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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zgbbrd (f08ls)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zgbbrd (f08ls) reduces a complex

m

n

band matrix to real upper bidiagonal form.

Syntax

[ab, d, e, q, pt, c, info] = f08ls(vect, m, kl, ku, ab, c, 'n', n, 'ncc', ncc)

[ab, d, e, q, pt, c, info] = nag_lapack_zgbbrd(vect, m, kl, ku, ab, c, 'n', n, 'ncc', ncc)

Description

nag_lapack_zgbbrd (f08ls) reduces a complex

m

n

band matrix to real upper bidiagonal form

B

by a unitary transformation:

A = Q B P^{H}

. The unitary matrices

Q

and

P^{H}

, 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^{H} C

The function uses a vectorizable form of the reduction.

References

None.

Parameters

Compulsory Input Parameters

1: $vect$ – string (length ≥ 1)

Indicates whether the matrices

Q

and/or

P^{H}

are generated.

$vect ='N'$: Neither $Q$ nor $P^{H}$ is generated.
$vect ='Q'$: $Q$ is generated.
$vect ='P'$: $P^{H}$ is generated.
$vect ='B'$: Both $Q$ and $P^{H}$ are generated.

Constraint:

vect ='N'

'Q'

'P'

'B'

2: $m$ – int64int32nag_int scalar

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $kl$ – int64int32nag_int scalar

The number of subdiagonals,

k_{l}

, within the band of

A

Constraint:

kl \geq 0

4: $ku$ – int64int32nag_int scalar

The number of superdiagonals,

k_{u}

, within the band of

A

Constraint:

ku \geq 0

5: $ab (ldab :)$ – complex array

The first dimension of the array ab must be at least

kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The original

m

n

band matrix

A

The matrix is stored in rows

1

k_{l} + k_{u} + 1

, more precisely, the element

A_{i j}

must be stored in

ab (k_{u} + 1 + i - j, j) for ​ \max (1, j - k_{u}) \leq i \leq \min (m, j + k_{l}) .

6: $c (ldc :)$ – complex array

The first dimension,

ldc

, of the array c must satisfy

if $ncc > 0$ , $ldc \geq \max (1, m)$ ;
if $ncc = 0$ , $ldc \geq 1$ .

The second dimension of the array c must be at least

\max (1, ncc)

m

n_{C}

matrix

C

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array ab.
$n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $ncc$ – int64int32nag_int scalar: Default: the second dimension of the array c.
$n_{C}$ , the number of columns of the matrix $C$ .

Constraint: $ncc \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – complex array

The first dimension of the array ab will be

kl + ku + 1

The second dimension of the array ab will be

\max (1, n)

ab stores values generated during the reduction.

2: $d (\min (m, n))$ – double array

The diagonal elements of the bidiagonal matrix

B

3: $e (\min (m, n) - 1)$ – double array

The superdiagonal elements of the bidiagonal matrix

B

4: $q (ldq :)$ – complex array

The first dimension,

ldq

, of the array q will be

if $vect ='Q'$ or $'B'$ , $ldq = \max (1, m)$ ;
otherwise $ldq = 1$ .

The second dimension of the array q will be

\max (1, m)

vect ='Q'

'B'

and

1

otherwise.

vect ='Q'

'B'

, contains the

m

m

unitary matrix

Q

vect ='N'

'P'

, q is not referenced.

5: $pt (ldpt :)$ – complex array

The first dimension,

ldpt

, of the array pt will be

if $vect ='P'$ or $'B'$ , $ldpt = \max (1, n)$ ;
otherwise $ldpt = 1$ .

The second dimension of the array pt will be

\max (1, n)

vect ='P'

'B'

and

1

otherwise.

The

n

n

unitary matrix

P^{H}

, if

vect ='P'

'B'

. If

vect ='N'

'Q'

, pt is not referenced.

6: $c (ldc :)$ – complex array

The first dimension,

ldc

, of the array c will be

if $ncc > 0$ , $ldc = \max (1, m)$ ;
if $ncc = 0$ , $ldc = 1$ .

The second dimension of the array c will be

\max (1, ncc)

c stores

Q^{H} C

. If

ncc = 0

, c is not referenced.

7: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: vect, 2: m, 3: n, 4: ncc, 5: kl, 6: ku, 7: ab, 8: ldab, 9: d, 10: e, 11: q, 12: ldq, 13: pt, 14: ldpt, 15: c, 16: ldc, 17: work, 18: rwork, 19: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed bidiagonal form

B

satisfies

Q B P^{H} = 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 unitary matrix by a matrix

F

such that

{‖F‖}_{2} = O (ε) .

A similar statement holds for the computed matrix

P^{H}

Further Comments

The total number of real floating-point operations is approximately the sum of:

$20 n^{2} k$ , if $vect ='N'$ and $ncc = 0$ , and
$10 n^{2} n_{C} (k - 1) / k$ , if $C$ is updated, and
$10 n^{3} (k - 1) / k$ , if either $Q$ or $P^{H}$ is generated (double this if both),

where

k = k_{l} + k_{u}

, assuming

n ≫ k

. For this section we assume that

m = n

The real analogue of this function is nag_lapack_dgbbrd (f08le).

Example

This example reduces the matrix

A

to upper bidiagonal form, where

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & 0.00 + 0.00 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ 0.00 + 0.00 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.26 + 0.26 i \end{array}) .

Open in the MATLAB editor: f08ls_example

function f08ls_example


fprintf('f08ls example results\n\n');

% Banded complex matrix A stored in banded format
m  = int64(6);
n  = int64(4);
kl = int64(2);
ku = int64(1);
ab = [ 0.00 + 0.00i, -0.03 + 0.96i, -0.66 + 0.42i, -1.11 + 0.60i;
       0.96 - 0.81i, -1.20 + 0.19i,  0.63 - 0.17i,  0.22 - 0.20i;
      -0.98 + 1.98i,  1.01 + 0.02i, -0.98 - 0.36i,  1.47 + 1.59i;
       0.62 - 0.46i,  0.19 - 0.54i, -0.17 - 0.46i,  0.26 + 0.26i];

% Reduce A to bidiagonal form
c = [];
vect = 'No Q or PT';
[~, d, e, ~, ~, ~, info] = f08ls( ...
                                  vect, m, kl, ku, ab, c);

fprintf('Diagonal:\n');
fprintf(' %8.4f',d);
fprintf('\nOff-diagonal (absolute values):\n');
fprintf(' %8.4f',abs(e));
fprintf('\n');

f08ls example results

Diagonal:
   2.6560   1.7501   2.0607   0.8658
Off-diagonal (absolute values):
   1.7033   1.2800   0.1467

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Chapter Contents

Chapter Introduction

NAG Toolbox