Container to encapsulate various operations for cubic equation of the forms with real coefficients: More...

#include <cubicEqn.H>

Inheritance diagram for cubicEqn:

Collaboration diagram for cubicEqn:

Public Types
enum	components { A , B , C , D }
	Component labeling enumeration. More...
Public Types inherited from VectorSpace< cubicEqn, scalar, 4 >
typedef VectorSpace< cubicEqn, scalar, Ncmpts >	vsType
	VectorSpace type.
typedef scalar	cmptType
	Component type.
typedef scalar	magType
	Magnitude type.
typedef direction	size_type
	The type to represent the size of a VectorSpace.
typedef scalar *	iterator
	Random access iterator for traversing VectorSpace.
typedef const scalar *	const_iterator
	Random access iterator for traversing VectorSpace.

Public Member Functions
	cubicEqn ()=default
	Default construct.
	cubicEqn (const Foam::zero)
	Construct initialized to zero.
	cubicEqn (const scalar a, const scalar b, const scalar c, const scalar d)
	Construct from components.
scalar	a () const noexcept
scalar	b () const noexcept
scalar	c () const noexcept
scalar	d () const noexcept
scalar &	a () noexcept
scalar &	b () noexcept
scalar &	c () noexcept
scalar &	d () noexcept
scalar	value (const scalar x) const
	Evaluate the cubic equation at x.
scalar	derivative (const scalar x) const
	Evaluate the derivative of the cubic equation at x.
scalar	error (const scalar x) const
	Estimate the error of evaluation of the cubic equation at x.
Roots< 3 >	roots () const
	Return the roots of the cubic equation with no particular order.
Public Member Functions inherited from VectorSpace< cubicEqn, scalar, 4 >
const Foam::VectorSpace< cubicEqn, scalar, Ncmpts >::template ConstBlock< SubVector, BStart >	block () const
	VectorSpace ()=default
	Default construct.
const scalar &	component (const direction) const
void	replace (const direction, const scalar &)
const scalar *	cdata () const noexcept
	Return const pointer to the first data element.
scalar *	data () noexcept
	Return pointer to the first data element.
void	fill (const scalar &s)
	Assign all components to given value.
const scalar &	operator[] (const direction) const
void	operator= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)
void	operator+= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)
void	operator-= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)
void	operator*= (const scalar)
void	operator/= (const scalar)
iterator	begin () noexcept
	Return an iterator (pointer) to begin of VectorSpace.
const_iterator	cbegin () const noexcept
	Return const_iterator (const pointer) to begin of VectorSpace.
iterator	end () noexcept
	Return an iterator (pointer) to end of VectorSpace.
const_iterator	cend () const noexcept
	Return const_iterator (const pointer) to end of VectorSpace.

Additional Inherited Members
Static Public Member Functions inherited from VectorSpace< cubicEqn, scalar, 4 >
static constexpr direction	size () noexcept
	The number of elements in the VectorSpace = Ncmpts.
static cubicEqn	uniform (const scalar &s)
	Return a VectorSpace with all elements = s.
Public Attributes inherited from VectorSpace< cubicEqn, scalar, 4 >
scalar	v_ [Ncmpts]
	The components of this vector space.
Static Public Attributes inherited from VectorSpace< cubicEqn, scalar, 4 >
static constexpr direction	dim
	Dimensionality of space.
static constexpr direction	nComponents
	Number of components in this vector space.
static constexpr direction	mRows
static constexpr direction	nCols
static const char *const	typeName
static const char *const	componentNames []
static const cubicEqn	zero
static const cubicEqn	one
static const cubicEqn	max
static const cubicEqn	min
static const cubicEqn	rootMax
static const cubicEqn	rootMin

Detailed Description

Container to encapsulate various operations for cubic equation of the forms with real coefficients:

$\‍[ a*x^3 + b*x^2 + c*x + d = 0 x^3 + B*x^2 + C*x + D = 0 \‍]$

The following two substitutions into the above forms are used:

$\‍[ x = t - B/3 t = w - P/3/w \‍]$

This reduces the problem to a quadratic in w^3:

$\‍[ w^6 + Q*w^3 - P = 0 \‍]$

where Q and P are given in the code.

The properties of the cubic can be related to the properties of this quadratic in w^3.

If the quadratic eqn has two identical real roots at zero, three identical real roots exist in the cubic eqn.

If the quadratic eqn has two identical real roots at non-zero, two identical and one distinct real roots exist in the cubic eqn.

If the quadratic eqn has two complex roots (a complex conjugate pair), three distinct real roots exist in the cubic eqn.

If the quadratic eqn has two distinct real roots, one real root and two complex roots (a complex conjugate pair) exist in the cubic eqn.

The quadratic eqn is solved for the most numerically accurate value of w^3. See the quadraticEqn.H for details on how to pick a value. This single value of w^3 can yield up to three cube roots for w, which relate to the three solutions for x.

Only a single root, or pair of conjugate roots, is directly evaluated; the one, or ones with the lowest relative numerical error. Root identities are then used to recover the remaining roots, possibly utilising a quadratic and/or linear solution. This seems to be a good way of maintaining the accuracy of roots at very different magnitudes.

Reference:

    Kahan's algo. to compute 'p' using fused multiply-adds (tag:JML):
        Jeannerod, C. P., Louvet, N., & Muller, J. M. (2013).
        Further analysis of Kahan's algorithm for the accurate
        computation of 2× 2 determinants.
        Mathematics of Computation, 82(284), 2245-2264.
        DOI:10.1090/S0025-5718-2013-02679-8

See also: Test-cubicEqn.C

Source files

Definition at line 108 of file cubicEqn.H.

Member Enumeration Documentation

◆ components

enum components

Component labeling enumeration.

Enumerator
A
B
C
D

Definition at line 117 of file cubicEqn.H.

Constructor & Destructor Documentation

◆ cubicEqn() [1/3]

cubicEqn ( )

default

Default construct.

References a(), b(), c(), and d().

Referenced by cubicEqn().

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◆ cubicEqn() [2/3]

cubicEqn ( const Foam::zero )

inline

Construct initialized to zero.

Definition at line 23 of file cubicEqnI.H.

References cubicEqn(), VectorSpace< cubicEqn, scalar, 4 >::VectorSpace(), and VectorSpace< cubicEqn, scalar, 4 >::zero.

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◆ cubicEqn() [3/3]

cubicEqn	(	const scalar	a,
		const scalar	b,
		const scalar	c,
		const scalar	d )

inline

Construct from components.

Definition at line 29 of file cubicEqnI.H.

References A, a(), B, b(), C, c(), D, d(), and VectorSpace< cubicEqn, scalar, 4 >::v_.

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Member Function Documentation

◆ a() [1/2]

scalar a ( ) const

inlinenoexcept

Definition at line 148 of file cubicEqn.H.

References A, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

Referenced by cubicEqn(), cubicEqn(), derivative(), error(), roots(), and value().

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◆ b() [1/2]

scalar b ( ) const

inlinenoexcept

Definition at line 149 of file cubicEqn.H.

References B, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

Referenced by cubicEqn(), cubicEqn(), and roots().

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◆ c() [1/2]

scalar c ( ) const

inlinenoexcept

Definition at line 150 of file cubicEqn.H.

References C, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

Referenced by cubicEqn(), cubicEqn(), derivative(), error(), roots(), and value().

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◆ d() [1/2]

scalar d ( ) const

inlinenoexcept

Definition at line 151 of file cubicEqn.H.

References D, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

Referenced by cubicEqn(), cubicEqn(), error(), roots(), and value().

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◆ a() [2/2]

scalar & a ( )

inlinenoexcept

Definition at line 153 of file cubicEqn.H.

References A, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

◆ b() [2/2]

scalar & b ( )

inlinenoexcept

Definition at line 154 of file cubicEqn.H.

References B, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

◆ c() [2/2]

scalar & c ( )

inlinenoexcept

Definition at line 155 of file cubicEqn.H.

References C, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

◆ d() [2/2]

scalar & d ( )

inlinenoexcept

Definition at line 156 of file cubicEqn.H.

References D, Foam::noexcept, and VectorSpace< cubicEqn, scalar, 4 >::v_.

◆ value()

Foam::scalar value ( const scalar x ) const

inline

Evaluate the cubic equation at x.

Definition at line 46 of file cubicEqnI.H.

References a(), b, c(), d(), and x.

Referenced by particle< Type >::trackToMovingTri().

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◆ derivative()

Foam::scalar derivative ( const scalar x ) const

inline

Evaluate the derivative of the cubic equation at x.

Definition at line 52 of file cubicEqnI.H.

References a(), b, c(), and x.

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◆ error()

Foam::scalar error ( const scalar x ) const

inline

Estimate the error of evaluation of the cubic equation at x.

Definition at line 58 of file cubicEqnI.H.

References a(), b, c(), d(), Foam::mag(), Foam::magSqr(), and x.

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◆ roots()

Foam::Roots< 3 > roots ( ) const

Return the roots of the cubic equation with no particular order.

if discriminant > 0: return three distinct real roots if discriminant < 0: return one real root and one complex root (one member of the complex conjugate pair) if discriminant = 0: return two identical roots, and one distinct real root if identical zero Hessian: return three identical real roots where the discriminant = - 4p^3 - 27q^2.

Definition at line 28 of file cubicEqn.C.

References a(), Foam::atan2(), b(), c(), Foam::cbrt(), Foam::roots::complex, Foam::cos(), d(), Foam::endl(), Foam::hypot(), Foam::Info, Foam::mag(), Foam::roots::nan, Foam::nl, p, Foam::roots::real, roots(), Foam::sign(), Foam::sin(), Foam::sqrt(), Foam::tab, Roots< N >::type(), and x.

Referenced by roots().

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

src/OpenFOAM/primitives/polynomialEqns/cubicEqn/cubicEqn.H
src/OpenFOAM/primitives/polynomialEqns/cubicEqn/cubicEqn.C
src/OpenFOAM/primitives/polynomialEqns/cubicEqn/cubicEqnI.H

Public Types

Public Member Functions

Additional Inherited Members

Detailed Description

Member Enumeration Documentation

◆ components

Constructor & Destructor Documentation

◆ cubicEqn() [1/3]

◆ cubicEqn() [2/3]

◆ cubicEqn() [3/3]

Member Function Documentation

◆ a() [1/2]

◆ b() [1/2]

◆ c() [1/2]

◆ d() [1/2]

◆ a() [2/2]

◆ b() [2/2]

◆ c() [2/2]

◆ d() [2/2]

◆ value()

◆ derivative()

◆ error()

◆ roots()