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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_eigen_complex_triang_svd (f02xu)

## Purpose

nag_eigen_complex_triang_svd (f02xu) returns all, or part, of the singular value decomposition of a complex upper triangular matrix.

## Syntax

[a, b, q, sv, rwork, ifail] = f02xu(a, b, wantq, wantp, 'n', n, 'ncolb', ncolb)
[a, b, q, sv, rwork, ifail] = nag_eigen_complex_triang_svd(a, b, wantq, wantp, 'n', n, 'ncolb', ncolb)

## Description

The n$n$ by n$n$ upper triangular matrix R$R$ is factorized as
 R = QSPH, $R=QSPH,$
where Q$Q$ and P$P$ are n$n$ by n$n$ unitary matrices and S$S$ is an n$n$ by n$n$ diagonal matrix with real non-negative diagonal elements, sv1,sv2,,svn$s{v}_{1},s{v}_{2},\dots ,s{v}_{n}$, ordered such that
 sv1 ≥ sv2 ≥ ⋯ ≥ svn ≥ 0. $sv1≥sv2≥⋯≥svn≥0.$
The columns of Q$Q$ are the left-hand singular vectors of R$R$, the diagonal elements of S$S$ are the singular values of R$R$ and the columns of P$P$ are the right-hand singular vectors of R$R$.
Either or both of Q$Q$ and PH${P}^{\mathrm{H}}$ may be requested and the matrix C$C$ given by
 C = QHB, $C=QHB,$
where B$B$ is an n$n$ by ncolb$\mathit{ncolb}$ given matrix, may also be requested.
nag_eigen_complex_triang_svd (f02xu) obtains the singular value decomposition by first reducing R$R$ to bidiagonal form by means of Givens plane rotations and then using the QR$QR$ algorithm to obtain the singular value decomposition of the bidiagonal form.
Good background descriptions to the singular value decomposition are given in Dongarra et al. (1979), Hammarling (1985) and Wilkinson (1978).
Note that if K$K$ is any unitary diagonal matrix so that
 KKH = I, $KKH=I,$
then
 A = (QK)S(PK)H $A=(QK)S(PK)H$
is also a singular value decomposition of A$A$.

## References

Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25
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, : $:$) – complex array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The leading n$n$ by n$n$ upper triangular part of the array a must contain the upper triangular matrix R$R$.
2:     b(ldb, : $:$) – complex array
The first dimension, ldb, of the array b must satisfy
• if ncolb > 0${\mathbf{ncolb}}>0$, ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldb1$\mathit{ldb}\ge 1$.
The second dimension of the array must be at least max (1,ncolb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncolb}}\right)$
If ncolb > 0${\mathbf{ncolb}}>0$, the leading n$n$ by ncolb$\mathit{ncolb}$ part of the array b must contain the matrix to be transformed.
3:     wantq – logical scalar
Must be true if the matrix Q$Q$ is required.
If wantq = false${\mathbf{wantq}}=\mathbf{false}$ then the array q is not referenced.
4:     wantp – logical scalar
Must be true if the matrix PH${P}^{\mathrm{H}}$ is required, in which case PH${P}^{\mathrm{H}}$ is returned in the array a, otherwise wantp must be false.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
n$n$, the order of the matrix R$R$.
If n = 0${\mathbf{n}}=0$, an immediate return is effected.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     ncolb – int64int32nag_int scalar
Default: The second dimension of the array b.
ncolb$\mathit{ncolb}$, the number of columns of the matrix B$B$.
If ncolb = 0${\mathbf{ncolb}}=0$, the array b is not referenced.
Constraint: ncolb0${\mathbf{ncolb}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb ldq cwork

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If wantp = true${\mathbf{wantp}}=\mathbf{true}$, the n$n$ by n$n$ part of a will contain the n$n$ by n$n$ unitary matrix PH${P}^{\mathrm{H}}$, otherwise the n$n$ by n$n$ upper triangular part of a is used as internal workspace, but the strictly lower triangular part of a is not referenced.
2:     b(ldb, : $:$) – complex array
The first dimension, ldb, of the array b will be
• if ncolb > 0${\mathbf{ncolb}}>0$, ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldb1$\mathit{ldb}\ge 1$.
The second dimension of the array will be max (1,ncolb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ncolb}}\right)$
Stores the n$n$ by ncolb$\mathit{ncolb}$ matrix QHB${Q}^{\mathrm{H}}B$.
3:     q(ldq, : $:$) – complex array
The first dimension, ldq, of the array q will be
• if wantq = true${\mathbf{wantq}}=\mathbf{true}$, ldqmax (1,n)$\mathit{ldq}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldq1$\mathit{ldq}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if wantq = true${\mathbf{wantq}}=\mathbf{true}$, and at least 1$1$ otherwise
If wantq = true${\mathbf{wantq}}=\mathbf{true}$, the leading n$n$ by n$n$ part of the array q will contain the unitary matrix Q$Q$. Otherwise the array q is not referenced.
4:     sv(n) – double array
The n$n$ diagonal elements of the matrix S$S$.
5:     rwork( : $:$) – double array
Note: the dimension of the array rwork must be at least max (1,2 × (n1))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×\left({\mathbf{n}}-1\right)\right)$ if ncolb = 0${\mathbf{ncolb}}=0$ and wantq = false${\mathbf{wantq}}=\mathbf{false}$ and wantp = false${\mathbf{wantp}}=\mathbf{false}$, max (1,3 × (n1))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×\left({\mathbf{n}}-1\right)\right)$ if ncolb = 0${\mathbf{ncolb}}=0$ and wantq = false${\mathbf{wantq}}=\mathbf{false}$ and wantp = true${\mathbf{wantp}}=\mathbf{true}$ or ncolb > 0${\mathbf{ncolb}}>0$ and wantp = false${\mathbf{wantp}}=\mathbf{false}$ or wantq = true${\mathbf{wantq}}=\mathbf{true}$ and wantp = false${\mathbf{wantp}}=\mathbf{false}$, and at least max (1,5 × (n1))$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,5×\left({\mathbf{n}}-1\right)\right)$ otherwise.
rwork(n) contains the total number of iterations taken by the QR$QR$ algorithm.
The rest of the array is used as workspace.
6:     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:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail = 1${\mathbf{ifail}}=-1$
 On entry, n < 0${\mathbf{n}}<0$, or lda < n$\mathit{lda}<{\mathbf{n}}$, or ncolb < 0${\mathbf{ncolb}}<0$, or ldb < n$\mathit{ldb}<{\mathbf{n}}$ and ncolb > 0${\mathbf{ncolb}}>0$, or ldq < n$\mathit{ldq}<{\mathbf{n}}$ and wantq = true${\mathbf{wantq}}=\mathbf{true}$
W ifail > 0${\mathbf{ifail}}>0$
The QR$QR$ algorithm has failed to converge in 50 × n$50×{\mathbf{n}}$ iterations. In this case sv(1),sv(2),,sv(ifail)${\mathbf{sv}}\left(1\right),{\mathbf{sv}}\left(2\right),\dots ,{\mathbf{sv}}\left({\mathbf{ifail}}\right)$ may not have been found correctly and the remaining singular values may not be the smallest. The matrix R$R$ will nevertheless have been factorized as R = QEPH$R=QE{P}^{\mathrm{H}}$, where E$E$ is a 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 the diagonal elements and rwork(1),rwork(2),,rwork(n1)${\mathbf{rwork}}\left(1\right),{\mathbf{rwork}}\left(2\right),\dots ,{\mathbf{rwork}}\left(n-1\right)$ as the superdiagonal elements.
This failure is not likely to occur.

## Accuracy

The computed factors Q$Q$, S$S$ and P$P$ satisfy the relation
 QSPH = A + E, $QSPH=A+E,$
where
 ‖E‖ ≤ cε ‖A‖, $‖E‖≤cε ‖A‖,$
ε$\epsilon$ is the machine precision, c$c$ is a modest function of n$n$ and . $‖.‖$ denotes the spectral (two) norm. Note that A = sv1$‖A‖=s{v}_{1}$.

For given values of ncolb, wantq and wantp, the number of floating point operations required is approximately proportional to n3${n}^{3}$.
>Following the use of nag_eigen_complex_triang_svd (f02xu) the rank of R$R$ may be estimated as follows:
```tol = eps;
irank = 1;
while (irank <= numel(sv) && sv(irank) >= tol*sv(1) )
irank = irank + 1;
end
```
returns the value k$k$ in irank, where k$k$ is the smallest integer for which sv(k) < tol × sv(1)${\mathbf{sv}}\left(k\right)<\mathit{tol}×{\mathbf{sv}}\left(1\right)$, where tol$\mathit{tol}$ is typically the machine precision, so that irank is an estimate of the rank of S$S$ and thus also of R$R$.

## Example

```function nag_eigen_complex_triang_svd_example
a = [1,  1 + 1i,  1 + 1i;
0 + 0i,  -2 + 0i,  -1 - 1i;
0 + 0i,  0 + 0i,  -3 + 0i];
b = [ 1 + 1i;  -1 + 0i;  -1 + 1i];
wantq = true;
wantp = true;
[aOut, bOut, q, sv, rwork, ifail] = nag_eigen_complex_triang_svd(a, b, wantq, wantp)
```
```

aOut =

0.1275 + 0.0000i   0.3899 + 0.2046i   0.5289 + 0.7142i
0.2265 + 0.0000i   0.3397 + 0.7926i   0.0000 - 0.4529i
0.9656 + 0.0000i  -0.1311 - 0.2129i  -0.0698 + 0.0119i

bOut =

1.9656 + 0.7935i
-0.1132 + 0.3397i
0.0915 + 0.6086i

q =

0.5005 + 0.0000i   0.4529 + 0.0000i   0.7378 + 0.0000i
-0.5152 + 0.1514i  -0.1132 + 0.5661i   0.4190 - 0.4502i
-0.4041 + 0.5457i  -0.0000 - 0.6794i   0.2741 + 0.0468i

sv =

3.9263
2.0000
0.7641

rwork =

0
0
1.0000
0.7071
0
0
0
0
0
0

ifail =

0

```
```function f02xu_example
a = [1,  1 + 1i,  1 + 1i;
0 + 0i,  -2 + 0i,  -1 - 1i;
0 + 0i,  0 + 0i,  -3 + 0i];
b = [ 1 + 1i;  -1 + 0i;  -1 + 1i];
wantq = true;
wantp = true;
[aOut, bOut, q, sv, rwork, ifail] = f02xu(a, b, wantq, wantp)
```
```

aOut =

0.1275 + 0.0000i   0.3899 + 0.2046i   0.5289 + 0.7142i
0.2265 + 0.0000i   0.3397 + 0.7926i   0.0000 - 0.4529i
0.9656 + 0.0000i  -0.1311 - 0.2129i  -0.0698 + 0.0119i

bOut =

1.9656 + 0.7935i
-0.1132 + 0.3397i
0.0915 + 0.6086i

q =

0.5005 + 0.0000i   0.4529 + 0.0000i   0.7378 + 0.0000i
-0.5152 + 0.1514i  -0.1132 + 0.5661i   0.4190 - 0.4502i
-0.4041 + 0.5457i  -0.0000 - 0.6794i   0.2741 + 0.0468i

sv =

3.9263
2.0000
0.7641

rwork =

0
0
1.0000
0.7071
0
0
0
0
0
0

ifail =

0

```