Integer, Intent (In)	::	n, ncvt, nru, ncc, ldvt, ldu, ldc
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	d(), e(), work(*)
Complex (Kind=nag_wp), Intent (Inout)	::	vt(ldvt,), u(ldu,), c(ldc,*)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f08msf_ (const char uplo, const Integer n, const Integer ncvt, const Integer nru, const Integer ncc, double d[], double e[], Complex vt[], const Integer ldvt, Complex u[], const Integer ldu, Complex c[], const Integer ldc, double work[], Integer *info, const Charlen length_uplo)

The routine may be called by the names f08msf, nagf_lapackeig_zbdsqr or its LAPACK name zbdsqr.

3 Description

f08msf 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 f08msf is called.

The routine 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 f08ktf and passed to f08msf in the arrays u and/or vt respectively.

f08msf 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.

f08msf 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: $uplo$ – Character(1) Input

On entry: indicates whether

B

is an upper or lower bidiagonal matrix.

$uplo ='U'$: $B$ is an upper bidiagonal matrix.
$uplo ='L'$: $B$ is a lower bidiagonal matrix.

Constraint:

uplo ='U'

'L'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

B

Constraint:

n \geq 0

3: $ncvt$ – 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

4: $nru$ – 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

5: $ncc$ – 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

6: $d (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension 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

info > 0

(in which case see Section 6).

7: $e (*)$ – Real (Kind=nag_wp) array Input/Output

Note: the dimension 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

info > 0

see Section 6.

8: $vt (ldvt, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array vt must be at least

\max (1, ncvt)

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 f08ktf with

vect ='P'

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.

9: $ldvt$ – Integer Input

On entry: the first dimension of the array vt as declared in the (sub)program from which f08msf is called.

Constraints:

if $ncvt > 0$ , $ldvt \geq \max (1, n)$ ;
otherwise $ldvt \geq 1$ .

10: $u (ldu, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array u must be at least

\max (1, n)

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 f08ktf with

vect ='Q'

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.

11: $ldu$ – Integer Input

On entry: the first dimension of the array u as declared in the (sub)program from which f08msf is called.

Constraint:

ldu \geq \max (1, nru)

12: $c (ldc, *)$ – Complex (Kind=nag_wp) array Input/Output

Note: the second dimension of the array c must be at least

\max (1, ncc)

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.

13: $ldc$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which f08msf is called.

Constraints:

if $ncc > 0$ , $ldc \geq \max (1, n)$ ;
otherwise $ldc \geq 1$ .

14: $work (*)$ – Real (Kind=nag_wp) array Workspace

Note: the dimension of the array work must be at least

\max (1, 4 \times n)

15: $info$ – Integer Output

On exit:

info = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: $⟨ value ⟩$ 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 $B$ .

7 Accuracy

Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the routine) 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

f08msf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08msf 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 routine. 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 routine is f08mef.