NAG Library Routine Document

f08kpf  (zgesvd)

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 $U$.

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

All $m$ columns of $U$ are returned in array u.

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

The first $\min \left(m,n\right)$ columns of $U$ (the left singular vectors) are returned in the array u.

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

The first $\min \left(m,n\right)$ columns of $U$ (the left singular vectors) are overwritten on the array a.

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

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

Constraint: $\mathbf{jobu}=\text{'A'}$, $\text{'S'}$, $\text{'O'}$ or $\text{'N'}$.

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

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

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

All $n$ rows of $V^{\mathrm{H}}$ are returned in the array vt.

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

The first $\min \left(m,n\right)$ rows of $V^{\mathrm{H}}$ (the right singular vectors) are returned in the array vt.

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

The first $\min \left(m,n\right)$ rows of $V^{\mathrm{H}}$ (the right singular vectors) are overwritten on the array a.

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

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

Constraints:

• $\mathbf{jobvt}=\text{'A'}$, $\text{'S'}$, $\text{'O'}$ or $\text{'N'}$;
• jobvt and jobu cannot both be $\text{'O'}$.

3:     $\mathbf{m}$ – IntegerInput

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

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

4:     $\mathbf{n}$ – IntegerInput

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

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

5:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Complex (Kind=nag_wp) arrayInput/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 $\mathbf{jobu}=\text{'O'}$, a is overwritten with the first $\min \left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise).

If $\mathbf{jobvt}=\text{'O'}$, a is overwritten with the first $\min \left(m,n\right)$ rows of $V^{\mathrm{H}}$ (the right singular vectors, stored row-wise).

If $\mathbf{jobu}\ne \text{'O'}$ and $\mathbf{jobvt}\ne \text{'O'}$, the contents of a are destroyed.

6:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08kpf (zgesvd) is called.

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

7:     $\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput

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

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

8:     $\mathbf{u}\left(\mathbf{ldu},*\right)$ – Complex (Kind=nag_wp) arrayOutput

Note: the second dimension of the array u must be at least $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobu}=\text{'A'}$, $\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$ if $\mathbf{jobu}=\text{'S'}$, and at least $1$ otherwise.

On exit: if $\mathbf{jobu}=\text{'A'}$, u contains the $m$ by $m$ unitary matrix $U$.

If $\mathbf{jobu}=\text{'S'}$, u contains the first $\min \left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise).

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

9:     $\mathbf{ldu}$ – IntegerInput

On entry: the first dimension of the array u as declared in the (sub)program from which f08kpf (zgesvd) is called.

Constraints:

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

10:   $\mathbf{vt}\left(\mathbf{ldvt},*\right)$ – Complex (Kind=nag_wp) arrayOutput

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

On exit: if $\mathbf{jobvt}=\text{'A'}$, vt contains the $n$ by $n$ unitary matrix $V^{\mathrm{H}}$.

If $\mathbf{jobvt}=\text{'S'}$, vt contains the first $\min \left(m,n\right)$ rows of $V^{\mathrm{H}}$ (the right singular vectors, stored row-wise).

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

11:   $\mathbf{ldvt}$ – IntegerInput

On entry: the first dimension of the array vt as declared in the (sub)program from which f08kpf (zgesvd) is called.

Constraints:

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

12:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – Complex (Kind=nag_wp) arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, the real part of $\mathbf{work}\left(1\right)$ contains the minimum value of lwork required for optimal performance.

13:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08kpf (zgesvd) 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. Consider increasing lwork by at least $\mathit{nb}\times \min \left(\mathbf{m},\mathbf{n}\right)$, where $\mathit{nb}$ is the optimal block size.

Constraint: $\mathbf{lwork}\ge \max \left(1,2\times \min \left(\mathbf{m},\mathbf{n}\right)+\max \left(\mathbf{m},\mathbf{n}\right)\right)$.

14:   $\mathbf{rwork}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace

Note: the dimension of the array rwork must be at least $\max \left(1,5\times \min \left(\mathbf{m},\mathbf{n}\right)\right)$.

On exit: if $\mathbf{info}>\mathbf{0}$, $\mathbf{rwork}\left(1:\min \left(\mathbf{m},\mathbf{n}\right)-1\right)$ contains the unconverged superdiagonal elements of an upper bidiagonal matrix $B$ whose diagonal is in $S$ (not necessarily sorted). $B$ satisfies $A=UB{V}^{\mathrm{H}}$, so it has the same singular values as $A$, and singular vectors related by $U$ and $V^{\mathrm{H}}$.

15:   $\mathbf{info}$ – IntegerOutput

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 f08kpf (zgesvd) did not converge, info specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero. See the description of rwork above for details.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08kpfe.f90)

10.2

Program Data

Program Data (f08kpfe.d)

10.3

Program Results

Program Results (f08kpfe.r)