NAG Library Routine Document

f08kaf (dgelss)

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:     $\mathbf{a}\left(\mathbf{lda},*\right)$ – 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: the first $\min \left(m,n\right)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.

5:     $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f08kaf (dgelss) is called.

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

6:     $\mathbf{b}\left(\mathbf{ldb},*\right)$ – Real (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 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 f08kaf (dgelss) 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:   $\mathbf{work}\left(\max \left(1,\mathbf{lwork}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if $\mathbf{info}=\mathbf{0}$, $\mathbf{work}\left(1\right)$ 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 f08kaf (dgelss) 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: $\mathbf{lwork}\ge 1$ and
$\mathbf{lwork}\ge 3\times \min \left(\mathbf{m},\mathbf{n}\right)+\max \left(2\times \min \left(\mathbf{m},\mathbf{n}\right),\max \left(\mathbf{m},\mathbf{n}\right),\mathbf{nrhs}\right)$.

13:   $\mathbf{info}$ – 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 (f08kafe.f90)

10.2

Program Data

Program Data (f08kafe.d)

10.3

Program Results

Program Results (f08kafe.r)