naginterfaces.library.lapackeig.dgels

naginterfaces.library.lapackeig.dgels(trans, a, b)

dgels solves linear least squares problems of the form

$${min}_x{\parallel b-Ax\parallel }_2\text{or}{min}_x{\Vert b-A^{T}x\Vert }_2\text{,}$$

where $A$ is an $m\times n$ real matrix of full rank, using a $QR$ or $LQ$ factorization of $A$.

For full information please refer to the NAG Library document for f08aa

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08aaf.html

Parameters

transstr, length 1

If $trans=\text{'N'}$, the linear system involves $A$.

If $trans=\text{'T'}$, the linear system involves $A^T$.

afloat, array-like, shape $(m,n)$

The $m\times n$ matrix $A$.

bfloat, array-like, shape $(max(m,n),\text{nrhs})$

The matrix $B$ of right-hand side vectors, stored in columns; $b$ is $m\times r$ if $trans=\text{'N'}$, or $n\times r$ if $trans=\text{'T'}$.

Returns

afloat, ndarray, shape $(m,n)$

If $m\ge n$, $a$ is overwritten by details of its $QR$ factorization, as returned by dgeqrf().

If $m<n$, $a$ is overwritten by details of its $LQ$ factorization, as returned by dgelqf().

bfloat, ndarray, shape $(max(m,n),\text{nrhs})$

$b$ is overwritten by the solution vectors, $x$, stored in columns:

if $trans=\text{'N'}$ and $m\ge n$, or $trans=\text{'T'}$ and $m<n$, elements $1$ to $min(m,n)$ in each column of $b$ contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $(min(m,n)+1)$ to $max(m,n)$ in that column;

otherwise, elements $1$ to $max(m,n)$ in each column of $b$ contain the minimum norm solution vectors.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $trans$.

Constraint: $trans=\text{'N'}$ or $\text{'T'}$.

(errno $-2$)

On entry, error in parameter $\text{m}$.

Constraint: $m\ge 0$.

(errno $-3$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-4$)

On entry, error in parameter $\text{nrhs}$.

Constraint: $\text{nrhs}\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Diagonal element $⟨value⟩$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

Notes

The following options are provided:

1. If $trans=\text{'N'}$ and $m\ge n$: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

   $${min}_x{\parallel b-Ax\parallel }_2\text{.}$$

2. If $trans=\text{'N'}$ and $m<n$: find the minimum norm solution of an underdetermined system $Ax=b$.

3. If $trans=\text{'T'}$ and $m\ge n$: find the minimum norm solution of an undetermined system $A^{T}x=b$.

4. If $trans=\text{'T'}$ and $m<n$: find the least squares solution of an overdetermined system, i.e., solve the least squares problem

   $${min}_x{\Vert b-A^{T}x\Vert }_2\text{.}$$

Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call; they are stored as the columns of the $m\times r$ right-hand side matrix $B$ and the $n\times r$ solution matrix $X$.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore