﻿ f08kb Method
f08kb computes the singular value decomposition (SVD) of a real $m$ by $n$ matrix $A$, optionally computing the left and/or right singular vectors.

# Syntax

C#
```public static void f08kb(
string jobu,
string jobvt,
int m,
int n,
double[,] a,
double[] s,
double[,] u,
double[,] vt,
out int info
)```
Visual Basic
```Public Shared Sub f08kb ( _
jobu As String, _
jobvt As String, _
m As Integer, _
n As Integer, _
a As Double(,), _
s As Double(), _
u As Double(,), _
vt As Double(,), _
<OutAttribute> ByRef info As Integer _
)```
Visual C++
```public:
static void f08kb(
String^ jobu,
String^ jobvt,
int m,
int n,
array<double,2>^ a,
array<double>^ s,
array<double,2>^ u,
array<double,2>^ vt,
[OutAttribute] int% info
)```
F#
```static member f08kb :
jobu : string *
jobvt : string *
m : int *
n : int *
a : float[,] *
s : float[] *
u : float[,] *
vt : float[,] *
info : int byref -> unit
```

#### Parameters

jobu
Type: System..::..String
On entry: specifies options for computing all or part of the matrix $U$.
${\mathbf{jobu}}=\text{"A"}$
All $m$ columns of $U$ are returned in array u.
${\mathbf{jobu}}=\text{"S"}$
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.
${\mathbf{jobu}}=\text{"O"}$
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.
${\mathbf{jobu}}=\text{"N"}$
No columns of $U$ (no left singular vectors) are computed.
Constraint: ${\mathbf{jobu}}=\text{"A"}$, $\text{"S"}$, $\text{"O"}$ or $\text{"N"}$.
jobvt
Type: System..::..String
On entry: specifies options for computing all or part of the matrix ${V}^{\mathrm{T}}$.
${\mathbf{jobvt}}=\text{"A"}$
All $n$ rows of ${V}^{\mathrm{T}}$ are returned in the array vt.
${\mathbf{jobvt}}=\text{"S"}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{T}}$ (the right singular vectors) are returned in the array vt.
${\mathbf{jobvt}}=\text{"O"}$
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of ${V}^{\mathrm{T}}$ (the right singular vectors) are overwritten on the array a.
${\mathbf{jobvt}}=\text{"N"}$
No rows of ${V}^{\mathrm{T}}$ (no right singular vectors) are computed.
Constraints:
• ${\mathbf{jobvt}}=\text{"A"}$, $\text{"S"}$, $\text{"O"}$ or $\text{"N"}$;
• jobvt and jobu cannot both be $\text{"O"}$.
m
Type: System..::..Int32
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
n
Type: System..::..Int32
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
a
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: if ${\mathbf{jobu}}=\text{"O"}$, 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 ${\mathbf{jobvt}}=\text{"O"}$, a is overwritten with 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 ${\mathbf{jobu}}\ne \text{"O"}$ and ${\mathbf{jobvt}}\ne \text{"O"}$, the contents of a are destroyed.
s
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension 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 ${\mathbf{s}}\left[i-1\right]\ge {\mathbf{s}}\left[i\right]$.
u
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint:
• if ${\mathbf{jobu}}=\text{"A"}$ or $\text{"S"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise $\mathrm{dim1}\ge 1$.
• if ${\mathbf{jobu}}=\text{"A"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{jobu}}=\text{"S"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• otherwise $\mathrm{dim1}\ge 1$.
Note: the second dimension of the array u must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobu}}=\text{"A"}$, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$ if ${\mathbf{jobu}}=\text{"S"}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobu}}=\text{"A"}$, u contains the $m$ by $m$ orthogonal matrix $U$.
If ${\mathbf{jobu}}=\text{"S"}$, 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 ${\mathbf{jobu}}=\text{"N"}$ or $\text{"O"}$, u is not referenced.
vt
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint:
• if ${\mathbf{jobvt}}=\text{"A"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{jobvt}}=\text{"S"}$, $\mathrm{dim1}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$;
• otherwise $\mathrm{dim1}\ge 1$.
Note: the second dimension of the array vt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvt}}=\text{"A"}$, $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvt}}=\text{"S"}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvt}}=\text{"A"}$, vt contains the $n$ by $n$ orthogonal matrix ${V}^{\mathrm{T}}$.
If ${\mathbf{jobvt}}=\text{"S"}$, 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 ${\mathbf{jobvt}}=\text{"N"}$ or $\text{"O"}$, vt is not referenced.
info
Type: System..::..Int32%
On exit: ${\mathbf{info}}=0$ unless the method detects an error (see [Error Indicators and Warnings]).

# Description

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 method returns ${V}^{\mathrm{T}}$, not $V$.

# 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

# Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA, LDU, LDVT) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
If f08kb did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero. See the description of work above for details.
${\mathbf{ifail}}=-9000$
An error occured, see message report.
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-4000$
Invalid dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-8000$
Negative dimension for array $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$
${\mathbf{ifail}}=-6000$
Invalid Parameters $〈\mathit{\text{value}}〉$

# Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix $\left(A+E\right)$, where
 $E2=OεA2,$
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.

# Parallelism and Performance

None.

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 method is f08kp.

# Example

This example finds the singular values and left and right singular vectors of the $6$ by $4$ matrix
 $A= 2.27 -1.54 1.15 -1.94 0.28 -1.67 0.94 -0.78 -0.48 -3.09 0.99 -0.21 1.07 1.22 0.79 0.63 -2.35 2.93 -1.45 2.30 0.62 -7.39 1.03 -2.57 ,$
together with approximate error bounds for the computed singular values and vectors.

Example program (C#): f08kbe.cs

Example program data: f08kbe.d

Example program results: f08kbe.r