void	nag_dgebrd (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double d[], double e[], double tauq[], double taup[], NagError *fail)

3

Description

nag_dgebrd (f08kec) reduces a real

m

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

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

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). 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_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

n \geq 0

4: $a [\dim]$ – doubleInput/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

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: $pda$ – IntegerInput

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

Constraints:

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

6: $d [\dim]$ – doubleOutput

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: $e [\dim]$ – doubleOutput

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: $tauq [\dim]$ – doubleOutput

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: $taup [\dim]$ – doubleOutput

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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

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

nag_dgebrd (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 nag_dgeqrf (f08aec) to perform a

Q R

factorization of

A

, and then to call nag_dgebrd (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 nag_dgelqf (f08ahc) to perform an

L Q

factorization of

A

, and then to call nag_dgebrd (f08kec) to reduce the factor

L

to bidiagonal form. This requires approximately

2 m^{2} (m + n)

operations.

To form the orthogonal matrices

P^{T}

and/or

Q

nag_dgebrd (f08kec) may be followed by calls to nag_dorgbr (f08kfc):

to form the

m

m

orthogonal matrix

Q

nag_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 nag_dgebrd (f08kec);

to form the

n

n

orthogonal matrix

P^{T}

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

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

To apply

Q

P

to a real rectangular matrix

C

, nag_dgebrd (f08kec) may be followed by a call to nag_dormbr (f08kgc).

The complex analogue of this function is nag_zgebrd (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}) .

NAG Library Function Document

nag_dgebrd (f08kec)

▸▿ 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

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