NAG Library Routine Document

f08kdf  (dgesdd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

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

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

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

All $m$ columns of $U$ and all $n$ rows of $V^{\mathrm{T}}$ are returned in the arrays u and vt.

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

The first $\min \left(m,n\right)$ columns of $U$ and the first $\min \left(m,n\right)$ rows of $V^{\mathrm{T}}$ are returned in the arrays u and vt.

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

If $\mathbf{m}\ge \mathbf{n}$, the first $n$ columns of $U$ are overwritten on the array a and all rows of $V^{\mathrm{T}}$ are returned in the array vt. Otherwise, all columns of $U$ are returned in the array u and the first $m$ rows of $V^{\mathrm{T}}$ are overwritten in the array vt.

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

No columns of $U$ or rows of $V^{\mathrm{T}}$ are computed.

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

2:     $\mathbf{m}$ – IntegerInput

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

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

3:     $\mathbf{n}$ – IntegerInput

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

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

4:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – Real (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 $\mathbf{jobz}=\text{'O'}$, a is overwritten with the first $n$ columns of $U$ (the left singular vectors, stored column-wise) if $\mathbf{m}\ge \mathbf{n}$; a is overwritten with the first $m$ rows of $V^{\mathrm{T}}$ (the right singular vectors, stored row-wise) otherwise.

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

5:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08kdf (dgesdd) is called.

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

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

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

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

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

On exit:

If $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, u contains the $m$ by $m$ orthogonal matrix $U$.

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

If $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$, or $\mathbf{jobz}=\text{'N'}$, u is not referenced.

8:     $\mathbf{ldu}$ – IntegerInput

On entry: the first dimension of the array u as declared in the (sub)program from which f08kdf (dgesdd) is called.

Constraints:

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

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

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

On exit: if $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$, vt contains the $n$ by $n$ orthogonal matrix $V^{\mathrm{T}}$.

If $\mathbf{jobz}=\text{'S'}$, vt contains the first $\min \left(m,n\right)$ rows of $V^{\mathrm{T}}$ (the right singular vectors, stored row-wise).

If $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, or $\mathbf{jobz}=\text{'N'}$, vt is not referenced.

10:   $\mathbf{ldvt}$ – IntegerInput

On entry: the first dimension of the array vt as declared in the (sub)program from which f08kdf (dgesdd) is called.

Constraints:

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

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

12:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08kdf (dgesdd) 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.

Constraints:

• if $\mathbf{jobz}=\text{'N'}$, $\mathbf{lwork}\ge 3\times \min \left(\mathbf{m},\mathbf{n}\right)+\max \left(1,\max \left(\mathbf{m},\mathbf{n}\right),7\times \min \left(\mathbf{m},\mathbf{n}\right)\right)$;
• if $\mathbf{jobz}=\text{'O'}$, $$\begin{gathered} \mathbf{lwork}\ge 3\times \min \left(\mathbf{m},\mathbf{n}\right)+ \\ \max \left(1,\max \left(\mathbf{m},\mathbf{n}\right),5\times \min \left(\mathbf{m},\mathbf{n}\right)\times \min \left(\mathbf{m},\mathbf{n}\right)+4\times \min \left(\mathbf{m},\mathbf{n}\right)\right) \end{gathered}$$;
• if $\mathbf{jobz}=\text{'S'}$ or $\text{'A'}$, $$\begin{gathered} \mathbf{lwork}\ge 3\times \min \left(\mathbf{m},\mathbf{n}\right)+ \\ \max \left(1,\max \left(\mathbf{m},\mathbf{n}\right),4\times \min \left(\mathbf{m},\mathbf{n}\right)\times \min \left(\mathbf{m},\mathbf{n}\right)+4\times \min \left(\mathbf{m},\mathbf{n}\right)\right) \end{gathered}$$;
• otherwise $\mathbf{lwork}\ge 1$.

13:   $\mathbf{iwork}\left(8\times \min \left(\mathbf{m},\mathbf{n}\right)\right)$ – Integer arrayWorkspace

14:   $\mathbf{info}$ – 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.

$\mathbf{info}>0$

f08kdf (dgesdd) did not converge, the updating process failed.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08kdfe.f90)

10.2

Program Data

Program Data (f08kdfe.d)

10.3

Program Results

Program Results (f08kdfe.r)