NAG Library Routine Document

F08MDF (DBDSDC)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{UPLO}$ – CHARACTER(1)Input

On entry: indicates whether $B$ is upper or lower bidiagonal.

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

$B$ is upper bidiagonal.

$\mathbf{UPLO}=\text{'L'}$

$B$ is lower bidiagonal.

Constraint: $\mathbf{UPLO}=\text{'U'}$ or $\text{'L'}$.

2:     $\mathrm{COMPQ}$ – CHARACTER(1)Input

On entry: specifies whether singular vectors are to be computed.

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

Compute singular values only.

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

Compute singular values and compute singular vectors in compact form.

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

Compute singular values and singular vectors.

Constraint: $\mathbf{COMPQ}=\text{'N'}$, $\text{'P'}$ or $\text{'I'}$.

3:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the matrix $B$.

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

4:     $\mathrm{D}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

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

On entry: the $n$ diagonal elements of the bidiagonal matrix $B$.

On exit: if $\mathbf{INFO}=\mathbf{0}$, the singular values of $B$.

5:     $\mathrm{E}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array E must be at least $\max \left(1,\mathbf{N}-1\right)$.

On entry: the $\left(n-1\right)$ off-diagonal elements of the bidiagonal matrix $B$.

On exit: the contents of E are destroyed.

6:     $\mathrm{U}\left(\mathbf{LDU},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array U must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{COMPQ}=\text{'I'}$, and at least $1$ otherwise.

On exit: if $\mathbf{COMPQ}=\text{'I'}$, then if $\mathbf{INFO}=\mathbf{0}$, U contains the left singular vectors of the bidiagonal matrix $B$.

If $\mathbf{COMPQ}\ne \text{'I'}$, U is not referenced.

7:     $\mathrm{LDU}$ – INTEGERInput

On entry: the first dimension of the array U as declared in the (sub)program from which F08MDF (DBDSDC) is called.

Constraints:

• if $\mathbf{COMPQ}=\text{'I'}$, $\mathbf{LDU}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDU}\ge 1$.

8:     $\mathrm{VT}\left(\mathbf{LDVT},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array VT must be at least $\max \left(1,\mathbf{N}\right)$ if $\mathbf{COMPQ}=\text{'I'}$, and at least $1$ otherwise.

On exit: if $\mathbf{COMPQ}=\text{'I'}$, then if $\mathbf{INFO}=\mathbf{0}$, the rows of VT contain the right singular vectors of the bidiagonal matrix $B$.

If $\mathbf{COMPQ}\ne \text{'I'}$, VT is not referenced.

9:     $\mathrm{LDVT}$ – INTEGERInput

On entry: the first dimension of the array VT as declared in the (sub)program from which F08MDF (DBDSDC) is called.

Constraints:

• if $\mathbf{COMPQ}=\text{'I'}$, $\mathbf{LDVT}\ge \max \left(1,\mathbf{N}\right)$;
• otherwise $\mathbf{LDVT}\ge 1$.

10:   $\mathrm{Q}\left(*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array Q must be at least $\max \left(1,{\mathbf{N}}^2+5\mathbf{N},\mathit{ldq}\right)$.

On exit: if $\mathbf{COMPQ}=\text{'P'}$, then if $\mathbf{INFO}=\mathbf{0}$, Q and IQ contain the left and right singular vectors in a compact form, requiring $\mathit{O}\left(\mathbf{N}{\log }_2\mathbf{N}\right)$ space instead of $2\times {\mathbf{N}}^2$. In particular, Q contains all the real data in the first $\mathit{ldq}=\mathbf{N}\times \left(11+2\times \mathit{smlsiz}+8\times \mathrm{int}\left({\log }_2\left(\mathbf{N}/\left(\mathit{smlsiz}+1\right)\right)\right)\right)$ elements of Q, where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$).

If $\mathbf{COMPQ}\ne \text{'P'}$, Q is not referenced.

11:   $\mathrm{IQ}\left(*\right)$ – INTEGER arrayOutput

Note: the dimension of the array IQ must be at least $\max \left(1,\mathit{ldiq}\right)$.

On exit: if $\mathbf{COMPQ}=\text{'P'}$, then if $\mathbf{INFO}=\mathbf{0}$, Q and IQ contain the left and right singular vectors in a compact form, requiring $\mathit{O}\left(\mathbf{N}{\log }_2\mathbf{N}\right)$ space instead of $2\times {\mathbf{N}}^2$. In particular, IQ contains all integer data in the first $\mathit{ldiq}=\mathbf{N}\times \left(3+3\times \mathrm{int}\left({\log }_2\left(\mathbf{N}/\left(\mathit{smlsiz}+1\right)\right)\right)\right)$ elements of IQ, where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$).

If $\mathbf{COMPQ}\ne \text{'P'}$, IQ is not referenced.

12:   $\mathrm{WORK}\left(*\right)$ – REAL (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least $\max \left(1,6\times \mathbf{N}-2\right)$ if $\mathbf{COMPQ}=\text{'N'}$, $\max \left(1,6\times \mathbf{N}\right)$ if $\mathbf{COMPQ}=\text{'P'}$, $\max \left(1,3\times {\mathbf{N}}^2+4\times \mathbf{N}\right)$ if $\mathbf{COMPQ}=\text{'I'}$, and at least $1$ otherwise.

13:   $\mathrm{IWORK}\left(8\times \mathbf{N}\right)$ – INTEGER arrayWorkspace

14:   $\mathrm{INFO}$ – 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.

$\mathbf{INFO}>0$

The algorithm failed to compute a singular value. The update process of divide-and-conquer failed.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08mdfe.f90)

10.2  Program Data

Program Data (f08mdfe.d)

10.3  Program Results

Program Results (f08mdfe.r)