NAG Library Routine Document

F08KEF (DGEBRD)

+− 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

1 Purpose

F08KEF (DGEBRD) reduces a real

m

n

matrix to bidiagonal form.

2 Specification

SUBROUTINE F08KEF (

M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO)

INTEGER	M, N, LDA, LWORK, INFO
REAL (KIND=nag_wp)	A(LDA,), D(), E(), TAUQ(), TAUP(*), WORK(max(1,LWORK))

The routine may be called by its LAPACK name dgebrd.

3 Description

F08KEF (DGEBRD) reduces a real

m

n

matrix

A

to bidiagonal form

B

by an orthogonal transformation:

A = Q B P^{T}

, where

Q

and

P^{T}

are orthogonal matrices of order

m

and

n

respectively.

m \geq n

, the reduction is given by:

A = Q (\begin{matrix} B_{1} \\ 0 \end{matrix}) P^{T} = Q_{1} B_{1} P^{T},

where

B_{1}

is an

n

n

upper bidiagonal matrix and

Q_{1}

consists of the first

n

columns of

Q

m < n

, the reduction is given by

A = Q (\begin{matrix} B_{1} & 0 \end{matrix}) P^{T} = Q B_{1} P_{1}^{T},

where

B_{1}

is an

m

m

lower bidiagonal matrix and

P_{1}^{T}

consists of the first

m

rows of

P^{T}

The orthogonal matrices

Q

and

P

are not formed explicitly but are represented as products of elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with

Q

and

P

in this representation (see Section 8).

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: M – INTEGERInput: On entry: $m$ , the number of rows of the matrix $A$ .
Constraint: $M \geq 0$ .
2: N – INTEGERInput: On entry: $n$ , the number of columns of the matrix $A$ .
Constraint: $N \geq 0$ .
3: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output: Note: the second dimension of the array A must be at least $\max (1, N)$ .
On entry: the $m$ by $n$ matrix $A$ .

On exit: if $m \geq n$ , the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix $B$ , elements below the diagonal are overwritten by details of the orthogonal matrix $Q$ and elements above the first superdiagonal are overwritten by details of the orthogonal matrix $P$ .
If $m < n$ , the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix $B$ , elements below the first subdiagonal are overwritten by details of the orthogonal matrix $Q$ and elements above the diagonal are overwritten by details of the orthogonal matrix $P$ .
4: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F08KEF (DGEBRD) is called.
Constraint: $LDA \geq \max (1, M)$ .
5: D( $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the dimension of the array D must be at least $\max (1, \min (M, N))$ .
On exit: the diagonal elements of the bidiagonal matrix $B$ .
6: E( $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the dimension of the array E must be at least $\max (1, \min (M, N) - 1)$ .
On exit: the off-diagonal elements of the bidiagonal matrix $B$ .
7: TAUQ( $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the dimension of the array TAUQ must be at least $\max (1, \min (M, N))$ .
On exit: further details of the orthogonal matrix $Q$ .
8: TAUP( $*$ ) – REAL (KIND=nag_wp) arrayOutput: Note: the dimension of the array TAUP must be at least $\max (1, \min (M, N))$ .
On exit: further details of the orthogonal matrix $P$ .
9: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , $WORK (1)$ contains the minimum value of LWORK required for optimal performance.
10: LWORK – INTEGERInput: On entry: the dimension of the array WORK as declared in the (sub)program from which F08KEF (DGEBRD) is called.
If $LWORK = - 1$ , a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

Suggested value: for optimal performance, $LWORK \geq (M + N) \times nb$ , where $nb$ is the optimal block size.
Constraint: $LWORK \geq \max (1, M, N)$ or $LWORK = - 1$ .
11: INFO – INTEGEROutput: On exit: $INFO = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: 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.

8 Further Comments

The total number of floating point operations is approximately

\frac{4}{3} n^{2} (3 m - n)

m \geq n

\frac{4}{3} m^{2} (3 n - m)

m < n

m ≫ n

, it can be more efficient to first call F08AEF (DGEQRF) to perform a

Q R

factorization of

A

, and then to call F08KEF (DGEBRD) to reduce the factor

R

to bidiagonal form. This requires approximately

2 n^{2} (m + n)

floating point operations.

m ≪ n

, it can be more efficient to first call F08AHF (DGELQF) to perform an

L Q

factorization of

A

, and then to call F08KEF (DGEBRD) to reduce the factor

L

to bidiagonal form. This requires approximately

2 m^{2} (m + n)

operations.

To form the orthogonal matrices

P^{T}

and/or

Q

F08KEF (DGEBRD) may be followed by calls to F08KFF (DORGBR):

to form the

m

m

orthogonal matrix

Q

CALL DORGBR('Q',M,M,N,A,LDA,TAUQ,WORK,LWORK,INFO)

but note that the second dimension of the array A must be at least M, which may be larger than was required by F08KEF (DGEBRD);

to form the

n

n

orthogonal matrix

P^{T}

CALL DORGBR('P',N,N,M,A,LDA,TAUP,WORK,LWORK,INFO)

but note that the first dimension of the array A, specified by the parameter LDA, must be at least N, which may be larger than was required by F08KEF (DGEBRD).

To apply

Q

P

to a real rectangular matrix

C

, F08KEF (DGEBRD) may be followed by a call to F08KGF (DORMBR).

The complex analogue of this routine is F08KSF (ZGEBRD).

9 Example

This example reduces the matrix

A

to bidiagonal form, where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array}) .

NAG Library Routine DocumentF08KEF (DGEBRD)