# NAG CL Interfacef08kzc (zgesvdx)

## 1Purpose

f08kzc 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

 #include
 void f08kzc (Nag_OrderType order, Nag_ComputeSingularVecsType jobu, Nag_ComputeSingularVecsType jobvt, Nag_RangeType range, Integer m, Integer n, Complex a[], Integer pda, double vl, double vu, Integer il, Integer iu, Integer *ns, double s[], Complex u[], Integer pdu, Complex vt[], Integer pdvt, double rwork[], Integer jfail[], NagError *fail)
The function may be called by the names: f08kzc, nag_lapackeig_zgesvdx or nag_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 function 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: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{jobu}$Nag_ComputeSingularVecsType Input
On entry: specifies options for computing all or part of the matrix $U$.
${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$
The ns columns of $U$, as specified by range, are returned in array u.
${\mathbf{jobu}}=\mathrm{Nag_NotSingularVecs}$
No columns of $U$ (no left singular vectors) are computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$ or $\mathrm{Nag_NotSingularVecs}$.
3: $\mathbf{jobvt}$Nag_ComputeSingularVecsType Input
On entry: specifies options for computing all or part of the matrix ${V}^{\mathrm{T}}$.
${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$
The ns rows of ${V}^{\mathrm{T}}$, as specified by range, are returned in the array vt.
${\mathbf{jobvt}}=\mathrm{Nag_NotSingularVecs}$
No rows of ${V}^{\mathrm{T}}$ (no right singular vectors) are computed.
Constraint: ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$ or $\mathrm{Nag_NotSingularVecs}$.
4: $\mathbf{range}$Nag_RangeType Input
On entry: indicates which singular values should be returned.
${\mathbf{range}}=\mathrm{Nag_AllValues}$
All singular values will be found.
${\mathbf{range}}=\mathrm{Nag_Interval}$
All singular values in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\mathrm{Nag_Indices}$
The ilth through iuth singular values will be found.
Constraint: ${\mathbf{range}}=\mathrm{Nag_AllValues}$, $\mathrm{Nag_Interval}$ or $\mathrm{Nag_Indices}$.
5: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
6: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
7: $\mathbf{a}\left[\mathit{dim}\right]$Complex Input/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: if ${\mathbf{jobu}}\ne \mathrm{Nag_NotSingularVecs}$ and ${\mathbf{jobvt}}\ne \mathrm{Nag_NotSingularVecs}$, the contents of a are destroyed.
8: $\mathbf{pda}$Integer Input
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{vl}$double Input
On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, the lower bound of the interval to be searched for singular values.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vl is not referenced.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, $0.0\le {\mathbf{vl}}$.
10: $\mathbf{vu}$double Input
On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, the upper bound of the interval to be searched for singular values.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vu is not referenced.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
11: $\mathbf{il}$Integer Input
12: $\mathbf{iu}$Integer Input
On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Interval}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
13: $\mathbf{ns}$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 ${\mathbf{range}}=\mathrm{Nag_AllValues}$, ${\mathbf{ns}}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
If ${\mathbf{range}}=\mathrm{Nag_Indices}$, ${\mathbf{ns}}={\mathbf{iu}}-{\mathbf{il}}+1$.
If ${\mathbf{range}}=\mathrm{Nag_Interval}$ 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., $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
14: $\mathbf{s}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$double Output
On exit: the singular values of $A$, sorted so that ${\mathbf{s}}\left[i-1\right]\ge {\mathbf{s}}\left[i\right]$.
15: $\mathbf{u}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array u must be at least
• ${\mathbf{pdu}}×\mathit{nsmax}$ when ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{m}}×{\mathbf{pdu}}$ when ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• otherwise u may be NULL;
where $\mathit{nsmax}$ is a value larger than the output value ns.
The $\left(i,j\right)$th element of the matrix $U$ is stored in
• ${\mathbf{u}}\left[\left(j-1\right)×{\mathbf{pdu}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{u}}\left[\left(i-1\right)×{\mathbf{pdu}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$, u contains the first ns columns of $U$ (the left singular vectors, stored column-wise).
If ${\mathbf{jobu}}=\mathrm{Nag_NotSingularVecs}$, u is not referenced.
16: $\mathbf{pdu}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge {\mathbf{m}}$;
• otherwise ${\mathbf{pdu}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge \mathit{nsmax}$;
• otherwise u may be NULL;
where $\mathit{nsmax}$ is a value larger than the output value ns.
17: $\mathbf{vt}\left[\mathit{dim}\right]$Complex Output
Note: the dimension, dim, of the array vt must be at least
• ${\mathbf{pdvt}}×{\mathbf{n}}$ when ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×{\mathbf{pdvt}}$ when ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• otherwise vt may be NULL.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vt}}\left[\left(j-1\right)×{\mathbf{pdvt}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vt}}\left[\left(i-1\right)×{\mathbf{pdvt}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$, vt contains the first ns rows of ${V}^{\mathrm{H}}$ (the right singular vectors, stored row-wise).
If ${\mathbf{jobvt}}=\mathrm{Nag_NotSingularVecs}$, vt is not referenced.
18: $\mathbf{pdvt}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge {\mathbf{n}}$;
• otherwise vt may be NULL.
19: $\mathbf{rwork}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$double Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE, ${\mathbf{RWORK}}\left(2:\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ (using the notation described in Section 3.1.4 in the Introduction to the NAG Library CL Interface) 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 ${V}^{\mathrm{H}}$.
20: $\mathbf{jfail}\left[2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$Integer Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE, jfail contains, in its first $k$ nonzero elements, the indices of the $k$ eigenvectors (associated with a left or right singular vector, see f08mbc) that failed to converge.
21: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
If f08kzc did not converge, ${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$ specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT
On entry, ${\mathbf{jobu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge \mathit{nsmax}$.
NE_ENUM_INT_2
On entry, ${\mathbf{jobu}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{jobvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge {\mathbf{n}}$.
NE_ENUM_INT_3
On entry, ${\mathbf{jobvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
NE_ENUM_INT_4
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{il}}=〈\mathit{\text{value}}〉$, ${\mathbf{iu}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
NE_ENUM_REAL_1
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$ and ${\mathbf{vl}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, $0.0\le {\mathbf{vl}}$.
NE_ENUM_REAL_2
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{vl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{vu}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdvt}}>0$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 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

f08kzc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kzc 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 function. 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 function is f08kmc.

## 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 f08krc illustrates finding a singular value decomposition for the case $m\le n$.

### 10.1Program Text

Program Text (f08kzce.c)

### 10.2Program Data

Program Data (f08kzce.d)

### 10.3Program Results

Program Results (f08kzce.r)