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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zungbr (f08kt)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_lapack_zungbr (f08kt) generates one of the complex unitary matrices Q or PH which were determined by nag_lapack_zgebrd (f08ks) when reducing a complex matrix to bidiagonal form.

Syntax

[a, info] = f08kt(vect, k, a, tau, 'm', m, 'n', n)
[a, info] = nag_lapack_zungbr(vect, k, a, tau, 'm', m, 'n', n)

Description

nag_lapack_zungbr (f08kt) is intended to be used after a call to nag_lapack_zgebrd (f08ks), which reduces a complex rectangular matrix A to real bidiagonal form B by a unitary transformation: A=QBPH. nag_lapack_zgebrd (f08ks) represents the matrices Q and PH as products of elementary reflectors.
This function may be used to generate Q or PH explicitly as square matrices, or in some cases just the leading columns of Q or the leading rows of PH.
The various possibilities are specified by the arguments vect, m, n and k. The appropriate values to cover the most likely cases are as follows (assuming that A was an m by n matrix):
1. To form the full m by m matrix Q: 
[a, info] = f08kt('Q', n, a, tau);
(note that the array a must have at least m columns).
2. If m>n, to form the n leading columns of Q: 
[a, info] = f08kt('Q', n, a(1:m,1:n), tau);
3. To form the full n by n matrix PH: 
[a, info] = f08kt('P', m, a, tau);
(note that the array a must have at least n rows).
4. If m<n, to form the m leading rows of PH: 
[a, info] = f08kt('P', m, a(1:m,1:n), tau);

References

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

Parameters

Compulsory Input Parameters

1:     vect – string (length ≥ 1)
Indicates whether the unitary matrix Q or PH is generated.
vect='Q'
Q is generated.
vect='P'
PH is generated.
Constraint: vect='Q' or 'P'.
2:     k – int64int32nag_int scalar
If vect='Q', the number of columns in the original matrix A.
If vect='P', the number of rows in the original matrix A.
Constraint: k≥0.
3:     alda: – complex array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgebrd (f08ks).
4:     tau: – complex array
The dimension of the array tau must be at least max1,minm,k if vect='Q' and at least max1,minn,k if vect='P'
Further details of the elementary reflectors, as returned by nag_lapack_zgebrd (f08ks) in its argument tauq if vect='Q', or in its argument taup if vect='P'.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the unitary matrix Q or PH to be returned.
Constraint: m≥0.
2:     n – int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the unitary matrix Q or PH to be returned.
Constraints:
  • n≥0;
  • if vect='Q' and m>k, m≥n≥k;
  • if vect='Q' and m≤k, m=n;
  • if vect='P' and n>k, n≥m≥k;
  • if vect='P' and n≤k, n=m.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
The unitary matrix Q or PH, or the leading rows or columns thereof, as specified by vect, m and n.
2:     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: k, 5: a, 6: lda, 7: tau, 8: work, 9: lwork, 10: 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 matrix Q differs from an exactly unitary matrix by a matrix E such that
E2 = Oε ,  
where ε is the machine precision. A similar statement holds for the computed matrix PH.

Further Comments

The total number of real floating-point operations for the cases listed in Description are approximately as follows:
1. To form the whole of Q:
  • 163n3m2-3mn+n2 if m>n,
  • 163m3 if m≤n;
2. To form the n leading columns of Q when m>n:
  • 83n23m-n;
3. To form the whole of PH:
  • 163n3 if m≥n,
  • 163m33n2-3mn+m2 if m<n;
4. To form the m leading rows of PH when m<n:
  • 83m23n-m.
The real analogue of this function is nag_lapack_dorgbr (f08kf).

Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix A, where
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  
in the first example and
A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i  
in the second. A must first be reduced to tridiagonal form by nag_lapack_zgebrd (f08ks). The program then calls nag_lapack_zungbr (f08kt) twice to form Q and PH, and passes these matrices to nag_lapack_zbdsqr (f08ms), which computes the singular value decomposition of A.

Open in the MATLAB editor: f08kt_example

function f08kt_example


fprintf('f08kt example results\n\n');

% Two cases of performing Singular Value Decomposition
% Case 1: m > n
ex1;
% Case 2: m < n
ex2;



function ex1

  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];
  
  % Reduce A to bidiagonal form
  [B, d, e, tauq, taup, info] =  ...
  f08ks(a);

  % Form PT explicitly (n-by-n, k = m)
  vect = 'P';
  [PT, info] = f08kt( ...
                      vect, m, B, taup, 'm', n, 'n', n);

  % Form Q explicitly (m-by-n, k = n)
  vect = 'Q';
  [Q, info] = f08kt( ...
                     vect, n, B, tauq, 'm', m, 'n', n);
  
  % Compute SVD of A using its bidiagonal form.
  c = [];
  uplo = 'Upper';
  [s, ~, VT, U, c, info] = f08ms( ...
                                  uplo, d, e, PT, Q, c);

  disp('Example 1: singular values');
  disp(s(1:n)');
  disp('Example 1: right singular vectors, by row');
  disp(VT(1:n,:));
  disp('Example 1: left singular vectors, by column');
  disp(U);

function ex2
  m = int64(3);
  n = int64(4);
  a = [0.28 - 0.36i  0.50 - 0.86i -0.77 - 0.48i  1.58 + 0.66i;
      -0.50 - 1.10i -1.21 + 0.76i -0.32 - 0.24i -0.27 - 1.15i;
       0.36 - 0.51i -0.07 + 1.33i -0.75 + 0.47i -0.08 + 1.01i];
 
  % Reduce A to bidiagonal form
  [B, d, e, tauq, taup, info] =  ...
  f08ks(a);

  % Form P^T explicitly (m-by-n, k = m)
  vect = 'P';
  [PT, info] = f08kt( ...
                      vect, m, B, taup, 'm', m, 'n', n);
  % Form Q explicitly (m-by-m, k = n)
  vect = 'Q';
  [Q, info] = f08kt( ...
                     vect, n, B, tauq, 'm', m, 'n', m);
  
  % Compute SVD of A using its bidiagonal form.
  c = [];
  uplo = 'Lower';
  [s, ~, VT, U, c, info] = f08ms( ...
                                  uplo, d, e, PT, Q, c, 'n', m);

  disp('Example 2: singular values');
  disp(s(1:m)');
  disp('Example 2: right singular vectors, by row');
  disp(VT);
  disp('Example 2: left singular vectors, by column');
  disp(U(:,1:m));

f08kt example results

Example 1: singular values
    3.9994    3.0003    1.9944    0.9995

Example 1: right singular vectors, by row
  -0.6971 + 0.0000i  -0.0867 - 0.3548i   0.0560 - 0.5400i  -0.1878 - 0.2253i
   0.2403 + 0.0000i   0.0725 - 0.2336i  -0.2477 - 0.5291i   0.7026 + 0.2177i
  -0.5123 + 0.0000i  -0.3030 - 0.1735i   0.0678 + 0.5162i   0.4418 + 0.3864i
  -0.4403 + 0.0000i   0.5294 + 0.6361i  -0.3027 - 0.0346i   0.1667 + 0.0258i

Example 1: left singular vectors, by column
  -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

Example 2: singular values
    3.0004    1.9967    0.9973

Example 2: right singular vectors, by row
   0.2454 - 0.0001i   0.2942 - 0.5843i   0.0162 - 0.0810i   0.6794 + 0.2083i
  -0.1692 + 0.5194i   0.1915 - 0.4374i   0.5205 - 0.0244i  -0.3149 - 0.3208i
  -0.5553 + 0.1403i   0.1438 - 0.1507i  -0.5684 - 0.5505i  -0.0318 - 0.0378i

Example 2: left singular vectors, by column
   0.6518 + 0.0000i  -0.4312 + 0.0000i   0.6239 + 0.0000i
  -0.4437 - 0.5027i  -0.3794 + 0.1026i   0.2014 + 0.5961i
  -0.2012 + 0.2916i  -0.8122 + 0.0030i  -0.3511 - 0.3026i
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