void f08lec (Nag_OrderType order, Nag_VectType vect, Integer m, Integer n, Integer ncc, Integer kl, Integer ku, double ab[], Integer pdab, double d[], double e[], double q[], Integer pdq, double pt[], Integer pdpt, double c[], Integer pdc, NagError *fail)

The function may be called by the names: f08lec, nag_lapackeig_dgbbrd or nag_dgbbrd.

3 Description

f08lec reduces a real

m \times 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 function if required. A matrix

C

may also be updated to give

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

The function uses a vectorizable form of the reduction.

4 References

None.

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: $vect$ – Nag_VectType Input

On entry: indicates whether the matrices

Q

and/or

P^{T}

are generated.

$vect = Nag_DoNotForm$: Neither $Q$ nor $P^{T}$ is generated.
$vect = Nag_FormQ$: $Q$ is generated.
$vect = Nag_FormP$: $P^{T}$ is generated.
$vect = Nag_FormBoth$: Both $Q$ and $P^{T}$ are generated.

Constraint:

vect = Nag_DoNotForm

Nag_FormQ

Nag_FormP

Nag_FormBoth

3: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: $ncc$ – Integer Input

On entry:

n_{C}

, the number of columns of the matrix

C

Constraint:

ncc \geq 0

6: $kl$ – Integer Input

On entry: the number of subdiagonals,

k_{l}

, within the band of

A

Constraint:

kl \geq 0

7: $ku$ – Integer Input

On entry: the number of superdiagonals,

k_{u}

, within the band of

A

Constraint:

ku \geq 0

8: $ab [\dim]$ – double Input/Output

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

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

On entry: the original

m \times n

band matrix

A

This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements

A_{i j}

, for row

i = 1, \dots, m

and column

j = \max (1, i - k_{l}), \dots, \min (n, i + k_{u})

, depends on the order argument as follows:

if $order = Nag_ColMajor$ , $A_{i j}$ is stored as $ab [(j - 1) \times pdab + ku + i - j]$ ;
if $order = Nag_RowMajor$ , $A_{i j}$ is stored as $ab [(i - 1) \times pdab + kl + j - i]$ .

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

9: $pdab$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) of the matrix

A

in the array ab.

Constraint:

pdab \geq kl + ku + 1

10: $d [\min (m, n)]$ – double Output

On exit: the diagonal elements of the bidiagonal matrix

B

11: $e [\min (m, n) - 1]$ – double Output

On exit: the superdiagonal elements of the bidiagonal matrix

B

12: $q [\dim]$ – double Output

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

$\max (1, pdq \times m)$ when $vect = Nag_FormQ$ or $Nag_FormBoth$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix

Q

is stored in

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

On exit: if

vect = Nag_FormQ

Nag_FormBoth

, contains the

m \times m

orthogonal matrix

Q

vect = Nag_DoNotForm

Nag_FormP

, q is not referenced.

13: $pdq$ – Integer Input

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

Constraints:

if $vect = Nag_FormQ$ or $Nag_FormBoth$ , $pdq \geq \max (1, m)$ ;
otherwise $pdq \geq 1$ .

14: $pt [\dim]$ – double Output

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

$\max (1, pdpt \times n)$ when $vect = Nag_FormP$ or $Nag_FormBoth$ ;
$1$ otherwise.

The

(i, j)

th element of the matrix is stored in

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

On exit: the

n \times n

orthogonal matrix

P^{T}

, if

vect = Nag_FormP

Nag_FormBoth

. If

vect = Nag_DoNotForm

Nag_FormQ

, pt is not referenced.

15: $pdpt$ – Integer Input

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

Constraints:

if $vect = Nag_FormP$ or $Nag_FormBoth$ , $pdpt \geq \max (1, n)$ ;
otherwise $pdpt \geq 1$ .

16: $c [\dim]$ – double Input/Output

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

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

The

(i, j)

th element of the matrix

C

is stored in

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

On entry: an

m \times n_{C}

matrix

C

On exit: c is overwritten by

Q^{T} C

. If

ncc = 0

, c is not referenced.

17: $pdc$ – Integer Input

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

Constraints:

if $order = Nag_ColMajor$ ,
- if $ncc > 0$ , $pdc \geq \max (1, m)$ ;
- if $ncc = 0$ , $pdc \geq 1$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, ncc)$ .

18: $fail$ – 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_ENUM_INT_2: On entry, $vect = ⟨ value ⟩$ , $pdpt = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: if $vect = Nag_FormP$ or $Nag_FormBoth$ , $pdpt \geq \max (1, n)$ ;
otherwise $pdpt \geq 1$ .

On entry, $vect = ⟨ value ⟩$ , $pdq = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $vect = Nag_FormQ$ or $Nag_FormBoth$ , $pdq \geq \max (1, m)$ ;
otherwise $pdq \geq 1$ .
NE_INT: On entry, $kl = ⟨ value ⟩$ .
Constraint: $kl \geq 0$ .

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

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

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

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

On entry, $pdab = ⟨ value ⟩$ .
Constraint: $pdab > 0$ .

On entry, $pdc = ⟨ value ⟩$ .
Constraint: $pdc > 0$ .

On entry, $pdpt = ⟨ value ⟩$ .
Constraint: $pdpt > 0$ .

On entry, $pdq = ⟨ value ⟩$ .
Constraint: $pdq > 0$ .
NE_INT_2: On entry, $pdc = ⟨ value ⟩$ and $ncc = ⟨ value ⟩$ .
Constraint: $pdc \geq \max (1, ncc)$ .
NE_INT_3: On entry, $ncc = ⟨ value ⟩$ , $pdc = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: if $ncc > 0$ , $pdc \geq \max (1, m)$ ;
if $ncc = 0$ , $pdc \geq 1$ .

On entry, $pdab = ⟨ value ⟩$ , $kl = ⟨ value ⟩$ and $ku = ⟨ value ⟩$ .
Constraint: $pdab \geq kl + ku + 1$ .
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.

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

Background information to multithreading can be found in the Multithreading documentation.

f08lec 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 real floating-point operations is approximately the sum of:

$6 n^{2} k$ , if $vect = Nag_DoNotForm$ 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 function is f08lsc.

10 Example

This example reduces the matrix

A

to upper bidiagonal form, where

A = (\begin{matrix} - 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{matrix}) .

f08le: FL CL CPP AD PY MB

NAG CL Interfacef08lec (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

NAG CL Interface
f08lec (dgbbrd)