# NAG FL Interfacef08bnf (zgelsy)

## 1Purpose

f08bnf computes the minimum norm solution to a complex linear least squares problem
 $minx b-Ax2$
using a complete orthogonal factorization of $A$. $A$ is an $m$ by $n$ matrix which may be rank-deficient. Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call.

## 2Specification

Fortran Interface
 Subroutine f08bnf ( m, n, nrhs, a, lda, b, ldb, jpvt, rank, work, info)
 Integer, Intent (In) :: m, n, nrhs, lda, ldb, lwork Integer, Intent (Inout) :: jpvt(*) Integer, Intent (Out) :: rank, info Real (Kind=nag_wp), Intent (In) :: rcond Real (Kind=nag_wp), Intent (Inout) :: rwork(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nag.h>
 void f08bnf_ (const Integer *m, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Integer jpvt[], const double *rcond, Integer *rank, Complex work[], const Integer *lwork, double rwork[], Integer *info)
The routine may be called by the names f08bnf, nagf_lapackeig_zgelsy or its LAPACK name zgelsy.

## 3Description

The right-hand side vectors are stored as the columns of the $m$ by $r$ matrix $B$ and the solution vectors in the $n$ by $r$ matrix $X$.
f08bnf first computes a $QR$ factorization with column pivoting
 $AP= Q R11 R12 0 R22 ,$
with ${R}_{11}$ defined as the largest leading sub-matrix whose estimated condition number is less than $1/{\mathbf{rcond}}$. 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
 $AP= Q T11 0 0 0 Z .$
The minimum norm solution is then
 $X = PZH T11-1 Q1H b 0$
where ${Q}_{1}$ consists of the first rank columns of $Q$.

## 4References

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

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: a has been overwritten by details of its complete orthogonal factorization.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08bnf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $m$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08bnf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
8: $\mathbf{jpvt}\left(*\right)$Integer array Input/Output
Note: the dimension of the array jpvt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{jpvt}}\left(i\right)\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.
On exit: if ${\mathbf{jpvt}}\left(i\right)=k$, the $i$th column of $AP$ was the $k$th column of $A$.
9: $\mathbf{rcond}$Real (Kind=nag_wp) Input
On entry: 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 $\text{}<1/{\mathbf{rcond}}$.
Suggested value: if the condition number of a is not known then ${\mathbf{rcond}}=\sqrt{\left(\epsilon \right)/2}$ (where $\epsilon$ is machine precision, see x02ajf) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective $\text{rank}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ that could be larger than its actual rank, leading to meaningless results.
10: $\mathbf{rank}$Integer Output
On exit: 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 $A$.
11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08bnf is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance,
 $lwork ≥ max k + 2 × n + nb × n+1 , 2 × k + nb × nrhs ,$
where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ and $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge k+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×k,{\mathbf{n}}+1,k+{\mathbf{nrhs}}\right)\text{, where ​}k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ or
${\mathbf{lwork}}=-1$.
13: $\mathbf{rwork}\left(*\right)$Real (Kind=nag_wp) array Workspace
Note: the dimension of the array rwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$.
14: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

## 8Parallelism and Performance

f08bnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bnf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The real analogue of this routine is f08baf.

## 10Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i$
and
 $b = -1.08-2.59i -2.61-1.49i 3.13-3.61i 7.33-8.01i 9.12+7.63i .$
A tolerance of $0.01$ is used to determine the effective rank of $A$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1Program Text

Program Text (f08bnfe.f90)

### 10.2Program Data

Program Data (f08bnfe.d)

### 10.3Program Results

Program Results (f08bnfe.r)