NAG CL Interface
f08bac (dgelsy)

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1 Purpose

f08bac computes the minimum norm solution to a real linear least squares problem
minx b-Ax2  
using a complete orthogonal factorization of A. A is an m×n matrix which may be rank-deficient. Several right-hand side vectors b and solution vectors x can be handled in a single call.

2 Specification

#include <nag.h>
void  f08bac (Nag_OrderType order, Integer m, Integer n, Integer nrhs, double a[], Integer pda, double b[], Integer pdb, Integer jpvt[], double rcond, Integer *rank, NagError *fail)
The function may be called by the names: f08bac, nag_lapackeig_dgelsy or nag_dgelsy.

3 Description

The right-hand side vectors are stored as the columns of the m×r matrix B and the solution vectors in the n×r matrix X.
f08bac first computes a QR factorization with column pivoting
AP= Q ( R11 R12 0 R22 ) ,  
with R11 defined as the largest leading sub-matrix whose estimated condition number is less than 1/rcond. The order of R11, rank, is the effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
AP= Q ( T11 0 0 0 ) Z .  
The minimum norm solution is then
X = PZT ( T11−1 Q1T b 0 )  
where Q1 consists of the first rank columns of Q.

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: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4: 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.
5: 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: a has been overwritten by details of its complete orthogonal factorization.
6: 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).
7: 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 m×r right-hand side matrix B.
On exit: the n×r solution matrix X.
8: 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).
9: jpvt[dim] Integer Input/Output
Note: the dimension, dim, of the array jpvt must be at least max(1,n).
On entry: if jpvt[i-1]0, the ith column of A is permuted to the front of AP, otherwise column i is a free column.
On exit: if jpvt[i-1]=k, the ith column of AP was the kth column of A.
10: rcond double Input
On entry: used to determine the effective rank of A, which is defined as the order of the largest leading triangular sub-matrix R11 in the QR factorization of A, whose estimated condition number is <1/rcond.
Suggested value: if the condition number of a is not known then rcond=(ε)/2 (where ε is machine precision, see X02AJC) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective rank=min(m,n) that could be larger than its actual rank, leading to meaningless results.
11: rank Integer * Output
On exit: the effective rank of A, i.e., the order of the sub-matrix R11. This is the same as the order of the sub-matrix T11 in the complete orthogonal factorization of A.
12: 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_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

f08bac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bac 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 complex analogue of this function is f08bnc.

10 Example

This example solves the linear least squares problem
minx b-Ax2  
for the solution, x, of minimum norm, where
A = ( -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 )   and   b= ( 7.4 4.2 -8.3 1.8 8.6 2.1 ) .  
A tolerance of 0.01 is used to determine the effective rank of A.

10.1 Program Text

Program Text (f08bace.c)

10.2 Program Data

Program Data (f08bace.d)

10.3 Program Results

Program Results (f08bace.r)