NAG Library Function Document

nag_dormbr (f08kgc)

void	nag_dormbr (Nag_OrderType order, Nag_VectType vect, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

3 Description

nag_dormbr (f08kgc) is intended to be used after a call to nag_dgebrd (f08kec), which reduces a real rectangular matrix

A

to bidiagonal form

B

by an orthogonal transformation:

A = Q B P^{T}

. nag_dgebrd (f08kec) represents the matrices

Q

and

P^{T}

as products of elementary reflectors.

This function may be used to form one of the matrix products

Q C, Q^{T} C, C Q, C Q^{T}, P C, P^{T} C, C P ​ or ​ C P^{T},

overwriting the result on

C

(which may be any real rectangular matrix).

4 References

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

5 Arguments

Note: in the descriptions below,

r

denotes the order of

Q

P^{T}

: if

side = Nag_LeftSide

r = m

and if

side = Nag_RightSide

r = n

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

Q

Q^{T}

P

P^{T}

is to be applied to

C

$vect = Nag_ApplyQ$: $Q$ or $Q^{T}$ is applied to $C$ .
$vect = Nag_ApplyP$: $P$ or $P^{T}$ is applied to $C$ .

Constraint:

vect = Nag_ApplyQ

Nag_ApplyP

3: side – Nag_SideTypeInput

On entry: indicates how

Q

Q^{T}

P

P^{T}

is to be applied to

C

$side = Nag_LeftSide$: $Q$ or $Q^{T}$ or $P$ or $P^{T}$ is applied to $C$ from the left.
$side = Nag_RightSide$: $Q$ or $Q^{T}$ or $P$ or $P^{T}$ is applied to $C$ from the right.

Constraint:

side = Nag_LeftSide

Nag_RightSide

4: trans – Nag_TransTypeInput

On entry: indicates whether

Q

P

Q^{T}

P^{T}

is to be applied to

C

$trans = Nag_NoTrans$: $Q$ or $P$ is applied to $C$ .
$trans = Nag_Trans$: $Q^{T}$ or $P^{T}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

5: m – IntegerInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

6: n – IntegerInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

7: k – IntegerInput

On entry: if

vect = Nag_ApplyQ

, the number of columns in the original matrix

A

vect = Nag_ApplyP

, the number of rows in the original matrix

A

Constraint:

k \geq 0

8: a[ $\dim$ ] – const doubleInput

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

$\max (1, pda \times \min (r, k))$ when $vect = Nag_ApplyQ$ and $order = Nag_ColMajor$ ;
$\max (1, r \times pda)$ when $vect = Nag_ApplyQ$ and $order = Nag_RowMajor$ ;
$\max (1, pda \times r)$ when $vect = Nag_ApplyP$ and $order = Nag_ColMajor$ ;
$\max (1, \min (r, k) \times pda)$ when $vect = Nag_ApplyP$ and $order = Nag_RowMajor$ .

On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgebrd (f08kec).

9: pda – IntegerInput

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

Constraints:

if order=Nag_ColMajor,
- if $vect = Nag_ApplyQ$ , $pda \geq \max (1, r)$ ;
- if $vect = Nag_ApplyP$ , $pda \geq \max (1, \min (r, k))$ ;
if order=Nag_RowMajor,
- if $vect = Nag_ApplyQ$ , $pda \geq \max (1, \min (r, k))$ ;
- if $vect = Nag_ApplyP$ , $pda \geq \max (1, r)$ .

10: tau[ $\dim$ ] – const doubleInput

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

\max (1, \min (r, k))

On entry: further details of the elementary reflectors, as returned by nag_dgebrd (f08kec) in its argument tauq if

vect = Nag_ApplyQ

, or in its argument taup if

vect = Nag_ApplyP

11: c[ $\dim$ ] – doubleInput/Output

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

$\max (1, pdc \times n)$ 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: the matrix

C

On exit: c is overwritten by

Q C

Q^{T} C

C Q

C^{T} Q

P C

P^{T} C

C P

C^{T} P

as specified by vect, side and trans.

12: pdc – IntegerInput

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

Constraints:

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

13: 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, $vect = ⟨value⟩$ , $pda = ⟨value⟩$ , $k = ⟨value⟩$ .
Constraint: if $vect = Nag_ApplyQ$ , $pda \geq \max (1, \min (r, k))$ ;
if $vect = Nag_ApplyP$ , $pda \geq \max (1, r)$ .
On entry, $vect = ⟨value⟩$ , $pda = ⟨value⟩$ and $k = ⟨value⟩$ .
Constraint: if $vect = Nag_ApplyQ$ , $pda \geq \max (1, r)$ ;
if $vect = Nag_ApplyP$ , $pda \geq \max (1, \min (r, k))$ .
NE_INT: On entry, $k = ⟨value⟩$ .
Constraint: $k \geq 0$ .
On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
On entry, $pdc = ⟨value⟩$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pdc = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, m)$ .
On entry, $pdc = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdc \geq \max (1, n)$ .
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.

7 Accuracy

The computed result differs from the exact result by a matrix

E

such that

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Parallelism and Performance

nag_dormbr (f08kgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dormbr (f08kgc) 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 floating-point operations is approximately

if $side = Nag_LeftSide$ and $m \geq k$ , $2 n k (2 m - k)$ ;
if $side = Nag_RightSide$ and $n \geq k$ , $2 m k (2 n - k)$ ;
if $side = Nag_LeftSide$ and $m < k$ , $2 m^{2} n$ ;
if $side = Nag_RightSide$ and $n < k$ , $2 m n^{2}$ ,

where

k

is the value of the argument k.

The complex analogue of this function is nag_zunmbr (f08kuc).

10 Example

For this function two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix

A

may be preceded by a

Q R

L Q

factorization of

A

In the first example,

m > n

, and

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}) .

The function first performs a

Q R

factorization of

A

A = Q_{a} R

and then reduces the factor

R

to bidiagonal form

B

R = Q_{b} B P^{T}

. Finally it forms

Q_{a}

and calls nag_dormbr (f08kgc) to form

Q = Q_{a} Q_{b}

In the second example,

m < n

, and

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array}) .

The function first performs an

L Q

factorization of

A

A = L P_{a}^{T}

and then reduces the factor

L

to bidiagonal form

B

L = Q B P_{b}^{T}

. Finally it forms

P_{b}^{T}

and calls nag_dormbr (f08kgc) to form

P^{T} = P_{b}^{T} P_{a}^{T}

NAG Library Function Documentnag_dormbr (f08kgc)

+− 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_dormbr (f08kgc)