NAG CL Interface
f08aac (dgels)

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1 Purpose

f08aac solves linear least squares problems of the form
minx b-Ax2   or   minx b-ATx2 ,  
where A is an m×n real matrix of full rank, using a QR or LQ factorization of A.

2 Specification

#include <nag.h>
void  f08aac (Nag_OrderType order, Nag_TransType trans, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, NagError *fail)
The function may be called by the names: f08aac, nag_lapackeig_dgels or nag_dgels.

3 Description

The following options are provided:
  1. 1.If trans=Nag_NoTrans and mn: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-Ax2 .  
  2. 2.If trans=Nag_NoTrans and m<n: find the minimum norm solution of an underdetermined system Ax=b.
  3. 3.If trans=Nag_Trans and mn: find the minimum norm solution of an undetermined system ATx=b.
  4. 4.If trans=Nag_Trans and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-ATx2 .  
Several right-hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m×r right-hand side matrix B and the n×r solution matrix X.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: order Nag_OrderType Input
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 order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: trans Nag_TransType Input
On entry: if trans=Nag_NoTrans, the linear system involves A.
If trans=Nag_Trans, the linear system involves AT.
Constraint: trans=Nag_NoTrans or Nag_Trans.
3: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
4: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
5: nrhs Integer Input
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrices B and X.
Constraint: nrhs0.
6: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
The (i,j)th element of the matrix A is stored in
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the m×n matrix A.
On exit: if mn, a is overwritten by details of its QR factorization, as returned by f08aec.
If m<n, a is overwritten by details of its LQ factorization, as returned by f08ahc.
7: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
8: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×nrhs) when order=Nag_ColMajor;
  • max(1,max(1,m,n)×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the matrix B of right-hand side vectors, stored in rows or columns; b is m×r if trans=Nag_NoTrans, or n×r if trans=Nag_Trans.
On exit: b is overwritten by the solution vectors, x, stored in rows or columns:
  • if trans=Nag_NoTrans and mn, or trans=Nag_Trans and m<n, elements 1 to min(m,n) 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 (min(m,n)+1) to max(m,n) in that column;
  • otherwise, elements 1 to max(m,n) in each column of b contain the minimum norm solution vectors.
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,m,n);
  • if order=Nag_RowMajor, pdbmax(1,nrhs).
10: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_FULL_RANK
Diagonal element value of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, nrhs=value.
Constraint: nrhs0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and nrhs=value.
Constraint: pdbmax(1,nrhs).
NE_INT_3
On entry, pdb=value, m=value and n=value.
Constraint: pdbmax(1,m,n).
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08aac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aac makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations required to factorize A is approximately 23 n2 (3m-n) if mn and 23 m2 (3n-m) otherwise. Following the factorization the solution for a single vector x requires O(min(m2,n2)) operations.
The complex analogue of this function is f08anc.

10 Example

This example solves the linear least squares problem
minx b-Ax2 ,  
where
A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 )   and   b= ( -2.67 -0.55 3.34 -0.77 0.48 4.10 ) .  
The square root of the residual sum of squares is also output.

10.1 Program Text

Program Text (f08aace.c)

10.2 Program Data

Program Data (f08aace.d)

10.3 Program Results

Program Results (f08aace.r)