void	f08kbc (Nag_OrderType order, Nag_ComputeUType jobu, Nag_ComputeVTType jobvt, Integer m, Integer n, double a[], Integer pda, double s[], double u[], Integer pdu, double vt[], Integer pdvt, double work[], NagError *fail)

The function may be called by the names: f08kbc, nag_lapackeig_dgesvd or nag_dgesvd.

3 Description

The SVD is written as

A = U Σ V^{T},

where

Σ

is an

m \times n

matrix which is zero except for its

\min (m, n)

diagonal elements,

U

is an

m \times m

orthogonal matrix, and

V

is an

n \times n

orthogonal matrix. The diagonal elements of

Σ

are the singular values of

A

; they are real and non-negative, and are returned in descending order. The first

\min (m, n)

columns of

U

and

V

are the left and right singular vectors of

A

Note that the function returns

V^{T}

, not

V

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 https://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: $order$ – Nag_OrderType Input

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

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $jobu$ – Nag_ComputeUType Input

On entry: specifies options for computing all or part of the matrix

U

$jobu = Nag_AllU$: All $m$ columns of $U$ are returned in array u.
$jobu = Nag_SingularVecsU$: The first $\min (m, n)$ columns of $U$ (the left singular vectors) are returned in the array u.
$jobu = Nag_Overwrite$: The first $\min (m, n)$ columns of $U$ (the left singular vectors) are overwritten on the array a.
$jobu = Nag_NotU$: No columns of $U$ (no left singular vectors) are computed.

Constraint:

jobu = Nag_AllU

Nag_SingularVecsU

Nag_Overwrite

Nag_NotU

3: $jobvt$ – Nag_ComputeVTType Input

On entry: specifies options for computing all or part of the matrix

V^{T}

$jobvt = Nag_AllVT$: All $n$ rows of $V^{T}$ are returned in the array vt.
$jobvt = Nag_SingularVecsVT$: The first $\min (m, n)$ rows of $V^{T}$ (the right singular vectors) are returned in the array vt.
$jobvt = Nag_OverwriteVT$: The first $\min (m, n)$ rows of $V^{T}$ (the right singular vectors) are overwritten on the array a.
$jobvt = Nag_NotVT$: No rows of $V^{T}$ (no right singular vectors) are computed.

Constraints:

$jobvt = Nag_AllVT$ , $Nag_SingularVecsVT$ , $Nag_OverwriteVT$ or $Nag_NotVT$ ;
If $jobu = Nag_Overwrite$ , jobvt cannot be $Nag_OverwriteVT$ .

4: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

5: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

6: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m \times n

matrix

A

On exit: if

jobu = Nag_Overwrite

, a is overwritten with the first

\min (m, n)

columns of

U

(the left singular vectors, stored column-wise).

jobvt = Nag_OverwriteVT

, a is overwritten with the first

\min (m, n)

rows of

V^{T}

(the right singular vectors, stored row-wise).

jobu \neq Nag_Overwrite

and

jobvt \neq Nag_OverwriteVT

, the contents of a are destroyed.

7: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

8: $s [\dim]$ – double Output

Note: the dimension, dim, of the array s must be at least

\max (1, \min (m, n))

On exit: the singular values of

A

, sorted so that

s [i - 1] \geq s [i]

9: $u [\dim]$ – double Output

Note: the dimension, dim, of the array u must be at least

$\max (1, pdu \times m)$ when $jobu = Nag_AllU$ ;
$\max (1, pdu \times \min (m, n))$ when $jobu = Nag_SingularVecsU$ and $order = Nag_ColMajor$ ;
$\max (1, m \times pdu)$ when $jobu = Nag_SingularVecsU$ and $order = Nag_RowMajor$ ;
$\max (1, m)$ otherwise.

The

(i, j)

th element of the matrix

U

is stored in

$u [(j - 1) \times pdu + i - 1]$ when $order = Nag_ColMajor$ ;
$u [(i - 1) \times pdu + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobu = Nag_AllU

, u contains the

m \times m

orthogonal matrix

U

jobu = Nag_SingularVecsU

, u contains the first

\min (m, n)

columns of

U

(the left singular vectors, stored column-wise).

jobu = Nag_NotU

Nag_Overwrite

, u is not referenced.

10: $pdu$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array u.

Constraints:

if $order = Nag_ColMajor$ ,
- if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
- if $jobu = Nag_SingularVecsU$ , $pdu \geq \max (1, m)$ ;
- otherwise $pdu \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
- if $jobu = Nag_SingularVecsU$ , $pdu \geq \max (1, \min (m, n))$ ;
- otherwise $pdu \geq 1$ .

11: $vt [\dim]$ – double Output

Note: the dimension, dim, of the array vt must be at least

$\max (1, pdvt \times n)$ when $jobvt = Nag_AllVT$ ;
$\max (1, pdvt \times n)$ when $jobvt = Nag_SingularVecsVT$ and $order = Nag_ColMajor$ ;
$\max (1, \min (m, n) \times pdvt)$ when $jobvt = Nag_SingularVecsVT$ and $order = Nag_RowMajor$ ;
$\max (1, \min (m, n))$ otherwise.

The

(i, j)

th element of the matrix is stored in

$vt [(j - 1) \times pdvt + i - 1]$ when $order = Nag_ColMajor$ ;
$vt [(i - 1) \times pdvt + j - 1]$ when $order = Nag_RowMajor$ .

On exit: if

jobvt = Nag_AllVT

, vt contains the

n \times n

orthogonal matrix

V^{T}

jobvt = Nag_SingularVecsVT

, vt contains the first

\min (m, n)

rows of

V^{T}

(the right singular vectors, stored row-wise).

jobvt = Nag_NotVT

Nag_OverwriteVT

, vt is not referenced.

12: $pdvt$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array vt.

Constraints:

if $order = Nag_ColMajor$ ,
- if $jobvt = Nag_AllVT$ , $pdvt \geq \max (1, n)$ ;
- if $jobvt = Nag_SingularVecsVT$ , $pdvt \geq \max (1, \min (m, n))$ ;
- otherwise $pdvt \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $jobvt = Nag_AllVT$ , $pdvt \geq \max (1, n)$ ;
- if $jobvt = Nag_SingularVecsVT$ , $pdvt \geq \max (1, n)$ ;
- otherwise $pdvt \geq 1$ .

13: $work [\min (m, n)]$ – double Output

On exit: if

fail . code =

NE_CONVERGENCE,

WORK (2 : \min (m, n))

(using the notation described in Section 3.1.4 in the Introduction to the NAG Library CL Interface) contains the unconverged superdiagonal elements of an upper bidiagonal matrix

B

whose diagonal is in s (not necessarily sorted).

B

satisfies

A = U B V^{T}

, so it has the same singular values as

A

, and singular vectors related by

U

and

V^{T}

14: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE: If f08kbc did not converge, $fail . errnum$ specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT_2: On entry, $jobu = ⟨ value ⟩$ , $pdu = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
if $jobu = Nag_SingularVecsU$ , $pdu \geq \max (1, m)$ ;
otherwise $pdu \geq 1$ .

On entry, $jobvt = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvt = Nag_AllVT$ , $pdvt \geq \max (1, n)$ ;
if $jobvt = Nag_SingularVecsVT$ , $pdvt \geq \max (1, n)$ ;
otherwise $pdvt \geq 1$ .
NE_ENUM_INT_3: On entry, $jobu = ⟨ value ⟩$ , $pdu = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobu = Nag_AllU$ , $pdu \geq \max (1, m)$ ;
if $jobu = Nag_SingularVecsU$ , $pdu \geq \max (1, \min (m, n))$ ;
otherwise $pdu \geq 1$ .

On entry, $jobvt = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvt = Nag_AllVT$ , $pdvt \geq \max (1, n)$ ;
if $jobvt = Nag_SingularVecsVT$ , $pdvt \geq \max (1, \min (m, n))$ ;
otherwise $pdvt \geq 1$ .
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .

On entry, $pdu = ⟨ value ⟩$ .
Constraint: $pdu > 0$ .

On entry, $pdvt = ⟨ value ⟩$ .
Constraint: $pdvt > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix

(A + E)

, where

{‖ E ‖}_{2} = O (ε) {‖ A ‖}_{2},

and

ε

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.

8 Parallelism and Performance

f08kbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kbc 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.

9 Further Comments

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 f08kpc.

10 Example

This example finds the singular values and left and right singular vectors of the

6 \times 4

matrix

A = (\begin{array}{r} 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 \end{array}),

together with approximate error bounds for the computed singular values and vectors.

The example program for f08kdc illustrates finding a singular value decomposition for the case

m \leq n

f08kb: FL CL CPP AD

NAG CL Interfacef08kbc (dgesvd)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f08kbc (dgesvd)