f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zungql (f08ctc)

## 1  Purpose

nag_zungql (f08ctc) generates all or part of the complex $m$ by $m$ unitary matrix $Q$ from a $QL$ factorization computed by nag_zgeqlf (f08csc).

## 2  Specification

 #include #include
 void nag_zungql (Nag_OrderType order, Integer m, Integer n, Integer k, Complex a[], Integer pda, const Complex tau[], NagError *fail)

## 3  Description

nag_zungql (f08ctc) is intended to be used after a call to nag_zgeqlf (f08csc), which performs a $QL$ factorization of a complex matrix $A$. The unitary matrix $Q$ is represented as a product of elementary reflectors.
This function may be used to generate $Q$ explicitly as a square matrix, or to form only its trailing columns.
Usually $Q$ is determined from the $QL$ factorization of an $m$ by $p$ matrix $A$ with $m\ge p$. The whole of $Q$ may be computed by:
```nag_zungql(order,m,m,p,&a,pda,tau,&fail)
```
(note that the array a must have at least $m$ columns) or its trailing $p$ columns by:
```nag_zungql(order,m,p,p,&a,pda,tau,&fail)
```
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus nag_zgeqlf (f08csc) followed by nag_zungql (f08ctc) can be used to orthogonalise the columns of $A$.
The information returned by nag_zgeqlf (f08csc) also yields the $QL$ factorization of the trailing $k$ columns of $A$, where $k. The unitary matrix arising from this factorization can be computed by:
```nag_zungql(order,m,m,k,&a,pda,tau,&fail)
```
or its trailing $k$ columns by:
```nag_zungql(order,m,k,k,&a,pda,tau,&fail)
```

## 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 $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     nIntegerInput
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
4:     kIntegerInput
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgeqlf (f08csc).
On exit: the $m$ by $n$ matrix $Q$.
If ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$.
If ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
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:     tau[$\mathit{dim}$]const ComplexInput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
On entry: further details of the elementary reflectors as returned by nag_zgeqlf (f08csc).
8:     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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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

The computed matrix $Q$ differs from an exactly unitary matrix by a matrix $E$ such that
 $E2 = Oε ,$
where $\epsilon$ is the machine precision.

The total number of real floating point operations is approximately $16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when $n=k$, the number is approximately $\frac{8}{3}{n}^{2}\left(3m-n\right)$.
The real analogue of this function is nag_dorgql (f08cfc).

## 9  Example

This example generates the first four columns of the matrix $Q$ of the $QL$ factorization of $A$ as returned by nag_zgeqlf (f08csc), where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$

### 9.1  Program Text

Program Text (f08ctce.c)

### 9.2  Program Data

Program Data (f08ctce.d)

### 9.3  Program Results

Program Results (f08ctce.r)