Integer, Intent (In)	::	m, n, lda, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), d(), e(), tauq(), taup(*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))

C Header Interface

#include <nag.h>

void	f08kef_ (const Integer m, const Integer n, double a[], const Integer lda, double d[], double e[], double tauq[], double taup[], double work[], const Integer lwork, Integer *info)

The routine may be called by the names f08kef, nagf_lapackeig_dgebrd or its LAPACK name dgebrd.

3 Description

f08kef reduces a real

m \times 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 \times 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 \times 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 9).

4 References

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

5 Arguments

1: $m$ – Integer Input: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $n$ – Integer Input: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .
3: $a (lda, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $m \times n$ 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: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08kef is called.

Constraint: $lda \geq \max (1, m)$ .
5: $d (*)$ – Real (Kind=nag_wp) array Output: 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) array Output: 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) array Output: 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) array Output: 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) array Workspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
10: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08kef 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: $info$ – 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.

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 Parallelism and Performance

f08kef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kef 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 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 to perform a

Q R

factorization of

A

, and then to call f08kef 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 to perform an

L Q

factorization of

A

, and then to call f08kef to reduce the factor

L

to bidiagonal form. This requires approximately

2 m^{2} (m + n)

operations.

To form the

m \times m

orthogonal matrix

Q

f08kef may be followed by a call to f08kff . For example

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.

To form the

n \times n

orthogonal matrix

P^{T}

another call to f08kff may be made . For example

Call dorgbr('P',n,n,m,a,lda,taup,work,lwork,info)

but note that the first dimension of the array a, must be at least n, which may be larger than was required by f08kef.

To apply

Q

P

to a real rectangular matrix

C

, f08kef may be followed by a call to f08kgf.

The complex analogue of this routine is f08ksf.

10 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}) .

f08ke: FL CL CPP AD

NAG FL Interfacef08kef (dgebrd)

▸▿ 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

NAG FL Interface
f08kef (dgebrd)