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#include <vtkPolynomialSolversUnivariate.h>

Inheritance diagram for vtkPolynomialSolversUnivariate:

Collaboration diagram for vtkPolynomialSolversUnivariate:

Public Types
typedef vtkObject	Superclass

Public Member Functions
virtual vtkTypeBool	IsA (const char *type)
	Return 1 if this class is the same type of (or a subclass of) the named class.

vtkPolynomialSolversUnivariate *	NewInstance () const

void	PrintSelf (ostream &os, vtkIndent indent) override
	Methods invoked by print to print information about the object including superclasses.

Public Member Functions inherited from vtkObject
	vtkBaseTypeMacro (vtkObject, vtkObjectBase)

virtual void	DebugOn ()
	Turn debugging output on.

virtual void	DebugOff ()
	Turn debugging output off.

bool	GetDebug ()
	Get the value of the debug flag.

void	SetDebug (bool debugFlag)
	Set the value of the debug flag.

virtual void	Modified ()
	Update the modification time for this object.

virtual vtkMTimeType	GetMTime ()
	Return this object's modified time.

void	PrintSelf (ostream &os, vtkIndent indent) override
	Methods invoked by print to print information about the object including superclasses.

void	RemoveObserver (unsigned long tag)

void	RemoveObservers (unsigned long event)

void	RemoveObservers (const char *event)

void	RemoveAllObservers ()

vtkTypeBool	HasObserver (unsigned long event)

vtkTypeBool	HasObserver (const char *event)

int	InvokeEvent (unsigned long event)

int	InvokeEvent (const char *event)

unsigned long	AddObserver (unsigned long event, vtkCommand *, float priority=0.0f)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

unsigned long	AddObserver (const char event, vtkCommand , float priority=0.0f)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

vtkCommand *	GetCommand (unsigned long tag)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

void	RemoveObserver (vtkCommand *)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

void	RemoveObservers (unsigned long event, vtkCommand *)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

void	RemoveObservers (const char event, vtkCommand )
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

vtkTypeBool	HasObserver (unsigned long event, vtkCommand *)
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

vtkTypeBool	HasObserver (const char event, vtkCommand )
	Allow people to add/remove/invoke observers (callbacks) to any VTK object.

template<class U , class T >
unsigned long	AddObserver (unsigned long event, U observer, void(T::*callback)(), float priority=0.0f)
	Overloads to AddObserver that allow developers to add class member functions as callbacks for events.

template<class U , class T >
unsigned long	AddObserver (unsigned long event, U observer, void(T::callback)(vtkObject , unsigned long, void *), float priority=0.0f)
	Overloads to AddObserver that allow developers to add class member functions as callbacks for events.

template<class U , class T >
unsigned long	AddObserver (unsigned long event, U observer, bool(T::callback)(vtkObject , unsigned long, void *), float priority=0.0f)
	Allow user to set the AbortFlagOn() with the return value of the callback method.

int	InvokeEvent (unsigned long event, void *callData)
	This method invokes an event and return whether the event was aborted or not.

int	InvokeEvent (const char event, void callData)
	This method invokes an event and return whether the event was aborted or not.

Public Member Functions inherited from vtkObjectBase
const char *	GetClassName () const
	Return the class name as a string.

virtual vtkTypeBool	IsA (const char *name)
	Return 1 if this class is the same type of (or a subclass of) the named class.

virtual vtkIdType	GetNumberOfGenerationsFromBase (const char *name)
	Given the name of a base class of this class type, return the distance of inheritance between this class type and the named class (how many generations of inheritance are there between this class and the named class).

virtual void	Delete ()
	Delete a VTK object.

virtual void	FastDelete ()
	Delete a reference to this object.

void	InitializeObjectBase ()

void	Print (ostream &os)
	Print an object to an ostream.

virtual void	Register (vtkObjectBase *o)
	Increase the reference count (mark as used by another object).

virtual void	UnRegister (vtkObjectBase *o)
	Decrease the reference count (release by another object).

int	GetReferenceCount ()
	Return the current reference count of this object.

void	SetReferenceCount (int)
	Sets the reference count.

bool	GetIsInMemkind () const
	A local state flag that remembers whether this object lives in the normal or extended memory space.

virtual void	PrintHeader (ostream &os, vtkIndent indent)
	Methods invoked by print to print information about the object including superclasses.

virtual void	PrintTrailer (ostream &os, vtkIndent indent)
	Methods invoked by print to print information about the object including superclasses.

Static Public Member Functions
static vtkPolynomialSolversUnivariate *	New ()

static vtkTypeBool	IsTypeOf (const char *type)

static vtkPolynomialSolversUnivariate *	SafeDownCast (vtkObjectBase *o)

static ostream &	PrintPolynomial (ostream &os, double *P, int degP)

static int	FilterRoots (double P, int d, double upperBnds, int rootcount, double diameter)
	This uses the derivative sequence to filter possible roots of a polynomial.

static int	LinBairstowSolve (double c, int d, double r, double &tolerance)
	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.

static int	FerrariSolve (double c, double r, int *m, double tol)
	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.

static int	TartagliaCardanSolve (double c, double r, int *m, double tol)
	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.

static double *	SolveCubic (double c0, double c1, double c2, double c3)
	Solves a cubic equation c0t^3 + c1t^2 + c2*t + c3 = 0 when c0, c1, c2, and c3 are REAL.

static double *	SolveQuadratic (double c0, double c1, double c2)
	Solves a quadratic equation c0t^2 + c1t + c2 = 0 when c0, c1, and c2 are REAL.

static double *	SolveLinear (double c0, double c1)
	Solves a linear equation c0*t + c1 = 0 when c0 and c1 are REAL.

static int	SolveCubic (double c0, double c1, double c2, double c3, double r1, double r2, double r3, int num_roots)
	Solves a cubic equation when c0, c1, c2, And c3 Are REAL.

static int	SolveQuadratic (double c0, double c1, double c2, double r1, double r2, int *num_roots)
	Solves a quadratic equation c0t^2 + c1t + c2 = 0 when c0, c1, and c2 are REAL.

static int	SolveQuadratic (double c, double r, int *m)
	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.

static int	SolveLinear (double c0, double c1, double r1, int num_roots)
	Solves a linear equation c0*t + c1 = 0 when c0 and c1 are REAL.


static int	HabichtBisectionSolve (double P, int d, double a, double *upperBnds, double tol)
	Finds all REAL roots (within tolerance tol) of the d -th degree polynomial.

static int	HabichtBisectionSolve (double P, int d, double a, double *upperBnds, double tol, int intervalType)
	Finds all REAL roots (within tolerance tol) of the d -th degree polynomial.

static int	HabichtBisectionSolve (double P, int d, double a, double *upperBnds, double tol, int intervalType, bool divideGCD)
	Finds all REAL roots (within tolerance tol) of the d -th degree polynomial.


static int	SturmBisectionSolve (double P, int d, double a, double *upperBnds, double tol)
	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.

static int	SturmBisectionSolve (double P, int d, double a, double *upperBnds, double tol, int intervalType)
	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.

static int	SturmBisectionSolve (double P, int d, double a, double *upperBnds, double tol, int intervalType, bool divideGCD)
	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.


static void	SetDivisionTolerance (double tol)
	Set/get the tolerance used when performing polynomial Euclidean division to find polynomial roots.

static double	GetDivisionTolerance ()
	Set/get the tolerance used when performing polynomial Euclidean division to find polynomial roots.

Static Public Member Functions inherited from vtkObject
static vtkObject *	New ()
	Create an object with Debug turned off, modified time initialized to zero, and reference counting on.

static void	BreakOnError ()
	This method is called when vtkErrorMacro executes.

static void	SetGlobalWarningDisplay (int val)
	This is a global flag that controls whether any debug, warning or error messages are displayed.

static void	GlobalWarningDisplayOn ()
	This is a global flag that controls whether any debug, warning or error messages are displayed.

static void	GlobalWarningDisplayOff ()
	This is a global flag that controls whether any debug, warning or error messages are displayed.

static int	GetGlobalWarningDisplay ()
	This is a global flag that controls whether any debug, warning or error messages are displayed.

Static Public Member Functions inherited from vtkObjectBase
static vtkTypeBool	IsTypeOf (const char *name)
	Return 1 if this class type is the same type of (or a subclass of) the named class.

static vtkIdType	GetNumberOfGenerationsFromBaseType (const char *name)
	Given a the name of a base class of this class type, return the distance of inheritance between this class type and the named class (how many generations of inheritance are there between this class and the named class).

static vtkObjectBase *	New ()
	Create an object with Debug turned off, modified time initialized to zero, and reference counting on.

static void	SetMemkindDirectory (const char *directoryname)
	The name of a directory, ideally mounted -o dax, to memory map an extended memory space within.

static bool	GetUsingMemkind ()
	A global state flag that controls whether vtkObjects are constructed in the usual way (the default) or within the extended memory space.

Protected Member Functions
virtual vtkObjectBase *	NewInstanceInternal () const

	vtkPolynomialSolversUnivariate ()=default

	~vtkPolynomialSolversUnivariate () override=default

Protected Member Functions inherited from vtkObject
	vtkObject ()

	~vtkObject () override

void	RegisterInternal (vtkObjectBase *, vtkTypeBool check) override

void	UnRegisterInternal (vtkObjectBase *, vtkTypeBool check) override

void	InternalGrabFocus (vtkCommand mouseEvents, vtkCommand keypressEvents=nullptr)
	These methods allow a command to exclusively grab all events.

void	InternalReleaseFocus ()
	These methods allow a command to exclusively grab all events.

Protected Member Functions inherited from vtkObjectBase
	vtkObjectBase ()

virtual	~vtkObjectBase ()

virtual void	RegisterInternal (vtkObjectBase *, vtkTypeBool check)

virtual void	UnRegisterInternal (vtkObjectBase *, vtkTypeBool check)

virtual void	ReportReferences (vtkGarbageCollector *)

	vtkObjectBase (const vtkObjectBase &)

void	operator= (const vtkObjectBase &)

Static Protected Attributes
static double	DivisionTolerance

Additional Inherited Members
Static Protected Member Functions inherited from vtkObjectBase
static vtkMallocingFunction	GetCurrentMallocFunction ()

static vtkReallocingFunction	GetCurrentReallocFunction ()

static vtkFreeingFunction	GetCurrentFreeFunction ()

static vtkFreeingFunction	GetAlternateFreeFunction ()

Protected Attributes inherited from vtkObject
bool	Debug

vtkTimeStamp	MTime

vtkSubjectHelper *	SubjectHelper

Protected Attributes inherited from vtkObjectBase
std::atomic< int32_t >	ReferenceCount

vtkWeakPointerBase **	WeakPointers

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.

Definition at line 55 of file vtkPolynomialSolversUnivariate.h.

Member Typedef Documentation

◆ Superclass

typedef vtkObject vtkPolynomialSolversUnivariate::Superclass

Definition at line 59 of file vtkPolynomialSolversUnivariate.h.

Constructor & Destructor Documentation

◆ vtkPolynomialSolversUnivariate()

vtkPolynomialSolversUnivariate::vtkPolynomialSolversUnivariate ( )

protecteddefault

◆ ~vtkPolynomialSolversUnivariate()

vtkPolynomialSolversUnivariate::~vtkPolynomialSolversUnivariate ( )

overrideprotecteddefault

Member Function Documentation

◆ New()

static vtkPolynomialSolversUnivariate * vtkPolynomialSolversUnivariate::New ( )

static

◆ IsTypeOf()

static vtkTypeBool vtkPolynomialSolversUnivariate::IsTypeOf ( const char * type )

static

◆ IsA()

virtual vtkTypeBool 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 vtkObjectBase.

◆ SafeDownCast()

static vtkPolynomialSolversUnivariate * vtkPolynomialSolversUnivariate::SafeDownCast ( vtkObjectBase * o )

static

◆ NewInstanceInternal()

virtual vtkObjectBase * vtkPolynomialSolversUnivariate::NewInstanceInternal ( ) const

protectedvirtual

◆ NewInstance()

vtkPolynomialSolversUnivariate * vtkPolynomialSolversUnivariate::NewInstance ( ) const

◆ PrintSelf()

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

overridevirtual

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.

◆ PrintPolynomial()

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

static

◆ HabichtBisectionSolve() [1/3]

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.

$P[0] X^d + ... + P[d-1] X + P[d]$

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ HabichtBisectionSolve() [2/3]

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.

$P[0] X^d + ... + P[d-1] X + P[d]$

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ HabichtBisectionSolve() [3/3]

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.

$P[0] X^d + ... + P[d-1] X + P[d]$

in ]a[0] ; a[1]] using the Habicht sequence (polynomial coefficients are REAL) and returns the count nr. All roots are bracketed in the nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ SturmBisectionSolve() [1/3]

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 nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ SturmBisectionSolve() [2/3]

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 nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ SturmBisectionSolve() [3/3]

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 nr 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 tol are lumped in the same bracket. array is large enough to contain the maximal number of expected upper bounds.

◆ FilterRoots()

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.

◆ LinBairstowSolve()

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.

◆ FerrariSolve()

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.

◆ TartagliaCardanSolve()

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.

◆ SolveCubic() [1/2]

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() ).

◆ SolveQuadratic() [1/3]

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

static

Solves a quadratic equation c0*t^2 + c1*t + c2 = 0 when c0, c1, and c2 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.

◆ SolveLinear() [1/2]

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

static

Solves a linear equation c0*t + c1 = 0 when c0 and c1 are REAL.

Solution is motivated by Numerical Recipes In C 2nd Ed. Return array contains number of roots followed by roots themselves.

◆ SolveCubic() [2/2]

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).

◆ SolveQuadratic() [2/3]

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

static

Solves a quadratic equation c0*t^2 + c1*t + c2 = 0 when c0, c1, and c2 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

◆ SolveQuadratic() [3/3]

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.

◆ SolveLinear() [2/2]

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

static

Solves a linear equation c0*t + c1 = 0 when c0 and c1 are REAL.

Solution is motivated by Numerical Recipes In C 2nd Ed. Root and number of (real) roots are stored in user provided variables r1 and num_roots.

◆ SetDivisionTolerance()

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.

◆ GetDivisionTolerance()

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

◆ DivisionTolerance

double vtkPolynomialSolversUnivariate::DivisionTolerance

staticprotected

Definition at line 284 of file vtkPolynomialSolversUnivariate.h.

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

Common/Math/vtkPolynomialSolversUnivariate.h

Public Types

Public Member Functions

Static Public Member Functions

Protected Member Functions

Static Protected Attributes

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ Superclass

Constructor & Destructor Documentation

◆ vtkPolynomialSolversUnivariate()

◆ ~vtkPolynomialSolversUnivariate()

Member Function Documentation

◆ New()

◆ IsTypeOf()

◆ IsA()

◆ SafeDownCast()

◆ NewInstanceInternal()

◆ NewInstance()

◆ PrintSelf()

◆ PrintPolynomial()

◆ HabichtBisectionSolve() [1/3]

◆ HabichtBisectionSolve() [2/3]

◆ HabichtBisectionSolve() [3/3]

◆ SturmBisectionSolve() [1/3]

◆ SturmBisectionSolve() [2/3]

◆ SturmBisectionSolve() [3/3]

◆ FilterRoots()

◆ LinBairstowSolve()

◆ FerrariSolve()

◆ TartagliaCardanSolve()

◆ SolveCubic() [1/2]

◆ SolveQuadratic() [1/3]

◆ SolveLinear() [1/2]

◆ SolveCubic() [2/2]

◆ SolveQuadratic() [2/3]

◆ SolveQuadratic() [3/3]

◆ SolveLinear() [2/2]

◆ SetDivisionTolerance()

◆ GetDivisionTolerance()

Member Data Documentation

◆ DivisionTolerance