NAG Library Routine Document

F08AAF (DGELS)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     TRANS – CHARACTER(1)Input

On entry: if $\mathbf{TRANS}=\text{'N'}$, the linear system involves $A$.

If $\mathbf{TRANS}=\text{'T'}$, the linear system involves $A^{\mathrm{T}}$.

Constraint: $\mathbf{TRANS}=\text{'N'}$ or $\text{'T'}$.

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(LDA,$*$) – 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: if $\mathbf{M}\ge \mathbf{N}$, A is overwritten by details of its $QR$ factorization, as returned by F08AEF (DGEQRF).

If $\mathbf{M}<\mathbf{N}$, A is overwritten by details of its $LQ$ factorization, as returned by F08AHF (DGELQF).

6:     LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08AAF (DGELS) is called.

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

7:     B(LDB,$*$) – 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 matrix $B$ of right-hand side vectors, stored in columns; B is $m$ by $r$ if $\mathbf{TRANS}=\text{'N'}$, or $n$ by $r$ if $\mathbf{TRANS}=\text{'T'}$.

On exit: B is overwritten by the solution vectors, $x$, stored in columns:

• if $\mathbf{TRANS}=\text{'N'}$ and $m\ge n$, or $\mathbf{TRANS}=\text{'T'}$ and $m<n$, elements $1$ to $\min \left(m,n\right)$ in each column of B contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $\left(\min \left(m,n\right)+1\right)$ to $\max \left(m,n\right)$ in that column;
• otherwise, elements $1$ to $\max \left(m,n\right)$ in each column of B contain the minimum norm solution vectors.

8:     LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08AAF (DGELS) is called.

Constraint: $\mathbf{LDB}\ge \max \left(1,\mathbf{M},\mathbf{N}\right)$.

9:     WORK($\max \left(1,\mathbf{LWORK}\right)$) – 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.

10:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08AAF (DGELS) 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, $\mathbf{LWORK}\ge \min \left(\mathbf{M},\mathbf{N}\right)+\max \left(1,\mathbf{M},\mathbf{N},\mathbf{NRHS}\right)\times \mathit{nb}$, where $\mathit{nb}$ is the optimal block size.

Constraint: $\mathbf{LWORK}\ge \min \left(\mathbf{M},\mathbf{N}\right)+\max \left(1,\mathbf{M},\mathbf{N},\mathbf{NRHS}\right)$ or $\mathbf{LWORK}=-1$.

11:   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.

$\mathbf{INFO}>0$

If $\mathbf{INFO}=i$, diagonal element $i$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08aafe.f90)

9.2  Program Data

Program Data (f08aafe.d)

9.3  Program Results

Program Results (f08aafe.r)