void	f08mbc (Nag_OrderType order, Nag_UploType uplo, Nag_ComputeSingularVecsType jobz, Nag_RangeType range, Integer n, const double d[], const double e[], double vl, double vu, Integer il, Integer iu, Integer ns, double s[], double z[], Integer pdz, Integer jfail[], NagError fail)

The function may be called by the names: f08mbc, nag_lapackeig_dbdsvdx or nag_dbdsvdx.

3 Description

f08mbc computes the singular value decomposition (SVD) of a real

n \times n

(upper or lower) bidiagonal matrix

B

B = U S V^{T},

where

S

is a diagonal matrix with non-negative diagonal elements (the singular values of

B

), and

U

and

V^{T}

are orthogonal matrices. The columns of

U

and

V

are the left and right singular vectors of

B

, respectively.

Given an upper bidiagonal matrix

B

with diagonal

d = (\begin{matrix} d_{1} & d_{2} & \dots & d_{n} \end{matrix})

and superdiagonal

e = (\begin{matrix} e_{1} & e_{2} & \dots & e_{N - 1} \end{matrix})

, f08mbc computes the singular value decomposition of

B

through the eigenvalues and eigenvectors of the

n \times 2

n \times 2

tridiagonal matrix

TGK = (\begin{matrix} 0 & d_{1} \\ d_{1} & 0 & e_{1} \\ e_{1} & 0 & d_{2} \\ d_{2} & . & . \\ . & . \end{matrix}) .

(s, u, v)

is a singular triplet of

B

with

‖ u ‖ = ‖ v ‖ = 1

, then

(s, q_{1})

and

(- s, q_{2})

‖ q_{1} ‖ = ‖ q_{2} ‖ = 1

, are eigenpairs of

TGK

, with

q_{1} = (v_{1}, u_{1}, v_{2}, u_{2}, \dots, v_{n}, u_{n}) / \sqrt{2}

, and

q_{2} = ({- v}_{1}, u_{1}, {- v}_{2}, u_{2}, \dots, {- v}_{n}, u_{n}) / \sqrt{2}

Given a

TGK

matrix, you can either

(i)compute $- s, - v$ and change signs so that the singular values (and corresponding vectors) are already in descending order (as in f08kbc) or
(ii)compute $s, v$ and reorder the values (and corresponding vectors).

f08mbc implements (i) by calling f08jbc (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See Williams and Lang (2013).)

Alternative to computing all singular values of

B

, 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

Williams P and Lang B (2013) A framework for the

M R^{3}

Algorithm: theory and implementation SIAM J. Sci. Comput. 35 740–766

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: $uplo$ – Nag_UploType Input

On entry: indicates whether

B

is upper or lower bidiagonal.

$uplo = Nag_Upper$: $B$ is upper bidiagonal.
$uplo = Nag_Lower$: $B$ is lower bidiagonal.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $jobz$ – Nag_ComputeSingularVecsType Input

On entry: indicates whether singular vectors are computed.

$jobz = Nag_NotSingularVecs$: Only singular values are computed.
$jobz = Nag_SingularVecs$: Singular values and singular vectors are computed.

Constraint:

jobz = Nag_NotSingularVecs

Nag_SingularVecs

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: $n$ – Integer Input

On entry:

n

, the order of the bidiagonal matrix

B

Constraint:

n \geq 0

6: $d [n]$ – const double Input

On entry: the diagonal elements

d

of the bidiagonal matrix

B

7: $e [n - 1]$ – const double Input

On entry: the

(n - 1)

off-diagonal elements

e

of the bidiagonal matrix

B

8: $vl$ – 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

9: $vu$ – 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

10: $il$ – Integer Input

11: $iu$ – 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 $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .

12: $ns$ – Integer * Output

On exit: the total number of singular values found.

0 \leq ns \leq n

range = Nag_AllValues

ns = n

range = Nag_Indices

ns = iu - il + 1

13: $s [n]$ – double Output

On exit: the first ns elements contain the selected singular values in ascending order.

14: $z [\dim]$ – double Output

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

$pdz \times (\min (n, ns + 1))$ when $jobz = Nag_SingularVecs$ and $order = Nag_ColMajor$ ;
$\max (1, \max (2, n \times 2) \times pdz)$ when $jobz = Nag_SingularVecs$ and $order = Nag_RowMajor$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Z

is stored in

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

On exit: if

jobz = Nag_SingularVecs

, then if

fail . errnum = 0

the first ns columns of z contain the singular vectors of the matrix

B

corresponding to the selected singular values, with

U

in rows

1

n

and

V

in rows

n + 1

n \times 2

, i.e.,

Z = (\begin{matrix} U \\ V \end{matrix}) .

jobz = Nag_NotSingularVecs

, then z is not referenced.

Note: the user must ensure that at least

K = \min (n, ns) + 1

columns are supplied in the array

Z

. If

range = Nag_Interval

, the exact value of ns is not known in advance and an upper bound of at least n must be used.

15: $pdz$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ ,
- if $jobz = Nag_SingularVecs$ , $pdz \geq \max (2, n \times 2)$ ;
- otherwise $pdz \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $jobz = Nag_SingularVecs$ , $pdz \geq \min (n, ns + 1)$ ;
- otherwise $pdz \geq 1$ .

16: $jfail [2 \times n]$ – Integer Output

On exit: if

jobz = Nag_SingularVecs

, then

if $fail . code =$ NE_NOERROR, the first ns elements of jfail are zero;
if $fail . code =$ NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge in f08jbc.

jobz = Nag_NotSingularVecs

, jfail is not referenced and may be NULL.

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_CONVERGENCE: The algorithm failed to converge; $⟨ value ⟩$ eigenvectors of the associated eigenproblem did not converge. Their indices are stored in array jfail.
NE_ENUM_INT: On entry, $jobz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $\min (n, ns + 1) > 0$ .
NE_ENUM_INT_2: On entry, $jobz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobz = Nag_SingularVecs$ , $pdz \geq \max (2, n \times 2)$ ;
otherwise $pdz \geq 1$ .

On entry, $jobz = ⟨ value ⟩$ , $pdz = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $jobz = Nag_SingularVecs$ , $pdz \geq \min (n, ns + 1)$ ;
otherwise $pdz \geq 1$ .
NE_ENUM_INT_3: On entry, $range = ⟨ value ⟩$ , $il = ⟨ value ⟩$ , $iu = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $range = Nag_Indices$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range = Nag_Indices$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .
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, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

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

Each computed singular value of

B

is accurate to nearly full relative precision, no matter how tiny the singular value. The

i

th computed singular value,

{\hat{s}}_{i}

, satisfies the bound

| {\hat{s}}_{i} - s_{i} | \leq p (n) ε s_{i}

where

ε

is the machine precision and

p (n)

is a modest function of

n

For bounds on the computed singular vectors, see Section 4.9.1 of Anderson et al. (1999). See also f08flc.

8 Parallelism and Performance

f08mbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08mbc 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

If only singular values are required, the total number of floating-point operations is approximately proportional to

n^{2}

. When singular vectors are required the number of operations is bounded above by approximately the same number of operations as f08mec, but for large matrices f08mbc is usually much faster.

There is no complex analogue of f08mbc.

10 Example

This example computes the singular value decomposition of the upper bidiagonal matrix

B = (\begin{array}{r} 3.62 & 1.26 & 0 & 0 \\ 0 & - 2.41 & - 1.53 & 0 \\ 0 & 0 & 1.92 & 1.19 \\ 0 & 0 & 0 & - 1.43 \end{array}) .

f08mb: FL CL CPP AD

NAG CL Interfacef08mbc (dbdsvdx)

▸▿ 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
f08mbc (dbdsvdx)