to upper triangular form. This factorization is usually used as a preprocessing step for computing the generalized singular value decomposition (GSVD).

Syntax

[a, b, k, l, u, v, q, info] = f08ve(jobu, jobv, jobq, a, b, tola, tolb, 'm', m, 'p', p, 'n', n)

[a, b, k, l, u, v, q, info] = nag_lapack_dggsvp(jobu, jobv, jobq, a, b, tola, tolb, 'm', m, 'p', p, 'n', n)

Description

nag_lapack_dggsvp (f08ve) computes orthogonal matrices

U

V

and

Q

such that

\begin{array}{l} U^{T} A Q = \{\begin{array}{r} \begin{array}{rc} n - k - l & k & l \\ k & 0 & A_{12} & A_{13} \\ l & 0 & 0 & A_{23} \\ m - k - l & 0 & 0 & 0 \\ ( & ) \end{array}, if ​ m - k - l \geq 0; \\ \begin{array}{rc} n - k - l & k & l \\ k & 0 & A_{12} & A_{13} \\ m - k & 0 & 0 & A_{23} \\ ( & ) \end{array}, if ​ m - k - l < 0; \end{array}  \\ V^{T} B Q = \begin{array}{rc} n - k - l & k & l \\ l & 0 & 0 & B_{13} \\ p - l & 0 & 0 & 0 \\ ( & ) \end{array} \end{array}

where the

k

k

matrix

A_{12}

and

l

l

matrix

B_{13}

are nonsingular upper triangular;

A_{23}

l

l

upper triangular if

m - k - l \geq 0

and is

(m - k)

l

upper trapezoidal otherwise.

(k + l)

is the effective numerical rank of the

(m + p)

n

matrix

{(\begin{matrix} A^{T} & B^{T} \end{matrix})}^{T}

This decomposition is usually used as the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), see function nag_lapack_dggsvd (f08va).

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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

Parameters

Compulsory Input Parameters

1: $jobu$ – string (length ≥ 1)

jobu ='U'

, the orthogonal matrix

U

is computed.

jobu ='N'

U

is not computed.

Constraint:

jobu ='U'

'N'

2: $jobv$ – string (length ≥ 1)

jobv ='V'

, the orthogonal matrix

V

is computed.

jobv ='N'

V

is not computed.

Constraint:

jobv ='V'

'N'

3: $jobq$ – string (length ≥ 1)

jobq ='Q'

, the orthogonal matrix

Q

is computed.

jobq ='N'

Q

is not computed.

Constraint:

jobq ='Q'

'N'

4: $a (lda :)$ – double array

The first dimension of the array a must be at least

\max (1, m)

The second dimension of the array a must be at least

\max (1, n)

The

m

n

matrix

A

5: $b (ldb :)$ – double array

The first dimension of the array b must be at least

\max (1, p)

The second dimension of the array b must be at least

\max (1, n)

The

p

n

matrix

B

6: $tola$ – double scalar

7: $tolb$ – double scalar

tola and tolb are the thresholds to determine the effective numerical rank of matrix

B

and a subblock of

A

. Generally, they are set to

\begin{matrix} tola = \max (m, n) ‖A‖ ε, \\ tolb = \max (p, n) ‖B‖ ε, \end{matrix}

where

ε

is the machine precision.

The size of tola and tolb may affect the size of backward errors of the decomposition.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $p$ – int64int32nag_int scalar: Default: the first dimension of the array b.
$p$ , the number of rows of the matrix $B$ .

Constraint: $p \geq 0$ .
3: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrices $A$ and $B$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array

The first dimension of the array a will be

\max (1, m)

The second dimension of the array a will be

\max (1, n)

Contains the triangular (or trapezoidal) matrix described in Description.

2: $b (ldb :)$ – double array

The first dimension of the array b will be

\max (1, p)

The second dimension of the array b will be

\max (1, n)

Contains the triangular matrix described in Description.

3: $k$ – int64int32nag_int scalar

4: $l$ – int64int32nag_int scalar

k and l specify the dimension of the subblocks

k

and

l

as described in Description;

(k + l)

is the effective numerical rank of

{(\begin{matrix} a^{T} & b^{T} \end{matrix})}^{T}

5: $u (ldu :)$ – double array

The first dimension,

ldu

, of the array u will be

if $jobu ='U'$ , $ldu = \max (1, m)$ ;
otherwise $ldu = 1$ .

The second dimension of the array u will be

\max (1, m)

jobu ='U'

and

1

otherwise.

jobu ='U'

, u contains the orthogonal matrix

U

jobu ='N'

, u is not referenced.

6: $v (ldv :)$ – double array

The first dimension,

ldv

, of the array v will be

if $jobv ='V'$ , $ldv = \max (1, p)$ ;
otherwise $ldv = 1$ .

The second dimension of the array v will be

\max (1, p)

jobv ='V'

and

1

otherwise.

jobv ='V'

, v contains the orthogonal matrix

V

jobv ='N'

, v is not referenced.

7: $q (ldq :)$ – double array

The first dimension,

ldq

, of the array q will be

if $jobq ='Q'$ , $ldq = \max (1, n)$ ;
otherwise $ldq = 1$ .

The second dimension of the array q will be

\max (1, n)

jobq ='Q'

and

1

otherwise.

jobq ='Q'

, q contains the orthogonal matrix

Q

jobq ='N'

, q is not referenced.

8: $info$ – 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: jobu, 2: jobv, 3: jobq, 4: m, 5: p, 6: n, 7: a, 8: lda, 9: b, 10: ldb, 11: tola, 12: tolb, 13: k, 14: l, 15: u, 16: ldu, 17: v, 18: ldv, 19: q, 20: ldq, 21: iwork, 22: tau, 23: work, 24: 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 factorization is nearly the exact factorization for nearby matrices

(A + E)

and

(B + F)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2} and {‖F‖}_{2} = O (ε) {‖B‖}_{2},

and

ε

is the machine precision.

Further Comments

The complex analogue of this function is nag_lapack_zggsvp (f08vs).

Example

This example finds the generalized factorization

A = U Σ_{1} (\begin{matrix} 0 & S \end{matrix}) Q^{T}, B = V Σ_{2} (\begin{matrix} 0 & T \end{matrix}) Q^{T},

of the matrix pair

(\begin{matrix} A & B \end{matrix})

, where

A = (\begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 8 \end{matrix}) and B = (\begin{matrix} - 2 & - 3 & 3 \\ 4 & 6 & 5 \end{matrix}) .

Open in the MATLAB editor: f08ve_example

function f08ve_example


fprintf('f08ve example results\n\n');

% Generalized singular values of (A,B) where:
m = 4;
n = 3;
a = [ 1  2  3;
      3  2  1;
      4  5  6;
      7  8  8]; 
p = 2;
b = [-2 -3  3;
      4  6  5]; 

% Reduce A and B to upper triangular form S = U^T A Q, T = V^T B Q
tola = max(m,n)*norm(a,1)*x02aj;
tolb = max(p,n)*norm(a,1)*x02aj;
[S, T, k, l, U, V, Q, info] = ...
  f08ve( ...
	 'U', 'V', 'Q', a, b, tola, tolb);

% Compute singular values
[S, T, alpha, beta, U, V, Q, ncycle, info] = ...
  f08ye( ...
	 'U', 'V', 'Q', k, l, S, T, tola, tolb, U, V, Q);

fprintf('Number of infinite generalized singular values = %3d\n',k);
fprintf('Number of   finite generalized singular values = %3d\n',l);
fprintf('Effective rank of the matrix pair  (A^T B^T)^T = %3d\n',k+l);
fprintf('Number of cycles of the Kogbetliantz method    = %3d\n\n',ncycle);
disp('Finite generalized singular values');
disp(alpha(k+1:k+l)./beta(k+1:k+l));

f08ve example results

Number of infinite generalized singular values =   1
Number of   finite generalized singular values =   2
Effective rank of the matrix pair  (A^T B^T)^T =   3
Number of cycles of the Kogbetliantz method    =   2

Finite generalized singular values
    1.3151
    0.0802

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox