The open source CFD toolbox
cubicEqn Class Reference

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

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## 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. More...

typedef scalar cmptType
Component type. More...

typedef scalar magType
Magnitude type. More...

typedef scalar * iterator
Random access iterator for traversing VectorSpace. More...

## Public Member Functions

cubicEqn ()
Construct null. More...

cubicEqn (const Foam::zero)
Construct initialized to zero. More...

cubicEqn (const scalar a, const scalar b, const scalar c, const scalar d)
Construct from components. More...

scalar a () const

scalar b () const

scalar c () const

scalar d () const

scalar & a ()

scalar & b ()

scalar & c ()

scalar & d ()

scalar value (const scalar x) const
Evaluate the cubic equation at x. More...

scalar derivative (const scalar x) const
Evaluate the derivative of the cubic equation at x. More...

scalar error (const scalar x) const
Estimate the error of evaluation of the cubic equation at x. More...

Roots< 3 > roots () const
Return the roots of the cubic equation with no particular order. More...

Public Member Functions inherited from VectorSpace< cubicEqn, scalar, 4 >
VectorSpace ()=default
Default construct. More...

VectorSpace (const Foam::zero)
Construct initialized to zero. More...

VectorSpace (const VectorSpace< cubicEqn, scalar, Ncmpts > &vs)
Copy construct. More...

VectorSpace (const VectorSpace< Form2, Cmpt2, Ncmpts > &)
Copy construct of a VectorSpace with the same size. More...

VectorSpace (Istream &is)
Construct from Istream. More...

const scalar & component (const direction) const

scalar & component (const direction)

void component (scalar &, const direction) const

void replace (const direction, const scalar &)

const scalar * cdata () const noexcept
Return const pointer to the first data element. More...

scalar * data () noexcept
Return pointer to the first data element. More...

const ConstBlock< SubVector, BStart > block () const

const Foam::VectorSpace< cubicEqn, scalar, Ncmpts >::template ConstBlock< SubVector, BStart > block () const

const scalar & operator[] (const direction) const

scalar & operator[] (const direction)

void operator= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)

void operator= (const Foam::zero)

void operator+= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)

void operator-= (const VectorSpace< cubicEqn, scalar, Ncmpts > &)

void operator*= (const scalar)

void operator/= (const scalar)

iterator begin ()
Return an iterator to begin of VectorSpace. More...

const_iterator begin () const
Return const_iterator to begin of VectorSpace. More...

iterator end ()
Return an iterator to end of VectorSpace. More...

const_iterator end () const
Return const_iterator to end of VectorSpace. More...

const_iterator cbegin () const
Return const_iterator to begin of VectorSpace. More...

const_iterator cend () const
Return const_iterator to end of VectorSpace. More...

Static Public Member Functions inherited from VectorSpace< cubicEqn, scalar, 4 >
static constexpr direction size () noexcept
The number of elements in the VectorSpace = Ncmpts. More...

static cubicEqn uniform (const scalar &s)
Return a VectorSpace with all elements = s. More...

Public Attributes inherited from VectorSpace< cubicEqn, scalar, 4 >
const typedef scalar * const_iterator
Random access iterator for traversing VectorSpace. More...

scalar v_ [Ncmpts]
The components of this vector space. More...

Static Public Attributes inherited from VectorSpace< cubicEqn, scalar, 4 >
static constexpr direction dim
Dimensionality of space. More...

static constexpr direction nComponents
Number of components in this vector space. More...

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

Test-cubicEqn.C
Source files

Definition at line 112 of file cubicEqn.H.

## ◆ components

 enum components

Component labeling enumeration.

Enumerator

Definition at line 119 of file cubicEqn.H.

## ◆ cubicEqn() [1/3]

 cubicEqn ( )
inline

Construct null.

Definition at line 30 of file cubicEqnI.H.

## ◆ cubicEqn() [2/3]

 cubicEqn ( const Foam::zero )
inline

Construct initialized to zero.

Definition at line 34 of file cubicEqnI.H.

## ◆ cubicEqn() [3/3]

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

Construct from components.

Definition at line 41 of file cubicEqnI.H.

References A, B, Foam::constant::physicoChemical::b, Foam::constant::universal::c, C, and D.

## ◆ a() [1/2]

 Foam::scalar a ( ) const
inline

Definition at line 57 of file cubicEqnI.H.

References A.

Referenced by cubicEqn::roots().

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## ◆ b() [1/2]

 Foam::scalar b ( ) const
inline

Definition at line 63 of file cubicEqnI.H.

References B.

Referenced by cubicEqn::roots().

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## ◆ c() [1/2]

 Foam::scalar c ( ) const
inline

Definition at line 69 of file cubicEqnI.H.

References C.

Referenced by cubicEqn::roots().

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## ◆ d() [1/2]

 Foam::scalar d ( ) const
inline

Definition at line 75 of file cubicEqnI.H.

References D.

Referenced by cubicEqn::roots().

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## ◆ a() [2/2]

 Foam::scalar & a ( )
inline

Definition at line 81 of file cubicEqnI.H.

References A.

## ◆ b() [2/2]

 Foam::scalar & b ( )
inline

Definition at line 87 of file cubicEqnI.H.

References B.

## ◆ c() [2/2]

 Foam::scalar & c ( )
inline

Definition at line 93 of file cubicEqnI.H.

References C.

## ◆ d() [2/2]

 Foam::scalar & d ( )
inline

Definition at line 99 of file cubicEqnI.H.

References D.

## ◆ value()

 Foam::scalar value ( const scalar x ) const
inline

Evaluate the cubic equation at x.

Definition at line 105 of file cubicEqnI.H.

References Foam::constant::physicoChemical::b, Foam::constant::universal::c, 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 111 of file cubicEqnI.H.

References Foam::constant::physicoChemical::b, Foam::constant::universal::c, and x.

## ◆ error()

 Foam::scalar error ( const scalar x ) const
inline

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

Definition at line 117 of file cubicEqnI.H.

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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 35 of file cubicEqn.C.

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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