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test: some refactoring in the Remez solver to prepare multiple function 

solving.
legacy
Sam Hocevar sam 13 years ago
parent
7aac9df337
commit
57510be2b0
4 changed files with 395 additions and 347 deletions
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  1. +1
    -1
      test/math/Makefile.am
  2. +95
    -0
      test/math/remez-matrix.h
  3. +295
    -0
      test/math/remez-solver.h
  4. +4
    -346
      test/math/remez.cpp

+ 1
- 1
test/math/Makefile.am View File

@@ -16,7 +16,7 @@ pi_CPPFLAGS = @LOL_CFLAGS@ @PIPI_CFLAGS@
pi_LDFLAGS = $(top_builddir)/src/liblol.a @LOL_LIBS@ @PIPI_LIBS@ pi_LDFLAGS = $(top_builddir)/src/liblol.a @LOL_LIBS@ @PIPI_LIBS@
pi_DEPENDENCIES = $(top_builddir)/src/liblol.a pi_DEPENDENCIES = $(top_builddir)/src/liblol.a


remez_SOURCES = remez.cpp
remez_SOURCES = remez.cpp remez-matrix.h remez-solver.h
remez_CPPFLAGS = @LOL_CFLAGS@ @PIPI_CFLAGS@ remez_CPPFLAGS = @LOL_CFLAGS@ @PIPI_CFLAGS@
remez_LDFLAGS = $(top_builddir)/src/liblol.a @LOL_LIBS@ @PIPI_LIBS@ remez_LDFLAGS = $(top_builddir)/src/liblol.a @LOL_LIBS@ @PIPI_LIBS@
remez_DEPENDENCIES = $(top_builddir)/src/liblol.a remez_DEPENDENCIES = $(top_builddir)/src/liblol.a


+ 95
- 0
test/math/remez-matrix.h View File

@@ -0,0 +1,95 @@
//
// Lol Engine - Sample math program: Chebyshev polynomials
//
// Copyright: (c) 2005-2011 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://sam.zoy.org/projects/COPYING.WTFPL for more details.
//

#if !defined __REMEZ_MATRIX_H__
#define __REMEZ_MATRIX_H__

template<int N> struct Matrix
{
inline Matrix() {}

Matrix(real 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 */
Matrix<N> inv() const
{
Matrix a = *this, b((real)1.0);

/* 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 ((double)a.m[i][i] == 0.0)
{
for (int j = i + 1; j < N; j++)
{
if ((double)a.m[i][j] == 0.0)
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 */
real x = (real)1.0 / a.m[i][i];
for (int j = 0; j < N; j++)
{
if (j == i)
continue;
real 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;
}

void print() const
{
for (int j = 0; j < N; j++)
{
for (int i = 0; i < N; i++)
printf("%9.5f ", (double)m[j][i]);
printf("\n");
}
}

real m[N][N];
};

#endif /* __REMEZ_MATRIX_H__ */


+ 295
- 0
test/math/remez-solver.h View File

@@ -0,0 +1,295 @@
//
// Lol Engine - Sample math program: Chebyshev polynomials
//
// Copyright: (c) 2005-2011 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://sam.zoy.org/projects/COPYING.WTFPL for more details.
//

#if !defined __REMEZ_SOLVER_H__
#define __REMEZ_SOLVER_H__

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

RemezSolver()
{
ChebyInit();
}

void Run(RealFunc *func, RealFunc *error, int steps)
{
m_func = func;
m_error = error;

Init();

ChebyCoeff();
for (int j = 0; j < ORDER + 1; j++)
printf("%s%14.12gx^%i", j && (bn[j] >= real::R_0) ? "+" : "", (double)bn[j], j);
printf("\n");

for (int n = 0; n < steps; n++)
{
FindError();
Step();

ChebyCoeff();
for (int j = 0; j < ORDER + 1; j++)
printf("%s%14.12gx^%i", j && (bn[j] >= real::R_0) ? "+" : "", (double)bn[j], j);
printf("\n");

FindZeroes();
}

FindError();
Step();

ChebyCoeff();
for (int j = 0; j < ORDER + 1; j++)
printf("%s%14.12gx^%i", j && (bn[j] >= real::R_0) ? "+" : "", (double)bn[j], j);
printf("\n");
}

/* Fill the Chebyshev tables */
void ChebyInit()
{
memset(cheby, 0, sizeof(cheby));

cheby[0][0] = 1;
cheby[1][1] = 1;

for (int i = 2; i < ORDER + 1; i++)
{
cheby[i][0] = -cheby[i - 2][0];
for (int j = 1; j < ORDER + 1; j++)
cheby[i][j] = 2 * cheby[i - 1][j - 1] - cheby[i - 2][j];
}
}

void ChebyCoeff()
{
for (int i = 0; i < ORDER + 1; i++)
{
bn[i] = 0;
for (int j = 0; j < ORDER + 1; j++)
if (cheby[j][i])
bn[i] += coeff[j] * (real)cheby[j][i];
}
}

real ChebyEval(real const &x)
{
real ret = 0.0, xn = 1.0;

for (int i = 0; i < ORDER + 1; i++)
{
real mul = 0;
for (int j = 0; j < ORDER + 1; j++)
if (cheby[j][i])
mul += coeff[j] * (real)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) */
real fxn[ORDER + 1];
for (int i = 0; i < ORDER + 1; i++)
{
zeroes[i] = (real)(2 * i - ORDER) / (real)(ORDER + 1);
fxn[i] = m_func(zeroes[i]);
}

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<ORDER + 1> mat;
for (int i = 0; i < ORDER + 1; i++)
{
/* Compute the powers of x_i */
real 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++)
{
real sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
if (cheby[n][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 < ORDER + 1; j++)
{
coeff[j] = 0;
for (int i = 0; i < ORDER + 1; i++)
coeff[j] += mat.m[j][i] * fxn[i];
}
}

void FindZeroes()
{
for (int i = 0; i < ORDER + 1; i++)
{
real a = control[i];
real ea = ChebyEval(a) - m_func(a);
real b = control[i + 1];
real eb = ChebyEval(b) - m_func(b);

while (fabs(a - b) > (real)1e-140)
{
real c = (a + b) * (real)0.5;
real ec = ChebyEval(c) - m_func(c);

if ((ea < (real)0 && ec < (real)0)
|| (ea > (real)0 && ec > (real)0))
{
a = c;
ea = ec;
}
else
{
b = c;
eb = ec;
}
}

zeroes[i] = a;
}
}

void FindError()
{
real final = 0;

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

printf("Error for [%g..%g]: ", (double)a, (double)b);
for (;;)
{
real c = a, delta = (b - a) / (real)10.0;
real maxerror = 0;
int best = -1;
for (int k = 0; k <= 10; k++)
{
real e = fabs(ChebyEval(c) - m_func(c));
if (e > maxerror)
{
maxerror = e;
best = k;
}
c += delta;
}

if (best == 0)
best = 1;
if (best == 10)
best = 9;

b = a + (real)(best + 1) * delta;
a = a + (real)(best - 1) * delta;

if (b - a < (real)1e-15)
{
if (maxerror > final)
final = maxerror;
control[i] = (a + b) * (real)0.5;
printf("%g (at %g)\n", (double)maxerror, (double)control[i]);
break;
}
}
}

printf("Final error: %g\n", (double)final);
}

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

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<ORDER + 2> mat;
for (int i = 0; i < ORDER + 2; i++)
{
/* Compute the powers of x_i */
real 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++)
{
real sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
if (cheby[n][k])
sum += (real)cheby[n][k] * powers[k];
mat.m[i][n] = sum;
}
if (i & 1)
mat.m[i][ORDER + 1] = fabs(m_error(control[i]));
else
mat.m[i][ORDER + 1] = -fabs(m_error(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 */
real error = 0;
for (int i = 0; i < ORDER + 2; i++)
error += mat.m[ORDER + 1][i] * fxn[i];
}

int cheby[ORDER + 1][ORDER + 1];

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

real bn[ORDER + 1];

private:
RealFunc *m_func;
RealFunc *m_error;
};

#endif /* __REMEZ_SOLVER_H__ */


+ 4
- 346
test/math/remez.cpp View File

@@ -20,367 +20,25 @@
using namespace lol; using namespace lol;
using namespace std; using namespace std;


/* The order of the approximation we're looking for */
static int const ORDER = 4;
#include "remez-matrix.h"
#include "remez-solver.h"


/* The function we want to approximate */ /* The function we want to approximate */
static real myfun(real const &x) static real myfun(real const &x)
{ {
return exp(x); return exp(x);
//if (!x)
// return real::R_PI_2;
//return sin(x * real::R_PI_2) / x;
} }


static real myerror(real const &x) static real myerror(real const &x)
{ {
return myfun(x); return myfun(x);
//return real::R_1;
}

/* Naive matrix inversion */
template<int N> struct Matrix
{
inline Matrix() {}

Matrix(real 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;
}

Matrix<N> inv() const
{
Matrix a = *this, b((real)1.0);

/* 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 ((double)a.m[i][i] == 0.0)
{
for (int j = i + 1; j < N; j++)
{
if ((double)a.m[i][j] == 0.0)
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 */
real x = (real)1.0 / a.m[i][i];
for (int j = 0; j < N; j++)
{
if (j == i)
continue;
real 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;
}

void print() const
{
for (int j = 0; j < N; j++)
{
for (int i = 0; i < N; i++)
printf("%9.5f ", (double)m[j][i]);
printf("\n");
}
}

real m[N][N];
};


static int cheby[ORDER + 1][ORDER + 1];

/* Fill the Chebyshev tables */
static void cheby_init()
{
memset(cheby, 0, sizeof(cheby));

cheby[0][0] = 1;
cheby[1][1] = 1;

for (int i = 2; i < ORDER + 1; i++)
{
cheby[i][0] = -cheby[i - 2][0];
for (int j = 1; j < ORDER + 1; j++)
cheby[i][j] = 2 * cheby[i - 1][j - 1] - cheby[i - 2][j];
}
}

static void cheby_coeff(real *coeff, real *bn)
{
for (int i = 0; i < ORDER + 1; i++)
{
bn[i] = 0;
for (int j = 0; j < ORDER + 1; j++)
if (cheby[j][i])
bn[i] += coeff[j] * (real)cheby[j][i];
}
}

static real cheby_eval(real *coeff, real const &x)
{
real ret = 0.0, xn = 1.0;

for (int i = 0; i < ORDER + 1; i++)
{
real mul = 0;
for (int j = 0; j < ORDER + 1; j++)
if (cheby[j][i])
mul += coeff[j] * (real)cheby[j][i];
ret += mul * xn;
xn *= x;
}

return ret;
}

static void remez_init(real *coeff, real *zeroes)
{
/* Pick up x_i where error will be 0 and compute f(x_i) */
real fxn[ORDER + 1];
for (int i = 0; i < ORDER + 1; i++)
{
zeroes[i] = (real)(2 * i - ORDER) / (real)(ORDER + 1);
fxn[i] = myfun(zeroes[i]);
}

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<ORDER + 1> mat;
for (int i = 0; i < ORDER + 1; i++)
{
/* Compute the powers of x_i */
real 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++)
{
real sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
if (cheby[n][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 < ORDER + 1; j++)
{
coeff[j] = 0;
for (int i = 0; i < ORDER + 1; i++)
coeff[j] += mat.m[j][i] * fxn[i];
}
}

static void remez_findzeroes(real *coeff, real *zeroes, real *control)
{
for (int i = 0; i < ORDER + 1; i++)
{
real a = control[i];
real ea = cheby_eval(coeff, a) - myfun(a);
real b = control[i + 1];
real eb = cheby_eval(coeff, b) - myfun(b);

while (fabs(a - b) > (real)1e-140)
{
real c = (a + b) * (real)0.5;
real ec = cheby_eval(coeff, c) - myfun(c);

if ((ea < (real)0 && ec < (real)0)
|| (ea > (real)0 && ec > (real)0))
{
a = c;
ea = ec;
}
else
{
b = c;
eb = ec;
}
}

zeroes[i] = a;
}
}

static void remez_finderror(real *coeff, real *zeroes, real *control)
{
real final = 0;

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

printf("Error for [%g..%g]: ", (double)a, (double)b);
for (;;)
{
real c = a, delta = (b - a) / (real)10.0;
real maxerror = 0;
int best = -1;
for (int k = 0; k <= 10; k++)
{
real e = fabs(cheby_eval(coeff, c) - myfun(c));
if (e > maxerror)
{
maxerror = e;
best = k;
}
c += delta;
}

if (best == 0)
best = 1;
if (best == 10)
best = 9;

b = a + (real)(best + 1) * delta;
a = a + (real)(best - 1) * delta;

if (b - a < (real)1e-15)
{
if (maxerror > final)
final = maxerror;
control[i] = (a + b) * (real)0.5;
printf("%g (at %g)\n", (double)maxerror, (double)control[i]);
break;
}
}
}

printf("Final error: %g\n", (double)final);
}

static void remez_step(real *coeff, real *control)
{
/* Pick up x_i where error will be 0 and compute f(x_i) */
real fxn[ORDER + 2];
for (int i = 0; i < ORDER + 2; i++)
fxn[i] = myfun(control[i]);

/* We build a matrix of Chebishev evaluations: row i contains the
* evaluations of x_i for polynomial order n = 0, 1, ... */
Matrix<ORDER + 2> mat;
for (int i = 0; i < ORDER + 2; i++)
{
/* Compute the powers of x_i */
real 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++)
{
real sum = 0.0;
for (int k = 0; k < ORDER + 1; k++)
if (cheby[n][k])
sum += (real)cheby[n][k] * powers[k];
mat.m[i][n] = sum;
}
if (i & 1)
mat.m[i][ORDER + 1] = fabs(myerror(control[i]));
else
mat.m[i][ORDER + 1] = -fabs(myerror(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 */
real error = 0;
for (int i = 0; i < ORDER + 2; i++)
error += mat.m[ORDER + 1][i] * fxn[i];
} }


int main(int argc, char **argv) int main(int argc, char **argv)
{ {
cheby_init();

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

real bn[ORDER + 1];

remez_init(coeff, zeroes);

cheby_coeff(coeff, bn);
for (int j = 0; j < ORDER + 1; j++)
printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j);
printf("\n");

for (int n = 0; n < 200; n++)
{
remez_finderror(coeff, zeroes, control);
remez_step(coeff, control);

cheby_coeff(coeff, bn);
for (int j = 0; j < ORDER + 1; j++)
printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j);
printf("\n");

remez_findzeroes(coeff, zeroes, control);
}

remez_finderror(coeff, zeroes, control);
remez_step(coeff, control);
RemezSolver<4> solver;


cheby_coeff(coeff, bn);
for (int j = 0; j < ORDER + 1; j++)
printf("%s%12.10gx^%i", j ? "+" : "", (double)bn[j], j);
printf("\n");
solver.Run(myfun, myerror, 10);


return EXIT_SUCCESS; return EXIT_SUCCESS;
} }


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