f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_zgelsy (f08bnc)

1  Purpose

nag_zgelsy (f08bnc) 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.

2  Specification

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

3  Description

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 $X$.
nag_zgelsy (f08bnc) 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 $Q$.

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 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: a has been overwritten by details of its complete orthogonal factorization.
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: the $n$ by $r$ solution matrix $X$.
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:     jpvt[$\mathit{dim}$]IntegerInput/Output
Note: the dimension, dim, of the array jpvt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{jpvt}}\left[i-1\right]\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.
On exit: if ${\mathbf{jpvt}}\left[i-1\right]=k$, then the $i$th column of $AP$ was the $k$th column of $A$.
10:   rconddoubleInput
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 nag_machine_precision (X02AJC)) 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.
11:   rankInteger *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 $A$.
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_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 of error bounds.

The real analogue of this function is nag_dgelsy (f08bac).

9  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 $A$.

9.1  Program Text

Program Text (f08bnce.c)

9.2  Program Data

Program Data (f08bnce.d)

9.3  Program Results

Program Results (f08bnce.r)