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LOCA::Hopf::MinimallyAugmented::Constraint Class Reference

Implementation of LOCA::MultiContinuation::ConstraintInterfaceMVDX for computing Hopf bifurcations for the minimally augmented Hopf formulation. More...

#include <LOCA_Hopf_MinimallyAugmented_Constraint.H>

Inheritance diagram for LOCA::Hopf::MinimallyAugmented::Constraint:
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Public Member Functions

 Constraint (const Teuchos::RCP< LOCA::GlobalData > &global_data, const Teuchos::RCP< LOCA::Parameter::SublistParser > &topParams, const Teuchos::RCP< Teuchos::ParameterList > &hpfParams, const Teuchos::RCP< LOCA::Hopf::MinimallyAugmented::AbstractGroup > &g, bool is_symmetric, const NOX::Abstract::Vector &a_real, const NOX::Abstract::Vector &a_imag, const NOX::Abstract::Vector *b_real, const NOX::Abstract::Vector *b_imag, int bif_param, double freq)
 Constructor.
 Constraint (const Constraint &source, NOX::CopyType type=NOX::DeepCopy)
 Copy constructor.
virtual ~Constraint ()
 Destructor.
virtual void setGroup (const Teuchos::RCP< LOCA::Hopf::MinimallyAugmented::AbstractGroup > &g)
 Set the group pointer.
virtual void setFrequency (double freq)
 Set Hopf frequency.
virtual Teuchos::RCP< const NOX::Abstract::Vector > getLeftNullVecReal () const
 Returns real component of left null vector w.
virtual Teuchos::RCP< const NOX::Abstract::Vector > getLeftNullVecImag () const
 Returns imaginary component of left null vector w.
virtual Teuchos::RCP< const NOX::Abstract::Vector > getRightNullVecReal () const
 Returns real component of right null vector v.
virtual Teuchos::RCP< const NOX::Abstract::Vector > getRightNullVecImag () const
 Returns imaginary component of right null vector v.
virtual double getSigmaReal () const
 Returns real component of sigma.
virtual double getSigmaImag () const
 Returns imaginary component of sigma.
Public Member Functions inherited from LOCA::MultiContinuation::ConstraintInterfaceMVDX
 ConstraintInterfaceMVDX ()
 Constructor.
virtual ~ConstraintInterfaceMVDX ()
 Destructor.
virtual NOX::Abstract::Group::ReturnType multiplyDX (double alpha, const NOX::Abstract::MultiVector &input_x, NOX::Abstract::MultiVector::DenseMatrix &result_p) const
 Compute result_p = alpha * dg/dx * input_x.
virtual NOX::Abstract::Group::ReturnType addDX (Teuchos::ETransp transb, double alpha, const NOX::Abstract::MultiVector::DenseMatrix &b, double beta, NOX::Abstract::MultiVector &result_x) const
 Compute result_x = alpha * dg/dx^T * op(b) + beta * result_x.
Public Member Functions inherited from LOCA::MultiContinuation::ConstraintInterface
 ConstraintInterface ()
 Constructor.
virtual ~ConstraintInterface ()
 Destructor.
virtual void preProcessContinuationStep (LOCA::Abstract::Iterator::StepStatus)
 Perform any preprocessing before a continuation step starts.

Implementation of LOCA::MultiContinuation::ConstraintInterface

virtual methods

Teuchos::RCP< LOCA::GlobalData > globalData
 Pointer LOCA global data object.
Teuchos::RCP< LOCA::Parameter::SublistParser > parsedParams
 Parsed top-level parameters.
Teuchos::RCP< Teuchos::ParameterList > hopfParams
 Bifurcation parameter list.
Teuchos::RCP< LOCA::Hopf::MinimallyAugmented::AbstractGroup > grpPtr
 Pointer to base group that defines $ F$.
Teuchos::RCP< NOX::Abstract::MultiVector > a_vector
 Vector for $ a$.
Teuchos::RCP< NOX::Abstract::MultiVector > b_vector
 Vector for $ b$.
Teuchos::RCP< NOX::Abstract::MultiVector > w_vector
 Stores left null vector.
Teuchos::RCP< NOX::Abstract::MultiVector > v_vector
 Stores right null vector.
Teuchos::RCP< NOX::Abstract::MultiVector > Cv_vector
 Stores C*v.
Teuchos::RCP< NOX::Abstract::MultiVector > sigma_x
 Stores sigma_x.
NOX::Abstract::MultiVector::DenseMatrix constraints
 Constraint values.
Teuchos::RCP< LOCA::BorderedSolver::AbstractStrategy > borderedSolver
 Stores bordered solver strategy.
double dn
 Stores vector length as a double.
double sigma_scale
 Stores scale factor on sigma.
bool isSymmetric
 Flag indicating whether Jacobian is symmetric.
bool isValidConstraints
 Flag indicating whether constraints are valid.
bool isValidDX
 Flag indicating whether sigma_x is valid.
std::vector< int > bifParamID
 Stores the bifurcation parameter index.
double omega
 Stores Hopf frequency.
bool updateVectorsEveryContinuationStep
 Flag indicating whether to update $ a$ and $ b$ every continuation step.
bool updateVectorsEveryIteration
 Flag indicating whether to update $ a$ and $ b$ every nonlinear iteration.
virtual void copy (const LOCA::MultiContinuation::ConstraintInterface &source)
 Copy.
virtual Teuchos::RCP< LOCA::MultiContinuation::ConstraintInterface > clone (NOX::CopyType type=NOX::DeepCopy) const
 Cloning function.
virtual int numConstraints () const
 Return number of constraints.
virtual void setX (const NOX::Abstract::Vector &y)
 Set the solution vector to y.
virtual void setParam (int paramID, double val)
 Sets parameter indexed by paramID.
virtual void setParams (const std::vector< int > &paramIDs, const NOX::Abstract::MultiVector::DenseMatrix &vals)
 Sets parameters indexed by paramIDs.
virtual NOX::Abstract::Group::ReturnType computeConstraints ()
 Compute continuation constraint equations.
virtual NOX::Abstract::Group::ReturnType computeDX ()
 Compute derivative of constraints w.r.t. solution vector x.
virtual NOX::Abstract::Group::ReturnType computeDP (const std::vector< int > &paramIDs, NOX::Abstract::MultiVector::DenseMatrix &dgdp, bool isValidG)
 Compute derivative of constraints w.r.t. supplied parameters.
virtual bool isConstraints () const
 Return true if constraint residuals are valid.
virtual bool isDX () const
 Return true if derivatives of constraints w.r.t. x are valid.
virtual const NOX::Abstract::MultiVector::DenseMatrix & getConstraints () const
 Return constraint residuals.
virtual const NOX::Abstract::MultiVector * getDX () const
 Return solution component of constraint derivatives.
virtual bool isDXZero () const
 Return true if solution component of constraint derivatives is zero.
virtual void postProcessContinuationStep (LOCA::Abstract::Iterator::StepStatus stepStatus)
 Perform any postprocessing after a continuation step finishes.
virtual NOX::Abstract::Group::ReturnType computeDOmega (NOX::Abstract::MultiVector::DenseMatrix &domega)
 Compute derivative of sigma w.r.t. frequency omega.

Detailed Description

Implementation of LOCA::MultiContinuation::ConstraintInterfaceMVDX for computing Hopf bifurcations for the minimally augmented Hopf formulation.

This class implements the turning point constraint equation $\sigma(x,p,\omega) = 0$ for the minimally augmented Hopf formulation where $\sigma$ is defined via

\‍[ \begin{bmatrix} J+i\omega M & a \\ b^H & 0 \end{bmatrix} \begin{bmatrix} v \\ \sigma_1 \end{bmatrix} = \begin{bmatrix} 0 \\ n \end{bmatrix}, \‍]

\‍[ \begin{bmatrix} J^T-i\omega M^T & b \\ a^H & 0 \end{bmatrix} \begin{bmatrix} w \\ \sigma_2 \end{bmatrix} = \begin{bmatrix} 0 \\ n \end{bmatrix}, \‍]

\‍[ \sigma = w^H J+i\omega M v/n \‍]

for any vectors $ a$ and $ b$ in $\mathbb{C}^n$. Using these relationships, it is easy to show

\‍[ \begin{split} \sigma_x &= (w^H(J+i\omega M)v)_x/n = w^H(J+i\omega M)_x v/n \\ \sigma_p &= (w^H(J+i\omega M)v)_p/n = w^H(J+i\omega M)_p v/n \end{split} \‍]

The class is intialized via the hpfParams parameter list argument to the constructor. The parameters this class recognizes are:

  • "Update Null Vectors Every Continuation Step" – [bool] (default: true) - Flag indicating whether to update $ a$ and $ b$ vectors via $ a = w$ and $ b = v$ every continuation step
  • "Update Null Vectors Every Nonlinear Iteration" – [bool] (default: false) - Flag indicating whether to update $ a$ and $ b$ vectors via $ a = w$ and $ b = v$ every nonlinear iteration

Constructor & Destructor Documentation

◆ Constraint() [1/2]

LOCA::Hopf::MinimallyAugmented::Constraint::Constraint ( const Teuchos::RCP< LOCA::GlobalData > & global_data,
const Teuchos::RCP< LOCA::Parameter::SublistParser > & topParams,
const Teuchos::RCP< Teuchos::ParameterList > & hpfParams,
const Teuchos::RCP< LOCA::Hopf::MinimallyAugmented::AbstractGroup > & g,
bool is_symmetric,
const NOX::Abstract::Vector & a_real,
const NOX::Abstract::Vector & a_imag,
const NOX::Abstract::Vector * b_real,
const NOX::Abstract::Vector * b_imag,
int bif_param,
double freq )

Constructor.

References a_vector, b_vector, bifParamID, borderedSolver, constraints, NOX::Abstract::Vector::createMultiVector(), Cv_vector, NOX::DeepCopy, dn, globalData, grpPtr, hopfParams, isSymmetric, isValidConstraints, isValidDX, omega, parsedParams, NOX::ShapeCopy, sigma_scale, sigma_x, updateVectorsEveryContinuationStep, updateVectorsEveryIteration, v_vector, and w_vector.

Referenced by clone().

◆ Constraint() [2/2]

LOCA::Hopf::MinimallyAugmented::Constraint::Constraint ( const Constraint & source,
NOX::CopyType type = NOX::DeepCopy )

Copy constructor.

References a_vector, b_vector, bifParamID, borderedSolver, clone(), constraints, Cv_vector, NOX::DeepCopy, dn, globalData, grpPtr, hopfParams, isSymmetric, isValidConstraints, isValidDX, omega, parsedParams, sigma_scale, sigma_x, updateVectorsEveryContinuationStep, updateVectorsEveryIteration, v_vector, and w_vector.

Member Function Documentation

◆ clone()

Teuchos::RCP< LOCA::MultiContinuation::ConstraintInterface > LOCA::Hopf::MinimallyAugmented::Constraint::clone ( NOX::CopyType type = NOX::DeepCopy) const
virtual

Cloning function.

Implements LOCA::MultiContinuation::ConstraintInterface.

References Constraint().

Referenced by Constraint().

◆ computeConstraints()

NOX::Abstract::Group::ReturnType LOCA::Hopf::MinimallyAugmented::Constraint::computeConstraints ( )
virtual

Compute continuation constraint equations.

Implements LOCA::MultiContinuation::ConstraintInterface.

References a_vector, b_vector, borderedSolver, constraints, Cv_vector, dn, globalData, grpPtr, isSymmetric, isValidConstraints, NOX::Abstract::Group::Ok, omega, NOX::Utils::OuterIteration, parsedParams, sigma_scale, updateVectorsEveryIteration, v_vector, and w_vector.

Referenced by computeDOmega(), computeDP(), and computeDX().

◆ computeDOmega()

NOX::Abstract::Group::ReturnType LOCA::Hopf::MinimallyAugmented::Constraint::computeDOmega ( NOX::Abstract::MultiVector::DenseMatrix & domega)
virtual

Compute derivative of sigma w.r.t. frequency omega.

References computeConstraints(), globalData, grpPtr, isValidConstraints, NOX::Abstract::Group::Ok, NOX::ShapeCopy, sigma_scale, v_vector, and w_vector.

◆ computeDP()

NOX::Abstract::Group::ReturnType LOCA::Hopf::MinimallyAugmented::Constraint::computeDP ( const std::vector< int > & paramIDs,
NOX::Abstract::MultiVector::DenseMatrix & dgdp,
bool isValidG )
virtual

Compute derivative of constraints w.r.t. supplied parameters.

The first column of dgdp should be filled with the constraint residuals $ g$ if isValidG is false. If isValidG is true, then the dgdp contains $ g$ on input.

Implements LOCA::MultiContinuation::ConstraintInterface.

References computeConstraints(), constraints, globalData, grpPtr, isValidConstraints, NOX::Abstract::Group::Ok, omega, sigma_scale, v_vector, and w_vector.

◆ computeDX()

NOX::Abstract::Group::ReturnType LOCA::Hopf::MinimallyAugmented::Constraint::computeDX ( )
virtual

Compute derivative of constraints w.r.t. solution vector x.

Implements LOCA::MultiContinuation::ConstraintInterface.

References computeConstraints(), globalData, grpPtr, isValidConstraints, isValidDX, NOX::Abstract::Group::Ok, omega, sigma_scale, sigma_x, v_vector, and w_vector.

◆ copy()

void LOCA::Hopf::MinimallyAugmented::Constraint::copy ( const LOCA::MultiContinuation::ConstraintInterface & source)
virtual

Copy.

Implements LOCA::MultiContinuation::ConstraintInterface.

References a_vector, b_vector, bifParamID, borderedSolver, constraints, Cv_vector, dn, globalData, hopfParams, isSymmetric, isValidConstraints, isValidDX, omega, parsedParams, sigma_scale, sigma_x, updateVectorsEveryContinuationStep, updateVectorsEveryIteration, v_vector, and w_vector.

◆ getConstraints()

const NOX::Abstract::MultiVector::DenseMatrix & LOCA::Hopf::MinimallyAugmented::Constraint::getConstraints ( ) const
virtual

Return constraint residuals.

Implements LOCA::MultiContinuation::ConstraintInterface.

References constraints.

◆ getDX()

const NOX::Abstract::MultiVector * LOCA::Hopf::MinimallyAugmented::Constraint::getDX ( ) const
virtual

Return solution component of constraint derivatives.

Implements LOCA::MultiContinuation::ConstraintInterfaceMVDX.

References sigma_x.

◆ getLeftNullVecImag()

Teuchos::RCP< const NOX::Abstract::Vector > LOCA::Hopf::MinimallyAugmented::Constraint::getLeftNullVecImag ( ) const
virtual

Returns imaginary component of left null vector w.

References w_vector.

◆ getLeftNullVecReal()

Teuchos::RCP< const NOX::Abstract::Vector > LOCA::Hopf::MinimallyAugmented::Constraint::getLeftNullVecReal ( ) const
virtual

Returns real component of left null vector w.

References w_vector.

◆ getRightNullVecImag()

Teuchos::RCP< const NOX::Abstract::Vector > LOCA::Hopf::MinimallyAugmented::Constraint::getRightNullVecImag ( ) const
virtual

Returns imaginary component of right null vector v.

References v_vector.

◆ getRightNullVecReal()

Teuchos::RCP< const NOX::Abstract::Vector > LOCA::Hopf::MinimallyAugmented::Constraint::getRightNullVecReal ( ) const
virtual

Returns real component of right null vector v.

References v_vector.

◆ getSigmaImag()

double LOCA::Hopf::MinimallyAugmented::Constraint::getSigmaImag ( ) const
virtual

Returns imaginary component of sigma.

References constraints.

◆ getSigmaReal()

double LOCA::Hopf::MinimallyAugmented::Constraint::getSigmaReal ( ) const
virtual

Returns real component of sigma.

References constraints.

◆ isConstraints()

bool LOCA::Hopf::MinimallyAugmented::Constraint::isConstraints ( ) const
virtual

Return true if constraint residuals are valid.

Implements LOCA::MultiContinuation::ConstraintInterface.

References isValidConstraints.

◆ isDX()

bool LOCA::Hopf::MinimallyAugmented::Constraint::isDX ( ) const
virtual

Return true if derivatives of constraints w.r.t. x are valid.

Implements LOCA::MultiContinuation::ConstraintInterface.

References isValidDX.

◆ isDXZero()

bool LOCA::Hopf::MinimallyAugmented::Constraint::isDXZero ( ) const
virtual

Return true if solution component of constraint derivatives is zero.

Implements LOCA::MultiContinuation::ConstraintInterface.

◆ numConstraints()

int LOCA::Hopf::MinimallyAugmented::Constraint::numConstraints ( ) const
virtual

Return number of constraints.

Implements LOCA::MultiContinuation::ConstraintInterface.

◆ postProcessContinuationStep()

void LOCA::Hopf::MinimallyAugmented::Constraint::postProcessContinuationStep ( LOCA::Abstract::Iterator::StepStatus stepStatus)
virtual

Perform any postprocessing after a continuation step finishes.

The stepStatus argument indicates whether the step was successful. Here we update the $ a$ and $ b$ vectors to $ w$ and $ v$ respectively if requested.

Reimplemented from LOCA::MultiContinuation::ConstraintInterface.

References a_vector, b_vector, dn, globalData, NOX::Utils::StepperDetails, LOCA::Abstract::Iterator::Successful, updateVectorsEveryContinuationStep, v_vector, and w_vector.

◆ setFrequency()

void LOCA::Hopf::MinimallyAugmented::Constraint::setFrequency ( double freq)
virtual

Set Hopf frequency.

References isValidConstraints, isValidDX, and omega.

◆ setGroup()

void LOCA::Hopf::MinimallyAugmented::Constraint::setGroup ( const Teuchos::RCP< LOCA::Hopf::MinimallyAugmented::AbstractGroup > & g)
virtual

Set the group pointer.

This method should be called when ever the constrained group is copied, since we don't explicitly copy the underlying group here.

References grpPtr.

◆ setParam()

void LOCA::Hopf::MinimallyAugmented::Constraint::setParam ( int paramID,
double val )
virtual

Sets parameter indexed by paramID.

Implements LOCA::MultiContinuation::ConstraintInterface.

References grpPtr, isValidConstraints, and isValidDX.

◆ setParams()

void LOCA::Hopf::MinimallyAugmented::Constraint::setParams ( const std::vector< int > & paramIDs,
const NOX::Abstract::MultiVector::DenseMatrix & vals )
virtual

Sets parameters indexed by paramIDs.

Implements LOCA::MultiContinuation::ConstraintInterface.

References grpPtr, isValidConstraints, and isValidDX.

◆ setX()

void LOCA::Hopf::MinimallyAugmented::Constraint::setX ( const NOX::Abstract::Vector & y)
virtual

Set the solution vector to y.

Implements LOCA::MultiContinuation::ConstraintInterface.

References grpPtr, isValidConstraints, and isValidDX.

Member Data Documentation

◆ a_vector

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::a_vector
protected

Vector for $ a$.

Referenced by computeConstraints(), Constraint(), Constraint(), copy(), and postProcessContinuationStep().

◆ b_vector

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::b_vector
protected

Vector for $ b$.

Referenced by computeConstraints(), Constraint(), Constraint(), copy(), and postProcessContinuationStep().

◆ bifParamID

std::vector<int> LOCA::Hopf::MinimallyAugmented::Constraint::bifParamID
protected

Stores the bifurcation parameter index.

Referenced by Constraint(), Constraint(), and copy().

◆ borderedSolver

Teuchos::RCP<LOCA::BorderedSolver::AbstractStrategy> LOCA::Hopf::MinimallyAugmented::Constraint::borderedSolver
protected

Stores bordered solver strategy.

Referenced by computeConstraints(), Constraint(), Constraint(), and copy().

◆ constraints

NOX::Abstract::MultiVector::DenseMatrix LOCA::Hopf::MinimallyAugmented::Constraint::constraints
protected

Constraint values.

Referenced by computeConstraints(), computeDP(), Constraint(), Constraint(), copy(), getConstraints(), getSigmaImag(), and getSigmaReal().

◆ Cv_vector

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::Cv_vector
protected

Stores C*v.

Referenced by computeConstraints(), Constraint(), Constraint(), and copy().

◆ dn

double LOCA::Hopf::MinimallyAugmented::Constraint::dn
protected

Stores vector length as a double.

Referenced by computeConstraints(), Constraint(), Constraint(), copy(), and postProcessContinuationStep().

◆ globalData

Teuchos::RCP<LOCA::GlobalData> LOCA::Hopf::MinimallyAugmented::Constraint::globalData
protected

Pointer LOCA global data object.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), copy(), and postProcessContinuationStep().

◆ grpPtr

Teuchos::RCP<LOCA::Hopf::MinimallyAugmented::AbstractGroup> LOCA::Hopf::MinimallyAugmented::Constraint::grpPtr
protected

Pointer to base group that defines $ F$.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), setGroup(), setParam(), setParams(), and setX().

◆ hopfParams

Teuchos::RCP<Teuchos::ParameterList> LOCA::Hopf::MinimallyAugmented::Constraint::hopfParams
protected

Bifurcation parameter list.

Referenced by Constraint(), Constraint(), and copy().

◆ isSymmetric

bool LOCA::Hopf::MinimallyAugmented::Constraint::isSymmetric
protected

Flag indicating whether Jacobian is symmetric.

Referenced by computeConstraints(), Constraint(), Constraint(), and copy().

◆ isValidConstraints

bool LOCA::Hopf::MinimallyAugmented::Constraint::isValidConstraints
protected

Flag indicating whether constraints are valid.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), copy(), isConstraints(), setFrequency(), setParam(), setParams(), and setX().

◆ isValidDX

bool LOCA::Hopf::MinimallyAugmented::Constraint::isValidDX
protected

Flag indicating whether sigma_x is valid.

Referenced by computeDX(), Constraint(), Constraint(), copy(), isDX(), setFrequency(), setParam(), setParams(), and setX().

◆ omega

double LOCA::Hopf::MinimallyAugmented::Constraint::omega
protected

Stores Hopf frequency.

Referenced by computeConstraints(), computeDP(), computeDX(), Constraint(), Constraint(), copy(), and setFrequency().

◆ parsedParams

Teuchos::RCP<LOCA::Parameter::SublistParser> LOCA::Hopf::MinimallyAugmented::Constraint::parsedParams
protected

Parsed top-level parameters.

Referenced by computeConstraints(), Constraint(), Constraint(), and copy().

◆ sigma_scale

double LOCA::Hopf::MinimallyAugmented::Constraint::sigma_scale
protected

Stores scale factor on sigma.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), and copy().

◆ sigma_x

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::sigma_x
protected

Stores sigma_x.

Referenced by computeDX(), Constraint(), Constraint(), copy(), and getDX().

◆ updateVectorsEveryContinuationStep

bool LOCA::Hopf::MinimallyAugmented::Constraint::updateVectorsEveryContinuationStep
protected

Flag indicating whether to update $ a$ and $ b$ every continuation step.

Referenced by Constraint(), Constraint(), copy(), and postProcessContinuationStep().

◆ updateVectorsEveryIteration

bool LOCA::Hopf::MinimallyAugmented::Constraint::updateVectorsEveryIteration
protected

Flag indicating whether to update $ a$ and $ b$ every nonlinear iteration.

Referenced by computeConstraints(), Constraint(), Constraint(), and copy().

◆ v_vector

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::v_vector
protected

Stores right null vector.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), copy(), getRightNullVecImag(), getRightNullVecReal(), and postProcessContinuationStep().

◆ w_vector

Teuchos::RCP<NOX::Abstract::MultiVector> LOCA::Hopf::MinimallyAugmented::Constraint::w_vector
protected

Stores left null vector.

Referenced by computeConstraints(), computeDOmega(), computeDP(), computeDX(), Constraint(), Constraint(), copy(), getLeftNullVecImag(), getLeftNullVecReal(), and postProcessContinuationStep().

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