Class xlifepp::SVQBOrthoManager

template<class ScalarType, class MV, class OP>
class SVQBOrthoManager : public xlifepp::MatOrthoManager<ScalarType, MV, OP>

Inheritance diagram for xlifepp::SVQBOrthoManager:

Collaboration diagram for xlifepp::SVQBOrthoManager:

An implementation of the xlifepp::MatOrthoManager that performs orthogonalization using the SVQB iterative orthogonalization technique described by Stathapoulos and Wu.

This orthogonalization routine, while not returning the upper triangular factors of the popular Gram-Schmidt method, has a communication cost (measured in number of communication calls) that is independent of the number of columns in the basis.

Error methods

virtual NumTraits<ScalarType>::RealScalar orthonormErrorMat(const MV &X, SmartPtr<const MV> MX = _smPtrNull) const

This method computes the error in orthonormality of a multivector, measured as the Frobenius norm of the difference innerProd(X,Y) - I.

The method has the option of exploiting a caller-provided MX.

virtual NumTraits<ScalarType>::RealScalar orthogErrorMat(const MV &X, const MV &Y, SmartPtr<const MV> MX = _smPtrNull, SmartPtr<const MV> MY = _smPtrNull) const

This method computes the error in orthogonality of two multivectors, measured as the Frobenius norm of innerProd(X,Y).

The method has the option of exploiting a caller-provided MX.

Constructor/Destructor

SVQBOrthoManager(SmartPtr<const OP> Op = _smPtrNull, bool debug = false)

Constructor specifying re-orthogonalization tolerance.

inline ~SVQBOrthoManager()

Destructor.

Methods implementing xlifepp::MatOrthoManager

virtual void projectMat(MV &X, std::vector<SmartPtr<const MV>> Q, std::vector<SmartPtr<MatrixEigenDense<ScalarType>>> C, SmartPtr<MV> MX, std::vector<SmartPtr<const MV>> MQ) const

Given a list of mutually orthogonal and internally orthonormal bases Q, this method projects a multivector X onto the space orthogonal to the individual Q[i], optionally returning the coefficients of X for the individual Q[i].

All of this is done with respect to the inner product innerProd().

After calling this routine, X will be orthogonal to each of the Q[i].

Parameters:

• X – [inout]

  The multivector to be modified.

  On output, the columns of

  X will be orthogonal to each Q[i], satisfying
  \[ X_{out} = X_{in} - \sum_i Q[i] \langle Q[i], X_{in} \rangle \]

• MX – [inout] The image of X under the inner product operator Op. If \( MX != 0\): On input, this is expected to be consistent with Op . X. On output, this is updated consistent with updates to X. If \( MX == 0\) or \( Op == 0\): MX is not referenced.

• C – [out] The coefficients of X in the bases Q[i]. If C[i] is a non-null pointer and C[i] matches the dimensions of X and Q[i], then the coefficients computed during the orthogonalization routine will be stored in the matrix C[i], similar to calling

  innerProd(Q[i], X, C[i]);
  

  If C[i] points to a MatrixEigenDense with size inconsistent with X and [i], then a std::invalid_argument exception will be thrown. Otherwise, if C.size() < i or C[i] is a null pointer, the caller will not have access to the computed coefficients.

• Q – [in] A list of multivector bases specifying the subspaces to be orthogonalized against, satisfying

  \[ \langle Q[i], Q[j] \rangle = I \quad\textrm{if}\quad i=j \]
  and
  \[ \langle Q[i], Q[j] \rangle = 0 \quad\textrm{if}\quad i \neq j\ . \]

• MQ –

virtual int normalizeMat(MV &X, SmartPtr<MatrixEigenDense<ScalarType>> B = _smPtrNull, SmartPtr<MV> MX = _smPtrNull) const

This method takes a multivector X and attempts to compute an orthonormal basis for \(colspan(X)\), with respect to innerProd().

This method does not compute an upper triangular coefficient matrix B.

This routine returns an integer rank stating the rank of the computed basis. If X does not have full rank and the normalize() routine does not attempt to augment the subspace, then rank may be smaller than the number of columns in X. In this case, only the first rank columns of output X and first rank rows of B will be valid.

The method attempts to find a basis with dimension equal to the number of columns in X. It does this by augmenting linearly dependent vectors in X with random directions. A finite number of these attempts will be made; therefore, it is possible that the dimension of the computed basis is less than the number of vectors in X.

Parameters:

• X – [inout]

  The multivector to be modified.

  On output, the first

  rank columns of X satisfy
  \[ \langle X[i], X[j] \rangle = \delta_{ij}\ . \]
  Also,
  \[ X_{in}(1:m,1:n) = X_{out}(1:m,1:rank) B(1:rank,1:n) \]
  where m is the number of rows in X and n is the number of columns in X.

• MX – [inout] The image of X under the inner product operator Op. If \( MX != 0\): On input, this is expected to be consistent with Op . X. On output, this is updated consistent with updates to X. If \( MX == 0\) or \( Op == 0\): MX is not referenced.

• B – [out] The coefficients of the original X with respect to the computed basis. If B is a non-null pointer and B matches the dimensions of B, then the coefficients computed during the orthogonalization routine will be stored in B, similar to calling

  innerProd(Xout, Xin, B);
  

  If B points to a MatrixEigenDense with size inconsistent with X, then a std::invalid_argument exception will be thrown. Otherwise, if B

  is null, the caller will not have access to the computed coefficients. This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.

  In general,

  B has no non-zero structure.

Returns:

Rank of the basis computed by this method, less than or equal to the number of columns in X. This specifies how many columns in the returned X and rows in the returned B are valid.

virtual int projectAndNormalizeMat(MV &X, std::vector<SmartPtr<const MV>> Q, std::vector<SmartPtr<MatrixEigenDense<ScalarType>>> C, SmartPtr<MatrixEigenDense<ScalarType>> B, SmartPtr<MV> MX = _smPtrNull, std::vector<SmartPtr<const MV>> MQ = std::vector<SmartPtr<const MV>>(1, _smPtrNull)) const

Given a set of bases Q[i] and a multivector X, this method computes an orthonormal basis for \(colspan(X) - \sum_i colspan(Q[i])\).

This routine returns an integer rank stating the rank of the computed basis. If the subspace \(colspan(X) - \sum_i colspan(Q[i])\) does not have dimension as large as the number of columns of X and the orthogonalization manager doe not attempt to augment the subspace, then rank may be smaller than the number of columns of X. In this case, only the first rank columns of output X and first rank rows of B will be valid.

The method attempts to find a basis with dimension the same as the number of columns in X. It does this by augmenting linearly dependent vectors with random directions. A finite number of these attempts will be made; therefore, it is possible that the dimension of the computed basis is less than the number of vectors in X.

Parameters:

• X – [inout]

  The multivector to be modified.

  On output, the first

  rank columns of X satisfy
  \[ \langle X[i], X[j] \rangle = \delta_{ij} \quad \textrm{and} \quad \langle X, Q[i] \rangle = 0\ . \]
  Also,
  \[ X_{in}(1:m,1:n) = X_{out}(1:m,1:rank) B(1:rank,1:n) + \sum_i Q[i] C[i] \]
  where m is the number of rows in X and n is the number of columns in X.

• MX – [inout] The image of X under the inner product operator Op. If \( MX != 0\): On input, this is expected to be consistent with Op . X. On output, this is updated consistent with updates to X. If \( MX == 0\) or \( Op == 0\): MX is not referenced.

• C – [out] The coefficients of X in the Q[i]. If C[i] is a non-null pointer and C[i] matches the dimensions of X and Q[i], then the coefficients computed during the orthogonalization routine will be stored in the matrix C[i], similar to calling

  innerProd(Q[i], X, C[i]);
  

  If C[i] points to a MatrixEigenDense with size inconsistent with X and [i], then a std::invalid_argument exception will be thrown. Otherwise, if C.size() < i or C[i] is a null pointer, the caller will not have access to the computed coefficients.

• B – [out] The coefficients of the original X with respect to the computed basis. If B is a non-null pointer and B matches the dimensions of B, then the coefficients computed during the orthogonalization routine will be stored in B, similar to calling

  innerProd(Xout, Xin, B);
  

  If B points to a MatrixEigenDense with size inconsistent with X, then a std::invalid_argument exception will be thrown. Otherwise, if B

  is null, the caller will not have access to the computed coefficients. This matrix is not necessarily triangular (as in a QR factorization); see the documentation of specific orthogonalization managers.

  In general,

  B has no non-zero structure.

• Q – [in] A list of multivector bases specifying the subspaces to be orthogonalized against, satisfying

  \[ \langle Q[i], Q[j] \rangle = I \quad\textrm{if}\quad i=j \]
  and
  \[ \langle Q[i], Q[j] \rangle = 0 \quad\textrm{if}\quad i \neq j\ . \]

• MQ –

Returns:

Rank of the basis computed by this method, less than or equal to the number of columns in X. This specifies how many columns in the returned X and rows in the returned B are valid.