NAG Library Routine Document

F08LEF (DGBBRD)

INTEGER	M, N, NCC, KL, KU, LDAB, LDQ, LDPT, LDC, INFO
REAL (KIND=nag_wp)	AB(LDAB,), D(min(M,N)), E(min(M,N)-1), Q(LDQ,), PT(LDPT,), C(LDC,), WORK(2*max(M,N))
CHARACTER(1)	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 Parameters

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 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).

9 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 DocumentF08LEF (DGBBRD)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08LEF (DGBBRD)