f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dbdsdc (f08mdc)

## 1  Purpose

nag_dbdsdc (f08mdc) computes the singular values and, optionally, the left and right singular vectors of a real $n$ by $n$ (upper or lower) bidiagonal matrix $B$.

## 2  Specification

 #include #include
 void nag_dbdsdc (Nag_OrderType order, Nag_UploType uplo, Nag_ComputeSingularVecsType compq, Integer n, double d[], double e[], double u[], Integer pdu, double vt[], Integer pdvt, double q[], Integer iq[], NagError *fail)

## 3  Description

nag_dbdsdc (f08mdc) computes the singular value decomposition (SVD) of the (upper or lower) bidiagonal matrix $B$ as
 $B = USVT ,$
where $S$ is a diagonal matrix with non-negative diagonal elements ${s}_{ii}={s}_{i}$, such that
 $s1 ≥ s2 ≥ ⋯ ≥ sn ≥ 0 ,$
and $U$ and $V$ are orthogonal matrices. The diagonal elements of $S$ are the singular values of $B$ and the columns of $U$ and $V$ are respectively the corresponding left and right singular vectors of $B$.
When only singular values are required the function uses the $QR$ algorithm, but when singular vectors are required a divide and conquer method is used. The singular values can optionally be returned in compact form, although currently no function is available to apply $U$ or $V$ when stored in compact form.

## 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:     orderNag_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:     uploNag_UploTypeInput
On entry: indicates whether $B$ is upper or lower bidiagonal.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
$B$ is upper bidiagonal.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
$B$ is lower bidiagonal.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     compqNag_ComputeSingularVecsTypeInput
On entry: specifies whether singular vectors are to be computed.
${\mathbf{compq}}=\mathrm{Nag_NotSingularVecs}$
Compute singular values only.
${\mathbf{compq}}=\mathrm{Nag_PackedSingularVecs}$
Compute singular values and compute singular vectors in compact form.
${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$
Compute singular values and singular vectors.
Constraint: ${\mathbf{compq}}=\mathrm{Nag_NotSingularVecs}$, $\mathrm{Nag_PackedSingularVecs}$ or $\mathrm{Nag_SingularVecs}$.
4:     nIntegerInput
On entry: $n$, the order of the matrix $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     d[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ diagonal elements of the bidiagonal matrix $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the singular values of $B$.
6:     e[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: the $\left(n-1\right)$ off-diagonal elements of the bidiagonal matrix $B$.
On exit: the contents of e are destroyed.
7:     u[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array u must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdu}}×{\mathbf{n}}\right)$ when ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$;
• $1$ 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 ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, u contains the left singular vectors of the bidiagonal matrix $B$.
If ${\mathbf{compq}}\ne \mathrm{Nag_SingularVecs}$, u is not referenced.
8:     pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
• if ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdu}}\ge 1$.
9:     vt[$\mathit{dim}$]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 ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$;
• $1$ 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 ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the rows of vt contain the right singular vectors of the bidiagonal matrix $B$.
If ${\mathbf{compq}}\ne \mathrm{Nag_SingularVecs}$, vt is not referenced.
10:   pdvtIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vt.
Constraints:
• if ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvt}}\ge 1$.
11:   q[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{{\mathbf{n}}}^{2}+5{\mathbf{n}},\mathit{ldq}\right)$.
On exit: if ${\mathbf{compq}}=\mathrm{Nag_PackedSingularVecs}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, q and iq contain the left and right singular vectors in a compact form, requiring $\mathit{O}\left({\mathbf{n}}{\mathrm{log}}_{2}{\mathbf{n}}\right)$ space instead of $2×{{\mathbf{n}}}^{2}$. In particular, q contains all the real data in the first $\mathit{ldq}={\mathbf{n}}×\left(11+2×\mathit{smlsiz}+8×\mathrm{int}\left({\mathrm{log}}_{2}\left({\mathbf{n}}/\left(\mathit{smlsiz}+1\right)\right)\right)\right)$ elements of q, where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$).
If ${\mathbf{compq}}\ne \mathrm{Nag_PackedSingularVecs}$, q is not referenced.
12:   iq[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array iq must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathit{ldiq}\right)$.
On exit: if ${\mathbf{compq}}=\mathrm{Nag_PackedSingularVecs}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, q and iq contain the left and right singular vectors in a compact form, requiring $\mathit{O}\left({\mathbf{n}}{\mathrm{log}}_{2}{\mathbf{n}}\right)$ space instead of $2×{{\mathbf{n}}}^{2}$. In particular, iq contains all integer data in the first $\mathit{ldiq}={\mathbf{n}}×\left(3+3×\mathrm{int}\left({\mathrm{log}}_{2}\left({\mathbf{n}}/\left(\mathit{smlsiz}+1\right)\right)\right)\right)$ elements of iq, where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$).
If ${\mathbf{compq}}\ne \mathrm{Nag_PackedSingularVecs}$, iq is not referenced.
13:   failNagError *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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{compq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdu}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdu}}\ge 1$.
On entry, ${\mathbf{compq}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvt}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{compq}}=\mathrm{Nag_SingularVecs}$, ${\mathbf{pdvt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvt}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdu}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdu}}>0$.
On entry, ${\mathbf{pdvt}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdvt}}>0$.
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.
NE_SINGULAR
The algorithm failed to compute a singular value. The update process of divide-and-conquer failed.

## 7  Accuracy

Each computed singular value of $B$ is accurate to nearly full relative precision, no matter how tiny the singular value. The $i$th computed singular value, ${\stackrel{^}{s}}_{i}$, satisfies the bound
 $s^i-si ≤ pnεsi$
where $\epsilon$ is the machine precision and $p\left(n\right)$ is a modest function of $n$.
For bounds on the computed singular values, see Section 4.9.1 of Anderson et al. (1999). See also nag_ddisna (f08flc).

## 8  Parallelism and Performance

nag_dbdsdc (f08mdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dbdsdc (f08mdc) 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.

If only singular values are required, the total number of floating-point operations is approximately proportional to ${n}^{2}$. When singular vectors are required the number of operations is bounded above by approximately the same number of operations as nag_dbdsqr (f08mec), but for large matrices nag_dbdsdc (f08mdc) is usually much faster.
There is no complex analogue of nag_dbdsdc (f08mdc).

## 10  Example

This example computes the singular value decomposition of the upper bidiagonal matrix
 $B = 3.62 1.26 0.00 0.00 0.00 -2.41 -1.53 0.00 0.00 0.00 1.92 1.19 0.00 0.00 0.00 -1.43 .$

### 10.1  Program Text

Program Text (f08mdce.c)

### 10.2  Program Data

Program Data (f08mdce.d)

### 10.3  Program Results

Program Results (f08mdce.r)