void	f10cac (Nag_ComputeUType jobu, Nag_ComputeVTType jobvt, Integer m, Integer n, const double a[], Integer pda, Integer k, double rtol_abs, double rtol_rel, Integer state[], double s[], double u[], Integer pdu, double vt[], Integer pdvt, Integer r, NagError fail)

The function may be called by the names: f10cac or nag_rnla_svd_rowext_real.

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

If the rank of

A

r < \min (m, n)

, then

σ

has

r

nonzero elements, and only

r

columns of

U

and

V

are well-defined. In this case we can reduce

σ

to an

r \times r

matrix,

U

to an

m \times r

matrix and

V^{T}

to an

r \times n

matrix.

f10cac is designed for efficiently computing the SVD in the case

r < \min (m, n)

. The input argument

k

should be greater than

r

by a small oversampling parameter,

δ

, such that

k = r + δ

. A reasonable value for

δ

, to compute the SVD to within machine precision, is

δ = 5

. The value of

δ

should not vary based on

m

n

. If

r

is not known then the function can be used iteratively to refine the estimate and accuracy of the computed SVD; using a larger value of

δ

than necessary increases the computational cost of the function.

As a by-product of computing the SVD, the function estimates

r

If the input argument

k

is less than

r

the accuracy depends on the

(k + 1)

th singular value,

σ_{k + 1}

. See Section 7 for more details.

A call to f10cac consists of the following:

1.A random projection is applied, $Y = A Ω$ , where $Ω$ is an $n \times k$ matrix. (Note that the product $A Ω$ is computed using a Fast Fourier Transform, so can be computed in $O (m n \log (k))$ time.) See f10dac for more details on the random projection.
2.A pivoted $Q R$ decomposition of $Y$ is calculated (see f08bfc for more details). The rank estimate is then $r$ such that, on the diagonal of $R$ ,

$| R_{r r} | > ε_{a} + ε_{r} \times R_{11},$

where $ε_{a}$ and $ε_{r}$ are the absolute and relative error tolerances, respectively, and $r$ is the largest diagonal index for which the above relation holds.
3.Obtain the SVD from the $Q R$ decomposition of $Y$ (or, depending on the rank, an approximation to the SVD) of $A$ . This is referred to as row extraction.

Further details of the randomized SVD procedure can be found in Sections 4 and 5 of Halko et al. (2011).

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

Halko N (2012) Randomized methods for computing low-rank approximations of matrices PhD thesis

Halko N, Martinsson P G and Tropp J A (2011) Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions SIAM Rev. 53(2) 217–288 https://epubs.siam.org/doi/abs/10.1137/090771806

5 Arguments

1: $jobu$ – Nag_ComputeUType Input

On entry: specifies options for computing part of or none of the matrix

U

$jobu = Nag_SingularVecsU$: The first $k$ columns of $U$ (the left singular vectors) are returned in the array u.
$jobu = Nag_NotU$: No columns of $U$ (no left singular vectors) are computed.

Constraint:

jobu = Nag_SingularVecsU

Nag_NotU

2: $jobvt$ – Nag_ComputeVTType Input

On entry: specifies options for computing part of or none of the matrix

V^{T}

$jobvt = Nag_SingularVecsVT$: The first $k$ rows of $V^{T}$ (the right singular vectors) are returned in the array vt.
$jobvt = Nag_NotVT$: No rows of $V^{T}$ (no right singular vectors) are computed.

Constraint:

jobvt = Nag_SingularVecsVT

Nag_NotVT

3: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m > 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n > 0

5: $a [\dim]$ – const double Input

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

pda \times n

The

(i, j)

th element of the matrix

A

is stored in

a [(j - 1) \times pda + i - 1]

On entry: the

m \times n

matrix

A

6: $pda$ – Integer Input

On entry: the stride separating matrix row elements in the array a.

Constraint:

pda \geq m

7: $k$ – Integer Input

On entry:

k

, number of columns in random projection,

Y = A Ω

Constraint:

0 < k \leq \min (m, n)

8: $rtol_abs$ – double Input

On entry: the absolute tolerance,

ε_{a}

used in defining the threshold on estimating the rank of

A

. If

rtol_abs < 0.0

then

0.0

is used unless

rtol_rel \leq 0.0

in which case

\sqrt{machine precision}

is used.

9: $rtol_rel$ – double Input

On entry: the relative tolerance,

ε_{r}

used in defining the threshold on estimating the rank of

A

. If

rtol_rel < 0.0

then

0.0

is used unless

rtol_abs \leq 0.0

in which case

\sqrt{machine precision}

is used.

10: $state [\dim]$ – Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

11: $s [k]$ – double Output

On exit: the first r elements of s contain the r largest singular values of

A

in descending order. The remaining values are set to zero.

12: $u [\dim]$ – double Output

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

k

when

jobu = Nag_SingularVecsU

The

(i, j)

th element of the matrix

U

is stored in

u [(j - 1) \times pdu + i - 1]

On exit: if

jobu = Nag_SingularVecsU

, u contains the first r columns of

U

(the left singular vectors, stored column-wise); the remaining elements of u are set to zero.

jobu = Nag_NotU

, u is not referenced.

13: $pdu$ – Integer Input

On entry: the stride separating matrix row elements in the array u.

Constraint: if

jobu = Nag_SingularVecsU

pdu \geq m

14: $vt [\dim]$ – double Output

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

$pdvt \times n$ when $jobvt = Nag_SingularVecsVT$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix is stored in

vt [(j - 1) \times pdvt + i - 1]

On exit: if

jobvt = Nag_SingularVecsVT

, vt contains the first r rows of

V^{T}

(the right singular vectors); the remaining elements of vt are set to zero.

jobvt = Nag_NotVT

, vt is not referenced.

15: $pdvt$ – Integer Input

On entry: the stride separating matrix row elements in the array vt.

Constraint: if

jobvt = Nag_SingularVecsVT

pdvt \geq k

16: $r$ – Integer * Output

On exit:

r

, contains estimated rank of array a.

17: $fail$ – 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_INT: On entry, $k = ⟨ value ⟩$ .
Constraint: $0 < k \leq \min (m, n)$ .

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

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

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda \geq m$ .

On entry, state vector has been corrupted or not initialized.
NE_INT_2: On entry, $jobu = ⟨ value ⟩$ and $pdu = ⟨ value ⟩$ .
Constraint: if $jobu = Nag_SingularVecsU$ , $pdu \geq m$ .

On entry, $jobvt = ⟨ value ⟩$ and $pdvt = ⟨ value ⟩$ .
Constraint: if $jobvt = Nag_SingularVecsVT$ , $pdvt \geq k$ .
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.
NE_REAL: $A$ has effective rank of zero.
First diagonal element of $R$ , from $Q R$ of $Y$ , $=$ $⟨ value ⟩$ .
Tolerance used to determine rank $=$ $⟨ value ⟩$ .

On exit, $r = k$ , the rank of $A$ may be larger than r.
Increase k to obtain a more accurate rank estimate.
Smallest diagonal element of $R$ , from $Q R$ of $Y$ , $=$ $⟨ value ⟩$ .
Tolerance used to determine rank $=$ $⟨ value ⟩$ .

7 Accuracy

The error is approximately,

‖ A - U σ V^{*} ‖ < [1 + \sqrt{1 + 4 k (n - k)}] ε,

where,

ε = \sqrt{1 + 7 n / k} \cdot σ_{k + 1} .

The norm on the left-hand side of the first equation is the spectral norm, and

σ_{k + 1}

is the

(k + 1)

th singular value of

A

. More details on the error bound can be found in Sections 5 and 11 of Halko et al. (2011).

8 Parallelism and Performance

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

f10cac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f10cac 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

O (\log (k) m n + k^{2} (m + n))

. The first term corresponds to applying the random projection, i.e., computing

Y = A Ω

. The second term corresponds to the

Q R

decomposition of

Y

and the steps required to obtain the SVD of the original matrix

A

Deterministic SVD solvers, such as f08kbc, require

O (m n^{2})

operations when

m \geq n

and

O (n m^{2})

operations when

n \geq m

The default values for rtol_abs and rtol_rel assume that you need an accurate approximation to

A

. If you only need to use a small number of singular values or singular vectors, larger values for these tolerances are appropriate. Increasing tolerances sufficiently will decrease r, the estimated rank. Decreasing r means that k can then be decreased to reduce the run-time of the routine.

10 Example

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

6 \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 \\ - 1.94 & - 0.78 & - 0.21 & 0.63 & 2.30 & - 2.57 \\ - 1.94 & - 0.78 & - 0.21 & 0.63 & 2.30 & - 2.57 \end{array}),

using the randomised solver, f10cac, and a deterministic solver, f08kbc for comparison.

f10ca: FL CL CPP AD PY MB

NAG CL Interfacef10cac (svd_​rowext_​real)

▸▿ 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
f10cac (svd_rowext_real)