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test_zero.cpp
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# include <cmath>
# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <ctime>
# include <complex>
# include <cstring>
using namespace std;
# include "test_zero.hpp"
//****************************************************************************80
void bisection ( double fatol, int step_max, int prob, double xatol,
double *xa, double *xb, double *fxa, double *fxb )
//****************************************************************************80
//
// Purpose:
//
// BISECTION carries out the bisection method to seek a root of F(X) = 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 04 May 2011
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double FATOL, an absolute error tolerance for
// the function value of the root. If an approximate root X satisfies
// ABS ( F ( X ) ) <= FATOL, then X will be accepted as the
// root and the iteration will be terminated.
//
// Input, int STEP_MAX, the maximum number of steps
// allowed for an iteration.
//
// Input, int PROB, the index of the function whose root is
// to be sought.
//
// Input, double XATOL, an absolute error tolerance for the root.
//
// Input/output, double *XA, *XB, two points at which the
// function differs in sign. On output, these values have been adjusted
// to a smaller interval.
//
// Input/output, double *FXA, *FXB, the value of the function
// at XA and XB.
//
{
double fxc;
int step_num;
double t;
double xc;
cout << "\n";
cout << "BISECTION\n";
cout << "\n";
cout << " Step XA XB F(XA) F(XB)\n";
cout << "\n";
//
// Make A the root with negative F, B the root with positive F.
//
if ( 0.0 < *fxa )
{
t = *xa;
*xa = *xb;
*xb = t;
t = *fxa;
*fxa = *fxb;
*fxb = t;
}
step_num = 0;
//
// Loop
//
for ( ; ; )
{
cout << " " << setw(4) << step_num
<< " " << setw(14) << *xa
<< " " << setw(14) << *xb
<< " " << setw(14) << *fxa
<< " " << setw(14) << *fxb << "\n";
step_num = step_num + 1;
if ( step_max < step_num )
{
cout << "\n";
cout << " Maximum number of steps taken without convergence.\n";
break;
}
if ( r8_abs ( *xa - *xb ) < xatol )
{
cout << "\n";
cout << " Interval small enough for convergence.\n";
break;
}
if ( r8_abs ( *fxa ) <= fatol || r8_abs ( *fxb ) <= fatol )
{
cout << "\n";
cout << " Function small enough for convergence.\n";
break;
}
//
// Compute the next iterate.
//
xc = 0.5 * ( *xa + *xb );
fxc = p00_fx ( prob, xc );
//
// Replace one of the old points.
//
if ( fxc < 0.0 )
{
*xa = xc;
*fxa = fxc;
}
else
{
*xb = xc;
*fxb = fxc;
}
}
return;
}
//****************************************************************************80
void brent ( double fatol, int step_max, int prob, double xatol, double xrtol,
double *xa, double *xb, double *fxa, double *fxb )
//****************************************************************************80
//
// Purpose:
//
// BRENT implements the Brent bisection-based zero finder.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 04 May 2011
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Richard Brent,
// Algorithms for Minimization without Derivatives,
// Prentice Hall, 1973.
//
// Parameters:
//
// Input, double FATOL, an absolute error tolerance for the
// function value of the root. If an approximate root X satisfies
// ABS ( F ( X ) ) <= FATOL, then X will be accepted as the
// root and the iteration will be terminated.
//
// Input, int STEP_MAX, the maximum number of steps allowed
// for an iteration.
//
// Input, int PROB, the index of the function whose root is
// to be sought.
//
// Input, double XATOL, XRTOL, absolute and relative error
// tolerances for the root.
//
// Input/output, double *XA, *XB, two points at which the
// function differs in sign. On output, these values have been adjusted
// to a smaller interval.
//
// Input/output, double *FXA, *FXB, the value of the function
// at XA and XB.
//
{
double d;
double e;
double fxc;
int step_num;
double p;
double q;
double r;
double s;
double xc;
double xm;
double xtol;
//
// Initialization.
//
cout << "\n";
cout << "BRENT\n";
cout << "\n";
cout << " Step XA XB F(XA) F(XB)\n";
cout << "\n";
step_num = 0;
*fxa = p00_fx ( prob, *xa );
*fxb = p00_fx ( prob, *xb );
//
// Check that f(ax) and f(bx) have different signs.
//
if ( ( *fxa < 0.0 && *fxb < 0.0 ) || ( 0.0 < *fxa && 0.0 < *fxb ) )
{
cerr << "\n";
cerr << "BRENT - Fatal error!\n";
cerr << " F(XA) and F(XB) have same sign.\n";
exit ( 1 );
}
xc = *xa;
fxc = *fxa;
d = *xb - *xa;
e = d;
for ( ; ; )
{
cout << " " << setw(4) << step_num
<< " " << setw(14) << *xb
<< " " << setw(14) << xc
<< " " << setw(14) << *fxb
<< " " << setw(14) << fxc << "\n";
step_num = step_num + 1;
if ( step_max < step_num )
{
cout << "\n";
cout << " Maximum number of steps taken.\n";
break;
}
if ( r8_abs ( fxc ) < r8_abs ( *fxb ) )
{
*xa = *xb;
*xb = xc;
xc = *xa;
*fxa = *fxb;
*fxb = fxc;
fxc = *fxa;
}
xtol = 2.0 * xrtol * r8_abs ( *xb ) + 0.5 * xatol;
//
// XM is the halfwidth of the current change-of-sign interval.
//
xm = 0.5 * ( xc - *xb );
if ( r8_abs ( xm ) <= xtol )
{
cout << "\n";
cout << " Interval small enough for convergence.\n";
break;
}
if ( r8_abs ( *fxb ) <= fatol )
{
cout << "\n";
cout << " Function small enough for convergence.\n";
break;
}
//
// See if a bisection is forced.
//
if ( r8_abs ( e ) < xtol || r8_abs ( *fxa ) <= r8_abs ( *fxb ) )
{
d = xm;
e = d;
}
else
{
s = *fxb / *fxa;
//
// Linear interpolation.
//
if ( *xa == xc )
{
p = 2.0 * xm * s;
q = 1.0 - s;
}
//
// Inverse quadratic interpolation.
//
else
{
q = *fxa / fxc;
r = *fxb / fxc;
p = s * ( 2.0 * xm * q * ( q - r ) - ( *xb - *xa ) * ( r - 1.0 ) );
q = ( q - 1.0 ) * ( r - 1.0 ) * ( s - 1.0 );
}
if ( 0.0 < p )
{
q = - q;
}
else
{
p = - p;
}
s = e;
e = d;
if ( 3.0 * xm * q - r8_abs ( xtol * q ) <= 2.0 * p ||
r8_abs ( 0.5 * s * q ) <= p )
{
d = xm;
e = d;
}
else
{
d = p / q;
}
}
//
// Save in XA, FXA the previous values of XB, FXB.
//
*xa = *xb;
*fxa = *fxb;
//
// Compute the new value of XB, and evaluate the function there.
//
if ( xtol < r8_abs ( d ) )
{
*xb = *xb + d;
}
else if ( 0.0 < xm )
{
*xb = *xb + xtol;
}
else if ( xm <= 0.0 )
{
*xb = *xb - xtol;
}
*fxb = p00_fx ( prob, *xb );
//
// If the new FXB has the same sign as FXC, then replace XC by XA.
//
if ( ( 0.0 < *fxb && 0.0 < fxc ) || ( *fxb < 0.0 && fxc < 0.0 ) )
{
xc = *xa;
fxc = *fxa;
d = *xb - *xa;
e = d;
}
}
return;
}
//****************************************************************************80
void muller ( double fatol, int step_max, int prob, double xatol, double xrtol,
double *xa, double *xb, double *xc, double *fxa, double *fxb, double *fxc )
//****************************************************************************80
//
// Purpose:
//
// MULLER carries out Muller's method for seeking a real root of a nonlinear function.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 04 May 2011
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double FATOL, an absolute error tolerance for the
// function value of the root. If an approximate root X satisfies7
// ABS ( F ( X ) ) <= FATOL, then X will be accepted as the
// root and the iteration will be terminated.
//
// Input, int STEP_MAX, the maximum number of steps allowed
// for an iteration.
//
// Input, int PROB, the index of the function whose root is
// to be sought.
//
// Input, double XATOL, XRTOL, absolute and relative error
// tolerances for the root.
//
// Input/output, double *XA, *XB, *XC, three points.
//
// Input/output, double *FXA, *FXB, *FXC, the value of the
// function at XA, XB, and XC.
//
{
double a;
double b;
double c;
int i;
double t;
double xd;
double z1;
double z2;
//
// Initialization.
//
cout << "\n";
cout << "MULLER\n";
cout << "\n";
cout << " Step XA XB XC\n";
cout << " F(XA) F(XB) F(XC)\n";
cout << "\n";
i = 0;
cout << " " << setw(4) << i
<< " " << setw(12) << *xa
<< " " << setw(12) << *xb
<< " " << setw(12) << *xc << "\n";
cout << " " << " "
<< " " << setw(12) << *fxa
<< " " << setw(12) << *fxb
<< " " << setw(12) << *fxc << "\n";
for ( i = 1; i <= step_max; i++ )
{
//
// Determine the coefficients
// A, B, C
// of the polynomial
// Y(X) = A * (X-X2)**2 + B * (X-X2) + C
// which goes through the data:
// (X1,Y1), (X2,Y2), (X3,Y3).
//
a = ( ( *fxa - *fxc ) * ( *xb - *xc )
- ( *fxb - *fxc ) * ( *xa - *xc ) ) /
( ( *xa - *xc ) * ( *xb - *xc ) * ( *xa - *xb ) );
b = ( ( *fxb - *fxc ) * ( *xa - *xc ) * ( *xa - *xc )
- ( *fxa - *fxc ) * ( *xb - *xc ) * ( *xb - *xc ) ) /
( ( *xa - *xc ) * ( *xb - *xc ) * ( *xa - *xb ) );
c = *fxc;
//
// Get the real roots of the polynomial,
// unless A = 0, in which case the algorithm is breaking down.
//
if ( a != 0.0 )
{
r8poly2_rroot ( a, b, c, &z1, &z2 );
}
else if ( b != 0.0 )
{
z2 = - c / b;
}
else
{
cout << "\n";
cout << " Polynomial fitting has failed.\n";
cout << " Muller''s algorithm breaks down.\n";
return;
}
xd = *xc + z2;
//
// Set XA, YA, based on which of XA and XB is closer to XD.
//
if ( r8_abs ( xd - *xb ) < r8_abs ( xd - *xa ) )
{
t = *xa;
*xa = *xb;
*xb = t;
t = *fxa;
*fxa = *fxb;
*fxb = t;
}
//
// Set XB, YB, based on which of XB and XC is closer to XD.
//
if ( r8_abs ( xd - *xc ) < r8_abs ( xd - *xb ) )
{
t = *xb;
*xb = *xc;
*xc = t;
t = *fxb;
*fxb = *fxc;
*fxc = t;
}
//
// Set XC, YC.
//
*xc = xd;
*fxc = p00_fx ( prob, *xc );
cout << " " << setw(4) << i
<< " " << setw(12) << *xa
<< " " << setw(12) << *xb
<< " " << setw(12) << *xc << "\n";
cout << " " << " "
<< " " << setw(12) << *fxa
<< " " << setw(12) << *fxb
<< " " << setw(12) << *fxc << "\n";
//
// Estimate the relative significance of the most recent correction.
//
if ( r8_abs ( z2 ) <= xrtol * r8_abs ( *xc ) + xatol )
{
cout << "\n";
cout << " Stepsize small enough for convergence.\n";
return;
}
if ( r8_abs ( *fxc ) < fatol )
{
cout << "\n";
cout << " Function small enough for convergence.\n";
return;
}
}
cout << "\n";
cout << " Took maximum number of steps without convergence.\n";
return;
}
//****************************************************************************80
void newton ( double fatol, int step_max, int prob, double xatol, double xmin,
double xmax, double *xa, double *fxa )
//****************************************************************************80
//
// Purpose:
//
// NEWTON carries out Newton's method to seek a root of F(X) = 0.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 14 May 2011
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, double FATOL, an absolute error tolerance for the
// function value of the root. If an approximate root X satisfies
// ABS ( F ( X ) ) <= FATOL, then X will be accepted as the
// root and the iteration will be terminated.
//
// Input, int STEP_MAX, the maximum number of steps allowed
// for an iteration.
//
// Input, int PROB, the index of the function whose root is
// to be sought.
//
// Input, double XATOL, an absolute error tolerance for the root.
//
// Input, double XMIN, XMAX, the interval in which the root should
// be sought.
//
// Input/output, double *XA, on input, the starting point. On output,
// the estimated root.
//
// Input/output, double *FXA, the value of the function at XA.
//
{
double fp;
int step_num;
double step;
step = 0.0;
cout << "\n";
cout << "NEWTON\n";
cout << "\n";
cout << " Step X F(X) FP(X)\n";
cout << "\n";
step_num = 0;
fp = p00_fx1 ( prob, *xa );
cout << " " << setw(4) << step_num
<< " " << setw(10) << *xa
<< " " << setw(10) << *fxa
<< " " << setw(10) << fp << "\n";
for ( step_num = 1; step_num <= step_max; step_num++ )
{
if ( *xa < xmin || xmax < *xa )
{
cout << "\n";
cout << " The iterate X = " << *xa << " has left the region [XMIN,XMAX].\n";
return;
}
if ( r8_abs ( *fxa ) <= fatol )
{
cout << "\n";
cout << " The function norm is small enough for convergence.\n";
return;
}
if ( 1 < step_num && r8_abs ( step ) <= xatol )
{
cout << "\n";
cout << " The stepsize is small enough for convergence.\n";
return;
}
if ( fp == 0.0 )
{
cout << "\n";
cout << " F''(X)=0, the algorithm fails.\n";
return;
}
step = *fxa / fp;
*xa = *xa - step;
*fxa = p00_fx ( prob, *xa );
fp = p00_fx1 ( prob, *xa );
cout << " " << setw(4) << step_num
<< " " << setw(10) << *xa
<< " " << setw(10) << *fxa
<< " " << setw(10) << fp << "\n";
}
cout << "\n";
cout << " Took maximum number of steps without convergence.\n";
return;
}
//****************************************************************************80
double p00_fx ( int prob, double x )
//****************************************************************************80
//
// Purpose:
//
// P00_FX evaluates a function specified by problem number.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 January 2013
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int PROB, the problem index.
//
// Input, double X, the point at which F is to be evaluated.
//
// Output, double P00_FX, the value of the function at X.
//
{
double fx;
if ( prob == 1 )
{
fx = p01_fx ( x );
}
else if ( prob == 2 )
{
fx = p02_fx ( x );
}
else if ( prob == 3 )
{
fx = p03_fx ( x );
}
else if ( prob == 4 )
{
fx = p04_fx ( x );
}
else if ( prob == 5 )
{
fx = p05_fx ( x );
}
else if ( prob == 6 )
{
fx = p06_fx ( x );
}
else if ( prob == 7 )
{
fx = p07_fx ( x );
}
else if ( prob == 8 )
{
fx = p08_fx ( x );
}
else if ( prob == 9 )
{
fx = p09_fx ( x );
}
else if ( prob == 10 )
{
fx = p10_fx ( x );
}
else if ( prob == 11 )
{
fx = p11_fx ( x );
}
else if ( prob == 12 )
{
fx = p12_fx ( x );
}
else if ( prob == 13 )
{
fx = p13_fx ( x );
}
else if ( prob == 14 )
{
fx = p14_fx ( x );
}
else if ( prob == 15 )
{
fx = p15_fx ( x );
}
else if ( prob == 16 )
{
fx = p16_fx ( x );
}
else if ( prob == 17 )
{
fx = p17_fx ( x );
}
else if ( prob == 18 )
{
fx = p18_fx ( x );
}
else if ( prob == 19 )
{
fx = p19_fx ( x );
}
else
{
cout << "\n";
cout << "P00_FX - Fatal error!\n";
cout << " Illegal problem number = " << prob << "\n";
exit ( 1 );
}
return fx;
}
//****************************************************************************80
double p00_fx1 ( int prob, double x )
//****************************************************************************80
//
// Purpose:
//
// P00_FX1 evaluates the first derivative of a function specified by problem number.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 January 2013
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int PROB, the problem index.
//
// Input, double X, the abscissa.
//
// Output, double P00_FX1, the first derivative of the function at X.
//
{
double fx1;
if ( prob == 1 )
{
fx1 = p01_fx1 ( x );
}
else if ( prob == 2 )
{
fx1 = p02_fx1 ( x );
}
else if ( prob == 3 )
{
fx1 = p03_fx1 ( x );
}
else if ( prob == 4 )
{
fx1 = p04_fx1 ( x );
}
else if ( prob == 5 )
{
fx1 = p05_fx1 ( x );
}
else if ( prob == 6 )
{
fx1 = p06_fx1 ( x );
}
else if ( prob == 7 )
{
fx1 = p07_fx1 ( x );
}
else if ( prob == 8 )
{
fx1 = p08_fx1 ( x );
}
else if ( prob == 9 )
{
fx1 = p09_fx1 ( x );
}
else if ( prob == 10 )
{
fx1 = p10_fx1 ( x );
}
else if ( prob == 11 )
{
fx1 = p11_fx1 ( x );
}
else if ( prob == 12 )
{
fx1 = p12_fx1 ( x );
}
else if ( prob == 13 )
{
fx1 = p13_fx1 ( x );
}
else if ( prob == 14 )
{
fx1 = p14_fx1 ( x );
}
else if ( prob == 15 )
{
fx1 = p15_fx1 ( x );
}
else if ( prob == 16 )
{
fx1 = p16_fx1 ( x );
}
else if ( prob == 17 )
{
fx1 = p17_fx1 ( x );
}
else if ( prob == 18 )
{
fx1 = p18_fx1 ( x );
}
else if ( prob == 19 )
{
fx1 = p19_fx1 ( x );
}
else
{
cout << "\n";
cout << "P00_FX1 - Fatal error!\n";
cout << " Illegal problem number = " << prob << "\n";
exit ( 1 );
}
return fx1;
}
//****************************************************************************80
double p00_fx2 ( int prob, double x )
//****************************************************************************80
//
// Purpose:
//
// P00_FX2 evaluates the second derivative of a function specified by problem number.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 16 January 2013
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int PROB, the problem index.
//
// Input, double X, the abscissa.
//
// Output, double P00_FX2, the second derivative of the function at X.
//
{
double fx2;
if ( prob == 1 )
{
fx2 = p01_fx2 ( x );
}
else if ( prob == 2 )
{
fx2 = p02_fx2 ( x );
}
else if ( prob == 3 )
{
fx2 = p03_fx2 ( x );
}
else if ( prob == 4 )
{
fx2 = p04_fx2 ( x );
}
else if ( prob == 5 )
{
fx2 = p05_fx2 ( x );
}
else if ( prob == 6 )
{
fx2 = p06_fx2 ( x );
}
else if ( prob == 7 )
{
fx2 = p07_fx2 ( x );
}
else if ( prob == 8 )
{
fx2 = p08_fx2 ( x );
}
else if ( prob == 9 )
{
fx2 = p09_fx2 ( x );
}
else if ( prob == 10 )
{
fx2 = p10_fx2 ( x );
}
else if ( prob == 11 )
{
fx2 = p11_fx2 ( x );
}
else if ( prob == 12 )
{
fx2 = p12_fx2 ( x );
}
else if ( prob == 13 )
{
fx2 = p13_fx2 ( x );
}
else if ( prob == 14 )
{
fx2 = p14_fx2 ( x );
}
else if ( prob == 15 )
{
fx2 = p15_fx2 ( x );
}
else if ( prob == 16 )
{
fx2 = p16_fx2 ( x );
}
else if ( prob == 17 )
{
fx2 = p17_fx2 ( x );
}
else if ( prob == 18 )
{
fx2 = p18_fx2 ( x );
}
else if ( prob == 19 )
{
fx2 = p19_fx2 ( x );
}