nag_lapack_dbdsqr (f08me) computes the singular value decomposition of a real upper or lower bidiagonal matrix, or of a real general matrix which has been reduced to bidiagonal form.

Syntax

[d, e, vt, u, c, info] = f08me(uplo, d, e, vt, u, c, 'n', n, 'ncvt', ncvt, 'nru', nru, 'ncc', ncc)

[d, e, vt, u, c, info] = nag_lapack_dbdsqr(uplo, d, e, vt, u, c, 'n', n, 'ncvt', ncvt, 'nru', nru, 'ncc', ncc)

Description

nag_lapack_dbdsqr (f08me) computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix

B

. In other words, it can compute the singular value decomposition (SVD) of

B

B = U Σ V^{T} .

Here

Σ

is a diagonal matrix with real diagonal elements

σ_{i}

(the singular values of

B

), such that

σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0;

U

is an orthogonal matrix whose columns are the left singular vectors

u_{i}

;

V

is an orthogonal matrix whose rows are the right singular vectors

v_{i}

. Thus

B u_{i} = σ_{i} v_{i} and B^{T} v_{i} = σ_{i} u_{i}, i = 1, 2, \dots, n .

To compute

U

and/or

V^{T}

, the arrays u and/or vt must be initialized to the unit matrix before nag_lapack_dbdsqr (f08me) is called.

The function may also be used to compute the SVD of a real general matrix

A

which has been reduced to bidiagonal form by an orthogonal transformation:

A = Q B P^{T}

. If

A

m

n

with

m \geq n

, then

Q

m

n

and

P^{T}

n

n

; if

A

n

p

with

n < p

, then

Q

n

n

and

P^{T}

n

p

. In this case, the matrices

Q

and/or

P^{T}

must be formed explicitly by nag_lapack_dorgbr (f08kf) and passed to nag_lapack_dbdsqr (f08me) in the arrays u and/or vt respectively.

nag_lapack_dbdsqr (f08me) also has the capability of forming

U^{T} C

, where

C

is an arbitrary real matrix; this is needed when using the SVD to solve linear least squares problems.

nag_lapack_dbdsqr (f08me) uses two different algorithms. If any singular vectors are required (i.e., if

ncvt > 0

nru > 0

ncc > 0

), the bidiagonal

Q R

algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between

Q R

and

Q L

variants in order to handle graded matrices effectively (see Demmel and Kahan (1990)). If only singular values are required (i.e., if

ncvt = nru = ncc = 0

), they are computed by the differential qd algorithm (see Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.

The singular vectors are normalized so that

‖u_{i}‖ = ‖v_{i}‖ = 1

, but are determined only to within a factor

\pm 1

References

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

Fernando K V and Parlett B N (1994) Accurate singular values and differential qd algorithms Numer. Math. 67 191–229

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

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Indicates whether

B

is an upper or lower bidiagonal matrix.

$uplo ='U'$: $B$ is an upper bidiagonal matrix.
$uplo ='L'$: $B$ is a lower bidiagonal matrix.

Constraint:

uplo ='U'

'L'

2: $d (:)$ – double array

The dimension of the array d must be at least

\max (1, n)

The diagonal elements of the bidiagonal matrix

B

3: $e (:)$ – double array

The dimension of the array e must be at least

\max (1, n - 1)

The off-diagonal elements of the bidiagonal matrix

B

4: $vt (ldvt :)$ – double array

The first dimension,

ldvt

, of the array vt must satisfy

if $ncvt > 0$ , $ldvt \geq \max (1, n)$ ;
otherwise $ldvt \geq 1$ .

The second dimension of the array vt must be at least

\max (1, ncvt)

ncvt > 0

, vt must contain an

n

ncvt

matrix. If the right singular vectors of

B

are required,

ncvt = n

and vt must contain the unit matrix; if the right singular vectors of

A

are required, vt must contain the orthogonal matrix

P^{T}

returned by nag_lapack_dorgbr (f08kf) with

vect ='P'

5: $u (ldu :)$ – double array

The first dimension of the array u must be at least

\max (1, nru)

The second dimension of the array u must be at least

\max (1, n)

nru > 0

, u must contain an

nru

n

matrix. If the left singular vectors of

B

are required,

nru = n

and u must contain the unit matrix; if the left singular vectors of

A

are required, u must contain the orthogonal matrix

Q

returned by nag_lapack_dorgbr (f08kf) with

vect ='Q'

6: $c (ldc :)$ – double array

The first dimension,

ldc

, of the array c must satisfy

if $ncc > 0$ , $ldc \geq \max (1, n)$ ;
otherwise $ldc \geq 1$ .

The second dimension of the array c must be at least

\max (1, ncc)

The

n

ncc

matrix

C

ncc > 0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the arrays d, u.
$n$ , the order of the matrix $B$ .

Constraint: $n \geq 0$ .
2: $ncvt$ – int64int32nag_int scalar: Default: the second dimension of the array vt.
$ncvt$ , the number of columns of the matrix $V^{T}$ of right singular vectors. Set $ncvt = 0$ if no right singular vectors are required.

Constraint: $ncvt \geq 0$ .
3: $nru$ – int64int32nag_int scalar: Default: the first dimension of the array u.
$nru$ , the number of rows of the matrix $U$ of left singular vectors. Set $nru = 0$ if no left singular vectors are required.

Constraint: $nru \geq 0$ .
4: $ncc$ – int64int32nag_int scalar: Default: the second dimension of the array c.
$ncc$ , the number of columns of the matrix $C$ . Set $ncc = 0$ if no matrix $C$ is supplied.

Constraint: $ncc \geq 0$ .

Output Parameters

1: $d (:)$ – double array

The dimension of the array d will be

\max (1, n)

The singular values in decreasing order of magnitude, unless

info > 0

(in which case see Error Indicators and Warnings).

2: $e (:)$ – double array

The dimension of the array e will be

\max (1, n - 1)

e is overwritten, but if

info > 0

see Error Indicators and Warnings.

3: $vt (ldvt :)$ – double array

The first dimension,

ldvt

, of the array vt will be

if $ncvt > 0$ , $ldvt = \max (1, n)$ ;
otherwise $ldvt = 1$ .

The second dimension of the array vt will be

\max (1, ncvt)

The

n

ncvt

matrix

V^{T}

V^{T} P^{T}

of right singular vectors, stored by rows.

ncvt = 0

, vt is not referenced.

4: $u (ldu :)$ – double array

The first dimension of the array u will be

\max (1, nru)

The second dimension of the array u will be

\max (1, n)

The

nru

n

matrix

U

Q U

of left singular vectors, stored as columns of the matrix.

nru = 0

, u is not referenced.

5: $c (ldc :)$ – double array

The first dimension,

ldc

, of the array c will be

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

The second dimension of the array c will be

\max (1, ncc)

c stores the matrix

U^{T} C

. If

ncc = 0

, c is not referenced.

6: $info$ – int64int32nag_int scalar

info = 0

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

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ncvt, 4: nru, 5: ncc, 6: d, 7: e, 8: vt, 9: ldvt, 10: u, 11: ldu, 12: c, 13: ldc, 14: work, 15: 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.

W $info > 0$: The algorithm failed to converge and info specifies how many off-diagonals did not converge. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to $B$ .

Accuracy

Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the function) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.

σ_{i}

is an exact singular value of

B

and

{\tilde{σ}}_{i}

is the corresponding computed value, then

|{\tilde{σ}}_{i} - σ_{i}| \leq p (m, n) ε σ_{i}

where

p (m, n)

is a modestly increasing function of

m

and

n

, and

ε

is the machine precision. If only singular values are computed, they are computed more accurately (i.e., the function

p (m, n)

is smaller), than when some singular vectors are also computed.

u_{i}

is the corresponding exact left singular vector of

B

, and

{\tilde{u}}_{i}

is the corresponding computed left singular vector, then the angle

θ ({\tilde{u}}_{i}, u_{i})

between them is bounded as follows:

θ ({\tilde{u}}_{i}, u_{i}) \leq \frac{p (m, n) ε}{{relgap}_{i}}

where

{relgap}_{i}

is the relative gap between

σ_{i}

and the other singular values, defined by

{relgap}_{i} = \min_{i \neq j} \frac{|σ_{i} - σ_{j}|}{(σ_{i} + σ_{j})} .

A similar error bound holds for the right singular vectors.

Further Comments

The total number of floating-point operations is roughly proportional to

n^{2}

if only the singular values are computed. About

6 n^{2} \times nru

additional operations are required to compute the left singular vectors and about

6 n^{2} \times ncvt

to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster.

The complex analogue of this function is nag_lapack_zbdsqr (f08ms).

Example

This example computes the singular value decomposition of the upper bidiagonal matrix

B

, where

B = (\begin{array}{r} 3.62 & 1.26 & 0.00 & 0.00 \\ 0.00 & - 2.41 & - 1.53 & 0.00 \\ 0.00 & 0.00 & 1.92 & 1.19 \\ 0.00 & 0.00 & 0.00 & - 1.43 \end{array}) .

See also the example for nag_lapack_dorgbr (f08kf), which illustrates the use of the function to compute the singular value decomposition of a general matrix.

Open in the MATLAB editor: f08me_example

function f08me_example


fprintf('f08me example results\n\n');

% Bidiagonal matrix B stored as diagonal and upper-diagonal
n = 4;
d = [3.62;     -2.41;     1.92;     -1.43];
e = [1.26;     -1.53;     1.19];

% SVD of B
vt = eye(4);
u  = vt;
c  = [];
uplo = 'U';
[s, ~, vt, u, c, info] = f08me( ...
                                uplo, d, e, vt, u, c);

disp('Singular values');
disp(s');
disp('Right singular vectors, by row');
disp(vt);
disp('Left singular vectors, by column');
disp(u);

f08me example results

Singular values
    4.0001    3.0006    1.9960    0.9998

Right singular vectors, by row
    0.8261    0.5246    0.2024    0.0369
    0.4512   -0.4056   -0.7350   -0.3030
    0.2823   -0.5644    0.1731    0.7561
    0.1852   -0.4916    0.6236   -0.5789

Left singular vectors, by column
    0.9129    0.3740    0.1556    0.0512
   -0.3935    0.7005    0.5489    0.2307
    0.1081   -0.5904    0.6173    0.5086
   -0.0132    0.1444   -0.5417    0.8280

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

Chapter Contents

Chapter Introduction

NAG Toolbox