vtkPolynomialSolversUnivariate Class Reference

polynomial solvers More...

#include <vtkPolynomialSolversUnivariate.h>

Inheritance diagram for vtkPolynomialSolversUnivariate:

Collaboration diagram for vtkPolynomialSolversUnivariate:

List of all members.

Public Types
typedef vtkObject	Superclass
Public Member Functions
virtual int	IsA (const char *type)
vtkPolynomialSolversUnivariate *	NewInstance () const
void	PrintSelf (ostream &os, vtkIndent indent)
Static Public Member Functions
static vtkPolynomialSolversUnivariate *	New ()
static int	IsTypeOf (const char *type)
static vtkPolynomialSolversUnivariate *	SafeDownCast (vtkObjectBase *o)
static ostream &	PrintPolynomial (ostream &os, double *P, int degP)
static int	LinBairstowSolve (double *c, int d, double *r, double &tolerance)
static int	FerrariSolve (double *c, double *r, int *m, double tol)
static int	TartagliaCardanSolve (double *c, double *r, int *m, double tol)
static double *	SolveCubic (double c0, double c1, double c2, double c3)
static double *	SolveQuadratic (double c0, double c1, double c2)
static double *	SolveLinear (double c0, double c1)
static int	SolveQuadratic (double *c, double *r, int *m)
static int	SolveLinear (double c0, double c1, double *r1, int *num_roots)
static int	HabichtBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol)
static int	HabichtBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol, int intervalType)
static int	HabichtBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol, int intervalType, bool divideGCD)
static int	SturmBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol)
static int	SturmBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol, int intervalType)
static int	SturmBisectionSolve (double *P, int d, double *a, double *upperBnds, double tol, int intervalType, bool divideGCD)
static int	FilterRoots (double *P, int d, double *upperBnds, int rootcount, double diameter)
static int	SolveCubic (double c0, double c1, double c2, double c3, double *r1, double *r2, double *r3, int *num_roots)
static int	SolveQuadratic (double c0, double c1, double c2, double *r1, double *r2, int *num_roots)
static void	SetDivisionTolerance (double tol)
static double	GetDivisionTolerance ()
Protected Member Functions
virtual vtkObjectBase *	NewInstanceInternal () const
	vtkPolynomialSolversUnivariate ()
	~vtkPolynomialSolversUnivariate ()
Static Protected Attributes
static double	DivisionTolerance

---

Detailed Description

polynomial solvers

vtkPolynomialSolversUnivariate provides solvers for univariate polynomial equations with real coefficients. The Tartaglia-Cardan and Ferrari solvers work on polynomials of fixed degree 3 and 4, respectively. The Lin-Bairstow and Sturm solvers work on polynomials of arbitrary degree. The Sturm solver is the most robust solver but only reports roots within an interval and does not report multiplicities. The Lin-Bairstow solver reports multiplicities.

For difficult polynomials, you may wish to use FilterRoots to eliminate some of the roots reported by the Sturm solver. FilterRoots evaluates the derivatives near each root to eliminate cases where a local minimum or maximum is close to zero.

Thanks:

Thanks to Philippe Pebay, Korben Rusek, David Thompson, and Maurice Rojas for implementing these solvers.

Tests:

vtkPolynomialSolversUnivariate (Tests)

Definition at line 57 of file vtkPolynomialSolversUnivariate.h.

---

Member Typedef Documentation

typedef vtkObject vtkPolynomialSolversUnivariate::Superclass

Reimplemented from vtkObject.

Definition at line 61 of file vtkPolynomialSolversUnivariate.h.

---

Constructor & Destructor Documentation

vtkPolynomialSolversUnivariate::vtkPolynomialSolversUnivariate

(

)

[inline, protected]

Definition at line 248 of file vtkPolynomialSolversUnivariate.h.

vtkPolynomialSolversUnivariate::~vtkPolynomialSolversUnivariate

(

)

[inline, protected]

Definition at line 249 of file vtkPolynomialSolversUnivariate.h.

---

Member Function Documentation

static vtkPolynomialSolversUnivariate* vtkPolynomialSolversUnivariate::New

(

)

[static]

Create an object with Debug turned off, modified time initialized to zero, and reference counting on.

Reimplemented from vtkObject.

static int vtkPolynomialSolversUnivariate::IsTypeOf

(

const char *

name

)

[static]

Return 1 if this class type is the same type of (or a subclass of) the named class. Returns 0 otherwise. This method works in combination with vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

virtual int vtkPolynomialSolversUnivariate::IsA

(

const char *

name

)

[virtual]

Return 1 if this class is the same type of (or a subclass of) the named class. Returns 0 otherwise. This method works in combination with vtkTypeMacro found in vtkSetGet.h.

Reimplemented from vtkObject.

static vtkPolynomialSolversUnivariate* vtkPolynomialSolversUnivariate::SafeDownCast

(

vtkObjectBase *

o

)

[static]

Reimplemented from vtkObject.

virtual vtkObjectBase* vtkPolynomialSolversUnivariate::NewInstanceInternal

(

)

const [protected, virtual]

Reimplemented from vtkObject.

vtkPolynomialSolversUnivariate* vtkPolynomialSolversUnivariate::NewInstance

(

)

const

Reimplemented from vtkObject.

void vtkPolynomialSolversUnivariate::PrintSelf	(	ostream &	os,
		vtkIndent	indent
	)		[virtual]

Methods invoked by print to print information about the object including superclasses. Typically not called by the user (use Print() instead) but used in the hierarchical print process to combine the output of several classes.

Reimplemented from vtkObject.

static ostream& vtkPolynomialSolversUnivariate::PrintPolynomial	(	ostream &	os,
		double *	P,
		int	degP
	)		[static]

static int vtkPolynomialSolversUnivariate::HabichtBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Habicht sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Compared to the Sturm solver the Habicht solver is slower, although both are O(d^2). The Habicht solver has the added benefit that it has a built in mechanism to keep the leading coefficients of the result from polynomial division bounded above and below in absolute value. This will tend to keep the coefficients of the polynomials in the sequence from zeroing out prematurely or becoming infinite. Constructing the Habicht sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::HabichtBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol,
		int	intervalType
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Habicht sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Compared to the Sturm solver the Habicht solver is slower, although both are O(d^2). The Habicht solver has the added benefit that it has a built in mechanism to keep the leading coefficients of the result from polynomial division bounded above and below in absolute value. This will tend to keep the coefficients of the polynomials in the sequence from zeroing out prematurely or becoming infinite. Constructing the Habicht sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::HabichtBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol,
		int	intervalType,
		bool	divideGCD
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Habicht sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Compared to the Sturm solver the Habicht solver is slower, although both are O(d^2). The Habicht solver has the added benefit that it has a built in mechanism to keep the leading coefficients of the result from polynomial division bounded above and below in absolute value. This will tend to keep the coefficients of the polynomials in the sequence from zeroing out prematurely or becoming infinite. Constructing the Habicht sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::SturmBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial P[0] X^d + ... + P[d-1] X + P[d] in ]a[0] ; a[1]] using Sturm's theorem ( polynomial coefficients are REAL ) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Sturm sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Constructing the Sturm sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::SturmBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol,
		int	intervalType
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial P[0] X^d + ... + P[d-1] X + P[d] in ]a[0] ; a[1]] using Sturm's theorem ( polynomial coefficients are REAL ) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Sturm sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Constructing the Sturm sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::SturmBisectionSolve	(	double *	P,
		int	d,
		double *	a,
		double *	upperBnds,
		double	tol,
		int	intervalType,
		bool	divideGCD
	)		[static]

Finds all REAL roots (within tolerance tol) of the d -th degree polynomial P[0] X^d + ... + P[d-1] X + P[d] in ]a[0] ; a[1]] using Sturm's theorem ( polynomial coefficients are REAL ) and returns the count nr. All roots are bracketed in the first ]upperBnds[i] - tol ; upperBnds[i]] intervals. Returns -1 if anything went wrong (such as: polynomial does not have degree d, the interval provided by the other is absurd, etc.). intervalType specifies the search interval as follows: 0 = 00 = ]a,b[ 1 = 10 = [a,b[ 2 = 01 = ]a,b] 3 = 11 = [a,b] This defaults to 0. The last non-zero item in the Sturm sequence is the gcd of P and P'. The parameter divideGCD specifies whether the program should attempt to divide by the gcd and run again. It works better with polynomials known to have high multiplicities. When divideGCD != 0 then it attempts to divide by the GCD, if applicable. This defaults to 0. Constructing the Sturm sequence is O(d^2) in both time and space. Warning: it is the user's responsibility to make sure the upperBnds array is large enough to contain the maximal number of expected roots. Note that nr is smaller or equal to the actual number of roots in ]a[0] ; a[1]] since roots within are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

static int vtkPolynomialSolversUnivariate::FilterRoots	(	double *	P,
		int	d,
		double *	upperBnds,
		int	rootcount,
		double	diameter
	)		[static]

This uses the derivative sequence to filter possible roots of a polynomial. First it sorts the roots and removes any duplicates. If the number of sign changes of the derivative sequence at a root at upperBnds[i] == that at upperBnds[i] - diameter then the i^th value is removed from upperBnds. It returns the new number of roots.

static int vtkPolynomialSolversUnivariate::LinBairstowSolve	(	double *	c,
		int	d,
		double *	r,
		double &	tolerance
	)		[static]

Seeks all REAL roots of the d -th degree polynomial c[0] X^d + ...

• c[d-1] X + c[d] = 0 equation Lin-Bairstow's method ( polynomial coefficients are REAL ) and stores the nr roots found ( multiple roots are multiply stored ) in r. tolerance is the user-defined solver tolerance; this variable may be relaxed by the iterative solver if needed. Returns nr. Warning: it is the user's responsibility to make sure the r array is large enough to contain the maximal number of expected roots.

static int vtkPolynomialSolversUnivariate::FerrariSolve	(	double *	c,
		double *	r,
		int *	m,
		double	tol
	)		[static]

Algebraically extracts REAL roots of the quartic polynomial with REAL coefficients X^4 + c[0] X^3 + c[1] X^2 + c[2] X + c[3] and stores them (when they exist) and their respective multiplicities in the r and m arrays, based on Ferrari's method. Some numerical noise can be filtered by the use of a tolerance tol instead of equality with 0 (one can use, e.g., VTK_DBL_EPSILON). Returns the number of roots. Warning: it is the user's responsibility to pass a non-negative tol.

static int vtkPolynomialSolversUnivariate::TartagliaCardanSolve	(	double *	c,
		double *	r,
		int *	m,
		double	tol
	)		[static]

Algebraically extracts REAL roots of the cubic polynomial with REAL coefficients X^3 + c[0] X^2 + c[1] X + c[2] and stores them (when they exist) and their respective multiplicities in the r and m arrays. Some numerical noise can be filtered by the use of a tolerance tol instead of equality with 0 (one can use, e.g., VTK_DBL_EPSILON). The main differences with SolveCubic are that (1) the polynomial must have unit leading coefficient, (2) complex roots are discarded upfront, (3) non-simple roots are stored only once, along with their respective multiplicities, and (4) some numerical noise is filtered by the use of relative tolerance instead of equality with 0. Returns the number of roots. In memoriam Niccolo Tartaglia (1500 - 1559), unfairly forgotten.

static double* vtkPolynomialSolversUnivariate::SolveCubic	(	double	c0,
		double	c1,
		double	c2,
		double	c3
	)		[static]

Solves a cubic equation c0*t^3 + c1*t^2 + c2*t + c3 = 0 when c0, c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of (real) roots (counting multiple roots as one) followed by roots themselves. The value in roots[4] is a integer giving further information about the roots (see return codes for int SolveCubic() ).

static double* vtkPolynomialSolversUnivariate::SolveQuadratic	(	double	c0,
		double	c1,
		double	c2
	)		[static]

Solves a quadratic equation c1*t^2 + c2*t + c3 = 0 when c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of (real) roots (counting multiple roots as one) followed by roots themselves. Note that roots[3] contains a return code further describing solution - see documentation for SolveCubic() for meaning of return codes.

static double* vtkPolynomialSolversUnivariate::SolveLinear	(	double	c0,
		double	c1
	)		[static]

Solves a linear equation c2*t + c3 = 0 when c2 and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of roots followed by roots themselves.

static int vtkPolynomialSolversUnivariate::SolveCubic	(	double	c0,
		double	c1,
		double	c2,
		double	c3,
		double *	r1,
		double *	r2,
		double *	r3,
		int *	num_roots
	)		[static]

Solves a cubic equation when c0, c1, c2, And c3 Are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Roots and number of real roots are stored in user provided variables r1, r2, r3, and num_roots. Note that the function can return the following integer values describing the roots: (0)-no solution; (-1)-infinite number of solutions; (1)-one distinct real root of multiplicity 3 (stored in r1); (2)-two distinct real roots, one of multiplicity 2 (stored in r1 & r2); (3)-three distinct real roots; (-2)-quadratic equation with complex conjugate solution (real part of root returned in r1, imaginary in r2); (-3)-one real root and a complex conjugate pair (real root in r1 and real part of pair in r2 and imaginary in r3).

static int vtkPolynomialSolversUnivariate::SolveQuadratic	(	double	c0,
		double	c1,
		double	c2,
		double *	r1,
		double *	r2,
		int *	num_roots
	)		[static]

Solves a quadratic equation c1*t^2 + c2*t + c3 = 0 when c1, c2, and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Roots and number of roots are stored in user provided variables r1, r2, num_roots

static int vtkPolynomialSolversUnivariate::SolveQuadratic	(	double *	c,
		double *	r,
		int *	m
	)		[static]

Algebraically extracts REAL roots of the quadratic polynomial with REAL coefficients c[0] X^2 + c[1] X + c[2] and stores them (when they exist) and their respective multiplicities in the r and m arrays. Returns either the number of roots, or -1 if ininite number of roots.

static int vtkPolynomialSolversUnivariate::SolveLinear	(	double	c0,
		double	c1,
		double *	r1,
		int *	num_roots
	)		[static]

Solves a linear equation c2*t + c3 = 0 when c2 and c3 are REAL. Solution is motivated by Numerical Recipes In C 2nd Ed. Root and number of (real) roots are stored in user provided variables r2 and num_roots.

static void vtkPolynomialSolversUnivariate::SetDivisionTolerance

(

double

tol

)

[static]

Set/get the tolerance used when performing polynomial Euclidean division to find polynomial roots. This tolerance is used to decide whether the coefficient(s) of a polynomial remainder are close enough to zero to be neglected.

static double vtkPolynomialSolversUnivariate::GetDivisionTolerance

(

)

[static]

Set/get the tolerance used when performing polynomial Euclidean division to find polynomial roots. This tolerance is used to decide whether the coefficient(s) of a polynomial remainder are close enough to zero to be neglected.

---

Member Data Documentation

double vtkPolynomialSolversUnivariate::DivisionTolerance [static, protected]

Definition at line 249 of file vtkPolynomialSolversUnivariate.h.

---

The documentation for this class was generated from the following file:

• dox/Common/Math/vtkPolynomialSolversUnivariate.h