NAG Library Function Document

nag_dgelss (f08kac)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:    $\mathbf{order}$ – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

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:    $\mathbf{nrhs}$ – IntegerInput

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

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

5:    $\mathbf{a}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

• $\max \left(1,\mathbf{pda}\times \mathbf{n}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\mathbf{m}\times \mathbf{pda}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $A$ is stored in

• $\mathbf{a}\left[\left(j-1\right)\times \mathbf{pda}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{a}\left[\left(i-1\right)\times \mathbf{pda}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

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.

6:    $\mathbf{pda}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array a.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pda}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

7:    $\mathbf{b}\left[\mathit{dim}\right]$ – doubleInput/Output

Note: the dimension, dim, of the array b must be at least

• $\max \left(1,\mathbf{pdb}\times \mathbf{nrhs}\right)$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\max \left(1,\max \left(1,\mathbf{m},\mathbf{n}\right)\times \mathbf{pdb}\right)$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

The $\left(i,j\right)$th element of the matrix $B$ is stored in

• $\mathbf{b}\left[\left(j-1\right)\times \mathbf{pdb}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{b}\left[\left(i-1\right)\times \mathbf{pdb}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

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.

8:    $\mathbf{pdb}$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

• if $\mathbf{order}=\mathrm{Nag\_ColMajor}$, $\mathbf{pdb}\ge \max \left(1,\mathbf{m},\mathbf{n}\right)$;
• if $\mathbf{order}=\mathrm{Nag\_RowMajor}$, $\mathbf{pdb}\ge \max \left(1,\mathbf{nrhs}\right)$.

9:    $\mathbf{s}\left[\mathit{dim}\right]$ – doubleOutput

Note: the dimension, dim, 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.

10:  $\mathbf{rcond}$ – doubleInput

On entry: used to determine the effective rank of $A$. Singular values $\mathbf{s}\left[i-1\right]\le \mathbf{rcond}\times \mathbf{s}\left[0\right]$ are treated as zero. If $\mathbf{rcond}<0$, machine precision is used instead.

11:  $\mathbf{rank}$ – Integer *Output

On exit: the effective rank of $A$, i.e., the number of singular values which are greater than $\mathbf{rcond}\times \mathbf{s}\left[0\right]$.

12:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

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.

NE_INT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{m}\ge 0$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 0$.

On entry, $\mathbf{nrhs}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nrhs}\ge 0$.

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}>0$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}>0$.

NE_INT_2

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{m}\right)$.

On entry, $\mathbf{pda}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pda}\ge \max \left(1,\mathbf{n}\right)$.

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$ and $\mathbf{nrhs}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}\ge \max \left(1,\mathbf{nrhs}\right)$.

NE_INT_3

On entry, $\mathbf{pdb}=〈\mathit{\text{value}}〉$, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdb}\ge \max \left(1,\mathbf{m},\mathbf{n}\right)$.

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08kace.c)

10.2  Program Data

Program Data (f08kace.d)

10.3  Program Results

Program Results (f08kace.r)