naginterfaces.library.lapackeig.dgelsy

naginterfaces.library.lapackeig.dgelsy(a, b, jpvt, rcond)

dgelsy computes the minimum norm solution to a real linear least squares problem

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

using a complete orthogonal factorization of $A$. $A$ is an $m\times n$ matrix which may be rank-deficient. Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call.

For full information please refer to the NAG Library document for f08ba

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08baf.html

Parameters

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

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

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

The $m\times r$ right-hand side matrix $B$.

jpvtint, array-like, shape $\left(n\right)$

If $jpvt[i-1]\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.

rcondfloat

Used to determine the effective rank of $A$, which is defined as the order of the largest leading triangular sub-matrix $R_{11}$ in the $QR$ factorization of $A$, whose estimated condition number is $<1/rcond$.

Suggested value: if the condition number of $a$ is not known then $rcond=\sqrt{\left(ϵ\right)/2}$ (where $ϵ$ is machine precision, see machine.precision) is a good choice. Negative values or values less than machine precision should be avoided since this will cause $a$ to have an effective $\text{rank}=min(m,n)$ that could be larger than its actual rank, leading to meaningless results.

Returns

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

$a$ has been overwritten by details of its complete orthogonal factorization.

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

The $n\times r$ solution matrix $X$.

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

If $jpvt[i-1]=k$, the $i$th column of $AP$ was the $k$th column of $A$.

rankint

The effective rank of $A$, i.e., the order of the sub-matrix $R_{11}$. This is the same as the order of the sub-matrix $T_{11}$ in the complete orthogonal factorization of $A$.

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: $n\ge 0$.

(errno $-3$)

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

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

Notes

The right-hand side vectors are stored as the columns of the $m\times r$ matrix $B$ and the solution vectors in the $n\times r$ matrix $X$.

dgelsy first computes a $QR$ factorization with column pivoting

$$\begin{array}{c}AP=Q\left(\begin{array}{cc}{R}_{11}& R_{12}\\ 0& R_{22}\end{array}\right)\text{,}\end{array}$$

with $R_{11}$ defined as the largest leading sub-matrix whose estimated condition number is less than $1/rcond$. The order of $R_{11}$, $rank$, is the effective rank of $A$.

Then, $R_{22}$ is considered to be negligible, and $R_{12}$ is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization

$$\begin{array}{c}AP=Q\left(\begin{array}{cc}{T}_{11}& 0\\ 0& 0\end{array}\right)Z\text{.}\end{array}$$

The minimum norm solution is then

$$\begin{array}{c}X=P{Z}^T\left(\begin{array}{c}{T}_{11}^{-1}{Q}_1^{T}b\\ 0\end{array}\right)\end{array}$$

where $Q_1$ consists of the first $rank$ columns of $Q$.

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