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lolremez: move some LolRemez matrix functions out of the engine.

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Sam Hocevar 12 年之前
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共有 7 個文件被更改，包括 447 次插入 和 463 次删除
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Show Stats Download Patch File Download Diff File
  1. +6
    -6
      src/lol/base/assert.h
  2. +0
    -75
      src/lol/math/vector.h
  3. +2
    -1
      tools/lolremez/Makefile.am
  4. +5
    -5
      tools/lolremez/lolremez.cpp
  5. +0
    -376
      tools/lolremez/lolremez.h
  6. +377
    -0
      tools/lolremez/solver.cpp
  7. +57
    -0
      tools/lolremez/solver.h

+ 6
- 6
src/lol/base/assert.h 查看文件

@@ -31,11 +31,11 @@ extern void DumpStack();
* on implementing __debugbreak() on POSIX systems. */ * on implementing __debugbreak() on POSIX systems. */
static inline void DebugAbort() static inline void DebugAbort()
{ {
DumpStack();
lol::DumpStack();
#if defined _WIN32 #if defined _WIN32
__debugbreak(); __debugbreak();
#endif #endif
Abort();
lol::Abort();
} }


#define LOL_CALL(macro, args) macro args #define LOL_CALL(macro, args) macro args
@@ -111,13 +111,13 @@ static inline void DebugAbort()
*/ */


#define LOL_ERROR_1(t) \ #define LOL_ERROR_1(t) \
Log::Error("assertion at %s:%d: %s\n", __FILE__, __LINE__, #t)
lol::Log::Error("assertion at %s:%d: %s\n", __FILE__, __LINE__, #t)


#define LOL_ERROR_2(t, s) \ #define LOL_ERROR_2(t, s) \
Log::Error("assertion at %s:%d: %s\n", __FILE__, __LINE__, s)
lol::Log::Error("assertion at %s:%d: %s\n", __FILE__, __LINE__, s)


#define LOL_ERROR_3(t, s, ...) \ #define LOL_ERROR_3(t, s, ...) \
Log::Error("assertion at %s:%d: " s "\n", __FILE__, __LINE__, __VA_ARGS__)
lol::Log::Error("assertion at %s:%d: " s "\n", __FILE__, __LINE__, __VA_ARGS__)


#if FINAL_RELEASE #if FINAL_RELEASE
# define ASSERT(...) UNUSED(LOL_CALL(LOL_1ST, (__VA_ARGS__))) # define ASSERT(...) UNUSED(LOL_CALL(LOL_1ST, (__VA_ARGS__)))
@@ -128,7 +128,7 @@ static inline void DebugAbort()
LOL_CALL(LOL_CAT(LOL_ERROR_, LOL_CALL(LOL_COUNT_TO_3, \ LOL_CALL(LOL_CAT(LOL_ERROR_, LOL_CALL(LOL_COUNT_TO_3, \
(__VA_ARGS__))), \ (__VA_ARGS__))), \
(__VA_ARGS__)); \ (__VA_ARGS__)); \
DebugAbort(); \
lol::DebugAbort(); \
} }
#endif #endif




+ 0
- 75
src/lol/math/vector.h 查看文件

@@ -1918,81 +1918,6 @@ template<typename T> Mat2<T> inverse(Mat2<T> const &);
template<typename T> Mat3<T> inverse(Mat3<T> const &); template<typename T> Mat3<T> inverse(Mat3<T> const &);
template<typename T> Mat4<T> inverse(Mat4<T> const &); template<typename T> Mat4<T> inverse(Mat4<T> const &);


/*
* Arbitrarily-sized square matrices; for now this only supports
* naive inversion and is used for the Remez inversion method.
*/

template<int N, typename T> struct Mat
{
inline Mat<N, T>() {}

Mat(T x)
{
for (int j = 0; j < N; j++)
for (int i = 0; i < N; i++)
if (i == j)
m[i][j] = x;
else
m[i][j] = 0;
}

/* Naive matrix inversion */
Mat<N, T> inv() const
{
Mat a = *this, b((T)1);

/* Inversion method: iterate through all columns and make sure
* all the terms are 1 on the diagonal and 0 everywhere else */
for (int i = 0; i < N; i++)
{
/* If the expected coefficient is zero, add one of
* the other lines. The first we meet will do. */
if (!a.m[i][i])
{
for (int j = i + 1; j < N; j++)
{
if (!a.m[i][j])
continue;
/* Add row j to row i */
for (int n = 0; n < N; n++)
{
a.m[n][i] += a.m[n][j];
b.m[n][i] += b.m[n][j];
}
break;
}
}

/* Now we know the diagonal term is non-zero. Get its inverse
* and use that to nullify all other terms in the column */
T x = (T)1 / a.m[i][i];
for (int j = 0; j < N; j++)
{
if (j == i)
continue;
T mul = x * a.m[i][j];
for (int n = 0; n < N; n++)
{
a.m[n][j] -= mul * a.m[n][i];
b.m[n][j] -= mul * b.m[n][i];
}
}

/* Finally, ensure the diagonal term is 1 */
for (int n = 0; n < N; n++)
{
a.m[n][i] *= x;
b.m[n][i] *= x;
}
}

return b;
}

T m[N][N];
};

} /* namespace lol */ } /* namespace lol */


#endif // __LOL_MATH_VECTOR_H__ #endif // __LOL_MATH_VECTOR_H__


+ 2
- 1
tools/lolremez/Makefile.am 查看文件

@@ -5,7 +5,8 @@ EXTRA_DIST += NEWS.txt lolremez.sln remez.vcxproj remez.vcxproj.filters


noinst_PROGRAMS = lolremez noinst_PROGRAMS = lolremez


lolremez_SOURCES = lolremez.cpp
lolremez_SOURCES = \
lolremez.cpp solver.cpp solver.h matrix.h
lolremez_CPPFLAGS = $(AM_CPPFLAGS) lolremez_CPPFLAGS = $(AM_CPPFLAGS)
lolremez_DEPENDENCIES = @LOL_DEPS@ lolremez_DEPENDENCIES = @LOL_DEPS@



+ 5
- 5
tools/lolremez/lolremez.cpp 查看文件

@@ -1,5 +1,5 @@
// //
// Lol Engine - Sample math program: Chebyshev polynomials
// LolRemez - Remez algorithm implementation
// //
// Copyright: (c) 2005-2013 Sam Hocevar <sam@hocevar.net> // Copyright: (c) 2005-2013 Sam Hocevar <sam@hocevar.net>
// This program is free software; you can redistribute it and/or // This program is free software; you can redistribute it and/or
@@ -16,10 +16,9 @@


#include <lol/math/real.h> #include <lol/math/real.h>


#include "lolremez.h"
#include "solver.h"


using lol::real; using lol::real;
using lol::RemezSolver;


/* See the tutorial at http://lolengine.net/wiki/doc/maths/remez */ /* See the tutorial at http://lolengine.net/wiki/doc/maths/remez */
real f(real const &x) real f(real const &x)
@@ -36,8 +35,9 @@ int main(int argc, char **argv)
{ {
UNUSED(argc, argv); UNUSED(argc, argv);


RemezSolver<4, real> solver;
solver.Run(-1, 1, f, g, 40);
RemezSolver<real> solver;
solver.Run(4, 40, -1, 1, f, g);

return 0; return 0;
} }



+ 0
- 376
tools/lolremez/lolremez.h 查看文件

@@ -1,376 +0,0 @@
//
// Lol Engine
//
// Copyright: (c) 2010-2011 Sam Hocevar <sam@hocevar.net>
// This program is free software; you can redistribute it and/or
// modify it under the terms of the Do What The Fuck You Want To
// Public License, Version 2, as published by Sam Hocevar. See
// http://www.wtfpl.net/ for more details.
//

//
// The RemezSolver class
// ---------------------
//

#if !defined __LOL_MATH_REMEZ_H__
#define __LOL_MATH_REMEZ_H__

#include <cstdio>

#include "lol/math/vector.h"

namespace lol
{

template<int ORDER, typename T> class RemezSolver
{
public:
typedef T RealFunc(T const &x);

RemezSolver()
{
}

void Run(T a, T b, RealFunc *func, RealFunc *weight, int decimals)
{
using std::printf;

m_func = func;
m_weight = weight;
m_k1 = (b + a) / 2;
m_k2 = (b - a) / 2;
m_invk2 = re(m_k2);
m_invk1 = -m_k1 * m_invk2;
m_decimals = decimals;
m_epsilon = pow((T)10, (T)-(decimals + 2));

Init();

PrintPoly();

T error = -1;

for (int n = 0; ; n++)
{
T newerror = FindExtrema();
printf("Step %i error: ", n);
newerror.print(m_decimals);
printf("\n");

Step();

if (error >= (T)0 && fabs(newerror - error) < error * m_epsilon)
break;
error = newerror;

PrintPoly();

FindZeroes();
}

PrintPoly();
}

inline void Run(T a, T b, RealFunc *func, int decimals)
{
Run(a, b, func, nullptr, decimals);
}

T EvalCheby(T const &x)
{
T ret = 0.0, xn = 1.0;

for (int i = 0; i < ORDER + 1; i++)
{
T mul = 0;
for (int j = 0; j < ORDER + 1; j++)
mul += coeff[j] * (T)Cheby(j, i);
ret += mul * xn;
xn *= x;
}

return ret;
}

void Init()
{
/* Pick up x_i where error will be 0 and compute f(x_i) */
T fxn[ORDER + 1];
for (int i = 0; i < ORDER + 1; i++)
{
zeroes[i] = (T)(2 * i - ORDER) / (T)(ORDER + 1);
fxn[i] = EvalFunc(zeroes[i]);
}

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
lol::Mat<ORDER + 1, T> mat;
for (int i = 0; i < ORDER + 1; i++)
{
/* Compute the powers of x_i */
T powers[ORDER + 1];
powers[0] = 1.0;
for (int n = 1; n < ORDER + 1; n++)
powers[n] = powers[n - 1] * zeroes[i];

/* Compute the Chebishev evaluations at x_i */
for (int n = 0; n < ORDER + 1; n++)
{
T sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
sum += (T)Cheby(n, k) * powers[k];
mat.m[i][n] = sum;
}
}

/* Solve the system */
mat = mat.inv();

/* Compute interpolation coefficients */
for (int j = 0; j < ORDER + 1; j++)
{
coeff[j] = 0;
for (int i = 0; i < ORDER + 1; i++)
coeff[j] += mat.m[j][i] * fxn[i];
}
}

void FindZeroes()
{
/* Find ORDER + 1 zeroes of the error function. No need to
* compute the relative error: its zeroes are at the same
* place as the absolute error! */
for (int i = 0; i < ORDER + 1; i++)
{
struct { T value, error; } left, right, mid;

left.value = control[i];
left.error = EvalCheby(left.value) - EvalFunc(left.value);
right.value = control[i + 1];
right.error = EvalCheby(right.value) - EvalFunc(right.value);

static T limit = ldexp((T)1, -500);
static T zero = (T)0;
while (fabs(left.value - right.value) > limit)
{
mid.value = (left.value + right.value) / 2;
mid.error = EvalCheby(mid.value) - EvalFunc(mid.value);

if ((left.error <= zero && mid.error <= zero)
|| (left.error >= zero && mid.error >= zero))
left = mid;
else
right = mid;
}

zeroes[i] = mid.value;
}
}

real FindExtrema()
{
using std::printf;

/* Find ORDER + 2 extrema of the error function. We need to
* compute the relative error, since its extrema are at slightly
* different locations than the absolute error’s. */
T final = 0;

for (int i = 0; i < ORDER + 2; i++)
{
T a = -1, b = 1;
if (i > 0)
a = zeroes[i - 1];
if (i < ORDER + 1)
b = zeroes[i];

for (int round = 0; ; round++)
{
T maxerror = 0, maxweight = 0;
int best = -1;

T c = a, delta = (b - a) / 4;
for (int k = 0; k <= 4; k++)
{
if (round == 0 || (k & 1))
{
T error = fabs(EvalCheby(c) - EvalFunc(c));
T weight = fabs(Weight(c));
/* if error/weight >= maxerror/maxweight */
if (error * maxweight >= maxerror * weight)
{
maxerror = error;
maxweight = weight;
best = k;
}
}
c += delta;
}

switch (best)
{
case 0:
b = a + delta * 2;
break;
case 4:
a = b - delta * 2;
break;
default:
b = a + delta * (best + 1);
a = a + delta * (best - 1);
break;
}

if (delta < m_epsilon)
{
T e = fabs(maxerror / maxweight);
if (e > final)
final = e;
control[i] = (a + b) / 2;
break;
}
}
}

return final;
}

void Step()
{
/* Pick up x_i where error will be 0 and compute f(x_i) */
T fxn[ORDER + 2];
for (int i = 0; i < ORDER + 2; i++)
fxn[i] = EvalFunc(control[i]);

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
lol::Mat<ORDER + 2, T> mat;
for (int i = 0; i < ORDER + 2; i++)
{
/* Compute the powers of x_i */
T powers[ORDER + 1];
powers[0] = 1.0;
for (int n = 1; n < ORDER + 1; n++)
powers[n] = powers[n - 1] * control[i];

/* Compute the Chebishev evaluations at x_i */
for (int n = 0; n < ORDER + 1; n++)
{
T sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
sum += (T)Cheby(n, k) * powers[k];
mat.m[i][n] = sum;
}
if (i & 1)
mat.m[i][ORDER + 1] = fabs(Weight(control[i]));
else
mat.m[i][ORDER + 1] = -fabs(Weight(control[i]));
}

/* Solve the system */
mat = mat.inv();

/* Compute interpolation coefficients */
for (int j = 0; j < ORDER + 1; j++)
{
coeff[j] = 0;
for (int i = 0; i < ORDER + 2; i++)
coeff[j] += mat.m[j][i] * fxn[i];
}

/* Compute the error */
T error = 0;
for (int i = 0; i < ORDER + 2; i++)
error += mat.m[ORDER + 1][i] * fxn[i];
}

int Cheby(int n, int k)
{
if (k > n || k < 0)
return 0;
if (n <= 1)
return (n ^ k ^ 1) & 1;
return 2 * Cheby(n - 1, k - 1) - Cheby(n - 2, k);
}

int Comb(int n, int k)
{
if (k == 0 || k == n)
return 1;
return Comb(n - 1, k - 1) + Comb(n - 1, k);
}

void PrintPoly()
{
using std::printf;

/* Transform Chebyshev polynomial weights into powers of X^i
* in the [-1..1] range. */
T bn[ORDER + 1];

for (int i = 0; i < ORDER + 1; i++)
{
bn[i] = 0;
for (int j = 0; j < ORDER + 1; j++)
bn[i] += coeff[j] * (T)Cheby(j, i);
}

/* Transform a polynomial in the [-1..1] range into a polynomial
* in the [a..b] range. */
T k1p[ORDER + 1], k2p[ORDER + 1];
T an[ORDER + 1];

for (int i = 0; i < ORDER + 1; i++)
{
k1p[i] = i ? k1p[i - 1] * m_invk1 : (T)1;
k2p[i] = i ? k2p[i - 1] * m_invk2 : (T)1;
}

for (int i = 0; i < ORDER + 1; i++)
{
an[i] = 0;
for (int j = i; j < ORDER + 1; j++)
an[i] += (T)Comb(j, i) * k1p[j - i] * bn[j];
an[i] *= k2p[i];
}

printf("Polynomial estimate: ");
for (int j = 0; j < ORDER + 1; j++)
{
if (j)
printf(" + x**%i * ", j);
an[j].print(m_decimals);
}
printf("\n\n");
}

T EvalFunc(T const &x)
{
return m_func(x * m_k2 + m_k1);
}

T Weight(T const &x)
{
if (m_weight)
return m_weight(x * m_k2 + m_k1);
return 1;
}

/* ORDER + 1 Chebyshev coefficients and 1 error value */
T coeff[ORDER + 2];
/* ORDER + 1 zeroes of the error function */
T zeroes[ORDER + 1];
/* ORDER + 2 control points */
T control[ORDER + 2];

private:
RealFunc *m_func, *m_weight;
T m_k1, m_k2, m_invk1, m_invk2, m_epsilon;
int m_decimals;
};

} /* namespace lol */

#endif /* __LOL_MATH_REMEZ_H__ */


+ 377
- 0
tools/lolremez/solver.cpp 查看文件

@@ -0,0 +1,377 @@
//
// LolRemez - Remez algorithm implementation
//
// Copyright: (c) 2005-2013 Sam Hocevar <sam@hocevar.net>
// This program is free software; you can redistribute it and/or
// modify it under the terms of the Do What The Fuck You Want To
// Public License, Version 2, as published by Sam Hocevar. See
// http://www.wtfpl.net/ for more details.
//

#if defined HAVE_CONFIG_H
# include "config.h"
#endif

#include "core.h"

#include <lol/math/real.h>

#include "matrix.h"
#include "solver.h"

using lol::real;

/* Some forward declarations first. */
template<> void RemezSolver<real>::Init();
template<> void RemezSolver<real>::FindZeroes();
template<> real RemezSolver<real>::FindExtrema();
template<> void RemezSolver<real>::Step();
template<> int RemezSolver<real>::Cheby(int n, int k);
template<> void RemezSolver<real>::PrintPoly();
template<> real RemezSolver<real>::EvalFunc(real const &x);
template<> real RemezSolver<real>::Weight(real const &x);


template<>
void RemezSolver<real>::Run(int order, int decimals, real a, real b,
RemezSolver<real>::RealFunc *func,
RemezSolver<real>::RealFunc *weight)
{
using std::printf;

m_order = order;
m_func = func;
m_weight = weight;
m_k1 = (b + a) / 2;
m_k2 = (b - a) / 2;
m_invk2 = re(m_k2);
m_invk1 = -m_k1 * m_invk2;
m_decimals = decimals;
m_epsilon = pow((real)10, (real)-(decimals + 2));

Init();

PrintPoly();

real error = -1;

for (int n = 0; ; n++)
{
real newerror = FindExtrema();
printf("Step %i error: ", n);
newerror.print(m_decimals);
printf("\n");

Step();

if (error >= (real)0 && fabs(newerror - error) < error * m_epsilon)
break;
error = newerror;

PrintPoly();

FindZeroes();
}

PrintPoly();
}

template<>
real RemezSolver<real>::EvalCheby(real const &x)
{
real ret = 0.0, xn = 1.0;

for (int i = 0; i < m_order + 1; i++)
{
real mul = 0;
for (int j = 0; j < m_order + 1; j++)
mul += m_coeff[j] * (real)Cheby(j, i);
ret += mul * xn;
xn *= x;
}

return ret;
}

template<>
void RemezSolver<real>::Init()
{
/* m_order + 1 Chebyshev coefficients, plus 1 error value */
m_coeff.Resize(m_order + 2);

/* m_order + 1 zeroes of the error function */
m_zeroes.Resize(m_order + 1);

/* m_order + 2 control points */
m_control.Resize(m_order + 2);

/* Pick up x_i where error will be 0 and compute f(x_i) */
real fxn[m_order + 1];
for (int i = 0; i < m_order + 1; i++)
{
m_zeroes[i] = (real)(2 * i - m_order) / (real)(m_order + 1);
fxn[i] = EvalFunc(m_zeroes[i]);
}

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<real> mat(m_order + 1, m_order + 1);
for (int i = 0; i < m_order + 1; i++)
{
/* Compute the powers of x_i */
real powers[m_order + 1];
powers[0] = 1.0;
for (int n = 1; n < m_order + 1; n++)
powers[n] = powers[n - 1] * m_zeroes[i];

/* Compute the Chebishev evaluations at x_i */
for (int n = 0; n < m_order + 1; n++)
{
real sum = 0.0;
for (int k = 0; k < m_order + 1; k++)
sum += (real)Cheby(n, k) * powers[k];
mat.m(i, n) = sum;
}
}

/* Solve the system */
mat = mat.inv();

/* Compute interpolation coefficients */
for (int j = 0; j < m_order + 1; j++)
{
m_coeff[j] = 0;
for (int i = 0; i < m_order + 1; i++)
m_coeff[j] += mat.m(j, i) * fxn[i];
}
}

template<>
void RemezSolver<real>::FindZeroes()
{
/* Find m_order + 1 zeroes of the error function. No need to
* compute the relative error: its zeroes are at the same
* place as the absolute error! */
for (int i = 0; i < m_order + 1; i++)
{
struct { real value, error; } left, right, mid;

left.value = m_control[i];
left.error = EvalCheby(left.value) - EvalFunc(left.value);
right.value = m_control[i + 1];
right.error = EvalCheby(right.value) - EvalFunc(right.value);

static real limit = ldexp((real)1, -500);
static real zero = (real)0;
while (fabs(left.value - right.value) > limit)
{
mid.value = (left.value + right.value) / 2;
mid.error = EvalCheby(mid.value) - EvalFunc(mid.value);

if ((left.error <= zero && mid.error <= zero)
|| (left.error >= zero && mid.error >= zero))
left = mid;
else
right = mid;
}

m_zeroes[i] = mid.value;
}
}

template<>
real RemezSolver<real>::FindExtrema()
{
using std::printf;

/* Find m_order + 2 extrema of the error function. We need to
* compute the relative error, since its extrema are at slightly
* different locations than the absolute error’s. */
real final = 0;

for (int i = 0; i < m_order + 2; i++)
{
real a = -1, b = 1;
if (i > 0)
a = m_zeroes[i - 1];
if (i < m_order + 1)
b = m_zeroes[i];

for (int round = 0; ; round++)
{
real maxerror = 0, maxweight = 0;
int best = -1;

real c = a, delta = (b - a) / 4;
for (int k = 0; k <= 4; k++)
{
if (round == 0 || (k & 1))
{
real error = fabs(EvalCheby(c) - EvalFunc(c));
real weight = fabs(Weight(c));
/* if error/weight >= maxerror/maxweight */
if (error * maxweight >= maxerror * weight)
{
maxerror = error;
maxweight = weight;
best = k;
}
}
c += delta;
}

switch (best)
{
case 0:
b = a + delta * 2;
break;
case 4:
a = b - delta * 2;
break;
default:
b = a + delta * (best + 1);
a = a + delta * (best - 1);
break;
}

if (delta < m_epsilon)
{
real e = fabs(maxerror / maxweight);
if (e > final)
final = e;
m_control[i] = (a + b) / 2;
break;
}
}
}

return final;
}

template<>
void RemezSolver<real>::Step()
{
/* Pick up x_i where error will be 0 and compute f(x_i) */
real fxn[m_order + 2];
for (int i = 0; i < m_order + 2; i++)
fxn[i] = EvalFunc(m_control[i]);

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<real> mat(m_order + 2, m_order + 2);
for (int i = 0; i < m_order + 2; i++)
{
/* Compute the powers of x_i */
real powers[m_order + 1];
powers[0] = 1.0;
for (int n = 1; n < m_order + 1; n++)
powers[n] = powers[n - 1] * m_control[i];

/* Compute the Chebishev evaluations at x_i */
for (int n = 0; n < m_order + 1; n++)
{
real sum = 0.0;
for (int k = 0; k < m_order + 1; k++)
sum += (real)Cheby(n, k) * powers[k];
mat.m(i, n) = sum;
}
if (i & 1)
mat.m(i, m_order + 1) = fabs(Weight(m_control[i]));
else
mat.m(i, m_order + 1) = -fabs(Weight(m_control[i]));
}

/* Solve the system */
mat = mat.inv();

/* Compute interpolation coefficients */
for (int j = 0; j < m_order + 1; j++)
{
m_coeff[j] = 0;
for (int i = 0; i < m_order + 2; i++)
m_coeff[j] += mat.m(j, i) * fxn[i];
}

/* Compute the error */
real error = 0;
for (int i = 0; i < m_order + 2; i++)
error += mat.m(m_order + 1, i) * fxn[i];
}

template<>
int RemezSolver<real>::Cheby(int n, int k)
{
if (k > n || k < 0)
return 0;
if (n <= 1)
return (n ^ k ^ 1) & 1;
return 2 * Cheby(n - 1, k - 1) - Cheby(n - 2, k);
}

template<>
int RemezSolver<real>::Comb(int n, int k)
{
if (k == 0 || k == n)
return 1;
return Comb(n - 1, k - 1) + Comb(n - 1, k);
}

template<>
void RemezSolver<real>::PrintPoly()
{
using std::printf;

/* Transform Chebyshev polynomial weights into powers of X^i
* in the [-1..1] range. */
real bn[m_order + 1];

for (int i = 0; i < m_order + 1; i++)
{
bn[i] = 0;
for (int j = 0; j < m_order + 1; j++)
bn[i] += m_coeff[j] * (real)Cheby(j, i);
}

/* Transform a polynomial in the [-1..1] range into a polynomial
* in the [a..b] range. */
real k1p[m_order + 1], k2p[m_order + 1];
real an[m_order + 1];

for (int i = 0; i < m_order + 1; i++)
{
k1p[i] = i ? k1p[i - 1] * m_invk1 : (real)1;
k2p[i] = i ? k2p[i - 1] * m_invk2 : (real)1;
}

for (int i = 0; i < m_order + 1; i++)
{
an[i] = 0;
for (int j = i; j < m_order + 1; j++)
an[i] += (real)Comb(j, i) * k1p[j - i] * bn[j];
an[i] *= k2p[i];
}

printf("Polynomial estimate: ");
for (int j = 0; j < m_order + 1; j++)
{
if (j)
printf(" + x**%i * ", j);
an[j].print(m_decimals);
}
printf("\n\n");
}

template<>
real RemezSolver<real>::EvalFunc(real const &x)
{
return m_func(x * m_k2 + m_k1);
}

template<>
real RemezSolver<real>::Weight(real const &x)
{
if (m_weight)
return m_weight(x * m_k2 + m_k1);
return 1;
}


+ 57
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tools/lolremez/solver.h 查看文件

@@ -0,0 +1,57 @@
//
// LolRemez - Remez algorithm implementation
//
// Copyright: (c) 2010-2013 Sam Hocevar <sam@hocevar.net>
// This program is free software; you can redistribute it and/or
// modify it under the terms of the Do What The Fuck You Want To
// Public License, Version 2, as published by Sam Hocevar. See
// http://www.wtfpl.net/ for more details.
//

#pragma once

//
// The RemezSolver class
// ---------------------
//

#include <cstdio>

template<typename NUMERIC_T> class RemezSolver
{
public:
typedef NUMERIC_T RealFunc(NUMERIC_T const &x);

inline RemezSolver()
: m_order(0)
{
}

void Run(int order, int decimals, NUMERIC_T a, NUMERIC_T b,
RealFunc *func, RealFunc *weight = nullptr);

NUMERIC_T EvalCheby(NUMERIC_T const &x);
void Init();
void FindZeroes();
NUMERIC_T FindExtrema();
void Step();

int Cheby(int n, int k);
int Comb(int n, int k);

void PrintPoly();
NUMERIC_T EvalFunc(NUMERIC_T const &x);
NUMERIC_T Weight(NUMERIC_T const &x);

private:
int m_order;

lol::Array<NUMERIC_T> m_coeff;
lol::Array<NUMERIC_T> m_zeroes;
lol::Array<NUMERIC_T> m_control;

RealFunc *m_func, *m_weight;
NUMERIC_T m_k1, m_k2, m_invk1, m_invk2, m_epsilon;
int m_decimals;
};


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