NAG Toolbox: nag_lapack_dgejsv (f08kh)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{joba}$ – string (length ≥ 1)

Specifies the form of pivoting for the $QR$ factorization stage; whether an estimate of the condition number of the scaled matrix is required; and the form of rank reduction that is performed.

$\mathbf{joba}=\text{'C'}$

The initial $QR$ factorization of the input matrix is performed with column pivoting; no estimate of condition number is computed; and, the rank is reduced by only the underflowed part of the triangular factor $R$. This option works well (high relative accuracy) if $A=BD$, with well-conditioned $B$ and arbitrary diagonal matrix $D$. The accuracy cannot be spoiled by column scaling. The accuracy of the computed output depends on the condition of $B$, and the procedure aims at the best theoretical accuracy.

$\mathbf{joba}=\text{'E'}$

Computation as with $\mathbf{joba}=\text{'C'}$ with an additional estimate of the condition number of $B$. It provides a realistic error bound.

$\mathbf{joba}=\text{'F'}$

The initial $QR$ factorization of the input matrix is performed with full row and column pivoting; no estimate of condition number is computed; and, the rank is reduced by only the underflowed part of the triangular factor $R$. If $A=D_1\times C\times D_2$ with ill-conditioned diagonal scalings $D_1$, $D_2$, and well-conditioned matrix $C$, this option gives higher accuracy than the $\mathbf{joba}=\text{'C'}$ option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable.

$\mathbf{joba}=\text{'G'}$

Computation as with $\mathbf{joba}=\text{'F'}$ with an additional estimate of the condition number of $B$, where $A=DB$ (i.e., $B=C\times D_2$). If $A$ has heavily weighted rows, then using this condition number gives too pessimistic an error bound.

$\mathbf{joba}=\text{'A'}$

Computation as with $\mathbf{joba}=\text{'C'}$ except in the treatment of rank reduction. In this case, small singular values are to be considered as noise and, if found, the matrix is treated as numerically rank deficient. The computed SVD $A=U\Sigma V^{\mathrm{T}}$ restores $A$ up to $f\left(m,n\right)\times \epsilon \times ‖A‖$, where $\epsilon$ is machine precision. This gives the procedure licence to discard (set to zero) all singular values below $\mathbf{n}\times \epsilon \times ‖A‖$.

$\mathbf{joba}=\text{'R'}$

Similar to $\mathbf{joba}=\text{'A'}$. The rank revealing property of the initial $QR$ factorization is used to reveal (using the upper triangular factor) a gap ${\sigma }_{r+1}<\epsilon {\sigma }_r$ in which case the numerical rank is declared to be $r$. The SVD is computed with absolute error bounds, but more accurately than with $\mathbf{joba}=\text{'A'}$.

Constraint: $\mathbf{joba}=\text{'C'}$, $\text{'E'}$, $\text{'F'}$, $\text{'G'}$, $\text{'A'}$ or $\text{'R'}$.

2:     $\mathrm{jobu}$ – string (length ≥ 1)

Specifies options for computing the left singular vectors $U$.

$\mathbf{jobu}=\text{'U'}$

The first $n$ left singular vectors (columns of $U$) are computed and returned in the array u.

$\mathbf{jobu}=\text{'F'}$

All $m$ left singular vectors are computed and returned in the array u.

$\mathbf{jobu}=\text{'W'}$

No left singular vectors are computed, but the array u (with $\mathit{ldu}\ge \mathbf{m}$ and second dimension at least n) is available as workspace for computing right singular values. See the description of u.

$\mathbf{jobu}=\text{'N'}$

No left singular vectors are computed. $\mathbf{u}$ is not referenced.

Constraint: $\mathbf{jobu}=\text{'U'}$, $\text{'F'}$, $\text{'W'}$ or $\text{'N'}$.

3:     $\mathrm{jobv}$ – string (length ≥ 1)

Specifies options for computing the right singular vectors $V$.

$\mathbf{jobv}=\text{'V'}$

the $n$ right singular vectors (columns of $V$) are computed and returned in the array v; Jacobi rotations are not explicitly accumulated.

$\mathbf{jobv}=\text{'J'}$

the $n$ right singular vectors (columns of $V$) are computed and returned in the array v, but they are computed as the product of Jacobi rotations. This option is allowed only if $\mathbf{jobu}=\text{'U'}$ or $\text{'F'}$, i.e., in computing the full SVD.

$\mathbf{jobv}=\text{'W'}$

No right singular values are computed, but the array v (with $\mathit{ldv}\ge \mathbf{n}$ and second dimension at least n) is available as workspace for computing left singular values. See the description of v.

$\mathbf{jobv}=\text{'N'}$

No right singular vectors are computed. $\mathbf{v}$ is not referenced.

Constraints:

• $\mathbf{jobv}=\text{'V'}$, $\text{'J'}$, $\text{'W'}$ or $\text{'N'}$;
• if $\mathbf{jobu}=\text{'W'}$ or $\text{'N'}$, $\mathbf{jobv}\ne \text{'J'}$.

4:     $\mathrm{jobr}$ – string (length ≥ 1)

Suggested value: $\mathbf{jobr}=\text{'R'}$.

Specifies the conditions under which columns of $A$ are to be set to zero. This effectively specifies a lower limit on the range of singular values; any singular values below this limit are (through column zeroing) set to zero. If $A\ne 0$ is scaled so that the largest column (in the Euclidean norm) of $cA$ is equal to the square root of the overflow threshold, then jobr allows the function to kill columns of $A$ whose norm in $cA$ is less than $\sqrt{\mathit{sfmin}}$ (for $\mathbf{jobr}=\text{'R'}$), or less than $\mathit{sfmin}/\epsilon$ (otherwise). $\mathit{sfmin}$ is the safe range argument, as returned by function nag_machine_real_safe (x02am).

$\mathbf{jobr}=\text{'N'}$

Only set to zero those columns of $A$ for which the norm of corresponding column of $cA<\mathit{sfmin}/\epsilon$, that is, those columns that are effectively zero (to machine precision) anyway. If the condition number of $A$ is greater than the overflow threshold $\lambda$, where $\lambda$ is the value returned by nag_machine_real_largest (x02al), you are recommended to use function nag_lapack_dgesvj (f08kj).

$\mathbf{jobr}=\text{'R'}$

Set to zero those columns of $A$ for which the norm of the corresponding column of $cA<\sqrt{\mathit{sfmin}}$. This approximately represents a restricted range for $\sigma \left(cA\right)$ of $\left[\sqrt{\mathit{sfmin}},\sqrt{\lambda }\right]$.

For computing the singular values in the full range from the safe minimum up to the overflow threshold use nag_lapack_dgesvj (f08kj)

Constraint: $\mathbf{jobr}=\text{'N'}$ or $\text{'R'}$.

5:     $\mathrm{jobt}$ – string (length ≥ 1)

Specifies, in the case $n=m$, whether the function is permitted to use the transpose of $A$ for improved efficiency. If the matrix is square then the procedure may use transposed $A$ if $A^{\mathrm{T}}$ seems to be better with respect to convergence. If the matrix is not square, jobt is ignored. The decision is based on two values of entropy over the adjoint orbit of $A^{\mathrm{T}}A$. See the descriptions of $\mathbf{work}\left(6\right)$ and $\mathbf{work}\left(7\right)$.

$\mathbf{jobt}=\text{'T'}$

If $n=m$, perform an entropy test and then transpose if the test indicates possibly faster convergence of the Jacobi process if $A^{\mathrm{T}}$ is taken as input. If $A$ is replaced with $A^{\mathrm{T}}$, then the row pivoting is included automatically.

$\mathbf{jobt}=\text{'N'}$

No entropy test and no transposition is performed.

The option $\mathbf{jobt}=\text{'T'}$ can be used to compute only the singular values, or the full SVD ($U$, $\Sigma$ and $V$). In the case where only one set of singular vectors ($U$ or $V$) is required, the caller must still provide both u and v, as one of the matrices is used as workspace if the matrix $A$ is transposed. See the descriptions of u and v

Constraint: $\mathbf{jobt}=\text{'T'}$ or $\text{'N'}$.

6:     $\mathrm{jobp}$ – string (length ≥ 1)

Specifies whether the function should be allowed to introduce structured perturbations to drown denormalized numbers. For details see Drmac and Veselic (2008a) and Drmac and Veselic (2008b). For the sake of simplicity, these perturbations are included only when the full SVD or only the singular values are requested.

$\mathbf{jobp}=\text{'P'}$

Introduce perturbation if $A$ is found to be very badly scaled (introducing denormalized numbers).

$\mathbf{jobp}=\text{'N'}$

Do not perturb.

Constraint: $\mathbf{jobp}=\text{'P'}$ or $\text{'N'}$.

7:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array a must be at least $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The $m$ by $n$ matrix $A$.

Optional Input Parameters

1:     $\mathrm{m}$ – int64int32nag_int scalar

Default: the first dimension of the array a.

$m$, the number of rows of the matrix $A$.

Constraint: $\mathbf{m}\ge 0$.

2:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the second dimension of the array a.

$n$, the number of columns of the matrix $A$.

Constraint: $\mathbf{m}\ge \mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array a will be $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The contents of a are overwritten.

2:     $\mathrm{sva}\left(\mathbf{n}\right)$ – double array

The, possibly scaled, singular values of $A$.

The singular values of $A$ are ${\sigma }_{\mathit{i}}=\alpha \mathbf{sva}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n$, where $\alpha =\mathbf{work}\left(1\right)/\mathbf{work}\left(2\right)$. Normally $\alpha =1$ and no scaling is required to obtain the singular values. However, if the largest singular value of $A$ overflows or if small singular values have been saved from underflow by scaling the input matrix $A$, then $\alpha \ne 1$.

If $\mathbf{jobr}=\text{'R'}$ then some of the singular values may be returned as exact zeros because they are below the numerical rank threshold or are denormalized numbers.

3:     $\mathrm{u}\left(\mathit{ldu},:\right)$ – double array

The first dimension, $\mathit{ldu}$, of the array u will be

• if $\mathbf{jobu}=\text{'F'}$, $\text{'U'}$ or $\text{'W'}$, $\mathit{ldu}=\max \left(1,\mathbf{m}\right)$;
• otherwise $\mathit{ldu}=1$.

The second dimension of the array u will be $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobu}=\text{'F'}$, $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobu}=\text{'U'}$ or $\text{'W'}$ and $1$ otherwise.

If $\mathbf{jobu}=\text{'U'}$, u contains the $m$ by $n$ matrix of the left singular vectors.

If $\mathbf{jobu}=\text{'F'}$, u contains the $m$ by $m$ matrix of the left singular vectors, including an orthonormal basis of the orthogonal complement of Range($A$).

If $\mathbf{jobu}=\text{'W'}$ and ($\mathbf{jobv}=\text{'V'}$ and $\mathbf{jobt}=\text{'T'}$ and $\mathbf{m}=\mathbf{n}$), then u is used as workspace if the procedure replaces $A$ with $A^{\mathrm{T}}$. In that case, $V$ is computed in u as left singular vectors of $A^{\mathrm{T}}$ and then copied back to the array v.

If $\mathbf{jobu}=\text{'N'}$, u is not referenced.

4:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – double array

The first dimension, $\mathit{ldv}$, of the array v will be

• if $\mathbf{jobv}=\text{'V'}$, $\text{'J'}$ or $\text{'W'}$, $\mathit{ldv}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldv}=1$.

The second dimension of the array v will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobv}=\text{'V'}$, $\text{'J'}$ or $\text{'W'}$ and $1$ otherwise.

If $\mathbf{jobv}=\text{'V'}$ or $\text{'J'}$, v contains the $n$ by $n$ matrix of the right singular vectors.

If $\mathbf{jobv}=\text{'W'}$ and ($\mathbf{jobu}=\text{'U'}$ and $\mathbf{jobt}=\text{'T'}$ and $\mathbf{m}=\mathbf{n}$), then v is used as workspace if the procedure replaces $A$ with $A^{\mathrm{T}}$. In that case, $U$ is computed in v as right singular vectors of $A^{\mathrm{T}}$ and then copied back to the array u.

If $\mathbf{jobv}=\text{'N'}$, v is not referenced.

5:     $\mathrm{work}\left(\mathit{lwork}\right)$ – double array

Contains information about the completed job.

$\mathbf{work}\left(1\right)$

$\alpha =\mathbf{work}\left(1\right)/\mathbf{work}\left(2\right)$ is the scaling factor such that ${\sigma }_{\mathit{i}}=\alpha \mathbf{sva}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,n$ are the computed singular values of $A$. (See the description of $\mathbf{sva}$.)

$\mathbf{work}\left(2\right)$

See the description of $\mathbf{work}\left(1\right)$.

$\mathbf{work}\left(3\right)$

sconda, an estimate for the condition number of column equilibrated $A$ (if $\mathbf{joba}=\text{'E'}$ or $\text{'G'}$). sconda is an estimate of $\sqrt{\left({‖{\left(R^{\mathrm{T}}R\right)}^{-1}‖}_1\right)}$. It is computed using nag_lapack_dpocon (f07fg). It satisfies $n^{-\frac{1}{4}}\times \mathit{sconda}\le {‖R^{-1}‖}_2\le n^{\frac{1}{4}}\times \mathit{sconda}$ where $R$ is the triangular factor from the $QR$ factorization of $A$. However, if $R$ is truncated and the numerical rank is determined to be strictly smaller than $n$, sconda is returned as $-1$, thus indicating that the smallest singular values might be lost.

If full SVD is needed, and you are familiar with the details of the method, the following two condition numbers are useful for the analysis of the algorithm.

$\mathbf{work}\left(4\right)$

An estimate of the scaled condition number of the triangular factor in the first $QR$ factorization.

$\mathbf{work}\left(5\right)$

An estimate of the scaled condition number of the triangular factor in the second $QR$ factorization.

The following two parameters are computed if $\mathbf{jobt}=\text{'T'}$.

$\mathbf{work}\left(6\right)$

The entropy of $A^{\mathrm{T}}A$: this is the Shannon entropy of $\mathrm{diag}{A}^{\mathrm{T}}A/\mathrm{trace}{A}^{\mathrm{T}}A$ taken as a point in the probability simplex.

$\mathbf{work}\left(7\right)$

The entropy of $A{A}^{\mathrm{T}}$.

6:     $\mathrm{iwork}\left(\mathbf{m}+3\times \mathbf{n}\right)$ – int64int32nag_int array

Contains information about the completed job.

$\mathbf{iwork}\left(1\right)$

The numerical rank of $A$ determined after the initial $QR$ factorization with pivoting. See the descriptions of joba and jobr.

$\mathbf{iwork}\left(2\right)$

The number of computed nonzero singular values.

$\mathbf{iwork}\left(3\right)$

If nonzero, a warning message: If $\mathbf{iwork}\left(3\right)=1$ then some of the column norms of $A$ were denormalized (tiny) numbers. The requested high accuracy is not warranted by the data.

7:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ 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.

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W  $\mathbf{info}>0$

nag_lapack_dgejsv (f08kh) did not converge in the allowed number of iterations ($30$). The computed values might be inaccurate.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08kh_example

function f08kh_example


fprintf('f08kh example results\n\n');

% General 6-by-4 real matrix A
m = int64(6);
n = int64(4);
a = [2.27, -1.54,  1.15, -1.94;
     0.28, -1.67,  0.94, -0.78;
    -0.48, -3.09,  0.99, -0.21;
     1.07,  1.22,  0.79,  0.63;
    -2.35,  2.93, -1.45,  2.30;
     0.62, -7.39,  1.03, -2.57];

% Compute the SVD of A (A = U*S*V^T, m >= n)
joba = 'estimated condition number';
jobu = 'U required';
jobv = 'V required';
jobr = 'restricted range';
jobt = 'No entropy test';
jobp = 'No perturbations';
[~, s, u, v, work, iwork, info] = ...
  f08kh( ...
         joba, jobu, jobv, jobr, jobt, jobp, a);

% Approximate error bound for S, s(1) = norm(A).
eps = x02aj;
serrbd = eps*s(1);

% Print solution
if (abs(work(1)-work(2)) < 2*eps)
  % No scaling required
  fprintf('Singular values:\n');
  disp(transpose(s));
else
  fprintf('Scaled singular values:\n');
  disp(transpose(s));
  fprintf('\nFor true singular values, multiply by a/b\n');
  fprintf('   where a=%13.5e and b=%13.5e.\n', work(1), work(2));
end

[ifail] = x04ca( ...
                 'Gen', ' ', u, 'Left singular vectors');
fprintf('\n');
[ifail] = x04ca( ...
                 'Gen', ' ', v, 'Right singular vectors');

% Estimate reciprocal condition numbers for S
[rcondu, info] = f08fl( ...
                        'Left', m, n, s);
[rcondv, info] = f08fl( ...
                        'Right', m, n, s);

% display approximate error bounds
fprintf('\nEstimate of the condition number of column equilibrated A\n');
fprintf('%11.1e\n', work(3));
fprintf('\nError estimate  for S:\n');
fprintf('%11.1e\n', serrbd);
fprintf('\nError estimates for U:\n');
fprintf('%11.1e ',serrbd./rcondu);
fprintf('\n\nError estimates for V:\n');
fprintf('%11.1e ',serrbd./rcondv);
fprintf('\n');

f08kh example results

Singular values:
    9.9966    3.6831    1.3569    0.5000

 Left singular vectors
          1       2       3       4
 1   0.2774 -0.6003 -0.1277  0.1323
 2   0.2020 -0.0301  0.2805  0.7034
 3   0.2918  0.3348  0.6453  0.1906
 4  -0.0938 -0.3699  0.6781 -0.5399
 5  -0.4213  0.5266  0.0413 -0.0575
 6   0.7816  0.3353 -0.1645 -0.3957

 Right singular vectors
          1       2       3       4
 1   0.1921 -0.8030  0.0041 -0.5642
 2  -0.8794 -0.3926 -0.0752  0.2587
 3   0.2140 -0.2980  0.7827  0.5027
 4  -0.3795  0.3351  0.6178 -0.6017

Estimate of the condition number of column equilibrated A
    9.0e+00

Error estimate  for S:
    1.1e-15

Error estimates for U:
    1.8e-16     4.8e-16     1.3e-15     2.2e-15 

Error estimates for V:
    1.8e-16     4.8e-16     1.3e-15     1.3e-15