# NAG Library Routine Document

## 1Purpose

f08kaf (dgelss) computes the minimum norm solution to a real linear least squares problem
 $minx b-Ax2 .$

## 2Specification

Fortran Interface
 Subroutine f08kaf ( m, n, nrhs, a, lda, b, ldb, s, rank, work, info)
 Integer, Intent (In) :: m, n, nrhs, lda, ldb, lwork Integer, Intent (Out) :: rank, info Real (Kind=nag_wp), Intent (In) :: rcond Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), s(*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nagmk26.h>
 void f08kaf_ (const Integer *m, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double b[], const Integer *ldb, double s[], const double *rcond, Integer *rank, double work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name dgelss.

## 3Description

f08kaf (dgelss) uses the singular value decomposition (SVD) of $A$, where $A$ 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; they are stored as the columns of the $m$ by $r$ right-hand side matrix $B$ and the $n$ by $r$ solution matrix $X$.
The effective rank of $A$ is determined by treating as zero those singular values which are less than rcond times the largest singular value.

## 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 http://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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/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: the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kaf (dgelss) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/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: b is overwritten by the $n$ by $r$ solution matrix $X$. If $m\ge n$ and ${\mathbf{rank}}=n$, the residual sum of squares for the solution in the $i$th column is given by the sum of squares of elements $n+1,\dots ,m$ in that column.
7:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08kaf (dgelss) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
8:     $\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array s must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the singular values of $A$ in decreasing order.
9:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Input
On entry: used to determine the effective rank of $A$. Singular values ${\mathbf{s}}\left(i\right)\le {\mathbf{rcond}}×{\mathbf{s}}\left(1\right)$ are treated as zero. If ${\mathbf{rcond}}<0$, machine precision is used instead.
10:   $\mathbf{rank}$ – IntegerOutput
On exit: the effective rank of $A$, i.e., the number of singular values which are greater than ${\mathbf{rcond}}×{\mathbf{s}}\left(1\right)$.
11:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kaf (dgelss) 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 should generally be larger. Consider increasing lwork by at least $\mathit{nb}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge 1$ and
${\mathbf{lwork}}\ge 3×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right),\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right),{\mathbf{nrhs}}\right)$.
13:   $\mathbf{info}$ – IntegerOutput
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.
${\mathbf{info}}>0$
The algorithm for computing the SVD failed to converge; $〈\mathit{\text{value}}〉$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

## 7Accuracy

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

## 8Parallelism and Performance

f08kaf (dgelss) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kaf (dgelss) 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 complex analogue of this routine is f08knf (zgelss).

## 10Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and b= 7.4 4.2 -8.3 1.8 8.6 2.1 .$
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 (f08kafe.f90)

### 10.2Program Data

Program Data (f08kafe.d)

### 10.3Program Results

Program Results (f08kafe.r)