NAG Library Routine Document

1Purpose

f08kqf (zgelsd) computes the minimum norm solution to a complex linear least squares problem
 $minx b-Ax2 .$

2Specification

Fortran Interface
 Subroutine f08kqf ( m, n, nrhs, a, lda, b, ldb, s, rank, work, info)
 Integer, Intent (In) :: m, n, nrhs, lda, ldb, lwork Integer, Intent (Inout) :: iwork(*) Integer, Intent (Out) :: rank, info Real (Kind=nag_wp), Intent (In) :: rcond Real (Kind=nag_wp), Intent (Inout) :: s(*), rwork(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
C Header Interface
#include <nagmk26.h>
 void f08kqf_ (const Integer *m, const Integer *n, const Integer *nrhs, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, double s[], const double *rcond, Integer *rank, Complex work[], const Integer *lwork, double rwork[], Integer iwork[], Integer *info)
The routine may be called by its LAPACK name zgelsd.

3Description

f08kqf (zgelsd) uses the singular value decomposition (SVD) of $A$, where $A$ is a complex $m$ 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 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.

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:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Complex (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 $n$ coefficient matrix $A$.
On exit: the contents of a are destroyed.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kqf (zgelsd) 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) 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 the modulus 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 f08kqf (zgelsd) 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:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace
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:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kqf (zgelsd) is called.
The exact minimum amount of workspace needed depends on m, n and nrhs. As long as lwork is at least
 $max1,m+n+r,2r+r×nrhs ,$
where $r=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$, the code will execute correctly.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array and the minimum size of the iwork array, and returns these values as the first entries of the work and iwork arrays, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork should generally be larger than the required minimum. 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: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}+{\mathbf{n}}+\mathit{r},2\mathit{r}+\mathit{r}×{\mathbf{nrhs}}\right)$ or ${\mathbf{lwork}}=-1$.
13:   $\mathbf{rwork}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{lrwork}\right)$, where $\mathit{lrwork}$ is at least
 $10× n+2× n× smlsiz+8× n× nlvl+3× smlsiz× nrhs+smlsiz+12 , if ​ m≥n$
or
 $10× m+2× m× smlsiz+8× m× nlvl+3× smlsiz× nrhs+smlsiz+12 , if ​ m
where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$), and $\mathit{nlvl}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(0,\mathrm{int}\left({\mathrm{log}}_{2}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)/\left(\mathit{smlsiz}+1\right)\right)\right)+1\right)$, the code will execute correctly.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{rwork}}\left(1\right)$ contains the required minimal size of $\mathit{lrwork}$.
14:   $\mathbf{iwork}\left(*\right)$ – Integer arrayWorkspace
Note: the dimension of the array iwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{liwork}\right)$, where $\mathit{liwork}$ is at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×\mathit{nlvl}+11×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ returns the minimum $\mathit{liwork}$.
15:   $\mathbf{info}$ – 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

f08kqf (zgelsd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kqf (zgelsd) 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.

9Further Comments

The real analogue of this routine is f08kcf (dgelsd).

10Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = 0.47-0.34i -0.32-0.23i 0.35-0.60i 0.89+0.71i -0.19+0.06i -0.40+0.54i -0.05+0.20i -0.52-0.34i -0.45-0.45i 0.11-0.85i 0.60+0.01i -0.26-0.44i 0.87-0.11i -0.02-0.57i 1.44+0.80i 0.80-1.02i -0.43+0.17i -0.34-0.09i 1.14-0.78i 0.07+1.14i$
and
 $b = 2.15-0.20i -2.24+1.82i 4.45-4.28i 5.70-6.25i .$
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 (f08kqfe.f90)

10.2Program Data

Program Data (f08kqfe.d)

10.3Program Results

Program Results (f08kqfe.r)