naginterfaces.library.lapackeig.dgglse

naginterfaces.library.lapackeig.dgglse(n, a, b, c, d)

dgglse solves a real linear equality-constrained least squares problem.

For full information please refer to the NAG Library document for f08za

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08zaf.html

Parameters

nint

$n$, the number of columns of the matrices $A$ and $B$.

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

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

bfloat, array-like, shape $(p,n)$

The $p\times n$ matrix $B$.

cfloat, array-like, shape $\left(m\right)$

The right-hand side vector $c$ for the least squares part of the LSE problem.

dfloat, array-like, shape $\left(p\right)$

The right-hand side vector $d$ for the equality constraints.

Returns

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

$a$ is overwritten.

bfloat, ndarray, shape $(p,n)$

$b$ is overwritten.

cfloat, ndarray, shape $\left(m\right)$

The residual sum of squares for the solution vector $x$ is given by the sum of squares of elements $c[n-p],c[n-p+1],\dots ,c[m-1]$; the remaining elements are overwritten.

dfloat, ndarray, shape $\left(p\right)$

$d$ is overwritten.

xfloat, ndarray, shape $\left(n\right)$

The solution vector $x$ of the LSE problem.

Raises

NagValueError

(errno $-1$)

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

Constraint: $m\ge 0$.

(errno $-2$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-3$)

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

Constraint: $0\le p\le n\le m+p$.

(errno $1$)

The upper triangular factor $R$ associated with $B$ in the generalized $RQ$ factorization of the pair $(B,A)$ is singular, so that $rank\left(B\right)<p$; the least squares solution could not be computed.

(errno $2$)

The $(N-P)\times (N-P)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $RQ$ factorization of the pair $(B,A)$ is singular, so that the rank of the matrix ($E$) comprising the rows of $A$ and $B$ is less than $n$; the least squares solutions could not be computed.

Notes

dgglse solves the real linear equality-constrained least squares (LSE) problem

$${minimize}_x{\parallel c-Ax\parallel }_2\text{subject to}Bx=d$$

where $A$ is an $m\times n$ matrix, $B$ is a $p\times n$ matrix, $c$ is an $m$ element vector and $d$ is a $p$ element vector. It is assumed that $p\le n\le m+p$, $rank\left(B\right)=p$ and $rank\left(E\right)=n$, where $E=\left(\begin{array}{c}A\\ B\end{array}\right)$. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized $RQ$ factorization of the matrices $B$ and $A$.

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

Anderson, E, Bai, Z and Dongarra, J, 1992, Generalized $QR$ factorization and its applications, Linear Algebra Appl. (Volume 162–164), 243–271

Eldèn, L, 1980, Perturbation theory for the least squares problem with linear equality constraints, SIAM J. Numer. Anal. (17), 338–350