f08ms:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

nag_lapack_zbdsqr (f08ms) 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_zbdsqr (f08ms) is called.

The function stores the real orthogonal matrices

U

and

V^{T}

in complex arrays u and vt, so that it may also be used to compute the SVD of a complex general matrix

A

which has been reduced to bidiagonal form by a unitary transformation:

A = Q B P^{H}

. If

A

m

n

with

m \geq n

, then

Q

m

n

and

P^{H}

n

n

; if

A

n

p

with

n < p

, then

Q

n

n

and

P^{H}

n

p

. In this case, the matrices

Q

and/or

P^{H}

must be formed explicitly by nag_lapack_zungbr (f08kt) and passed to nag_lapack_zbdsqr (f08ms) in the arrays u and/or vt respectively.

nag_lapack_zbdsqr (f08ms) 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.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

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.

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 an 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 real floating-point operations is roughly proportional to

n^{2}

if only the singular values are computed. About

12 n^{2} \times nru

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

12 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.

Example

function f08ms_example


fprintf('f08ms example results\n\n');

m = int64(6);
n = int64(4);
a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
     -0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, -1.11 + 0.60i;
     -0.37 + 0.38i,  0.19 - 0.54i, -0.98 - 0.36i,  0.22 - 0.20i;
      0.83 + 0.51i,  0.20 + 0.01i, -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i,  0.26 + 0.26i];

% Factorize A = QR
[QR, tau, info] = f08as(a);

% Generate Q from QR
[Q, info] = f08at(QR, tau);

% Extract R from QR
R = triu(QR(1:n,1:n));

% Bidiagonalize R = Q1 B P^H
[B, d, e, tauq, taup, info] = f08ks(R);

% Form P^H explicitly
vect = 'P';
[PH, info] = f08kt(vect, m, B, taup);
% Form Q1 explicitly
vect = 'Q';
[Q1, info] = f08kt(vect, n, B, tauq);

% Update Q: Q2 = Q*Q1 (so A = QR = Q2 B PH)
vect = 'Q';
side = 'Right';
trans = 'No transpose';
[Q2, info] = f08ku(vect, side, trans, n, B, tauq, Q);

% Compute SVD of A from bidiagonal form
uplo = 'Upper';
c = [];
[S, ~, VH, U, ~, info] = f08ms(uplo, d, e, PH, Q2, c);

disp('Singular values:');
disp(S);
disp('Left Singular Vectors:');
disp(U);
disp('Right Singular Vectors:');
disp(VH');

f08ms example results

Singular values:
    3.9994
    3.0003
    1.9944
    0.9995

Left Singular Vectors:
  -0.5634 + 0.0016i   0.2687 + 0.2749i  -0.2451 - 0.4657i  -0.3787 - 0.2987i
   0.1205 - 0.6108i   0.2909 - 0.1085i  -0.4329 + 0.1758i   0.0182 + 0.0437i
  -0.0816 + 0.1613i   0.1660 - 0.3885i   0.4667 - 0.3821i   0.0800 + 0.2276i
   0.1441 - 0.1532i  -0.1984 + 0.1737i   0.0034 - 0.1555i  -0.2608 + 0.5382i
  -0.2487 - 0.0926i  -0.6253 - 0.3304i  -0.2643 + 0.0194i  -0.1002 - 0.0140i
  -0.3758 + 0.0793i   0.0307 + 0.0816i  -0.1266 - 0.1747i   0.4175 + 0.4058i

Right Singular Vectors:
  -0.6971 + 0.0000i  -0.2403 + 0.0000i   0.5123 + 0.0000i   0.4403 + 0.0000i
  -0.0867 + 0.3548i  -0.0725 - 0.2336i   0.3030 - 0.1735i  -0.5294 + 0.6361i
   0.0560 + 0.5400i   0.2477 - 0.5291i  -0.0678 + 0.5162i   0.3027 - 0.0346i
  -0.1878 + 0.2253i  -0.7026 + 0.2177i  -0.4418 + 0.3864i  -0.1667 + 0.0258i

NAG Toolbox: nag_lapack_zbdsqr (f08ms)

▸▿ Contents

Purpose

Syntax

Description