void	f08kdc (Nag_OrderType order, Nag_JobType job, Integer m, Integer n, double a[], Integer pda, double s[], double u[], Integer pdu, double vt[], Integer pdvt, NagError *fail)

The function may be called by the names: f08kdc, nag_lapackeig_dgesdd or nag_dgesdd.

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: $job$ – Nag_JobType Input

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

U

$job = Nag_DoAll$: All $m$ columns of $U$ and all $n$ rows of $V^{T}$ are returned in the arrays u and vt.
$job = Nag_DoSquare$: The first $\min (m, n)$ columns of $U$ and the first $\min (m, n)$ rows of $V^{T}$ are returned in the arrays u and vt.
$job = Nag_DoOverwrite$: If $m \geq n$ , the first $n$ columns of $U$ are overwritten on the array a and all rows of $V^{T}$ are returned in the array vt. Otherwise, all columns of $U$ are returned in the array u and the first $m$ rows of $V^{T}$ are overwritten in the array vt.
$job = Nag_DoNothing$: No columns of $U$ or rows of $V^{T}$ are computed.

Constraint:

job = Nag_DoAll

Nag_DoSquare

Nag_DoOverwrite

Nag_DoNothing

3: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: $a [\dim]$ – 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

job = Nag_DoOverwrite

, a is overwritten with the first

n

columns of

U

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

m \geq n

; a is overwritten with the first

m

rows of

V^{T}

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

job \neq Nag_DoOverwrite

, the contents of a are destroyed.

6: $pda$ – 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 $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

7: $s [\min (m, n)]$ – double Output

On exit: the singular values of

A

, sorted so that

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

8: $u [\dim]$ – double Output

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

$\max (1, pdu \times m)$ when $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m < n$ );
$\max (1, pdu \times \min (m, n))$ when $job = Nag_DoSquare$ and $order = Nag_ColMajor$ ;
$\max (1, m \times pdu)$ when $job = Nag_DoSquare$ 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:

job = Nag_DoAll

job = Nag_DoOverwrite

and

m < n

, u contains the

m \times m

orthogonal matrix

U

job = Nag_DoSquare

, u contains the first

\min (m, n)

columns of

U

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

job = Nag_DoOverwrite

and

m \geq n

, or

job = Nag_DoNothing

, u is not referenced.

9: $pdu$ – 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 $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m < n$ ), $pdu \geq \max (1, m)$ ;
- if $job = Nag_DoSquare$ , $pdu \geq \max (1, m)$ ;
- otherwise $pdu \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m < n$ ), $pdu \geq \max (1, m)$ ;
- if $job = Nag_DoSquare$ , $pdu \geq \max (1, \min (m, n))$ ;
- otherwise $pdu \geq 1$ .

10: $vt [\dim]$ – double Output

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

$\max (1, pdvt \times n)$ when $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m \geq n$ );
$\max (1, pdvt \times n)$ when $job = Nag_DoSquare$ and $order = Nag_ColMajor$ ;
$\max (1, \min (m, n) \times pdvt)$ when $job = Nag_DoSquare$ 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

job = Nag_DoAll

job = Nag_DoOverwrite

and

m \geq n

, vt contains the

n \times n

orthogonal matrix

V^{T}

job = Nag_DoSquare

, vt contains the first

\min (m, n)

rows of

V^{T}

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

job = Nag_DoOverwrite

and

m < n

, or

job = Nag_DoNothing

, vt is not referenced.

11: $pdvt$ – 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 $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m \geq n$ ), $pdvt \geq \max (1, n)$ ;
- if $job = Nag_DoSquare$ , $pdvt \geq \max (1, \min (m, n))$ ;
- otherwise $pdvt \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m \geq n$ ), $pdvt \geq \max (1, n)$ ;
- if $job = Nag_DoSquare$ , $pdvt \geq \max (1, n)$ ;
- otherwise $pdvt \geq 1$ .

12: $fail$ – 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: f08kdc did not converge, the updating process failed.
NE_ENUM_INT_3: On entry, $job = ⟨ value ⟩$ , $pdu = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m < n$ ), $pdu \geq \max (1, m)$ ;
if $job = Nag_DoSquare$ , $pdu \geq \max (1, m)$ ;
otherwise $pdu \geq 1$ .

On entry, $job = ⟨ value ⟩$ , $pdu = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m < n$ ), $pdu \geq \max (1, m)$ ;
if $job = Nag_DoSquare$ , $pdu \geq \max (1, \min (m, n))$ ;
otherwise $pdu \geq 1$ .

On entry, $job = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m \geq n$ ), $pdvt \geq \max (1, n)$ ;
if $job = Nag_DoSquare$ , $pdvt \geq \max (1, \min (m, n))$ ;
otherwise $pdvt \geq 1$ .

On entry, $job = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $job = Nag_DoAll$ or ( $job = Nag_DoOverwrite$ and $m \geq n$ ), $pdvt \geq \max (1, n)$ ;
if $job = Nag_DoSquare$ , $pdvt \geq \max (1, n)$ ;
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

f08kdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kdc 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 f08krc.

10 Example

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

4 \times 6

matrix

A = (\begin{array}{r} 2.27 & 0.28 & - 0.48 & 1.07 & - 2.35 & 0.62 \\ - 1.54 & - 1.67 & - 3.09 & 1.22 & 2.93 & - 7.39 \\ 1.15 & 0.94 & 0.99 & 0.79 & - 1.45 & 1.03 \\ - 1.94 & - 0.78 & - 0.21 & 0.63 & 2.30 & - 2.57 \end{array}),

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

The example program for f08kbc illustrates finding a singular value decomposition for the case

m \geq n

f08kd: FL CL CPP AD

NAG CL Interfacef08kdc (dgesdd)

▸▿ 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
f08kdc (dgesdd)