f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zgelss (f08knc)

## 1  Purpose

nag_zgelss (f08knc) computes the minimum norm solution to a complex linear least squares problem
 $minx b-Ax2 .$

## 2  Specification

 #include #include
 void nag_zgelss (Nag_OrderType order, Integer m, Integer n, Integer nrhs, Complex a[], Integer pda, Complex b[], Integer pdb, double s[], double rcond, Integer *rank, NagError *fail)

## 3  Description

nag_zgelss (f08knc) 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.

## 4  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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     mIntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     nIntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     nrhsIntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     b[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
9:     s[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, 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.
10:   rconddoubleInput
On entry: used to determine the effective rank of $A$. Singular values ${\mathbf{s}}\left[i-1\right]\le {\mathbf{rcond}}×{\mathbf{s}}\left[0\right]$ are treated as zero. If ${\mathbf{rcond}}<0$, machine precision is used instead.
11:   rankInteger *Output
On exit: the effective rank of $A$, i.e., the number of singular values which are greater than ${\mathbf{rcond}}×{\mathbf{s}}\left[0\right]$.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
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.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INT_3
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

## 7  Accuracy

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

## 8  Parallelism and Performance

nag_zgelss (f08knc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgelss (f08knc) 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 Users' Note for your implementation for any additional implementation-specific information.

## 9  Further Comments

The real analogue of this function is nag_dgelss (f08kac).

## 10  Example

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 $A$.

### 10.1  Program Text

Program Text (f08knce.c)

### 10.2  Program Data

Program Data (f08knce.d)

### 10.3  Program Results

Program Results (f08knce.r)