NAG Toolbox: nag_lapack_zgelsd (f08kq)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 $n$ coefficient matrix $A$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

The $m$ by $r$ right-hand side matrix $B$.

3:     $\mathrm{rcond}$ – double scalar

Used to determine the effective rank of $A$. Singular values $\mathbf{s}\left(i\right)\le \mathbf{rcond}\times \mathbf{s}\left(1\right)$ are treated as zero. If $\mathbf{rcond}<0$, machine precision is used instead.

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

3:     $\mathrm{nrhs\_p}$ – int64int32nag_int scalar

Default: the second dimension of the array b.

$r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.

Constraint: $\mathbf{nrhs\_p}\ge 0$.

4:     $\mathrm{lwork}$ – int64int32nag_int scalar

Suggested value: for optimal performance, lwork should generally be larger than the required minimum. Consider increasing lwork by at least $\mathit{nb}\times \min \left(\mathbf{m},\mathbf{n}\right)$, where $\mathit{nb}$ is the optimal block size.

Default: $\max \left(1,64\min \left(\mathbf{m},\mathbf{n}\right)\max \left(\mathbf{m}+\mathbf{n}+\mathit{r},2\mathit{r}+\mathit{r}\times \mathbf{nrhs\_p}\right)\right)$ 

The dimension of the array work.

The exact minimum amount of workspace needed depends on m, n and nrhs_p. As long as lwork is at least

$$\max \left(1,\mathbf{m}+\mathbf{n}+r,2r+r\times \mathbf{nrhs\_p}\right)\text{,}$$

where $r=\min \left(\mathbf{m},\mathbf{n}\right)$, the code will execute correctly.

Constraint: lwork must be at least $\max \left(1,\mathbf{m}+\mathbf{n}+r,2r+r\times \mathbf{nrhs\_p}\right)$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 destroyed.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

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

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

b stores the $n$ by $r$ solution matrix $X$. If $m\ge n$ and $\mathbf{rank}=n$, the residual sum of squares for the solution in the $i$th column is given by the sum of squares of the modulus of elements $n+1,\dots ,m$ in that column.

3:     $\mathrm{s}\left(:\right)$ – double array

The dimension of the array s will be $\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$

The singular values of $A$ in decreasing order.

4:     $\mathrm{rank}$ – int64int32nag_int scalar

The effective rank of $A$, i.e., the number of singular values which are greater than $\mathbf{rcond}\times \mathbf{s}\left(1\right)$.

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

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: m, 2: n, 3: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: s, 9: rcond, 10: rank, 11: work, 12: lwork, 13: rwork, 14: iwork, 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.

   $\mathbf{info}>0$

The algorithm for computing the SVD failed to converge; if $\mathbf{info}=i$, $i$ off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08kq_example

function f08kq_example


fprintf('f08kq example results\n\n');

% Least squares problem min ||b - Ax|| where A and b are:
a = [ 0.47 - 0.34i, -0.32 - 0.23i,  0.35 - 0.60i,  0.89 + 0.71i, -0.19 + 0.06i;
     -0.40 + 0.54i, -0.05 + 0.20i, -0.52 - 0.34i, -0.45 - 0.45i,  0.11 - 0.85i;
      0.60 + 0.01i, -0.26 - 0.44i,  0.87 - 0.11i, -0.02 - 0.57i,  1.44 + 0.80i;
      0.80 - 1.02i, -0.43 + 0.17i, -0.34 - 0.09i,  1.14 - 0.78i,  0.07 + 1.14i];
[m,n] = size(a);
b = [ 2.15 - 0.20i;
     -2.24 + 1.82i;
      4.45 - 4.28i;
      5.70 - 6.25i;
      0    + 0i];

% Treat singular values less than 0.01 as zero
rcond = 0.01;
[vr, x, s, rank, info] = f08kq( ...
                                a, b, rcond);

disp('Least squares solution');
disp(x(1:n));
disp('Tolerance used to estimate the rank of A');
fprintf('%12.2e\n',rcond);
disp('Estimated rank of A');
fprintf('%5d\n\n',rank);
disp('Singular values of A');
disp(s');

f08kq example results

Least squares solution
   3.9747 - 1.8377i
  -0.9186 + 0.8253i
  -0.3105 + 0.1477i
   1.0050 + 0.8626i
  -0.2256 - 1.9425i

Tolerance used to estimate the rank of A
    1.00e-02
Estimated rank of A
    3

Singular values of A
    2.9979    1.9983    1.0044    0.0064