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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zbdsqr (f08ms)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zbdsqr (f08ms) computes the singular value decomposition of a complex general matrix which has been reduced to bidiagonal form.

Syntax

[d, e, vt, u, c, info] = f08ms(uplo, d, e, vt, u, c, 'n', n, 'ncvt', ncvt, 'nru', nru, 'ncc', ncc)
[d, e, vt, u, c, info] = nag_lapack_zbdsqr(uplo, d, e, vt, u, c, 'n', n, 'ncvt', ncvt, 'nru', nru, 'ncc', ncc)

Description

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 as
B = U Σ VT .  
Here Σ is a diagonal matrix with real diagonal elements σi (the singular values of B), such that
σ1 σ2 σn 0 ;  
U is an orthogonal matrix whose columns are the left singular vectors ui; V is an orthogonal matrix whose rows are the right singular vectors vi. Thus
Bui = σi vi   and   BT vi = σi ui ,   i = 1,2,,n .  
To compute U and/or VT, 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 VT 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=QBPH. If A is m by n with mn, then Q is m by n and PH is n by n; if A is n by p with n<p, then Q is n by n and PH is n by p. In this case, the matrices Q and/or PH 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) also has the capability of forming UHC, where C is an arbitrary complex matrix; this is needed when using the SVD to solve linear least squares problems.
nag_lapack_zbdsqr (f08ms) uses two different algorithms. If any singular vectors are required (i.e., if ncvt>0 or nru>0 or ncc>0), the bidiagonal QR algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between QR and QL 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 ui=vi=1, but are determined only to within a complex factor of absolute value 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' or 'L'.
2:     d: – double array
The dimension of the array d must be at least max1,n
The diagonal elements of the bidiagonal matrix B.
3:     e: – double array
The dimension of the array e must be at least max1,n-1
The off-diagonal elements of the bidiagonal matrix B.
4:     vtldvt: – complex array
The first dimension, ldvt, of the array vt must satisfy
  • if ncvt>0, ldvt max1,n ;
  • otherwise ldvt1.
The second dimension of the array vt must be at least max1,ncvt.
If ncvt>0, vt must contain an n by 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 unitary matrix PH returned by nag_lapack_zungbr (f08kt) with vect='P'.
5:     uldu: – complex array
The first dimension of the array u must be at least max1,nru.
The second dimension of the array u must be at least max1,n.
If nru>0, u must contain an nru by 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 unitary matrix Q returned by nag_lapack_zungbr (f08kt) with vect='Q'.
6:     cldc: – complex array
The first dimension, ldc, of the array c must satisfy
  • if ncc>0, ldc max1,n ;
  • otherwise ldc1.
The second dimension of the array c must be at least max1,ncc.
The n by ncc matrix C if 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: n0.
2:     ncvt int64int32nag_int scalar
Default: the second dimension of the array vt.
ncvt, the number of columns of the matrix VH of right singular vectors. Set ncvt=0 of right singular vectors. Set ncvt=0 if no right singular vectors are required.
Constraint: ncvt0.
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: nru0.
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: ncc0.

Output Parameters

1:     d: – double array
The dimension of the array d will be max1,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 max1,n-1
e is overwritten, but if info>0 see Error Indicators and Warnings.
3:     vtldvt: – complex array
The first dimension, ldvt, of the array vt will be
  • if ncvt>0, ldvt= max1,n ;
  • otherwise ldvt=1.
The second dimension of the array vt will be max1,ncvt.
The n by ncvt matrix VH or VH of right singular vectors, stored by rows.
If ncvt=0, vt is not referenced.
4:     uldu: – complex array
The first dimension of the array u will be max1,nru.
The second dimension of the array u will be max1,n.
The nru by n matrix U or QU of left singular vectors, stored as columns of the matrix.
If nru=0, u is not referenced.
5:     cldc: – complex array
The first dimension, ldc, of the array c will be
  • if ncc>0, ldc= max1,n ;
  • otherwise ldc=1.
The second dimension of the array c will be max1,ncc.
c stores the matrix UHC. 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.
If σi is an exact singular value of B and σ~i is the corresponding computed value, then
σ~i - σi p m,n ε σi  
where pm,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 pm,n is smaller), than when some singular vectors are also computed.
If ui is an exact left singular vector of B, and u~i is the corresponding computed left singular vector, then the angle θu~i,ui between them is bounded as follows:
θ u~i,ui p m,n ε relgapi  
where relgapi is the relative gap between σi and the other singular values, defined by
relgapi = min ij σi - σj σi + σj .  
A similar error bound holds for the right singular vectors.

Further Comments

The total number of real floating-point operations is roughly proportional to n2 if only the singular values are computed. About 12n2×nru additional operations are required to compute the left singular vectors and about 12n2×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 real analogue of this function is nag_lapack_dbdsqr (f08me).

Example

See Example in nag_lapack_zungbr (f08kt), which illustrates the use of the function to compute the singular value decomposition of a general matrix.
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


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