naginterfaces.library.lapackeig.dggglm

naginterfaces.library.lapackeig.dggglm(p, a, b, d)

dggglm solves a real general Gauss–Markov linear (least squares) model problem.

For full information please refer to the NAG Library document for f08zb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08zbf.html

Parameters

pint

$p$, the number of columns of the matrix $B$.

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

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

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

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

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

The left-hand side vector $d$ of the GLM equation.

Returns

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

$a$ is overwritten.

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

$b$ is overwritten.

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

$d$ is overwritten.

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

The solution vector $x$ of the GLM problem.

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

The solution vector $y$ of the GLM problem.

Raises

NagValueError

(errno $-1$)

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

Constraint: $m\ge 0$.

(errno $-2$)

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

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

(errno $-3$)

On entry, error in parameter $p$.

Constraint: $p\ge m-n$.

(errno $1$)

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

(errno $2$)

The bottom $(m-n)\times (m-n)$ part of the upper trapezoidal factor $T$ associated with $B$ in the generalized $QR$ factorization of the pair $(A,B)$ is singular, so that $rank\left(\begin{array}{cc}A& B\end{array}\right)<n$; the least squares solutions could not be computed.

Notes

dggglm solves the real general Gauss–Markov linear model (GLM) problem

$${minimize}_x{\parallel y\parallel }_2\text{subject to}d=Ax+By$$

where $A$ is an $m\times n$ matrix, $B$ is an $m\times p$ matrix and $d$ is an $m$ element vector. It is assumed that $n\le m\le n+p$, $rank\left(A\right)=n$ and $rank\left(E\right)=m$, where $E=\left(\begin{array}{cc}A& B\end{array}\right)$. Under these assumptions, the problem has a unique solution $x$ and a minimal $2$-norm solution $y$, which is obtained using a generalized $QR$ factorization of the matrices $A$ and $B$.

In particular, if the matrix $B$ is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem

$${minimize}_x{\Vert B^{-1}(d-Ax)\Vert }_2\text{.}$$

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