# NAG FL Interfacef08ctf (zungql)

## 1Purpose

f08ctf generates all or part of the complex $m$ by $m$ unitary matrix $Q$ from a $QL$ factorization computed by f08csf.

## 2Specification

Fortran Interface
 Subroutine f08ctf ( m, n, k, a, lda, tau, work, info)
 Integer, Intent (In) :: m, n, k, lda, lwork Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: tau(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nag.h>
 void f08ctf_ (const Integer *m, const Integer *n, const Integer *k, Complex a[], const Integer *lda, const Complex tau[], Complex work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08ctf, nagf_lapackeig_zungql or its LAPACK name zungql.

## 3Description

f08ctf is intended to be used after a call to f08csf, which performs a $QL$ factorization of a complex matrix $A$. The unitary matrix $Q$ is represented as a product of elementary reflectors.
This routine 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 :
`Call zungql(m,m,p,a,lda,tau,work,lwork,info)`
(note that the array a must have at least $m$ columns) or its trailing $p$ columns by :
`Call zungql(m,p,p,a,lda,tau,work,lwork,info)`
The columns of $Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of $A$; thus f08csf followed by f08ctf can be used to orthogonalize the columns of $A$.
The information returned by f08csf 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 :
`Call zungql(m,m,k,a,lda,tau,work,lwork,info)`
or its trailing $k$ columns by :
`Call zungql(m,k,k,a,lda,tau,work,lwork,info)`

## 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 https://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: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $Q$.
Constraint: ${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
3: $\mathbf{k}$Integer Input
On entry: $k$, the number of elementary reflectors whose product defines the matrix $Q$.
Constraint: ${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/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: details of the vectors which define the elementary reflectors, as returned by f08csf.
On exit: the $m$ by $n$ matrix $Q$.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08ctf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6: $\mathbf{tau}\left(*\right)$Complex (Kind=nag_wp) array Input
Note: the dimension 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 f08csf.
7: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
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.
8: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08ctf 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, ${\mathbf{lwork}}\ge {\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{info}$Integer Output
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.

## 7Accuracy

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

## 8Parallelism and Performance

f08ctf 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 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 routine is f08cff.

## 10Example

This example generates the first four columns of the matrix $Q$ of the $QL$ factorization of $A$ as returned by f08csf, where
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i .$
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 (f08ctfe.f90)

### 10.2Program Data

Program Data (f08ctfe.d)

### 10.3Program Results

Program Results (f08ctfe.r)