# NAG Library Function Document

## 1Purpose

nag_dgesdd (f08kdc) computes the singular value decomposition (SVD) of a real $m$ by $n$ matrix $A$, optionally computing the left and/or right singular vectors, by using a divide-and-conquer method.

## 2Specification

 #include #include
 void nag_dgesdd (Nag_OrderType order, Nag_JobType job, Integer m, Integer n, double a[], Integer pda, double s[], double u[], Integer pdu, double vt[], Integer pdvt, NagError *fail)

## 3Description

The SVD is written as
 $A = UΣVT ,$
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 $m$ orthogonal matrix, and $V$ is an $n$ by $n$ orthogonal 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 ${V}^{\mathrm{T}}$, 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 http://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_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{job}$Nag_JobTypeInput
On entry: specifies options for computing all or part of the matrix $U$.
${\mathbf{job}}=\mathrm{Nag_DoAll}$
All $m$ columns of $U$ and all $n$ rows of ${V}^{\mathrm{T}}$ are returned in the arrays u and vt.
${\mathbf{job}}=\mathrm{Nag_DoSquare}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ columns of $U$ and the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{T}}$ are returned in the arrays u and vt.
${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$
If ${\mathbf{m}}\ge {\mathbf{n}}$, the first $n$ columns of $U$ are overwritten on the array a and all rows of ${V}^{\mathrm{T}}$ are returned in the array vt. Otherwise, all columns of $U$ are returned in the array u and the first $m$ rows of ${V}^{\mathrm{T}}$ are overwritten in the array vt.
${\mathbf{job}}=\mathrm{Nag_DoNothing}$
No columns of $U$ or rows of ${V}^{\mathrm{T}}$ are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_DoAll}$, $\mathrm{Nag_DoSquare}$, $\mathrm{Nag_DoOverwrite}$ or $\mathrm{Nag_DoNothing}$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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 ${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$, a is overwritten with the first $n$ columns of $U$ (the left singular vectors, stored column-wise) if ${\mathbf{m}}\ge {\mathbf{n}}$; a is overwritten with the first $m$ rows of ${V}^{\mathrm{T}}$ (the right singular vectors, stored row-wise) otherwise.
If ${\mathbf{job}}\ne \mathrm{Nag_DoOverwrite}$, the contents of a are destroyed.
6:    $\mathbf{pda}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:    $\mathbf{s}\left[\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right]$doubleOutput
On exit: the singular values of $A$, sorted so that ${\mathbf{s}}\left[i-1\right]\ge {\mathbf{s}}\left[i\right]$.
8:    $\mathbf{u}\left[\mathit{dim}\right]$doubleOutput
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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$);
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoSquare}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdu}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoSquare}$ 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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or ${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$, u contains the $m$ by $m$ orthogonal matrix $U$.
If ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, 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 ${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$, or ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, u is not referenced.
9:    $\mathbf{pdu}$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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$), ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$), ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
10:  $\mathbf{vt}\left[\mathit{dim}\right]$doubleOutput
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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$);
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvt}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoSquare}$ 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 ${\mathbf{job}}=\mathrm{Nag_DoSquare}$ 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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or ${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$, vt contains the $n$ by $n$ orthogonal matrix ${V}^{\mathrm{T}}$.
If ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, vt contains the first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{T}}$ (the right singular vectors, stored row-wise).
If ${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$, or ${\mathbf{job}}=\mathrm{Nag_DoNothing}$, vt is not referenced.
11:  $\mathbf{pdvt}$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 ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$), ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$), ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
nag_dgesdd (f08kdc) did not converge, the updating process failed.
NE_ENUM_INT_3
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$), ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdu}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}<{\mathbf{n}}$), ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$), ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
otherwise ${\mathbf{pdvt}}\ge 1$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvt}}=〈\mathit{\text{value}}〉$, ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoAll}$ or (${\mathbf{job}}=\mathrm{Nag_DoOverwrite}$ and ${\mathbf{m}}\ge {\mathbf{n}}$), ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{job}}=\mathrm{Nag_DoSquare}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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 orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

nag_dgesdd (f08kdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgesdd (f08kdc) 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 complex analogue of this function is nag_zgesdd (f08krc).

## 10Example

This example finds the singular values and left and right singular vectors of the $4$ by $6$ matrix
 $A = 2.27 0.28 -0.48 1.07 -2.35 0.62 -1.54 -1.67 -3.09 1.22 2.93 -7.39 1.15 0.94 0.99 0.79 -1.45 1.03 -1.94 -0.78 -0.21 0.63 2.30 -2.57 ,$
together with approximate error bounds for the computed singular values and vectors.
The example program for nag_dgesvd (f08kbc) illustrates finding a singular value decomposition for the case $m\ge n$.

### 10.1Program Text

Program Text (f08kdce.c)

### 10.2Program Data

Program Data (f08kdce.d)

### 10.3Program Results

Program Results (f08kdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017