NAG Library Routine Document

f08bnf (zgelsy)

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 $n$ matrix $A$.

On exit: a has been overwritten by details of its complete orthogonal factorization.

5:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08bnf (zgelsy) 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: the $n$ by $r$ solution matrix $X$.

7:     $\mathbf{ldb}$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f08bnf (zgelsy) is called.

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

8:     $\mathbf{jpvt}\left(*\right)$ – Integer arrayInput/Output

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

On entry: if $\mathbf{jpvt}\left(i\right)\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.

On exit: if $\mathbf{jpvt}\left(i\right)=k$, the $i$th column of $AP$ was the $k$th column of $A$.

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

On entry: used to determine the effective rank of $A$, which is defined as the order of the largest leading triangular sub-matrix $R_{11}$ in the $QR$ factorization of $A$, whose estimated condition number is $<1/\mathbf{rcond}$.

Suggested value: if the condition number of a is not known then $\mathbf{rcond}=\sqrt{\left(\epsilon \right)/2}$ (where $\epsilon$ is machine precision, see x02ajf) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective $\text{rank}=\min \left(\mathbf{m},\mathbf{n}\right)$ that could be larger than its actual rank, leading to meaningless results.

10:   $\mathbf{rank}$ – IntegerOutput

On exit: the effective rank of $A$, i.e., the order of the sub-matrix $R_{11}$. This is the same as the order of the sub-matrix $T_{11}$ in the complete orthogonal factorization of $A$.

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 f08bnf (zgelsy) 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,

$$\mathbf{lwork}\ge \max \left(k+2\times \mathbf{n}+\mathit{nb}\times \left(\mathbf{n}+1\right),2\times k+\mathit{nb}\times \mathbf{nrhs}\right)\text{,}$$

where $k=\min \left(\mathbf{m},\mathbf{n}\right)$ and $\mathit{nb}$ is the optimal block size.

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

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f08bnfe.f90)

10.2

Program Data

Program Data (f08bnfe.d)

10.3

Program Results

Program Results (f08bnfe.r)