f08kf:: Least Squares and Eigenvalue Problems (LAPACK) (NAG Toolbox)

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

n

matrix):

1.	To form the full $m$ by $m$ matrix $Q$ : [a, info] = f08kf('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] = f08kf('Q', n, a, tau);
3.	To form the full $n$ by $n$ matrix $P^{T}$ : [a, info] = f08kf('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 $P^{T}$ : [a, info] = f08kf('P', m, a, tau);

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

The total number of floating-point operations for the cases listed in Description are approximately as follows:

1.	To form the whole of $Q$ : $\frac{4}{3} n (3 m^{2} - 3 m n + n^{2})$ if $m > n$ , $\frac{4}{3} m^{3}$ if $m \leq n$ ;
2.	To form the $n$ leading columns of $Q$ when $m > n$ : $\frac{2}{3} n^{2} (3 m - n)$ ;
3.	To form the whole of $P^{T}$ : $\frac{4}{3} n^{3}$ if $m \geq n$ , $\frac{4}{3} m (3 n^{2} - 3 m n + m^{2})$ if $m < n$ ;
4.	To form the $m$ leading rows of $P^{T}$ when $m < n$ : $\frac{2}{3} m^{2} (3 n - m)$ .

Example

For this function two examples are presented, both of which involve computing the singular value decomposition of a matrix

A

, where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array})

in the first example and

A = (\begin{array}{r} - 5.42 & 3.28 & - 3.68 & 0.27 & 2.06 & 0.46 \\ - 1.65 & - 3.40 & - 3.20 & - 1.03 & - 4.06 & - 0.01 \\ - 0.37 & 2.35 & 1.90 & 4.31 & - 1.76 & 1.13 \\ - 3.15 & - 0.11 & 1.99 & - 2.70 & 0.26 & 4.50 \end{array})

in the second.

A

must first be reduced to tridiagonal form by nag_lapack_dgebrd (f08ke). The program then calls nag_lapack_dorgbr (f08kf) twice to form

Q

and

P^{T}

, and passes these matrices to nag_lapack_dbdsqr (f08me), which computes the singular value decomposition of

A

function f08kf_example


fprintf('f08kf 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 = [-0.57  -1.28  -0.39   0.25;
       -1.93   1.08  -0.31  -2.14;
        2.30   0.24   0.40  -0.35;
       -1.93   0.64  -0.66   0.08;
        0.15   0.30   0.15  -2.13;
       -0.02   1.03  -1.43   0.50];

  % Reduce A to bidiagonal form
  [B, d, e, tauq, taup, info] =  ...
  f08ke(a);

  % Form P^T explicitly (n-by-n, k = m)
  vect = 'P';
  [PT, info] = f08kf( ...
                      vect, m, B, taup, 'm', n, 'n', n);
  % Form Q explicitly (m-by-n, k = n)
  vect = 'Q';
  [Q, info] = f08kf( ...
                     vect, n, B, tauq, 'm', m, 'n', n);
  
  % Compute SVD of A using its bidiagonal form.
  c = [];
  uplo = 'Upper';
  [s, ~, VT, U, c, info] = f08me( ...
                                  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(4);
  n = int64(6);
  a = [-5.42   3.28  -3.68   0.27   2.06   0.46;
       -1.65  -3.40  -3.20  -1.03  -4.06  -0.01;
       -0.37   2.35   1.90   4.31  -1.76   1.13;
       -3.15  -0.11   1.99  -2.70   0.26   4.50];

  % Reduce A to bidiagonal form
  [B, d, e, tauq, taup, info] =  ...
  f08ke(a);

  % Form P^T explicitly (m-by-n, k = m)
  vect = 'P';
  [PT, info] = f08kf( ...
                      vect, m, B, taup, 'm', m, 'n', n);
  % Form Q explicitly (m-by-m, k = n)
  vect = 'Q';
  [Q, info] = f08kf( ...
                     vect, n, B, tauq, 'm', m, 'n', m);
  
  % Compute SVD of A using its bidiagonal form.
  c = [];
  uplo = 'Lower';
  [s, ~, VT, U, c, info] = f08me( ...
                                  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));

f08kf example results

Example 1: singular values
    3.9987    3.0005    1.9967    0.9999

Example 1: right singular vectors, by row
    0.8251   -0.2794    0.2048    0.4463
   -0.4530   -0.2121   -0.2622    0.8252
   -0.2829   -0.7961    0.4952   -0.2026
    0.1841   -0.4931   -0.8026   -0.2807

Example 1: left singular vectors, by column
   -0.0203    0.2794    0.4690    0.7692
   -0.7284   -0.3464   -0.0169   -0.0383
    0.4393   -0.4955   -0.2868    0.0822
   -0.4678    0.3258   -0.1536   -0.1636
   -0.2200   -0.6428    0.1125    0.3572
   -0.0935    0.1927   -0.8132    0.4957

Example 2: singular values
    7.9987    7.0059    5.9952    4.9989

Example 2: right singular vectors, by row
   -0.7933    0.3163   -0.3342   -0.1514    0.2142    0.3001
    0.1002    0.6442    0.4371    0.4890    0.3771    0.0501
    0.0111    0.1724   -0.6367    0.4354   -0.0430   -0.6111
    0.2361    0.0216   -0.1025   -0.5286    0.7460   -0.3120

Example 2: left singular vectors, by column
    0.8884    0.1275    0.4331    0.0838
    0.0733   -0.8264    0.1943   -0.5234
   -0.0361    0.5435    0.0756   -0.8352
    0.4518   -0.0733   -0.8769   -0.1466

NAG Toolbox: nag_lapack_dorgbr (f08kf)

▸▿ Contents

Purpose

Syntax

Description