NAG FL Interfacef08vuf (zggsvp3)

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1Purpose

f08vuf uses unitary transformations to simultaneously reduce the $m×n$ matrix $A$ and the $p×n$ matrix $B$ to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD). For sufficiently large problems, a blocked algorithm is used to make best use of Level 3 BLAS.

2Specification

Fortran Interface
 Subroutine f08vuf ( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola, tolb, k, l, u, ldu, v, ldv, q, ldq, tau, work, info)
 Integer, Intent (In) :: m, p, n, lda, ldb, ldu, ldv, ldq, lwork Integer, Intent (Out) :: k, l, iwork(n), info Real (Kind=nag_wp), Intent (In) :: tola, tolb Real (Kind=nag_wp), Intent (Out) :: rwork(2*n) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), u(ldu,*), v(ldv,*), q(ldq,*) Complex (Kind=nag_wp), Intent (Out) :: tau(n), work(max(1,lwork)) Character (1), Intent (In) :: jobu, jobv, jobq
#include <nag.h>
 void f08vuf_ (const char *jobu, const char *jobv, const char *jobq, const Integer *m, const Integer *p, const Integer *n, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, const double *tola, const double *tolb, Integer *k, Integer *l, Complex u[], const Integer *ldu, Complex v[], const Integer *ldv, Complex q[], const Integer *ldq, Integer iwork[], double rwork[], Complex tau[], Complex work[], const Integer *lwork, Integer *info, const Charlen length_jobu, const Charlen length_jobv, const Charlen length_jobq)
The routine may be called by the names f08vuf, nagf_lapackeig_zggsvp3 or its LAPACK name zggsvp3.

3Description

f08vuf computes unitary matrices $U$, $V$ and $Q$ such that
where the $k×k$ matrix ${A}_{12}$ and $l×l$ matrix ${B}_{13}$ are nonsingular upper triangular; ${A}_{23}$ is $l×l$ upper triangular if $m-k-l\ge 0$ and is $\left(m-k\right)×l$ upper trapezoidal otherwise. $\left(k+l\right)$ is the effective numerical rank of the $\left(m+p\right)×n$ matrix ${\left(\begin{array}{cc}{A}^{\mathrm{H}}& {B}^{\mathrm{H}}\end{array}\right)}^{\mathrm{H}}$.
This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see routine f08ysf; the two steps are combined in f08vqf.

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 (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore

5Arguments

1: $\mathbf{jobu}$Character(1) Input
On entry: if ${\mathbf{jobu}}=\text{'U'}$, the unitary matrix $U$ is computed.
If ${\mathbf{jobu}}=\text{'N'}$, $U$ is not computed.
Constraint: ${\mathbf{jobu}}=\text{'U'}$ or $\text{'N'}$.
2: $\mathbf{jobv}$Character(1) Input
On entry: if ${\mathbf{jobv}}=\text{'V'}$, the unitary matrix $V$ is computed.
If ${\mathbf{jobv}}=\text{'N'}$, $V$ is not computed.
Constraint: ${\mathbf{jobv}}=\text{'V'}$ or $\text{'N'}$.
3: $\mathbf{jobq}$Character(1) Input
On entry: if ${\mathbf{jobq}}=\text{'Q'}$, the unitary matrix $Q$ is computed.
If ${\mathbf{jobq}}=\text{'N'}$, $Q$ is not computed.
Constraint: ${\mathbf{jobq}}=\text{'Q'}$ or $\text{'N'}$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5: $\mathbf{p}$Integer Input
On entry: $p$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{p}}\ge 0$.
6: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{a}\left({\mathbf{lda}},*\right)$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: the $m×n$ matrix $A$.
On exit: contains the triangular (or trapezoidal) matrix described in Section 3.
8: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08vuf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
9: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $p×n$ matrix $B$.
On exit: contains the triangular matrix described in Section 3.
10: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08vuf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$.
11: $\mathbf{tola}$Real (Kind=nag_wp) Input
12: $\mathbf{tolb}$Real (Kind=nag_wp) Input
On entry: tola and tolb are the thresholds to determine the effective numerical rank of matrix $B$ and a subblock of $A$. Generally, they are set to
 $tola=max(m,n)‖A‖ε, tolb=max(p,n)‖B‖ε,$
where $\epsilon$ is the machine precision.
The size of tola and tolb may affect the size of backward errors of the decomposition.
13: $\mathbf{k}$Integer Output
14: $\mathbf{l}$Integer Output
On exit: k and l specify the dimension of the subblocks $k$ and $l$ as described in Section 3; $\left(k+l\right)$ is the effective numerical rank of ${\left(\begin{array}{cc}{{\mathbf{a}}}^{\mathrm{T}}& {{\mathbf{b}}}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.
15: $\mathbf{u}\left({\mathbf{ldu}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobu}}=\text{'U'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobu}}=\text{'U'}$, u contains the unitary matrix $U$.
If ${\mathbf{jobu}}=\text{'N'}$, u is not referenced.
16: $\mathbf{ldu}$Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08vuf is called.
Constraints:
• if ${\mathbf{jobu}}=\text{'U'}$, ${\mathbf{ldu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{ldu}}\ge 1$.
17: $\mathbf{v}\left({\mathbf{ldv}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array v must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$ if ${\mathbf{jobv}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobv}}=\text{'V'}$, v contains the unitary matrix $V$.
If ${\mathbf{jobv}}=\text{'N'}$, v is not referenced.
18: $\mathbf{ldv}$Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08vuf is called.
Constraints:
• if ${\mathbf{jobv}}=\text{'V'}$, ${\mathbf{ldv}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{p}}\right)$;
• otherwise ${\mathbf{ldv}}\ge 1$.
19: $\mathbf{q}\left({\mathbf{ldq}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobq}}=\text{'Q'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobq}}=\text{'Q'}$, q contains the unitary matrix $Q$.
If ${\mathbf{jobq}}=\text{'N'}$, q is not referenced.
20: $\mathbf{ldq}$Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which f08vuf is called.
Constraints:
• if ${\mathbf{jobq}}=\text{'Q'}$, ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldq}}\ge 1$.
21: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Workspace
22: $\mathbf{rwork}\left(2×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
23: $\mathbf{tau}\left({\mathbf{n}}\right)$Complex (Kind=nag_wp) array Workspace
24: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$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.
25: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08vuf 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, lwork must generally be larger than the minimum; increase workspace by, say, $\mathit{nb}×\left({\mathbf{n}}+1\right)$, where $\mathit{nb}$ is the optimal block size
Constraints:
if ${\mathbf{lwork}}\ne -1$,
• if ${\mathbf{jobv}}=\text{'V'}$, ${\mathbf{lwork}}=-1$ or ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{p}},{\mathbf{m}}\right)$;
• if ${\mathbf{jobv}}=\text{'N'}$, ${\mathbf{lwork}}=-1$ or ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}}+1,{\mathbf{m}}\right)$.
26: $\mathbf{info}$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 factorization is nearly the exact factorization for nearby matrices $\left(A+E\right)$ and $\left(B+F\right)$, where
 $‖E‖2 = O(ε)‖A‖2 and ‖F‖2= O(ε)‖B‖2,$
and $\epsilon$ is the machine precision.

8Parallelism and Performance

f08vuf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08vuf 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.

This routine replaces the deprecated routine f08vsf which used an unblocked algorithm and, therefore, did not make best use of Level 3 BLAS routines.
The real analogue of this routine is f08vgf.

10Example

This example finds the generalized factorization
 $A = UΣ1 ( 0 S ) QH , B= VΣ2 ( 0 T ) QH ,$
of the matrix pair $\left(\begin{array}{cc}A& B\end{array}\right)$, where
 $A = ( 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i 0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i )$
and
 $B = ( 10−10 010−1 ) .$

10.1Program Text

Program Text (f08vufe.f90)

10.2Program Data

Program Data (f08vufe.d)

10.3Program Results

Program Results (f08vufe.r)