NAG Library Function Document

nag_dgbbrd (f08lec)

void	nag_dgbbrd (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)

3 Description

nag_dgbbrd (f08lec) 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 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_OrderTypeInput

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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: vect – Nag_VectTypeInput

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 – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

4: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

5: ncc – IntegerInput

On entry:

n_{C}

, the number of columns of the matrix

C

Constraint:

ncc \geq 0

6: kl – IntegerInput

On entry: the number of subdiagonals,

k_{l}

, within the band of

A

Constraint:

kl \geq 0

7: ku – IntegerInput

On entry: the number of superdiagonals,

k_{u}

, within the band of

A

Constraint:

ku \geq 0

8: ab[ $\dim$ ] – doubleInput/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

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 – IntegerInput

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[ $\min (m, n)$ ] – doubleOutput

On exit: the diagonal elements of the bidiagonal matrix

B

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

On exit: the superdiagonal elements of the bidiagonal matrix

B

12: q[ $\dim$ ] – doubleOutput

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

m

orthogonal matrix

Q

vect = Nag_DoNotForm

Nag_FormP

, q is not referenced.

13: pdq – IntegerInput

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[ $\dim$ ] – doubleOutput

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

n

orthogonal matrix

P^{T}

, if

vect = Nag_FormP

Nag_FormBoth

. If

vect = Nag_DoNotForm

Nag_FormQ

, pt is not referenced.

15: pdpt – IntegerInput

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[ $\dim$ ] – doubleInput/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

n_{C}

matrix

C

On exit: c is overwritten by

Q^{T} C

. If

ncc = 0

, c is not referenced.

17: pdc – IntegerInput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.

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

nag_dgbbrd (f08lec) is not threaded by NAG in any implementation.

nag_dgbbrd (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 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 nag_zgbbrd (f08lsc).

10 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 Function Documentnag_dgbbrd (f08lec)

+− 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 Library Function Document

nag_dgbbrd (f08lec)