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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_eigen_real_gen_qu_svd (f02wd)

Purpose

nag_eigen_real_gen_qu_svd (f02wd) returns the Householder QU$QU$ factorization of a real rectangular m$m$ by n$n$ (mn)$\left(m\ge n\right)$ matrix A$A$. Further, on request or if A$A$ is not of full rank, part or all of the singular value decomposition of A$A$ is returned.

Syntax

[a, b, svd, irank, z, sv, r, pt, work, ifail] = f02wd(a, wantb, b, tol, svd, wantr, wantpt, lwork, 'm', m, 'n', n)
[a, b, svd, irank, z, sv, r, pt, work, ifail] = nag_eigen_real_gen_qu_svd(a, wantb, b, tol, svd, wantr, wantpt, lwork, 'm', m, 'n', n)

Description

The real m$m$ by n$n$ (mn)$\left(m\ge n\right)$ matrix A$A$ is first factorized as
A = Q
 ( U ) 0
,
$A=Q U 0 ,$
where Q$Q$ is an m$m$ by m$m$ orthogonal matrix and U$U$ is an n$n$ by n$n$ upper triangular matrix.
If either U$U$ is singular or svd is supplied as true, then the singular value decomposition (SVD) of U$U$ is obtained so that U$U$ is factorized as
 U = RDPT, $U=RDPT,$
where R$R$ and P$P$ are n$n$ by n$n$ orthogonal matrices and D$D$ is the n$n$ by n$n$ diagonal matrix
 D = diag(sv1,sv2, … ,svn), $D=diag(sv1,sv2,…,svn),$
with sv1sv2svn0.$s{v}_{1}\ge s{v}_{2}\ge \cdots \ge s{v}_{n}\ge 0\text{.}$
Note that the SVD of A$A$ is then given by
A = Q1
 ( D ) 0
PT  where  Q1 = Q
 ( R 0 ) 0 I
,
$A=Q1 D 0 PT where Q1=Q R 0 0 I ,$
the diagonal elements of D$D$ being the singular values of A$A$.
The option to form a vector QTb${Q}^{\mathrm{T}}b$, or if appropriate Q1T b${Q}_{1}^{\mathrm{T}}b$, is also provided.
The rank of the matrix A$A$, based upon a user-supplied parameter tol, is also returned.
The QU$QU$ factorization of A$A$ is obtained by Householder transformations. To obtain the SVD of U$U$ the matrix is first reduced to bidiagonal form by means of plane rotations and then the QR$QR$ algorithm is used to obtain the SVD of the bidiagonal form.

References

Wilkinson J H (1978) Singular Value Decomposition – Basic Aspects Numerical Software – Needs and Availability (ed D A H Jacobs) Academic Press

Parameters

Compulsory Input Parameters

1:     a(lda,n) – double array
lda, the first dimension of the array, must satisfy the constraint ldam$\mathit{lda}\ge {\mathbf{m}}$.
The leading m$m$ by n$n$ part of a must contain the matrix to be factorized.
2:     wantb – logical scalar
Must be true if QTb${Q}^{\mathrm{T}}b$ or Q1Tb${Q}_{1}^{\mathrm{T}}b$ is required.
If on entry wantb = false${\mathbf{wantb}}=\mathbf{false}$, b is not referenced.
3:     b(m) – double array
m, the dimension of the array, must satisfy the constraint mn${\mathbf{m}}\ge {\mathbf{n}}$.
If wantb is supplied as true, b must contain the m$m$ element vector b$b$. Otherwise, b is not referenced.
4:     tol – double scalar
Must specify a relative tolerance to be used to determine the rank of A$A$. tol should be chosen as approximately the largest relative error in the elements of A$A$. For example, if the elements of A$A$ are correct to about 4$4$ significant figures, tol should be set to about 5 × 104$5×{10}^{-4}$. See Section [Determining the Rank of ] for a description of how tol is used to determine rank.
If tol is outside the range (ε,1.0)$\left(\epsilon ,1.0\right)$, where ε$\epsilon$ is the machine precision, the value ε$\epsilon$ is used in place of tol. For most problems this is unreasonably small.
5:     svd – logical scalar
Must be true if the singular values are to be found even if A$A$ is of full rank.
If before entry, svd = false${\mathbf{svd}}=\mathbf{false}$ and A$A$ is determined to be of full rank, only the QU$QU$ factorization of A$A$ is computed.
6:     wantr – logical scalar
Must be true if the orthogonal matrix R$R$ is required when the singular values are computed.
If on entry wantr = false${\mathbf{wantr}}=\mathbf{false}$, r is not referenced.
7:     wantpt – logical scalar
Must be true if the orthogonal matrix PT${P}^{\mathrm{T}}$ is required when the singular values are computed.
Note that if svd is returned as true, pt is referenced even if wantpt is supplied as false, but see parameter pt.
8:     lwork – int64int32nag_int scalar
The dimension of the array work as declared in the (sub)program from which nag_eigen_real_gen_qu_svd (f02wd) is called.
Constraint: lwork3 × n${\mathbf{lwork}}\ge 3×{\mathbf{n}}$.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array b and the first dimension of the array a. (An error is raised if these dimensions are not equal.)
m$m$, the number of rows of the matrix A$A$.
Constraint: mn${\mathbf{m}}\ge {\mathbf{n}}$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: 1nm$1\le {\mathbf{n}}\le {\mathbf{m}}$.

lda ldr ldpt

Output Parameters

1:     a(lda,n) – double array
ldam$\mathit{lda}\ge {\mathbf{m}}$.
The leading m$m$ by n$n$ part of a, together with the n$n$-element vector z, contains details of the Householder QU$QU$ factorization.
Details of the storage of the QU$QU$ factorization are given in Section [Storage Details of the Factorization].
2:     b(m) – double array
Contains Q1Tb${Q}_{1}^{\mathrm{T}}b$ if svd is returned as true and QTb${Q}^{\mathrm{T}}b$ if svd is returned as false.
3:     svd – logical scalar
Is returned as false if only the QU$QU$ factorization of A$A$ has been obtained and is returned as true if the singular values of A$A$ have been obtained.
4:     irank – int64int32nag_int scalar
Returns the rank of the matrix A$A$. (It should be noted that it is possible for irank to be returned as n$n$ and svd to be returned as true, even if svd was supplied as false. This means that the matrix U$U$ only just failed the test for nonsingularity.)
5:     z(n) – double array
The n$n$-element vector z contains some details of the Householder transformations. See Section [Storage Details of the Factorization] for further information.
6:     sv(n) – double array
If svd is returned as true, sv contains the n$n$ singular values of A$A$ arranged in descending order.
7:     r(ldr,n) – double array
The first dimension, ldr, of the array r will be
• if wantr = true${\mathbf{wantr}}=\mathbf{true}$, ldrn$\mathit{ldr}\ge {\mathbf{n}}$;
• otherwise 1$1$.
The second dimension of the array will be n${\mathbf{n}}$ if wantr = true${\mathbf{wantr}}=\mathbf{true}$, and at least 1$1$ otherwise
If svd is returned as true and wantr was supplied as true, the leading n$n$ by n$n$ part of r will contain the left-hand orthogonal matrix of the svd of U$U$.
8:     pt(ldpt,n) – double array
ldptn$\mathit{ldpt}\ge {\mathbf{n}}$.
If svd is returned as true and wantpt was supplied as true, the leading n$n$ by n$n$ part of pt contains the orthogonal matrix PT${P}^{\mathrm{T}}$.
If svd is returned as true, but wantpt was supplied as false, the leading n$n$ by n$n$ part of pt is used for internal workspace.
9:     work(lwork) – double array
If svd is returned as false, work(1)${\mathbf{work}}\left(1\right)$ contains the condition number UEU1E${‖U‖}_{E}{‖{U}^{-1}‖}_{E}$ of the upper triangular matrix U$U$.
If svd is returned as true, work(1)${\mathbf{work}}\left(1\right)$ will contain the total number of iterations taken by the QR$QR$ algorithm.
The rest of the array is used as workspace and so contains no meaningful information.
10:   ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$, or m < n${\mathbf{m}}<{\mathbf{n}}$, or lda < m$\mathit{lda}<{\mathbf{m}}$, or ldr < n$\mathit{ldr}<{\mathbf{n}}$ when wantr = true${\mathbf{wantr}}=\mathbf{true}$, or ldpt < n$\mathit{ldpt}<{\mathbf{n}}$ or lwork < 3 × n${\mathbf{lwork}}<3×{\mathbf{n}}$.
(The function only checks ldr if wantr is supplied as true.)
ifail > 1${\mathbf{ifail}}>1$
The QR$QR$ algorithm has failed to converge to the singular values in 50 × n$50×{\mathbf{n}}$ iterations. In this case sv(1),sv(2),,sv(ifail1)${\mathbf{sv}}\left(1\right),{\mathbf{sv}}\left(2\right),\dots ,{\mathbf{sv}}\left({\mathbf{ifail}}-1\right)$ may not have been correctly found and the remaining singular values may not be the smallest singular values. The matrix A$A$ has nevertheless been factorized as A = Q1CPT$A={Q}_{1}C{P}^{\mathrm{T}}$, where C$C$ is an upper bidiagonal matrix with sv(1),sv(2),,sv(n)${\mathbf{sv}}\left(1\right),{\mathbf{sv}}\left(2\right),\dots ,{\mathbf{sv}}\left(n\right)$ as its diagonal elements and work(2),work(3),,work(n)${\mathbf{work}}\left(2\right),{\mathbf{work}}\left(3\right),\dots ,{\mathbf{work}}\left(n\right)$ as its superdiagonal elements.
This failure cannot occur if svd is returned as false and in any case is extremely rare.

Accuracy

The computed factors Q$Q$, U$U$, R$R$, D$D$ and PT${P}^{\mathrm{T}}$ satisfy the relations
Q
 ( U ) 0
= A + E,
$Q U 0 =A+E,$
Q
 ( R 0 ) 0 I
 ( D ) 0
PT = A + F
$Q R 0 0 I D 0 PT=A+F$
where E2c1ε A2${‖E‖}_{2}\le {c}_{1}\epsilon {‖A‖}_{2}$, F2c2ε A2${‖F‖}_{2}\le {c}_{2}\epsilon {‖A‖}_{2}$,
ε$\epsilon$ being the machine precision and c1${c}_{1}$ and c2${c}_{2}$ are modest functions of m$m$ and n$n$. Note that A2 = sv1${‖A‖}_{2}=s{v}_{1}$.

Timing

The time taken by nag_eigen_real_gen_qu_svd (f02wd) to obtain the Householder QU$QU$ factorization is approximately proportional to n2(3mn)${n}^{2}\left(3m-n\right)$.
The additional time taken to obtain the singular value decomposition is approximately proportional to n3${n}^{3}$, where the constant of proportionality depends upon whether or not the orthogonal matrices R$R$ and PT${P}^{\mathrm{T}}$ are required.

General Remarks

Singular vectors associated with a zero or multiple singular value, are not uniquely determined, even in exact arithmetic, and very different results may be obtained if they are computed on different machines.
This function is called by the least squares function nag_linsys_real_gen_solve (f04jg).

Determining the Rank of A

Following the QU$QU$ factorization of A$A$, if svd is supplied as false, then the condition number of U$U$ given by
 C(U) = ‖U‖F ‖U − 1‖F $C(U)=‖U‖F ‖U-1‖F$
is found, where .F${‖.‖}_{F}$ denotes the Frobenius norm, and if C(U)$C\left(U\right)$ is such that
 C(U) × tol > 1.0 $C(U)×tol>1.0$
then U$U$ is regarded as singular and the singular values of A$A$ are computed. If this test is not satisfied, then the rank of A$A$ is set to n$n$. Note that if svd is supplied as true then this test is omitted.
When the singular values are computed, then the rank of A$A$, r$r$, is returned as the largest integer such that
 svr > tol × sv1, $svr>tol×sv1,$
unless sv1 = 0$s{v}_{1}=0$ in which case r$r$ is returned as zero. That is, singular values which satisfy svitol × sv1$s{v}_{i}\le {\mathbf{tol}}×s{v}_{1}$ are regarded as negligible because relative perturbations of order tol can make such singular values zero.

Storage Details of the QU Factorization

The k$k$th Householder transformation matrix, Tk${T}_{k}$, used in the QU$QU$ factorization is chosen to introduce the zeros into the k$k$th column and has the form
Tk = I2
 ( 0 ) u
 ( 0 uT )
,  uTu = 1,
$Tk=I-2 0 u 0 uT , uTu=1,$
where u$u$ is an (mk + 1)$\left(m-k+1\right)$ element vector.
In place of u$u$ the function actually computes the vector z$z$ given by
 z = 2u1u. $z=2u1u.$
The first element of z$z$ is stored in z(k)${\mathbf{z}}\left(k\right)$ and the remaining elements of z$z$ are overwritten on the subdiagonal elements of the k$k$th column of a. The upper triangular matrix U$U$ is overwritten on the n$n$ by n$n$ upper triangular part of a.

Example

```function nag_eigen_real_gen_qu_svd_example
a = [22.25, 31.75, -38.25, 65.5;
20, 26.75, 28.5, -26.5;
-15.25, 24.25, 27.75, 18.5;
27.25, 10, 3, 2;
-17.25, -30.75, 11.25, 7.5;
17.25, 30.75, -11.25, -7.5];
wantb = false;
b = [0;
0;
0;
0;
0;
0];
tol = 0.0005;
svd = true;
wantr = true;
wantpt = true;
lwork = int64(24);
[aOut, bOut, svdOut, irank, z, sv, r, pt, work, ifail] = ...
nag_eigen_real_gen_qu_svd(a, wantb, b, tol, svd, wantr, wantpt, lwork)
```
```

aOut =

-49.6519  -44.4092   20.3542   -8.8818
0.4028  -48.2767   -9.5887  -20.3761
-0.3071    0.8369   52.9270  -48.8806
0.5488   -0.3907   -0.8364  -50.6742
-0.3474   -0.2585   -0.1851    0.6321
0.3474    0.2585    0.1851   -0.6321

bOut =

0
0
0
0
0
0

svdOut =

1

irank =

4

z =

1.4481
1.1153
1.4817
1.4482

sv =

91.0000
68.2500
45.5000
22.7500

r =

0.5639   -0.6344    0.4229    0.3172
0.3512   -0.3951   -0.6791   -0.5093
0.6403    0.5692    0.3095   -0.4126
0.3855    0.3427   -0.5140    0.6854

pt =

-0.3077   -0.4615    0.4615   -0.6923
0.4615    0.6923    0.3077   -0.4615
-0.4615    0.3077    0.6923    0.4615
-0.6923    0.4615   -0.4615   -0.3077

work =

1.0000
-49.6523
44.2299
28.9880
0
0
0.8037
0.9584
0
0
-0.5951
-0.2854
0
0
0
0
0
0
0
0
0
0
0
0

ifail =

0

```
```function f02wd_example
a = [22.25, 31.75, -38.25, 65.5;
20, 26.75, 28.5, -26.5;
-15.25, 24.25, 27.75, 18.5;
27.25, 10, 3, 2;
-17.25, -30.75, 11.25, 7.5;
17.25, 30.75, -11.25, -7.5];
wantb = false;
b = [0;
0;
0;
0;
0;
0];
tol = 0.0005;
svd = true;
wantr = true;
wantpt = true;
lwork = int64(24);
[aOut, bOut, svdOut, irank, z, sv, r, pt, work, ifail] = ...
f02wd(a, wantb, b, tol, svd, wantr, wantpt, lwork)
```
```

aOut =

-49.6519  -44.4092   20.3542   -8.8818
0.4028  -48.2767   -9.5887  -20.3761
-0.3071    0.8369   52.9270  -48.8806
0.5488   -0.3907   -0.8364  -50.6742
-0.3474   -0.2585   -0.1851    0.6321
0.3474    0.2585    0.1851   -0.6321

bOut =

0
0
0
0
0
0

svdOut =

1

irank =

4

z =

1.4481
1.1153
1.4817
1.4482

sv =

91.0000
68.2500
45.5000
22.7500

r =

0.5639   -0.6344    0.4229    0.3172
0.3512   -0.3951   -0.6791   -0.5093
0.6403    0.5692    0.3095   -0.4126
0.3855    0.3427   -0.5140    0.6854

pt =

-0.3077   -0.4615    0.4615   -0.6923
0.4615    0.6923    0.3077   -0.4615
-0.4615    0.3077    0.6923    0.4615
-0.6923    0.4615   -0.4615   -0.3077

work =

1.0000
-49.6523
44.2299
28.9880
0
0
0.8037
0.9584
0
0
-0.5951
-0.2854
0
0
0
0
0
0
0
0
0
0
0
0

ifail =

0

```