NAG Library Routine Document

f08kqf (zgelsd)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{m}$ – IntegerInput

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

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

2:     $\mathbf{n}$ – IntegerInput

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

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

3:     $\mathbf{nrhs}$ – IntegerInput

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

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

4:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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 $n$ coefficient matrix $A$.

On exit: 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 f08kqf (zgelsd) is called.

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

6:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output

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

On entry: the $m$ by $r$ right-hand side matrix $B$.

On exit: b is overwritten by 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.

7:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08kqf (zgelsd) is called.

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

8:     $\mathbf{s}\left(*\right)$ – 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 $A$ in decreasing order.

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

On entry: 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.

10:   $\mathbf{rank}$ – IntegerOutput

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

11:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – 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.

12:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08kqf (zgelsd) is called.

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

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

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

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

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.

Constraint: $\mathbf{lwork}\ge \max \left(1,\mathbf{m}+\mathbf{n}+\mathit{r},2\mathit{r}+\mathit{r}\times \mathbf{nrhs}\right)$ or $\mathbf{lwork}=-1$.

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

Note: the dimension of the array rwork must be at least $\max \left(1,\mathit{lrwork}\right)$, where $\mathit{lrwork}$ is at least

$$10\times \mathbf{n}+2\times \mathbf{n}\times \mathit{smlsiz}+8\times \mathbf{n}\times \mathit{nlvl}+3\times \mathit{smlsiz}\times \mathbf{nrhs}+{\left(\mathit{smlsiz}+1\right)}^2\text{, \hspace{1em} if }\mathbf{m}\ge \mathbf{n}$$

or

$$10\times \mathbf{m}+2\times \mathbf{m}\times \mathit{smlsiz}+8\times \mathbf{m}\times \mathit{nlvl}+3\times \mathit{smlsiz}\times \mathbf{nrhs}+{\left(\mathit{smlsiz}+1\right)}^2\text{, \hspace{1em} if }\mathbf{m}<\mathbf{n}$$

where $\mathit{smlsiz}$ is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about $25$), and $\mathit{nlvl}=\max \left(0,\mathrm{int}\left({\log }_2\left(\min \left(\mathbf{m},\mathbf{n}\right)/\left(\mathit{smlsiz}+1\right)\right)\right)+1\right)$, the code will execute correctly.

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{rwork}\left(1\right)$ contains the required minimal size of $\mathit{lrwork}$.

14:   $\mathbf{iwork}\left(*\right)$ – Integer arrayWorkspace

Note: the dimension of the array iwork must be at least $\max \left(1,\mathit{liwork}\right)$, where $\mathit{liwork}$ is at least $\max \left(1,3\times \min \left(\mathbf{m},\mathbf{n}\right)\times \mathit{nlvl}+11\times \min \left(\mathbf{m},\mathbf{n}\right)\right)$.

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{iwork}\left(1\right)$ returns the minimum $\mathit{liwork}$.

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.

$\mathbf{info}>0$

The algorithm for computing the SVD failed to converge; $〈\mathit{\text{value}}〉$ off-diagonal elements 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

Program Text (f08kqfe.f90)

10.2

Program Data

Program Data (f08kqfe.d)

10.3

Program Results

Program Results (f08kqfe.r)