# NAG CL Interfacef08kpc (zgesvd)

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## 1Purpose

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

## 2Specification

 #include
 void f08kpc (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)
The function may be called by the names: f08kpc, nag_lapackeig_zgesvd or nag_zgesvd.

## 3Description

The SVD is written as
 $A = UΣVH ,$
where $\Sigma$ is an $m×n$ matrix which is zero except for its $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ diagonal elements, $U$ is an $m×m$ unitary matrix, and $V$ is an $n×n$ 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 $V$.

## 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_ComputeUType Input
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_ComputeVTType Input
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: $\mathbf{m}$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: $\mathbf{a}\left[\mathit{dim}\right]$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 $m×n$ 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: $\mathbf{pda}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{s}\left[\mathit{dim}\right]$double Output
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: $\mathbf{u}\left[\mathit{dim}\right]$Complex Output
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 $m×m$ 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: $\mathbf{pdu}$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 ${\mathbf{jobu}}=\mathrm{Nag_AllU}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• 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: $\mathbf{vt}\left[\mathit{dim}\right]$Complex Output
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 $n×n$ 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: $\mathbf{pdvt}$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 ${\mathbf{jobvt}}=\mathrm{Nag_AllVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• 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: $\mathbf{rwork}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$double Output
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 3.1.4 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 $A=UB{V}^{\mathrm{T}}$, 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 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.
NE_BAD_PARAM
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE
If 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, ${\mathbf{jobvt}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvt}}=⟨\mathit{\text{value}}⟩$ 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 ${\mathbf{jobvt}}=\mathrm{Nag_SingularVecsVT}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
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 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
 $‖E‖2 = O(ε) ‖A‖2 ,$
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.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08kpc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

## 9Further Comments

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 f08kbc.

## 10Example

This example finds the singular values and left and right singular vectors of the $6×4$ 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 (f08kpce.c)

### 10.2Program Data

Program Data (f08kpce.d)

### 10.3Program Results

Program Results (f08kpce.r)