*xc = xcap;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Posneg2D
//
// Purpose: Set bi-exponential sampler for Posneg2 in case a < 0, g <= v
//
// | exp - x^2 / 2 + f x - u x + (g + v + a x)^2 / 2
// x2 = (v+g)/-a |
// | exp - x^2 / 2 + f x - u x
// x = 0 |
// | exp - x^2 / 2 + f x + u x
// x1 = -(v-g)/-a |
// | exp - x^2 / 2 + f x + u x + (g - v + a x)^2 / 2
//
// History: JS 31 Dec 2001, 3 Oct 2002
//-----------------------------------------------------------------------------
//
static void Posneg2D(
double fpu, // I f + u > 0
double fmu, // I f - u <= gmv
double gpv, // I g + v > 0
double gmv, // I g - v <= 0
double a, // I < 0
double height, // I height of sampler cusp above target maximum
double* xc, // O target maximum
double* p0, // O sampler exp( yc + p0 * (x-xc) ), x < xc (p0 > 0)
double* q0) // O sampler exp( yc + q0 * (x-xc) ), x > xc (q0 < 0)
{
double x1, x2, y1, y2, m1, m2;
double d, p, q, t;
d = 1.0 - a * a;
if ( d < DBL_EPSILON )
d = DBL_EPSILON;
// Set boundaries ...(x1,y1)...(0,y0)...(x2,y2)...
// and slopes m1 +ve,-ve m2
x1 = -gmv / a;
y1 = x1 * (fpu - 0.5 * x1);
m1 = fpu - x1;
x2 = -gpv / a;
y2 = x2 * (fmu - 0.5 * x2);
m2 = fmu - x2;
// Apex necessarily at 0
t = m1 * x1 - (y1 - height);
if ( t <= 0.0 ) // Lslope(0, height, -1/2., fpu, 0)
p = fpu + sqrt(2.0 * height);
else // Lslope(0,height,-d/2,fpu+a*gmv,gmv*gmv/2)
p = (fpu + a * gmv) + sqrt(d * (2.0 * t + d * x1 * x1));
t = m2 * x2 - (y2 - height);
if ( t <= 0.0 ) // Rslope(0, height, -1/2., fmu, 0)
q = fmu - sqrt(2.0 * height);
else // Rslope(0,height,-d/2,fmu+a*gpv,gpv*gpv/2)
q = (fmu + a * gpv) - sqrt(d * (2.0 * t + d * x2 * x2));
*xc = 0.0;
*p0 = p;
*q0 = q;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: RanPos1
//
// Purpose: Draw sample in x >= 0 from normalised form of p(x) exp(-L(x))
//
// p(x) = s * delta(x) + 1
//
// L(x) = g * x + A * x * x / 2
//
// where g = gradL(0)
// A = gradgradL
//
// History: JS 29 Oct 1997, 30 Nov 1997, 3 Oct 2002
//-----------------------------------------------------------------------------
//
double RanPos1( // O Sample value on exit
Rand_t Rand, // I O Random generator state
double s, // I Spike amplitude
double g, // I Gaussian gradient at origin
double A) // I Gaussian curvature >= 0
{
double acc, mu, prob0, z;
acc = sqrt(A);
if ( 0.3 * acc > g ) // Reject from completed normal curve
{
mu = -g / A;
prob0 = exp(g * mu / 2.0) * sqrt(A / TwoxPi);
prob0 = 1.0 / (1.0 + s * prob0);
do
{
if ( Randouble(Rand) > prob0 )
z = 0.0;
else
z = mu + Rangauss(Rand) / acc;
}
while ( z < 0.0 );
}
else // Reject from bounding exponential
{
prob0 = 1.0 / (1.0 + s * g);
do
{
if ( Randouble(Rand) > prob0 )
{
z = 0.0;
break;
}
else
{
z = -log(Randouble(Rand)) / g;
}
}
while ( -log(Randouble(Rand)) < A * z * z / 2.0 );
}
return z;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: RanPos01
//
// Purpose: Draw sample in 0 < x < 1 from normalised form of
//
// exp( - g * x - A * x * x / 2 )
//
// History: JS 9 Jun 1998
//-----------------------------------------------------------------------------
//
double RanPos01( // O Sample value on exit
Rand_t Rand, // I O Random generator state
double g, // I Gaussian gradient at origin
double A) // I Gaussian curvature >= 0
{
double a, b, E, x, x0;
int flag = 0;
if ( g < -A / 2.0 )
{
g = -A - g;
flag = 1;
}
if ( A <= sqrt(DBL_EPSILON) )
{
if ( g * g < DBL_EPSILON )
x = Randouble(Rand);
else
{
E = 1.0 - exp(-g);
do x = - log(1.0 - E * Randouble(Rand)) / g;
while ( x >= 1.0 );
}
}
else
{
A = sqrt(A);
a = g / A;
if ( g > 0.0 )
{
x0 = 1.0 / (sqrt(1.0 + 0.25 * a * a) + 0.5 * a);
if ( x0 > A )
x0 = A;
b = x0 + a;
E = 1.0 - exp(-A * b);
if ( E * E < DBL_EPSILON )
do
{
do x = A * Randouble(Rand);
while ( log(Randouble(Rand)) > -0.5 * (x - x0) * (x - x0) );
}
while ( x >= A );
else
do
{
do x = - log(1.0 - E * Randouble(Rand)) / b;
while ( log(Randouble(Rand)) > -0.5 * (x - x0) * (x - x0) );
}
while ( x >= A );
}
else
{
b = a + A;
x0 = (b < 1.0) ? b : 1.0;
E = 1.0 - exp(-b * x0);
if ( E * E < DBL_EPSILON )
do
{
do x = b * Randouble(Rand);
while ( log(Randouble(Rand)) > -0.5 * (x - x0) * (x - x0) );
x = (Ranint(Rand) < 0) ? - x - a : x - a;
}
while ( x * (x - A) > 0.0 );
else
do
{
do x = -log(1.0 - E * Randouble(Rand)) / x0;
while ( log(Randouble(Rand)) > -0.5 * (x - x0) * (x - x0) );
x = (Ranint(Rand) < 0) ? - x - a : x - a;
}
while ( x * (x - A) > 0.0 );
}
x /= A;
}
return flag ? 1.0 - x : x;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: Ran1posX
//
// Purpose: Draw sample in x >= 0 from normalised form of
//
// x * exp( - g * x - A * x * x / 2 )
//
// History: JS 14 Jan 2003
//-----------------------------------------------------------------------------
//
double Ran1posX( // O Sample value on exit
Rand_t Rand, // I O Random generator state
double g, // I Gaussian gradient at origin
double A) // I Gaussian curvature >= 0
{
double a, x, y, xcap, accept;
if ( A < 0.0 )
return 0.0;
if ( A == 0.0 && g <= 0.0 )
return 0.0;
xcap = (g >= 0.0) ? 2.0 / (sqrt(g * g + 4.0 * A) + g)
: (sqrt(g * g + 4.0 * A) - g) / (2.0 * A); // Apex
a = 1.7000 / sqrt(A + 1.0 / (xcap * xcap)); // Adequate Cauchy width
do // Reject from bounding Cauchy
{
do
{
y = Rancauchy(Rand);
x = xcap + y * a;
}
while ( x <= 0.0 );
accept = log((1.0 + y * y) * x / xcap)
+ (xcap - x) * (g + 0.5 * A * (xcap + x));
}
while ( accept < log(Randouble(Rand)) );
return x;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function:
// infinity - g*x - A*x*x/2
// logerf := log INTEGRAL e dx
// x=0
//
// = log( sqrt(pi/2A) exp(g*g/2A) (1 - erf(g/sqrt(2A)) )
//
// History: JS 25 Nov 1997 From "erf" code
// 31 Mar 1998, 9 Jun 1998
// 23 Jun 1998 Polish
//-----------------------------------------------------------------------------
//
double logerf( // O Output value
double g, // I Gradient at origin
double A) // I Curvature at origin
{
double a, a0, a1, b0, b1, f, q, r, t, y;
if ( A <= 0.0 )
return -log(g);
a = sqrt(g * g / A);
if ( a < 2.0 ) // before which continued fraction too slow
{
// Power series, proportional error = DBL_EPSILON * exp(a*a/2)
t = a * a / 2.0;
q = r = a;
for ( f = 1.5; q > r * DBL_EPSILON; f += 1.0 )
r += q *= t / f;
if ( g >= 0.0 )
y = log(SqrPi2 * exp(t) - r);
else
y = log(SqrPi2 * exp(t) + r);
}
else if ( a < 8.6 ) // after which asymptotic series is fully accurate
{
// Continued fraction, proportional error = DBL_EPSILON
b0 = q = 0.0;
a0 = b1 = f = 1.0;
a1 = t = a * a / 2.0;
for ( f = 1.0; a0 / a1 - b0 / b1 > DBL_EPSILON; f = 1.0 / a1 )
{
q += 0.5;
a0 = (a1 + a0 * q) * f;
b0 = (b1 + b0 * q) * f;
q += 0.5;
r = q * f;
a1 = t * a0 + r * a1;
b1 = t * b0 + r * b1;
}
y = b1 * a / (2.0 * a1);
if ( g >= 0.0 )
y = log(y);
else
y = log(2.0 * SqrPi2 * exp(t) - y);
}
else
{
// Asymptotic series, proportional error = MAX(e(-a*a/2), DBL_EPSILON)
if ( g >= 0.0 )
{
t = 2.0 / (a * a);
q = r = 1.0;
for ( f = 0.5 * t; fabs(q) > DBL_EPSILON; f += t )
{
r += q *= -f;
if ( f > 1.0 ) break; // unnecessary if range OK
}
y = log((r - q * f / 2.0) / a);
}
else
y = log(2.0 * SqrPi2) + a * a / 2.0;
}
return y - log(A) / 2.0;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function:
// inf, inf - g.x - x.A.x/2
// logerf2 := log INTEGRAL dx dy e (A non-negative)
// x=0, y=0
//
// where g.x = g1*x + g2*y, x.A.x = A11*x*x + 2*A12*x*y + A22*y*y
//
// History: JS 12 Jul 1998
//-----------------------------------------------------------------------------
//
double logerf2( // O Output value
double g1, // I x gradient at origin
double g2, // I y gradient at origin
double A11, // I xx curvature >= 0
double A12, // I xy curvature, A12*A12 <= A11*A22
double A22) // I yy curvature >= 0
{
double ctheta; // x coord of ivec = y coord of jvec
double stheta; // y coord of ivec = -x coord of jvec
double val_i; // Eigenvalue for ivec
double val_j; // Eigenvalue for jvec
double origi, origj; // Origin relative to eigencentre
double xveci, xvecj; // x-axis relative to eigencoords
double yveci, yvecj; // y-axis relative to eigencoords
double u, v; // Eigencoords
double a; // Radius to wedge corner in eigencoords
double s, t; // local
double TINY = pow(DBL_EPSILON, 2.0 / 3.0);
// Separable if |A12| * <x(separable)> * <y(separable)> <= sqrt(EPSILON)
u = (g1 > 0.0) ? 1.0 / (A11 + g1 * g1) : (A11 + g1 * g1) / (A11 * A11);
v = (g2 > 0.0) ? 1.0 / (A22 + g2 * g2) : (A22 + g2 * g2) / (A22 * A22);
if ( A12 * A12 * u * v <= DBL_EPSILON )
return logerf(g1, A11) + logerf(g2, A22);
// Generic case ....
// Scale Gaussian to standard form exp(-u*u/2-v*v/2)
if ( A11 >= A22 )
{
t = (A11 - A22) / 2.0;
t += sqrt(t * t + A12 * A12);
s = sqrt(t * t + A12 * A12);
val_i = t + A22;
ctheta = t / s;
stheta = A12 / s;
}
else
{
t = (A22 - A11) / 2.0;
t += sqrt(t * t + A12 * A12);
s = sqrt(t * t + A12 * A12);
val_i = t + A11;
ctheta = A12 / s;
stheta = t / s;
}
val_j = (A11 * A22 - A12 * A12) / val_i;
if ( val_j < TINY * sqrt(A11 * A22) )
val_j = TINY * sqrt(A11 * A22); // Protect singular eval
val_i = sqrt(val_i);
val_j = sqrt(val_j);
origi = ( g1 * ctheta + g2 * stheta) / val_i;
origj = (-g1 * stheta + g2 * ctheta) / val_j;
xveci = val_i * ctheta;
xvecj = -val_j * stheta;
yveci = val_i * stheta;
yvecj = val_j * ctheta;
a = sqrt(origi * origi + origj * origj);
if ( a > 0.0 )
{
origi /= a;
origj /= a;
}
s = eigenerf2(a,
origi * xveci + origj * xvecj, -origj * xveci + origi * xvecj,
origi * yveci + origj * yvecj, -origj * yveci + origi * yvecj);
s = s + a * a / 2.0 - log(val_i * val_j);
return s;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function:
// -r*r/2
// eigenerf2 := log INTEGRAL e dArea
// wedge
// yyy..
// yyy.....
// yyy........
// yyy...........
// | yyy..............
// --O--> a..................
// | xxxxx............
// xxxxx.......
// xxxxx..
//
// 0 < Angle(xxxx, yyyy) < Pi
//
// History: JS 12 Jul 1998
//-----------------------------------------------------------------------------
//
static double eigenerf2(// Output value
double a, // I Radius to corner of wedge
double xcos, // I x-edge direction is (xcos,xsin) relative to radius a
double xsin, // I x-edge direction is (xcos,xsin) relative to radius a
double ycos, // I y-edge direction is (ycos,ysin) relative to radius a
double ysin) // I y-edge direction is (ycos,ysin) relative to radius a
{
#undef PLUS
#define PLUS(x,y) ((x>y) ? x+log(1.0+exp(y-x)) : y+log(1.0+exp(x-y)))
#undef MINUS
#define MINUS(x,y) ((x>y) ? x+log(1.0-exp(y-x)) : log(DBL_MIN))
#undef erfint
#define erfint(x) (logerf((x), 1.0) - (x) * (x) / 2.0)
double t, u, v;
// Normalisation to unit vectors
t = sqrt(xcos * xcos + xsin * xsin);
xcos /= t;
xsin /= t;
t = sqrt(ycos * ycos + ysin * ysin);
ycos /= t;
ysin /= t;
// Force upward orientation
if ( xsin + ysin < 0.0 )
{
t = xcos;
xcos = ycos;
ycos = t;
t = xsin;
xsin = -ysin;
ysin = -t;
}
// Select case
if ( ycos > 0.5 )
{ // case P
u = wedge(a, xsin, ysin);
}
else if ( xcos < -0.5 )
{ // case R
u = wedge(a, -xsin, -ysin);
t = LogSqr2Pi + erfint2(a * ysin, a * xsin);
u = PLUS(u, t);
}
else if ( ycos + 1.732 * ysin < 0.0 )
{
if ( ysin < -0.866 )
{ // case W
v = wedge(a, xcos, -ycos);
t = erfint(a * xsin) + erfint(a * xcos);
v = PLUS(v, t);
t = erfint(-a * ysin) + erfint(a * ycos);
v = PLUS(v, t);
u = 2.0 * LogSqr2Pi;
u = MINUS(u, v);
}
else if ( 1.732 * xsin < xcos )
{ // case V
u = LogSqr2Pi + erfint(a * ysin);
t = wedge(a, -xsin, ysin);
u = MINUS(u, t);
}
else
{ // case T
u = LogSqr2Pi + erfint(a * ysin);
v = erfint(a * xsin) + erfint(a * xcos);
if ( -ysin > -xcos )
{
t = wedge(a, -xcos, -ysin);
u = PLUS(u, t);
}
else
{
t = wedge(a, -ysin, -xcos);
v = PLUS(v, t);
}
u = MINUS(u, v);
}
}
else
{
if ( xsin < -0.866 )
{ // case U
u = erfint(a * ysin) + erfint(a * ycos);
t = erfint(-a * xsin) + erfint(a * xcos);
u = PLUS(u, t);
t = wedge(a, -xcos, ycos);
u = MINUS(u, t);
}
else if ( 1.732 * xsin > xcos )
{ // case Q
u = wedge(a, -xcos, -ycos);
t = erfint(a * ysin) + erfint2(a * ycos, a * xcos);
u = PLUS(u, t);
if ( ysin > xsin )
{
t = erfint(a * xcos) + erfint2(a * xsin, a * ysin);
u = MINUS(u, t);
}
else
{
t = erfint(a * xcos) + erfint2(a * ysin, a * xsin);
u = PLUS(u, t);
}
}
else
{ // case S
u = erfint(a * ysin) + erfint(a * ycos);
if ( -ycos >= xsin )
{
t = wedge(a, xsin, -ycos);
u = PLUS(u, t);
}
else
{
t = wedge(a, -ycos, xsin);
u = MINUS(u, t);
}
}
}
return u;
#undef erfint
#undef MINUS
#undef PLUS
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function:
// x=b -x*x/2
// erfint2 := log INTEGRAL e dx, a < b
// x=a
//
// History: JS 12 Jul 1998
//-----------------------------------------------------------------------------
//
static double erfint2( // O Output value
double a, // I Lower limit
double b) // I Upper limit
{
double aa, bb, e1, e2, ee, s, t;
aa = fabs(a);
bb = fabs(b);
if ( aa < bb )
{
s = aa;
aa = bb;
bb = s;
}
e1 = DBL_EPSILON / (aa + 1.0);
t = b - a;
ee = (aa + fabs(t)) * t;
e2 = ee * ee * fabs(t);
if ( e2 < e1 )
return log((1.0 - (a + t / 3.0) * t / 2.0) * t) - a * a / 2.0;
s = logerf(aa, 1.0) - aa * aa / 2.0;
t = logerf(bb, 1.0) - bb * bb / 2.0;
if ( a * b >= 0.0 )
return t + log(1.0 - exp(s - t));
else
return log(Sqr2Pi - exp(s) - exp(t));
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function:
//
// wedge := log(w(a,t) - w(a,s)) (a >= 0, -1 < s < t < 1) where
//
// infinity
// w(a,x) := INTEGRAL (arcsin(x) - arcsin(a*x/r)) exp(-r*r/2) r dr
// r=a
//
// Notes: Series abs cgt for all |s| < 1, |t| < 1, but cgce slow
// and more storage would be needed if |s| ~ 1 or |t| ~ 1 .
// This version allows values up to sqrt(3)/2, as from eigenerf2.
//
// History: JS 12 Jul 1998, 4 Sep 1999, 12 Sep 1999
//-----------------------------------------------------------------------------
//
static double wedge( // O Output value
double a, // I Input radius
double s, // I Lower angle (as sine)
double t) // I Upper angle (as sine)
{
#define STORE 120 // Want s^STORE < DBL_EPSILON for full accuracy
double w[STORE]; // arcsin_terms(s,t) = SUM w[k] (a/r)^(k+k+1)
double p; // p[k] = INT[a,inf] (1 - (a/r)^(k+k+1)) exp(a*a/2-r*r/2) r dr
double q; // max p[k] when recurrence relation for p started internally
double b, m, r, u, v, sum;
int j, k, n;
// Series {x, (1/2)x^3/3, (1*3/2*4)x^5/5, (1*3*5/2*4*6)x^7/7),..} difference
n = 0;
m = 1.0;
if ( s * t > 0.0 )
{ // Implicit difference between s and t
r = u = v = 1.0;
while ( (w[n++] = r * u * (t - s) / m) > DBL_EPSILON )
{
if ( n == STORE )
break; // Memory overflow trap
r *= 1.0 - 0.5 / n;
v *= t;
u = u * s + v;
v *= t;
u = u * s + v;
m += 2.0;
}
}
else
{ // Explicit difference between s and t
r = 1.0;
u = s;
v = t;
while ( (w[n++] = r * (v - u) / m) > DBL_EPSILON )
{
if ( n == STORE )
break; // Memory overflow trap
r *= 1.0 - 0.5 / n;
u *= s * s;
v *= t * t;
m += 2.0;
}
}
// Initialise recurrence relation for exponential parts
b = a * a / 2.0;
sum = 0.0;
if ( a < 7.0 ) // Balance point for error control
{
// Start at beginning unless errors would amplify too far
p = 1.0 - a * exp(logerf(a, 1.0));
for ( k = 0; k < n; ++k )
{
sum += w[k] * p;
p = 1.0 - b * p / (k + 0.5);
}
}
else
{
// Start in middle at maximum term if errors would otherwise amplify
if ( b < STORE ) // (= dimension limit)
{
k = (int)b;
u = k * k;
p = 0.5 + (0.031 - 0.4 / u) / u; // Phenomenological asymptote
j = k + 1;
u = j * j;
q = 0.5 + (0.031 - 0.4 / u) / u; // Phenomenological asymptote
q = (j - 0.5) * (1.0 - q) / j;
u = b - k;
q = u * q + (1.0 - u) * p; // Interpolated maximum term
}
else
{
j = STORE;
q = (j + 0.5) / (j + b); // Phenomenological approximant
}
// Count down from maximum term
p = q;
for ( k = j - 1; k >= n; --k )
p = (k - 0.5) * (1.0 - p) / b;
for ( ; k >= 0; --k )
{
sum += w[k] * p;
p = (k - 0.5) * (1.0 - p) / b;
}
// Count up from maximum term
p = q;
for ( k = j; k < n; ++k )
{
p = 1.0 - b * p / (k - 0.5);
sum += w[k] * p;
}
}
return log(sum) - b;
#undef STORE
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: logfactorial
//
// Purpose: log( k! ) = lookup table up to 99, then logGamma(k+1)
//
// History: JS 15 May 1998, 24 Mar 2001, 3 Oct 2002, 18 Aug 2003
//-----------------------------------------------------------------------------
//
double logfactorial( // O Value = log(k!)
unsigned k) // I Argument
{
static const double logfact[100] = { 0.0000000000000000,
0.0000000000000000, 0.6931471805599453, 1.7917594692280550,
3.1780538303479456, 4.7874917427820460, 6.5792512120101010,
8.5251613610654143, 10.6046029027452502, 12.8018274800814696,
15.1044125730755153, 17.5023078458738858, 19.9872144956618861,
22.5521638531234229, 25.1912211827386815, 27.8992713838408916,
30.6718601060806728, 33.5050734501368889, 36.3954452080330536,
39.3398841871994940, 42.3356164607534850, 45.3801388984769080,
48.4711813518352239, 51.6066755677643736, 54.7847293981123192,
58.0036052229805199, 61.2617017610020020, 64.5575386270063311,
67.8897431371815350, 71.2570389671680090, 74.6582363488301644,
78.0922235533153106, 81.5579594561150372, 85.0544670175815174,
88.5808275421976788, 92.1361756036870925, 95.7196945421432025,
99.3306124547874269, 102.9681986145138126, 106.6317602606434591,
110.3206397147573954, 114.0342117814617032, 117.7718813997450715,
121.5330815154386340, 125.3172711493568951, 129.1239336391272149,
132.9525750356163099, 136.8027226373263685, 140.6739236482342594,
144.5657439463448860, 148.4777669517730321, 152.4095925844973578,
156.3608363030787852, 160.3311282166309070, 164.3201122631951814,
168.3274454484276523, 172.3527971391628016, 176.3958484069973517,
180.4562914175437711, 184.5338288614494905, 188.6281734236715912,
192.7390472878449024, 196.8661816728899940, 201.0093163992815267,
205.1681994826411985, 209.3425867525368356, 213.5322414945632612,
217.7369341139542273, 221.9564418191303340, 226.1905483237275933,
230.4390435657769523, 234.7017234428182677, 238.9783895618343230,
243.2688490029827142, 247.5729140961868839, 251.8904022097231944,
256.2211355500095255, 260.5649409718632093, 264.9216497985528010,
269.2910976510198225, 273.6731242856937041, 278.0675734403661429,
282.4742926876303960, 286.8931332954269940, 291.3239500942703076,
295.7666013507606240, 300.2209486470141318, 304.6868567656687155,
309.1641935801469219, 313.6528299498790618, 318.1526396202093268,
322.6634991267261769, 327.1852877037752172, 331.7178871969284731,
336.2611819791984770, 340.8150588707990179, 345.3794070622668541,
349.9541180407702369, 354.5390855194408088, 359.1342053695753988
};
double s, y;
if ( k < 100 )
return logfact[k];
y = k + 1.0;
s = y + 2.269488974204959960;
s = y + 1.517473649153287398 / s;
s = y + 1.011523068126841711 / s;
s = y + 0.525606469002695417 / s;
s = y + 0.252380952380952380 / s;
s = y + 0.033333333333333333 / s;
s = 0.083333333333333333 / s;
s = s + 0.91893853320467 - y + (y - 0.5) * log(y);
return s;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: logGamma
//
// Purpose: log( Gamma(x) )
//
// History: JS 18 Aug 2003 From gammln 8 Dec 1994
// Continued fraction (Abramowitz and Stegun)
//-----------------------------------------------------------------------------
//
double logGamma( // O Value = log(Gamma(x))
double x) // I Argument
{
int k;
double s, t, y;
y = x;
for ( k = 16; y < 16.0; --k )
{
y += 1.0;
}
// Continued fraction for s = log(Gamma(y))
s = y + 2.269488974204959960;
s = y + 1.517473649153287398 / s;
s = y + 1.011523068126841711 / s;
s = y + 0.525606469002695417 / s;
s = y + 0.252380952380952380 / s;
s = y + 0.033333333333333333 / s;
s = 0.083333333333333333 / s;
s = s + 0.91893853320467 - y + (y - 0.5) * log(y);
// Return to s = log(Gamma(original x))
t = 1.0;
for ( ++k; k <= 16; ++k )
{
y -= 1.0;
t *= y;
}
s -= log(t);
// Required output
return s;
}
//+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
// Function: InvNorm
//
// Purpose: Inverse of normal distribution, returning x(y) where
// y(x) = Normal(x)
// x
// = INTEGRAL exp(-u*u/2) du / sqrt(2*pi)
// -inf
// = ( 1 + erf(x/sqrt(2)) ) / 2
//
// Rational approximations claimed to be correct to 1.15e-9
//
// History: JS 13 Sep 2003
// From Peter J Acklam www//home.online.no/~pjacklam/notes/invnorm
//-----------------------------------------------------------------------------
//
double InvNorm( // O Inverse normal (= number of standard deviations)
double y) // I Argument (= cumulative probability)
{
double r, s;
if ( y < 0.02425 )
{
s = sqrt(-2.0 * log(y));
return + (2.938163982698783 +
(4.374664141464968 -
(2.549732539343734 +
(2.400758277161838 +
(0.3223964580411365 +
0.007784894002430293 * s) * s) * s) * s) * s)
/ (1.0 +
(3.754408661907416 +
(2.445134137142996 +
(0.3224671290700398 +
0.007784695709041462 * s) * s) * s) * s);
}
else if ( y > 0.97575 )
{
s = sqrt(-2.0 * log(1.0 - y));
return - (2.938163982698783 +
(4.374664141464968 -
(2.549732539343734 +
(2.400758277161838 +
(0.3223964580411365 +
0.007784894002430293 * s) * s) * s) * s) * s)
/ (1.0 +
(3.754408661907416 +
(2.445134137142996 +
(0.3224671290700398 +
0.007784695709041462 * s) * s) * s) * s);
}
else
{
s = y - 0.5;
r = s * s;
return s * (2.506628277459239 -
(30.66479806614716 -
(138.3577518672690 -
(275.9285104469687 -
(220.9460984245205 -
39.69683028665376 * r) * r) * r) * r) * r)
/ (1.0 -
(13.28068155288572 -
(66.80131188771972 -
(155.6989798598866 -
(161.5858368580409 -
54.47609879822406 * r) * r) * r) * r) * r);
}
}

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