naginterfaces.library.linsys.real_​gen_​lsqsol

naginterfaces.library.linsys.real_gen_lsqsol(a, b, eps)

real_gen_lsqsol calculates the accurate least squares solution of a set of $m$ linear equations in $n$ unknowns, $m\ge n$ and rank $=n$, with multiple right-hand sides, $AX=B$, using a $QR$ factorization and iterative refinement.

For full information please refer to the NAG Library document for f04am

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/f04/f04amf.html

Parameters

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

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

bfloat, array-like, shape $(m,\text{ir})$

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

epsfloat

Must be set to the value of the machine precision.

Returns

xfloat, ndarray, shape $(n,\text{ir})$

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

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

Details of the $QR$ factorization.

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

The diagonal elements of the upper triangular matrix $R$.

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

Details of the column interchanges.

Warns

NagAlgorithmicWarning

(errno $1$)

The rank of $a$ is less than $\text{n}$. The problem does not have a unique solution.

(errno $2$)

The iterative refinement fails to converge. The matrix $A$ is too ill-conditioned.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

To compute the least squares solution to a set of $m$ linear equations in $n$ unknowns $(m\ge n)$ $AX=B$, real_gen_lsqsol first computes a $QR$ factorization of $A$ with column pivoting, $AP=QR$, where $R$ is upper triangular, $Q$ is an $m\times m$ orthogonal matrix, and $P$ is a permutation matrix. $Q^T$ is applied to the $m\times r$ right-hand side matrix $B$ to give $C=Q^{T}B$, and the $n\times r$ solution matrix $X$ is calculated, to a first approximation, by back-substitution in $RX=C$. The residual matrix $S=B-AX$ is calculated using additional precision, and a correction $D$ to $X$ is computed as the least squares solution to $AD=S$. $X$ is replaced by $X+D$ and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.

References

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag