f08bac computes the minimum norm solution to a real linear least squares problem
using a complete orthogonal factorization of . is an matrix which may be rank-deficient. Several right-hand side vectors and solution vectors can be handled in a single call.
The function may be called by the names: f08bac, nag_lapackeig_dgelsy or nag_dgelsy.
3Description
The right-hand side vectors are stored as the columns of the matrix and the solution vectors in the matrix .
f08bac first computes a factorization with column pivoting
with defined as the largest leading sub-matrix whose estimated condition number is less than . The order of , rank, is the effective rank of .
Then, is considered to be negligible, and is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
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
5Arguments
1: – 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 . 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:
or .
2: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
5: – doubleInput/Output
Note: the dimension, dim, of the array a
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the matrix .
On exit: a has been overwritten by details of its complete orthogonal factorization.
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
if ,
;
if , .
7: – doubleInput/Output
Note: the dimension, dim, of the array b
must be at least
when
;
when
.
The th element of the matrix is stored in
when ;
when .
On entry: the right-hand side matrix .
On exit: the solution matrix .
8: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
if ,
;
if , .
9: – IntegerInput/Output
Note: the dimension, dim, of the array jpvt
must be at least
.
On entry: if , the th column of is permuted to the front of , otherwise column is a free column.
On exit: if , the th column of was the th column of .
10: – doubleInput
On entry: used to determine the effective rank of , which is defined as the order of the largest leading triangular sub-matrix in the factorization of , whose estimated condition number is .
Suggested value:
if the condition number of a is not known then (where 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 that could be larger than its actual rank, leading to meaningless results.
11: – Integer *Output
On exit: the effective rank of , i.e., the order of the sub-matrix . This is the same as the order of the sub-matrix in the complete orthogonal factorization of .
12: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error 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 had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, . Constraint: .
On entry, . Constraint: .
NE_INT_2
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
NE_INT_3
On entry, , and .
Constraint: .
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.
Background information to multithreading can be found in the Multithreading documentation.
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.