NAG FL Interface
f08kmf (dgesvdx)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{jobu}$ – Character(1) Input

On entry: specifies options for computing all or part of the matrix $U$.

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

The ns columns of $U$, as specified by range, are returned in array u.

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

No columns of $U$ (no left singular vectors) are computed.

Constraint: $\mathbf{jobu}=\text{'V'}$ or $\text{'N'}$.

2: $\mathbf{jobvt}$ – Character(1) Input

On entry: specifies options for computing all or part of the matrix $V^{\mathrm{T}}$.

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

The ns rows of $V^{\mathrm{T}}$, as specified by range, are returned in the array vt.

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

No rows of $V^{\mathrm{T}}$ (no right singular vectors) are computed.

Constraint: $\mathbf{jobvt}=\text{'V'}$ or $\text{'N'}$.

3: $\mathbf{range}$ – Character(1) Input

On entry: indicates which singular values should be returned.

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

All singular values will be found.

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

All singular values in the half-open interval $\left(\mathbf{vl},\mathbf{vu}\right]$ will be found.

$\mathbf{range}=\text{'I'}$

The ilth through iuth singular values will be found.

Constraint: $\mathbf{range}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

4: $\mathbf{m}$ – Integer Input

On entry: $m$, the number of rows of the matrix $A$.

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

5: $\mathbf{n}$ – Integer Input

On entry: $n$, the number of columns of the matrix $A$.

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

6: $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (Kind=nag_wp) array Input/Output

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

On entry: the $m$ by $n$ matrix $A$.

On exit: if $\mathbf{jobu}\ne \text{'N'}$ and $\mathbf{jobvt}\ne \text{'N'}$, the contents of a are destroyed.

7: $\mathbf{lda}$ – Integer Input

On entry: the first dimension of the array a as declared in the (sub)program from which f08kmf is called.

Constraint: $\mathbf{lda}\ge \max \left(1,\mathbf{m}\right)$.

8: $\mathbf{vl}$ – Real (Kind=nag_wp) Input

On entry: if $\mathbf{range}=\text{'V'}$, the lower bound of the interval to be searched for singular values.

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vl is not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $0.0\le \mathbf{vl}$.

9: $\mathbf{vu}$ – Real (Kind=nag_wp) Input

On entry: if $\mathbf{range}=\text{'V'}$, the upper bound of the interval to be searched for singular values.

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vu is not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

10: $\mathbf{il}$ – Integer Input

11: $\mathbf{iu}$ – Integer Input

On entry: if $\mathbf{range}=\text{'I'}$, il and iu specify the indices (in ascending order) of the smallest and largest singular values to be returned, respectively.

If $\mathbf{range}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\text{'I'}$ and $\min \left(\mathbf{m},\mathbf{n}\right)=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\text{'I'}$ and $\min \left(\mathbf{m},\mathbf{n}\right)>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

12: $\mathbf{ns}$ – Integer Output

On exit: the total number of singular values found. $0\le \mathbf{ns}\le \min \left(\mathbf{m},\mathbf{n}\right)$.

If $\mathbf{range}=\text{'A'}$, $\mathbf{ns}=\min \left(\mathbf{m},\mathbf{n}\right)$.

If $\mathbf{range}=\text{'I'}$, $\mathbf{ns}=\mathbf{iu}-\mathbf{il}+1$.

If $\mathbf{range}=\text{'V'}$ then the value of ns is not known in advance and so an upper limit should be used when specifying the dimensions of array u, e.g., $\min \left(\mathbf{m},\mathbf{n}\right)$.

13: $\mathbf{s}\left(\min \left(\mathbf{m},\mathbf{n}\right)\right)$ – Real (Kind=nag_wp) array Output

On exit: the singular values of $A$, sorted so that $\mathbf{s}\left(i\right)\ge \mathbf{s}\left(i+1\right)$.

14: $\mathbf{u}\left(\mathbf{ldu},*\right)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array u must be at least $\max \left(1,\mathit{nsmax}\right)$ if $\mathbf{jobu}=\text{'V'}$, where $\mathit{nsmax}$ is a value larger than the output value ns..

On exit: if $\mathbf{jobu}=\text{'V'}$, u contains the first ns columns of $U$ (the left singular vectors, stored column-wise).

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

15: $\mathbf{ldu}$ – Integer Input

On entry: the first dimension of the array u as declared in the (sub)program from which f08kmf is called.

Constraints:

• if $\mathbf{jobu}=\text{'V'}$, $\mathbf{ldu}\ge \max \left(1,\mathbf{m}\right)$;
• otherwise $\mathbf{ldu}\ge 1$.

16: $\mathbf{vt}\left(\mathbf{ldvt},*\right)$ – Real (Kind=nag_wp) array Output

Note: the second dimension of the array vt must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvt}=\text{'V'}$.

On exit: if $\mathbf{jobvt}=\text{'V'}$, vt contains the first ns rows of $V^{\mathrm{T}}$ (the right singular vectors, stored row-wise).

If $\mathbf{jobvt}=\text{'N'}$, vt is not referenced.

17: $\mathbf{ldvt}$ – Integer Input

Note: if $\mathbf{jobvt}=\text{'V'}$ and $\mathbf{range}=\text{'V'}$ then the value of ns is not known in advance and so an upper limit should be used, e.g., $\min \left(\mathbf{m},\mathbf{n}\right)$.

On entry: the first dimension of the array vt as declared in the (sub)program from which f08kmf is called.

Constraints:

• if $\mathbf{jobvt}=\text{'V'}$, $\mathbf{ldvt}\ge \max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$;
• otherwise $\mathbf{ldvt}\ge 1$.

18: $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – Real (Kind=nag_wp) array Workspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ returns the optimal lwork.

If $\mathbf{info}>\mathbf{0}$, $\mathbf{work}\left(2:\min \left(\mathbf{m},\mathbf{n}\right)\right)$ contains the unconverged superdiagonal elements of an upper bidiagonal matrix $B$ whose diagonal is in s (not necessarily sorted). $B$ satisfies $A=UB{V}^{\mathrm{T}}$, so it has the same singular values as $A$, and left and right singular vectors that are those of $A$ pre-multiplied by $U^{\mathrm{T}}$ and $V^{\mathrm{T}}$.

19: $\mathbf{lwork}$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08kmf is called.

If $\mathbf{lwork}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, lwork should generally be larger than the constrained minimum value. Consider increasing lwork beyond the minimum requirement.

Constraint: $\mathbf{lwork}\ge \max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\times \left(3\times \min \left(\mathbf{m},\mathbf{n}\right)+20\right),4\times \min \left(\mathbf{m},\mathbf{n}\right)+\max \left(\mathbf{m},\mathbf{n}\right)\right)$.

20: $\mathbf{iwork}\left(12\times \min \left(\mathbf{n},\mathbf{m}\right)\right)$ – Integer array Workspace

On exit:

• if $\mathbf{info}=\mathbf{0}$, the first ns elements of iwork are zero;
• if $\mathbf{info}>\mathbf{0}$, iwork contains the indices of the eigenvectors that failed to converge in f08jbf and f08mbf, see iwork in f08mbf.

21: $\mathbf{info}$ – Integer Output

On exit: $\mathbf{info}=0$ unless the routine detects an error (see Section 6).

6 Error Indicators and 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.

$\mathbf{info}>0$

If f08kmf did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results