are not formed explicitly but are represented as products of elementary reflectors (see the F08 Chapter Introduction for details). Functions 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: $order$ – Nag_OrderType Input

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – double Input/Output

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pda)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

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

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

5: $pda$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

if $order = Nag_ColMajor$ , $pda \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pda \geq \max (1, n)$ .

6: $d [\dim]$ – double Output

Note: the dimension, dim, of the array d must be at least

\max (1, \min (m, n))

On exit: the diagonal elements of the bidiagonal matrix

B

7: $e [\dim]$ – double Output

Note: the dimension, dim, 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

8: $tauq [\dim]$ – double Output

Note: the dimension, dim, of the array tauq must be at least

\max (1, \min (m, n))

On exit: further details of the orthogonal matrix

Q

9: $taup [\dim]$ – double Output

Note: the dimension, dim, of the array taup must be at least

\max (1, \min (m, n))

On exit: further details of the orthogonal matrix

P

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .

On entry, $pda = ⟨ value ⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, m)$ .

On entry, $pda = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pda \geq \max (1, n)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

f08kec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08kec 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 function. 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 f08aec to perform a

Q R

factorization of

A

, and then to call f08kec 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 f08ahc to perform an

L Q

factorization of

A

, and then to call f08kec 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

f08kec may be followed by a call to f08kfc . For example

nag_lapackeig_dorgbr(order,Nag_FormQ,m,m,n,&a,pda,tauq,&fail)

but note that the second dimension of the array a must be at least m, which may be larger than was required by f08kec.

To form the

n \times n

orthogonal matrix

P^{T}

another call to f08kfc may be made . For example

nag_lapackeig_dorgbr(order,Nag_FormP,n,n,m,&a,pda,taup,&fail)

but note that the first dimension of the array a, must be at least n, which may be larger than was required by f08kec.

To apply

Q

P

to a real rectangular matrix

C

, f08kec may be followed by a call to f08kgc.

The complex analogue of this function is f08ksc.

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 CL Interfacef08kec (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 CL Interface
f08kec (dgebrd)