naginterfaces.library.matop.real_​gen_​pseudinv

naginterfaces.library.matop.real_gen_pseudinv(t, a)

real_gen_pseudinv calculates the rank and pseudo-inverse of an $m\times n$ real matrix, $m\ge n$, using a $QR$ factorization with column interchanges.

For full information please refer to the NAG Library document for f01bl

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f01/f01blf.html

Parameters

tfloat

The tolerance used to decide when elements can be regarded as zero (see Further Comments).

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

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

Returns

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

The transpose of the pseudo-inverse of $A$.

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

$aijmax[i-1]$ contains the element of largest modulus in the reduced matrix at the $i$th stage. If $r<n$, then only the first $r+1$ elements of $aijmax$ have values assigned to them; the remaining elements are unused. The ratio $aijmax\left[0\right]/aijmax[r-1]$ usually gives an indication of the condition number of the original matrix (see Further Comments).

irankint

$r$, the rank of $A$ as determined using the tolerance $t$.

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

The record of the column interchanges in the Householder factorization.

Raises

NagValueError

(errno $1$)

Inverse not found. Incorrect $irank$.

(errno $2$)

Invalid tolerance, $t$ too large: $t=⟨value⟩$.

(errno $2$)

Invalid tolerance, $t<0.0$: $t=⟨value⟩$.

(errno $3$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m\ge n$.

Notes

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

Householder’s factorization with column interchanges is used in the decomposition $F=QU$, where $F$ is $A$ with its columns permuted, $Q$ is the first $r$ columns of an $m\times m$ orthogonal matrix and $U$ is an $r\times n$ upper-trapezoidal matrix of rank $r$. The pseudo-inverse of $F$ is given by $X$ where

$$X=U^T{\left(U{U}^T\right)}^{-1}{Q}^T\text{.}$$

If the matrix is found to be of maximum rank, $r=n$, $U$ is a nonsingular $n\times n$ upper-triangular matrix and the pseudo-inverse of $F$ simplifies to $X=U^{-1}{Q}^T$. The transpose of the pseudo-inverse of $A$ is overwritten on $A$.

References

Peters, G and Wilkinson, J H, 1970, The least squares problem and pseudo-inverses, Comput. J. (13), 309–316

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