Eigen
3.2.2

Householder rankrevealing QR decomposition of a matrix with full pivoting.
MatrixType  the type of the matrix of which we are computing the QR decomposition 
This class performs a rankrevealing QR decomposition of a matrix A into matrices P, Q and R such that
by using Householder transformations. Here, P is a permutation matrix, Q a unitary matrix and R an upper triangular matrix.
This decomposition performs a very prudent full pivoting in order to be rankrevealing and achieve optimal numerical stability. The tradeoff is that it is slower than HouseholderQR and ColPivHouseholderQR.
Public Member Functions  
MatrixType::RealScalar  absDeterminant () const 
const PermutationType &  colsPermutation () const 
FullPivHouseholderQR &  compute (const MatrixType &matrix) 
Index  dimensionOfKernel () const 
FullPivHouseholderQR ()  
Default Constructor. More...  
FullPivHouseholderQR (Index rows, Index cols)  
Default Constructor with memory preallocation. More...  
FullPivHouseholderQR (const MatrixType &matrix)  
Constructs a QR factorization from a given matrix. More...  
const HCoeffsType &  hCoeffs () const 
const internal::solve_retval < FullPivHouseholderQR, typename MatrixType::IdentityReturnType >  inverse () const 
bool  isInjective () const 
bool  isInvertible () const 
bool  isSurjective () const 
MatrixType::RealScalar  logAbsDeterminant () const 
MatrixQReturnType  matrixQ (void) const 
const MatrixType &  matrixQR () const 
RealScalar  maxPivot () const 
Index  nonzeroPivots () const 
Index  rank () const 
const IntDiagSizeVectorType &  rowsTranspositions () const 
FullPivHouseholderQR &  setThreshold (const RealScalar &threshold) 
FullPivHouseholderQR &  setThreshold (Default_t) 
template<typename Rhs >  
const internal::solve_retval < FullPivHouseholderQR, Rhs >  solve (const MatrixBase< Rhs > &b) const 
RealScalar  threshold () const 

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Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).

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Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.

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Constructs a QR factorization from a given matrix.
This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:
References FullPivHouseholderQR< MatrixType >::compute().
MatrixType::RealScalar absDeterminant  (  )  const 

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FullPivHouseholderQR< MatrixType > & compute  (  const MatrixType &  matrix  ) 
Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this
, and a reference to *this
is returned.
Referenced by FullPivHouseholderQR< MatrixType >::FullPivHouseholderQR().

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References FullPivHouseholderQR< MatrixType >::rank().

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Q
.For advanced uses only.

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References FullPivHouseholderQR< MatrixType >::rank().
Referenced by FullPivHouseholderQR< MatrixType >::isInvertible().

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References FullPivHouseholderQR< MatrixType >::isInjective(), and FullPivHouseholderQR< MatrixType >::isSurjective().

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References FullPivHouseholderQR< MatrixType >::rank().
Referenced by FullPivHouseholderQR< MatrixType >::isInvertible().
MatrixType::RealScalar logAbsDeterminant  (  )  const 

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References FullPivHouseholderQR< MatrixType >::threshold().
Referenced by FullPivHouseholderQR< MatrixType >::dimensionOfKernel(), FullPivHouseholderQR< MatrixType >::isInjective(), and FullPivHouseholderQR< MatrixType >::isSurjective().

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Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the QR decomposition itself.
When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.
threshold  The new value to use as the threshold. 
A pivot will be considered nonzero if its absolute value is strictly greater than where maxpivot is the biggest pivot.
If you want to come back to the default behavior, call setThreshold(Default_t)
References FullPivHouseholderQR< MatrixType >::threshold().

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Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.
You should pass the special object Eigen::Default as parameter here.
See the documentation of setThreshold(const RealScalar&).

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This method finds a solution x to the equation Ax=b, where A is the matrix of which *this
is the QR decomposition.
b  the righthandside of the equation to solve. 
This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:
This method avoids dividing by zero, so that the nonexistence of a solution doesn't by itself mean that you'll get inf
or nan
values.
If there exists more than one solution, this method will arbitrarily choose one.
Example:
Output:
Here is the matrix m: 0.68 0.597 0.33 0.211 0.823 0.536 0.566 0.605 0.444 Here is the matrix y: 0.108 0.27 0.832 0.0452 0.0268 0.271 0.258 0.904 0.435 Here is a solution x to the equation mx=y: 0.609 2.68 1.67 0.231 1.57 0.0713 0.51 3.51 1.05

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Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).
Referenced by FullPivHouseholderQR< MatrixType >::rank(), and FullPivHouseholderQR< MatrixType >::setThreshold().