VTK
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polynomial solvers More...
#include <vtkPolynomialSolversUnivariate.h>
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) |
vtkTypeBool | InvokeEvent (unsigned long event) |
vtkTypeBool | InvokeEvent (const char *event) |
std::string | GetObjectDescription () const override |
The object description printed in messages and PrintSelf output. | |
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. | |
vtkTypeBool | InvokeEvent (unsigned long event, void *callData) |
This method invokes an event and return whether the event was aborted or not. | |
vtkTypeBool | InvokeEvent (const char *event, void *callData) |
This method invokes an event and return whether the event was aborted or not. | |
virtual void | SetObjectName (const std::string &objectName) |
Set/get the name of this object for reporting purposes. | |
virtual std::string | GetObjectName () const |
Set/get the name of this object for reporting purposes. | |
Public Member Functions inherited from vtkObjectBase | |
const char * | GetClassName () const |
Return the class name as a string. | |
virtual std::string | GetObjectDescription () const |
The object description printed in messages and PrintSelf output. | |
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. | |
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. | |
virtual bool | UsesGarbageCollector () const |
Indicate whether the class uses vtkGarbageCollector or not. | |
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 c0*t^3 + c1*t^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 c0*t^2 + c1*t + 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 c0*t^2 + c1*t + 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 (vtkTypeBool 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 vtkTypeBool | 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 *) |
virtual void | ObjectFinalize () |
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 |
std::string | ObjectName |
Protected Attributes inherited from vtkObjectBase | |
std::atomic< int32_t > | ReferenceCount |
vtkWeakPointerBase ** | WeakPointers |
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.
Definition at line 38 of file vtkPolynomialSolversUnivariate.h.
Definition at line 42 of file vtkPolynomialSolversUnivariate.h.
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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.
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vtkPolynomialSolversUnivariate * vtkPolynomialSolversUnivariate::NewInstance | ( | ) | const |
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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() ).
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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.
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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.
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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).
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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
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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.
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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.
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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.
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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.
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Definition at line 267 of file vtkPolynomialSolversUnivariate.h.