NAG Library Routine Document

Integer, Intent (In)	::	m, n, ncc, kl, ku, ldab, ldq, ldpt, ldc
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	ab(ldab,), q(ldq,), pt(ldpt,), c(ldc,)
Real (Kind=nag_wp), Intent (Out)	::	d(min(m,n)), e(min(m,n)-1), work(2*max(m,n))
Character (1), Intent (In)	::	vect

C Header Interface

#include <nagmk26.h>

void

f08lef_ (const char *vect, const Integer *m, const Integer *n, const Integer *ncc, const Integer *kl, const Integer *ku, double ab[], const Integer *ldab, double d[], double e[], double q[], const Integer *ldq, double pt[], const Integer *ldpt, double c[], const Integer *ldc, double work[], Integer *info, const Charlen length_vect)

The routine may be called by its LAPACK name dgbbrd.

3

Description

f08lef (dgbbrd) reduces a real

m

n

band matrix to upper bidiagonal form

B

by an orthogonal transformation:

A = Q B P^{T}

. The orthogonal matrices

Q

and

P^{T}

, of order

m

and

n

respectively, are determined as a product of Givens rotation matrices, and may be formed explicitly by the routine if required. A matrix

C

may also be updated to give

\tilde{C} = Q^{T} C

The routine uses a vectorizable form of the reduction.

4

References

None.

5

Arguments

1: $vect$ – Character(1)Input

On entry: indicates whether the matrices

Q

and/or

P^{T}

are generated.

$vect ='N'$: Neither $Q$ nor $P^{T}$ is generated.
$vect ='Q'$: $Q$ is generated.
$vect ='P'$: $P^{T}$ is generated.
$vect ='B'$: Both $Q$ and $P^{T}$ are generated.

Constraint:

vect ='N'

'Q'

'P'

'B'

2: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $ncc$ – IntegerInput

On entry:

n_{C}

, the number of columns of the matrix

C

Constraint:

ncc \geq 0

5: $kl$ – IntegerInput

On entry: the number of subdiagonals,

k_{l}

, within the band of

A

Constraint:

kl \geq 0

6: $ku$ – IntegerInput

On entry: the number of superdiagonals,

k_{u}

, within the band of

A

Constraint:

ku \geq 0

7: $ab (ldab *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, n)

On entry: the original

m

n

band matrix

A

The matrix is stored in rows

1

k_{l} + k_{u} + 1

, more precisely, the element

A_{i j}

must be stored in

ab (k_{u} + 1 + i - j, j) for ​ \max (1, j - k_{u}) \leq i \leq \min (m, j + k_{l}) .

On exit: ab is overwritten by values generated during the reduction.

8: $ldab$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f08lef (dgbbrd) is called.

Constraint:

ldab \geq kl + ku + 1

9: $d (\min (m, n))$ – Real (Kind=nag_wp) arrayOutput

On exit: the diagonal elements of the bidiagonal matrix

B

10: $e (\min (m, n) - 1)$ – Real (Kind=nag_wp) arrayOutput

On exit: the superdiagonal elements of the bidiagonal matrix

B

11: $q (ldq *)$ – Real (Kind=nag_wp) arrayOutput

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

\max (1, m)

vect ='Q'

'B'

, and at least

1

otherwise.

On exit: if

vect ='Q'

'B'

, contains the

m

m

orthogonal matrix

Q

vect ='N'

'P'

, q is not referenced.

12: $ldq$ – IntegerInput

On entry: the first dimension of the array q as declared in the (sub)program from which f08lef (dgbbrd) is called.

Constraints:

if $vect ='Q'$ or $'B'$ , $ldq \geq \max (1, m)$ ;
otherwise $ldq \geq 1$ .

13: $pt (ldpt *)$ – Real (Kind=nag_wp) arrayOutput

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

\max (1, n)

vect ='P'

'B'

, and at least

1

otherwise.

On exit: the

n

n

orthogonal matrix

P^{T}

, if

vect ='P'

'B'

. If

vect ='N'

'Q'

, pt is not referenced.

14: $ldpt$ – IntegerInput

On entry: the first dimension of the array pt as declared in the (sub)program from which f08lef (dgbbrd) is called.

Constraints:

if $vect ='P'$ or $'B'$ , $ldpt \geq \max (1, n)$ ;
otherwise $ldpt \geq 1$ .

15: $c (ldc *)$ – Real (Kind=nag_wp) arrayInput/Output

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

\max (1, ncc)

On entry: an

m

n_{C}

matrix

C

On exit: c is overwritten by

Q^{T} C

. If

ncc = 0

, c is not referenced.

16: $ldc$ – IntegerInput

On entry: the first dimension of the array c as declared in the (sub)program from which f08lef (dgbbrd) is called.

Constraints:

if $ncc > 0$ , $ldc \geq \max (1, m)$ ;
if $ncc = 0$ , $ldc \geq 1$ .

17: $work (2 \times \max (m, n))$ – Real (Kind=nag_wp) arrayWorkspace

18: $info$ – IntegerOutput

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.

7

Accuracy

The computed bidiagonal form

B

satisfies

Q B P^{T} = A + E

, where

{‖E‖}_{2} \leq c (n) ε {‖A‖}_{2},

c (n)

is a modestly increasing function of

n

, and

ε

is the machine precision.

The elements of

B

themselves may be sensitive to small perturbations in

A

or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.

The computed matrix

Q

differs from an exactly orthogonal matrix by a matrix

F

such that

{‖F‖}_{2} = O (ε) .

A similar statement holds for the computed matrix

P^{T}

8

Parallelism and Performance

f08lef (dgbbrd) 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 approximately the sum of:

$6 n^{2} k$ , if $vect ='N'$ and $ncc = 0$ , and
$3 n^{2} n_{C} (k - 1) / k$ , if $C$ is updated, and
$3 n^{3} (k - 1) / k$ , if either $Q$ or $P^{T}$ is generated (double this if both),

where

k = k_{l} + k_{u}

, assuming

n ≫ k

. For this section we assume that

m = n

The complex analogue of this routine is f08lsf (zgbbrd).

10

Example

This example reduces the matrix

A

to upper bidiagonal form, where

A = (\begin{array}{r} - 0.57 & - 1.28 & 0.00 & 0.00 \\ - 1.93 & 1.08 & - 0.31 & 0.00 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ 0.00 & 0.64 & - 0.66 & 0.08 \\ 0.00 & 0.00 & 0.15 & - 2.13 \\ - 0.00 & 0.00 & 0.00 & 0.50 \end{array}) .

NAG Library Routine Document

f08lef (dgbbrd)

▸▿ 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

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