void	f08msc (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer ncvt, Integer nru, Integer ncc, double d[], double e[], Complex vt[], Integer pdvt, Complex u[], Integer pdu, Complex c[], Integer pdc, NagError *fail)

The function may be called by the names: f08msc, nag_lapackeig_zbdsqr or nag_zbdsqr.

3 Description

f08msc computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix

B

. In other words, it can compute the singular value decomposition (SVD) of

B

B = U Σ V^{T} .

Here

Σ

is a diagonal matrix with real diagonal elements

σ_{i}

(the singular values of

B

), such that

σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0;

U

is an orthogonal matrix whose columns are the left singular vectors

u_{i}

;

V

is an orthogonal matrix whose rows are the right singular vectors

v_{i}

. Thus

B u_{i} = σ_{i} v_{i} and B^{T} v_{i} = σ_{i} u_{i}, i = 1, 2, \dots, n .

To compute

U

and/or

V^{T}

, the arrays u and/or vt must be initialized to the unit matrix before f08msc is called.

The function stores the real orthogonal matrices

U

and

V^{T}

in complex arrays u and vt, so that it may also be used to compute the SVD of a complex general matrix

A

which has been reduced to bidiagonal form by a unitary transformation:

A = Q B P^{H}

. If

A

m \times n

with

m \geq n

, then

Q

m \times n

and

P^{H}

n \times n

; if

A

n \times p

with

n < p

, then

Q

n \times n

and

P^{H}

n \times p

. In this case, the matrices

Q

and/or

P^{H}

must be formed explicitly by f08ktc and passed to f08msc in the arrays u and/or vt respectively.

f08msc also has the capability of forming

U^{H} C

, where

C

is an arbitrary complex matrix; this is needed when using the SVD to solve linear least squares problems.

f08msc uses two different algorithms. If any singular vectors are required (i.e., if

ncvt > 0

nru > 0

ncc > 0

), the bidiagonal

Q R

algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between

Q R

and

Q L

variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (i.e., if

ncvt = nru = ncc = 0

), they are computed by the differential qd algorithm (see Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.

The singular vectors are normalized so that

‖ u_{i} ‖ = ‖ v_{i} ‖ = 1

, but are determined only to within a complex factor of absolute value

1

4 References

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

Fernando K V and Parlett B N (1994) Accurate singular values and differential qd algorithms Numer. Math. 67 191–229

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: $uplo$ – Nag_UploType Input

On entry: indicates whether

B

is an upper or lower bidiagonal matrix.

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

Constraint:

uplo = Nag_Upper

Nag_Lower

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

B

Constraint:

n \geq 0

4: $ncvt$ – Integer Input

On entry:

ncvt

, the number of columns of the matrix

V^{H}

of right singular vectors. Set

ncvt = 0

if no right singular vectors are required.

Constraint:

ncvt \geq 0

5: $nru$ – Integer Input

On entry:

nru

, the number of rows of the matrix

U

of left singular vectors. Set

nru = 0

if no left singular vectors are required.

Constraint:

nru \geq 0

6: $ncc$ – Integer Input

On entry:

ncc

, the number of columns of the matrix

C

. Set

ncc = 0

if no matrix

C

is supplied.

Constraint:

ncc \geq 0

7: $d [\dim]$ – double Input/Output

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

\max (1, n)

On entry: the diagonal elements of the bidiagonal matrix

B

On exit: the singular values in decreasing order of magnitude, unless

fail . code =

NE_CONVERGENCE (in which case see Section 6).

8: $e [\dim]$ – double Input/Output

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

\max (1, n - 1)

On entry: the off-diagonal elements of the bidiagonal matrix

B

On exit: e is overwritten, but if

fail . code =

NE_CONVERGENCE see Section 6.

9: $vt [\dim]$ – Complex Input/Output

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

\max (1, pdvt \times ncvt)

when

order = Nag_ColMajor

and at least

\max (1, pdvt \times n)

when

order = Nag_RowMajor

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 entry: if

ncvt > 0

, vt must contain an

n \times ncvt

matrix. If the right singular vectors of

B

are required,

ncvt = n

and vt must contain the unit matrix; if the right singular vectors of

A

are required, vt must contain the unitary matrix

P^{H}

returned by f08ktc with

vect = Nag_ApplyP

On exit: the

n \times ncvt

matrix

V^{H}

V^{H} P^{H}

of right singular vectors, stored by rows.

ncvt = 0

, vt is not referenced.

10: $pdvt$ – 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 $ncvt > 0$ , $pdvt \geq \max (1, n)$ ;
- otherwise $pdvt \geq 1$ ;
if $order = Nag_RowMajor$ ,
- if $ncvt > 0$ , $pdvt \geq ncvt$ ;
- otherwise $pdvt \geq 1$ .

11: $u [\dim]$ – Complex Input/Output

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

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

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 entry: if

nru > 0

, u must contain an

nru \times n

matrix. If the left singular vectors of

B

are required,

nru = n

and u must contain the unit matrix; if the left singular vectors of

A

are required, u must contain the unitary matrix

Q

returned by f08ktc with

vect = Nag_ApplyQ

On exit: the

nru \times n

matrix

U

Q U

of left singular vectors, stored as columns of the matrix.

nru = 0

, u is not referenced.

12: $pdu$ – Integer Input

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

Constraints:

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

13: $c [\dim]$ – Complex Input/Output

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

\max (1, pdc \times ncc)

when

order = Nag_ColMajor

and at least

\max (1, pdc \times n)

when

order = Nag_RowMajor

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: the

n \times ncc

matrix

C

ncc > 0

On exit: c is overwritten by the matrix

U^{H} C

. If

ncc = 0

, c is not referenced.

14: $pdc$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ ,
- if $ncc > 0$ , $pdc \geq \max (1, n)$ ;
- otherwise $pdc \geq 1$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, ncc)$ .

15: $fail$ – 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: $⟨ value ⟩$ off-diagonals did not converge. The arrays d and e contain the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to $B$ .
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

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

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

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

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

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

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

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

On entry, $pdu = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdu \geq \max (1, n)$ .

On entry, $pdu = ⟨ value ⟩$ and $nru = ⟨ value ⟩$ .
Constraint: $pdu \geq \max (1, nru)$ .

On entry, $pdvt = ⟨ value ⟩$ and $ncvt = ⟨ value ⟩$ .
Constraint: if $ncvt > 0$ , $pdvt \geq ncvt$ ;
otherwise $pdvt \geq 1$ .
NE_INT_3: On entry, $ncc = ⟨ value ⟩$ , $pdc = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $ncc > 0$ , $pdc \geq \max (1, n)$ ;
otherwise $pdc \geq 1$ .

On entry, $pdvt = ⟨ value ⟩$ , $ncvt = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $ncvt > 0$ , $pdvt \geq \max (1, n)$ ;
otherwise $pdvt \geq 1$ .
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 singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.

σ_{i}

is an exact singular value of

B

and

{\tilde{σ}}_{i}

is the corresponding computed value, then

| {\tilde{σ}}_{i} - σ_{i} | \leq p (m, n) ε σ_{i}

where

p (m, n)

is a modestly increasing function of

m

and

n

, and

ε

is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function

p (m, n)

is smaller), than when some singular vectors are also computed.

u_{i}

is an exact left singular vector of

B

, and

{\tilde{u}}_{i}

is the corresponding computed left singular vector, then the angle

θ ({\tilde{u}}_{i}, u_{i})

between them is bounded as follows:

θ ({\tilde{u}}_{i}, u_{i}) \leq \frac{p (m, n) ε}{{relgap}_{i}}

where

{relgap}_{i}

is the relative gap between

σ_{i}

and the other singular values, defined by

{relgap}_{i} = \min_{i \neq j} \frac{| σ_{i} - σ_{j} |}{(σ_{i} + σ_{j})} .

A similar error bound holds for the right singular vectors.

8 Parallelism and Performance

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

f08msc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08msc 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 real floating-point operations is roughly proportional to

n^{2}

if only the singular values are computed. About

12 n^{2} \times nru

additional operations are required to compute the left singular vectors and about

12 n^{2} \times ncvt

to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster.

The real analogue of this function is f08mec.