f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zgesvd (f08kpc)

## 1  Purpose

nag_zgesvd (f08kpc) computes the singular value decomposition (SVD) of a complex $m$ by $n$ matrix $A$, optionally computing the left and/or right singular vectors.

## 2  Specification

 #include #include
 void nag_zgesvd (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVTType jobvt, Integer m, Integer n, Complex a[], Integer pda, double s[], Complex u[], Integer pdu, Complex vt[], Integer pdvt, double rwork[], NagError *fail)

## 3  Description

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 real 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 $A$.
Note that the function returns ${V}^{\mathrm{H}}$, not $V$.

## 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:    $\mathbf{order}$Nag_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 $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{jobu}$Nag_ComputeUTypeInput
On entry: specifies options for computing all or part of the matrix $U$.
${\mathbf{jobu}}=\mathrm{Nag_AllU}$
All $m$ columns of $U$ are returned in array u.
${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ (the left singular vectors) are returned in the array u.
${\mathbf{jobu}}=\mathrm{Nag_Overwrite}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ (the left singular vectors) are overwritten on the array a.
${\mathbf{jobu}}=\mathrm{Nag_NotU}$
No columns of $U$ (no left singular vectors) are computed.
Constraint: ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, $\mathrm{Nag_SingularVecsU}$, $\mathrm{Nag_Overwrite}$ or $\mathrm{Nag_NotU}$.
3:    $\mathbf{jobvt}$Nag_ComputeVTTypeInput
On entry: specifies options for computing all or part of the matrix ${V}^{\mathrm{H}}$.
${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$
All $n$ rows of ${V}^{\mathrm{H}}$ are returned in the array vt.
${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{H}}$ (the right singular vectors) are returned in the array vt.
${\mathbf{jobvt}}=\mathrm{Nag_OverwriteVT}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{H}}$ (the right singular vectors) are overwritten on the array a.
${\mathbf{jobvt}}=\mathrm{Nag_NotVT}$
No rows of ${V}^{\mathrm{H}}$ (no right singular vectors) are computed.
Constraints:
• ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, $\mathrm{Nag_SingularVecsVT}$, $\mathrm{Nag_OverwriteVT}$ or $\mathrm{Nag_NotVT}$;
• If ${\mathbf{jobu}}=\mathrm{Nag_Overwrite}$, jobvt cannot be $\mathrm{Nag_OverwriteVT}$.
4:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6:    $\mathbf{a}\left[\mathit{dim}\right]$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 ${\mathbf{jobu}}=\mathrm{Nag_Overwrite}$, a is overwritten with the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise).
If ${\mathbf{jobvt}}=\mathrm{Nag_OverwriteVT}$, a is overwritten with the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{H}}$ (the right singular vectors, stored row-wise).
If ${\mathbf{jobu}}\ne \mathrm{Nag_Overwrite}$ and ${\mathbf{jobvt}}\ne \mathrm{Nag_OverwriteVT}$, the contents of a are destroyed.
7:    $\mathbf{pda}$IntegerInput
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:    $\mathbf{s}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array s must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: the singular values of $A$, sorted so that ${\mathbf{s}}\left[i-1\right]\ge {\mathbf{s}}\left[i\right]$.
9:    $\mathbf{u}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{m}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_AllU}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdu}}\right)$ when ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ otherwise.
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 ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, u contains the $m$ by $m$ unitary matrix $U$.
If ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$, u contains the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise).
If ${\mathbf{jobu}}=\mathrm{Nag_NotU}$ or $\mathrm{Nag_Overwrite}$, u is not referenced.
10:  $\mathbf{pdu}$IntegerInput
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 ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
11:  $\mathbf{vt}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array vt must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvt}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvt}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)×{\mathbf{pdvt}}\right)$ when ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ otherwise.
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 ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, vt contains the $n$ by $n$ unitary matrix ${V}^{\mathrm{H}}$.
If ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, vt contains the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{H}}$ (the right singular vectors, stored row-wise).
If ${\mathbf{jobvt}}=\mathrm{Nag_NotVT}$ or $\mathrm{Nag_OverwriteVT}$, vt is not referenced.
12:  $\mathbf{pdvt}$IntegerInput
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 ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$.
13:  $\mathbf{rwork}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$doubleOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE, ${\mathbf{RWORK}}\left(1:\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)-1\right)$ (using the notation described in Section 3.2.1.4 in the Essential Introduction) contains the unconverged superdiagonal elements of an upper bidiagonal matrix $B$ whose diagonal is in $S$ (not necessarily sorted). $B$ satisfies $A=UB{V}^{\mathrm{T}}$, so it has the same singular values as $A$, and singular vectors related by $U$ and ${V}^{\mathrm{T}}$.
14:  $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
If nag_zgesvd (f08kpc) 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_2
On entry, ${\mathbf{jobu}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvt}}\ge 1$.
NE_ENUM_INT_3
On entry, ${\mathbf{jobu}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{jobu}}=\mathrm{Nag_SingularVecsU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{jobvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
otherwise ${\mathbf{pdvt}}\ge 1$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

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 orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

## 8  Parallelism and Performance

nag_zgesvd (f08kpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgesvd (f08kpc) 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 nag_dgesvd (f08kbc).

## 10  Example

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

### 10.1  Program Text

Program Text (f08kpce.c)

### 10.2  Program Data

Program Data (f08kpce.d)

### 10.3  Program Results

Program Results (f08kpce.r)