NAG Library Function Document

nag_zgelss (f08knc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     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:     m – IntegerInput

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

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

3:     n – IntegerInput

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

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

4:     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:     a[$\mathit{dim}$] – ComplexInput/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:     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:     b[$\mathit{dim}$] – ComplexInput/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 the modulus of elements $n+1,\dots ,m$ in that column.

8:     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:     s[$\mathit{dim}$] – 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:   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:   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:   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.

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.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (f08knce.c)

10.2  Program Data

Program Data (f08knce.d)

10.3  Program Results

Program Results (f08knce.r)