naginterfaces.library.lapackeig.zgelsd

naginterfaces.library.lapackeig.zgelsd(a, b, rcond)

zgelsd computes the minimum norm solution to a complex linear least squares problem

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

For full information please refer to the NAG Library document for f08kq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08kqf.html

Parameters

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

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

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

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

rcondfloat

Used to determine the effective rank of $A$. Singular values $s[i-1]\le rcond\times s\left[0\right]$ are treated as zero. If $rcond<0$, machine precision is used instead.

Returns

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

The contents of $a$ are destroyed.

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

$b$ is overwritten by the $n\times r$ solution matrix $X$. If $m\ge n$ and $rank=n$, the residual sum of squares for the solution in the $i$th column is given by the sum of squares of the modulus of elements $n+1,\dots ,m$ in that column.

sfloat, ndarray, shape $(min(m,n))$

The singular values of $A$ in decreasing order.

rankint

The effective rank of $A$, i.e., the number of singular values which are greater than $rcond\times s\left[0\right]$.

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

(errno $i>0$)

The algorithm for computing the SVD failed to converge; $⟨value⟩$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Notes

zgelsd uses the singular value decomposition (SVD) of $A$, where $A$ is a complex $m\times n$ matrix which may be rank-deficient.

Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call; they are stored as the columns of the $m\times r$ right-hand side matrix $B$ and the $n\times r$ solution matrix $X$.

The problem is solved in three steps:

1. reduce the coefficient matrix $A$ to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS);

2. solve the BLS using a divide-and-conquer approach;

3. apply back all the Householder transformations to solve the original least squares problem.

The effective rank of $A$ is determined by treating as zero those singular values which are less than $rcond$ times the largest singular value.

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