Least Squares

Implementation

Given \(A \in \mathbb{F}^{m \times n}\) and a right-hand side \(b \in \mathbb{F}^m\), a least-squares method solves \(A x \approx b\) differently depending upon whether \(m \ge n\).

When \(m \ge n\), there are at least as many constraints as degrees of freedom, and so a solution is sought for

\[\min_x \| A x - b \|_2\]

This problem is solved through the use of QR().

When \(m < n\), the problem is under-constrained and a solution is sought for the problem

\[\min_x \| x \|_2 \;\;\; \text{such that } A x = b.\]

This problem is solved through the use of LQ().

The above optimization problems can be readily generalized to multiple right-hand sides by switching to Frobenius norms.

Note

If orientation is set to NORMAL, then solve \(AX=B\), otherwise orientation must be equal to ADJOINT and \(A^H X=B\) will be solved. Upon completion, \(A\) is overwritten with its QR or LQ factorization, and each column of \(X\) is overwritten with a solution vector.

C++ API

void LeastSquares(Orientation orientation, Matrix<F> &A, const Matrix<F> &B, Matrix<F> &X)

void LeastSquares(Orientation orientation, AbstractDistMatrix<F> &A, const AbstractDistMatrix<F> &B, AbstractDistMatrix<F> &X)

C API

ElError ElLeastSquares_s(ElOrientation orientation, ElMatrix_s A, ElConstMatrix_s B, ElMatrix_s X)

ElError ElLeastSquares_d(ElOrientation orientation, ElMatrix_d A, ElConstMatrix_d B, ElMatrix_d X)

ElError ElLeastSquares_c(ElOrientation orientation, ElMatrix_c A, ElConstMatrix_c B, ElMatrix_c X)

ElError ElLeastSquares_z(ElOrientation orientation, ElMatrix_z A, ElConstMatrix_z B, ElMatrix_z X)

ElError ElLeastSquaresDist_s(ElOrientation orientation, ElDistMatrix_s A, ElConstDistMatrix_s B, ElDistMatrix_s X)

ElError ElLeastSquaresDist_d(ElOrientation orientation, ElDistMatrix_d A, ElConstDistMatrix_d B, ElDistMatrix_d X)

ElError ElLeastSquaresDist_c(ElOrientation orientation, ElDistMatrix_c A, ElConstDistMatrix_c B, ElDistMatrix_c X)

ElError ElLeastSquaresDist_z(ElOrientation orientation, ElDistMatrix_z A, ElConstDistMatrix_z B, ElDistMatrix_z X)