# NAG FL Interfacef08kzf (zgesvdx)

## 1Purpose

f08kzf computes the singular value decomposition (SVD) of a complex $m$ by $n$ matrix $A$, optionally computing the left and/or right singular vectors. All singular values or a selected set of singular values may be computed.

## 2Specification

Fortran Interface
 Subroutine f08kzf ( jobu, m, n, a, lda, vl, vu, il, iu, ns, s, u, ldu, vt, ldvt, work, info)
 Integer, Intent (In) :: m, n, lda, il, iu, ldu, ldvt, lwork Integer, Intent (Out) :: ns, iwork(12*min(n,m)), info Real (Kind=nag_wp), Intent (In) :: vl, vu Real (Kind=nag_wp), Intent (Inout) :: rwork(*) Real (Kind=nag_wp), Intent (Out) :: s(min(m,n)) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), u(ldu,*), vt(ldvt,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobu, jobvt, range
#include <nag.h>
 void f08kzf_ (const char *jobu, const char *jobvt, const char *range, const Integer *m, const Integer *n, Complex a[], const Integer *lda, const double *vl, const double *vu, const Integer *il, const Integer *iu, Integer *ns, double s[], Complex u[], const Integer *ldu, Complex vt[], const Integer *ldvt, Complex work[], const Integer *lwork, double rwork[], Integer iwork[], Integer *info, const Charlen length_jobu, const Charlen length_jobvt, const Charlen length_range)
The routine may be called by the names f08kzf, nagf_lapackeig_zgesvdx or its LAPACK name zgesvdx.

## 3Description

The SVD is written as
 $A = UΣVH ,$
where $\Sigma$ is an $m$ by $n$ matrix which is zero except for its $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ diagonal elements, $U$ is an $m$ by $m$ unitary matrix, and $V$ is an $n$ by $n$ unitary matrix. The diagonal elements of $\Sigma$ are the singular values of $A$; they are complex and non-negative, and are returned in descending order. The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ and $V$ are the left and right singular vectors of $A$, respectively.
Note that the routine returns ${V}^{\mathrm{H}}$, not $V$.
Alternative to computing all singular values of $A$, a selected set can be computed. The set is either those singular values lying in a given interval, $\sigma \in \left({v}_{l},{v}_{u}\right]$, or those whose index (counting from largest to smallest in magnitude) lies in a given range $1\le {i}_{l},\dots ,{i}_{u}\le n$. In these cases, the corresponding left and right singular vectors can optionally be computed.

## 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: $\mathbf{jobu}$Character(1) Input
On entry: specifies options for computing all or part of the matrix $U$.
${\mathbf{jobu}}=\text{'V'}$
The ns columns of $U$, as specified by range, are returned in array u.
${\mathbf{jobu}}=\text{'N'}$
No columns of $U$ (no left singular vectors) are computed.
Constraint: ${\mathbf{jobu}}=\text{'V'}$ or $\text{'N'}$.
2: $\mathbf{jobvt}$Character(1) Input
On entry: specifies options for computing all or part of the matrix ${V}^{\mathrm{T}}$.
${\mathbf{jobvt}}=\text{'V'}$
The ns rows of ${V}^{\mathrm{T}}$, as specified by range, are returned in the array vt.
${\mathbf{jobvt}}=\text{'N'}$
No rows of ${V}^{\mathrm{T}}$ (no right singular vectors) are computed.
Constraint: ${\mathbf{jobvt}}=\text{'V'}$ or $\text{'N'}$.
3: $\mathbf{range}$Character(1) Input
On entry: indicates which singular values should be returned.
${\mathbf{range}}=\text{'A'}$
All singular values will be found.
${\mathbf{range}}=\text{'V'}$
All singular values in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\text{'I'}$
The ilth through iuth singular values will be found.
Constraint: ${\mathbf{range}}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{a}\left({\mathbf{lda}},*\right)$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 $m$ by $n$ matrix $A$.
On exit: if ${\mathbf{jobu}}\ne \text{'N'}$ and ${\mathbf{jobvt}}\ne \text{'N'}$, the contents of a are destroyed.
7: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kzf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
8: $\mathbf{vl}$Real (Kind=nag_wp) Input
On entry: if ${\mathbf{range}}=\text{'V'}$, the lower bound of the interval to be searched for singular values.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'I'}$, vl is not referenced.
Constraint: if ${\mathbf{range}}=\text{'V'}$, $0.0\le {\mathbf{vl}}$.
9: $\mathbf{vu}$Real (Kind=nag_wp) Input
On entry: if ${\mathbf{range}}=\text{'V'}$, the upper bound of the interval to be searched for singular values.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'I'}$, vu is not referenced.
Constraint: if ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
10: $\mathbf{il}$Integer Input
11: $\mathbf{iu}$Integer Input
On entry: if ${\mathbf{range}}=\text{'I'}$, il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{'I'}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\text{'I'}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
12: $\mathbf{ns}$Integer Output
On exit: the total number of singular values found. $0\le {\mathbf{ns}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
If ${\mathbf{range}}=\text{'A'}$, ${\mathbf{ns}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
If ${\mathbf{range}}=\text{'I'}$, ${\mathbf{ns}}={\mathbf{iu}}-{\mathbf{il}}+1$.
If ${\mathbf{range}}=\text{'V'}$ then the value of ns is not known in advance and so an upper limit should be used when specifying the dimensions of array u, e.g., $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
13: $\mathbf{s}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$Real (Kind=nag_wp) array Output
On exit: the singular values of $A$, sorted so that ${\mathbf{s}}\left(i\right)\ge {\mathbf{s}}\left(i+1\right)$.
14: $\mathbf{u}\left({\mathbf{ldu}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array u must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{nsmax}\right)$ if ${\mathbf{jobu}}=\text{'V'}$, where $\mathit{nsmax}$ is a value larger than the output value ns..
On exit: if ${\mathbf{jobu}}=\text{'V'}$, u contains the first ns columns of $U$ (the left singular vectors, stored column-wise).
If ${\mathbf{jobu}}=\text{'N'}$, u is not referenced.
15: $\mathbf{ldu}$Integer Input
On entry: the first dimension of the array u as declared in the (sub)program from which f08kzf is called.
Constraints:
• if ${\mathbf{jobu}}=\text{'V'}$, ${\mathbf{ldu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{ldu}}\ge 1$.
16: $\mathbf{vt}\left({\mathbf{ldvt}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array vt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvt}}=\text{'V'}$.
On exit: if ${\mathbf{jobvt}}=\text{'V'}$, vt contains the first ns rows of ${V}^{\mathrm{H}}$ (the right singular vectors, stored row-wise).
If ${\mathbf{jobvt}}=\text{'N'}$, vt is not referenced.
17: $\mathbf{ldvt}$Integer Input
Note: if ${\mathbf{jobvt}}=\text{'V'}$ and ${\mathbf{range}}=\text{'V'}$ then the value of ns is not known in advance and so an upper limit should be used, e.g., $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
On entry: the first dimension of the array vt as declared in the (sub)program from which f08kzf is called.
Constraints:
• if ${\mathbf{jobvt}}=\text{'V'}$, ${\mathbf{ldvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise ${\mathbf{ldvt}}\ge 1$.
18: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
19: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08kzf 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 should generally be larger than the constrained minimum value. Consider increasing lwork beyond the minimum requirement.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+5\right)\right)$.
20: $\mathbf{rwork}\left(*\right)$Real (Kind=nag_wp) array Workspace
Note: the dimension of the array rwork must be at least $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×\left(2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+17\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{rwork}}\left(1\right)$ returns the optimal lwork.
If ${\mathbf{info}}>{\mathbf{0}}$, ${\mathbf{rwork}}\left(2:\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ contains the unconverged superdiagonal elements of an upper bidiagonal matrix $B$ whose diagonal is in s (not necessarily sorted). $B$ satisfies $A=UB{V}^{\mathrm{H}}$, so it has the same singular values as $A$, and left and right singular vectors that are those of $A$ pre-multiplied by ${U}^{\mathrm{H}}$ and ${V}^{\mathrm{H}}$.
21: $\mathbf{iwork}\left(12×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},{\mathbf{m}}\right)\right)$Integer array Workspace
On exit:
• if ${\mathbf{info}}={\mathbf{0}}$, the first ns elements of iwork are zero;
• if ${\mathbf{info}}>{\mathbf{0}}$, iwork contains the indices of the eigenvectors that failed to converge in f08jbf and f08mbf, see iwork in f08mbf.
22: $\mathbf{info}$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.
${\mathbf{info}}>0$
If f08kzf did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.

## 7Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. In addition, the computed singular vectors are nearly unitary to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08kzf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kzf 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 floating-point operations is approximately proportional to $m{n}^{2}$ when $m>n$ and ${m}^{2}n$ otherwise.
The singular values are returned in descending order.
The real analogue of this routine is f08kmf.

## 10Example

This example finds the singular values and left and right singular vectors of the $6$ by $4$ matrix
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i ,$
together with approximate error bounds for the computed singular values and vectors.
The example program for f08krf illustrates finding a singular value decomposition for the case $m\le n$.

### 10.1Program Text

Program Text (f08kzfe.f90)

### 10.2Program Data

Program Data (f08kzfe.d)

### 10.3Program Results

Program Results (f08kzfe.r)