f08kmc (Nag_OrderType order, Nag_ComputeSingularVecsType jobu, Nag_ComputeSingularVecsType jobvt, Nag_RangeType range, Integer m, Integer n, double a[], Integer pda, double vl, double vu, Integer il, Integer iu, Integer *ns, double s[], double u[], Integer pdu, double vt[], Integer pdvt, double work[], Integer jfail[], NagError *fail)

The function may be called by the names: f08kmc, nag_lapackeig_dgesvdx or nag_dgesvdx.

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

, respectively.

Note that the function returns

V^{T}

, not

V

Alternative to computing all singular values of

A

, a selected set can be computed. The set is either those singular values lying in a given interval,

σ \in (v_{l}, v_{u}]

, or those whose index (counting from largest to smallest in magnitude) lies in a given range

1 \leq i_{l}, \dots, i_{u} \leq n

. In these cases, the corresponding left and right singular vectors can optionally be computed.

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_ComputeSingularVecsType Input

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

U

$jobu = Nag_SingularVecs$: The ns columns of $U$ , as specified by range, are returned in array u.
$jobu = Nag_NotSingularVecs$: No columns of $U$ (no left singular vectors) are computed.

Constraint:

jobu = Nag_SingularVecs

Nag_NotSingularVecs

3: $jobvt$ – Nag_ComputeSingularVecsType Input

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

V^{T}

$jobvt = Nag_SingularVecs$: The ns rows of $V^{T}$ , as specified by range, are returned in the array vt.
$jobvt = Nag_NotSingularVecs$: No rows of $V^{T}$ (no right singular vectors) are computed.

Constraint:

jobvt = Nag_SingularVecs

Nag_NotSingularVecs

4: $range$ – Nag_RangeType Input

On entry: indicates which singular values should be returned.

$range = Nag_AllValues$: All singular values will be found.
$range = Nag_Interval$: All singular values in the half-open interval $(vl, vu]$ will be found.
$range = Nag_Indices$: The ilth through iuth singular values will be found.

Constraint:

range = Nag_AllValues

Nag_Interval

Nag_Indices

5: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

6: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

7: $a [\dim]$ – 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 \neq Nag_NotSingularVecs

and

jobvt \neq Nag_NotSingularVecs

, the contents of a are destroyed.

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

9: $vl$ – double Input

On entry: if

range = Nag_Interval

, the lower bound of the interval to be searched for singular values.

range = Nag_AllValues

Nag_Indices

, vl is not referenced.

Constraint: if

range = Nag_Interval

0.0 \leq vl

10: $vu$ – double Input

On entry: if

range = Nag_Interval

, the upper bound of the interval to be searched for singular values.

range = Nag_AllValues

Nag_Indices

, vu is not referenced.

Constraint: if

range = Nag_Interval

vl < vu

11: $il$ – Integer Input

12: $iu$ – Integer Input

On entry: if

range = Nag_Indices

, il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.

range = Nag_AllValues

Nag_Interval

, il and iu are not referenced.

Constraints:

if $range = Nag_Indices$ and $\min (m, n) = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $\min (m, n) > 0$ , $1 \leq il \leq iu \leq \min (m, n)$ .

13: $ns$ – Integer * Output

On exit: the total number of singular values found.

0 \leq ns \leq \min (m, n)

range = Nag_AllValues

ns = \min (m, n)

range = Nag_Indices

ns = iu - il + 1

range = Nag_Interval

then the value of ns is not known in advance and so an upper limit should be used when specifying the dimensions of array u, e.g.,

\min (m, n)

14: $s [\min (m, n)]$ – double Output

On exit: the singular values of

A

, sorted so that

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

15: $u [\dim]$ – double Output

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

$pdu \times nsmax$ when $jobu = Nag_SingularVecs$ and $order = Nag_ColMajor$ ;
$m \times pdu$ when $jobu = Nag_SingularVecs$ and $order = Nag_RowMajor$ ;
otherwise u may be NULL;

where

nsmax

is a value larger than the output value ns.

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_SingularVecs

, u contains the first ns columns of

U

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

jobu = Nag_NotSingularVecs

, u is not referenced.

16: $pdu$ – 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 $jobu = Nag_SingularVecs$ , $pdu \geq m$ ;
- otherwise $pdu \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $jobu = Nag_SingularVecs$ , $pdu \geq nsmax$ ;
- otherwise u may be NULL;
where $nsmax$ is a value larger than the output value ns.

17: $vt [\dim]$ – double Output

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

$pdvt \times n$ when $jobvt = Nag_SingularVecs$ and $order = Nag_ColMajor$ ;
$\min (m, n) \times pdvt$ when $jobvt = Nag_SingularVecs$ and $order = Nag_RowMajor$ ;
otherwise vt may be NULL.

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_SingularVecs

, vt contains the first ns rows of

V^{T}

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

jobvt = Nag_NotSingularVecs

, vt is not referenced.

18: $pdvt$ – 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 $jobvt = Nag_SingularVecs$ , $pdvt \geq \min (m, n)$ ;
- otherwise $pdvt \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $jobvt = Nag_SingularVecs$ , $pdvt \geq n$ ;
- otherwise vt may be NULL.

19: $work [\min (m, n)]$ – 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}

20: $jfail [2 \times \min (m, n)]$ – Integer Output

On exit: if

fail . code =

NE_CONVERGENCE, jfail contains, in its first

k

nonzero elements, the indices of the

k

eigenvectors (associated with a left or right singular vector, see f08mbc) that failed to converge.

21: $fail$ – 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 f08kmc did not converge, $fail . errnum$ specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
NE_ENUM_INT: On entry, $jobu = ⟨ value ⟩$ and $pdu = ⟨ value ⟩$ .
Constraint: if $jobu = Nag_SingularVecs$ , $pdu \geq nsmax$ .
NE_ENUM_INT_2: On entry, $jobu = ⟨ value ⟩$ , $pdu = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $jobu = Nag_SingularVecs$ , $pdu \geq m$ .

On entry, $jobvt = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobvt = Nag_SingularVecs$ , $pdvt \geq n$ .
NE_ENUM_INT_3: On entry, $jobvt = ⟨ value ⟩$ , $pdvt = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $jobvt = Nag_SingularVecs$ , $pdvt \geq \min (m, n)$ .
NE_ENUM_INT_4: On entry, $range = ⟨ value ⟩$ , $il = ⟨ value ⟩$ , $iu = ⟨ value ⟩$ , $n = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $range = Nag_Indices$ and $\min (m, n) = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $\min (m, n) > 0$ , $1 \leq il \leq iu \leq \min (m, n)$ .
NE_ENUM_REAL_1: On entry, $range = ⟨ value ⟩$ and $vl = ⟨ value ⟩$ .
Constraint: if $range = Nag_Interval$ , $0.0 \leq vl$ .
NE_ENUM_REAL_2: On entry, $range = ⟨ value ⟩$ , $vl = ⟨ value ⟩$ and $vu = ⟨ value ⟩$ .
Constraint: if $range = Nag_Interval$ , $vl < vu$ .
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

Background information to multithreading can be found in the Multithreading documentation.

f08kmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kmc 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 f08kzc.

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.

f08km: FL CL CPP AD PY MB

NAG CL Interfacef08kmc (dgesvdx)

▸▿ 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
f08kmc (dgesvdx)