NAG Library Routine Document
F08AAF (DGELS)
1 Purpose
F08AAF (DGELS) solves linear least squares problems of the form
where
A is an
m by
n matrix of full rank, using a
QR or
LQ factorization of
A.
2 Specification
| SUBROUTINE F08AAF ( |
TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) |
| INTEGER |
M, N, NRHS, LDA, LDB, LWORK, INFO |
| REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), WORK(max(1,LWORK)) |
| CHARACTER(1) |
TRANS |
|
The routine may be called by its
LAPACK
name dgels.
3 Description
The following options are provided:
- If TRANS='N' and m≥n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
- If TRANS='N' and m<n: find the minimum norm solution of an underdetermined system Ax=b.
- If TRANS='T' and m≥n: find the minimum norm solution of an undetermined system ATx=b.
- If TRANS='T' and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
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 by r right-hand side matrix B and the n by 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
http://www.netlib.org/lapack/lugGolub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: TRANS – CHARACTER(1)Input
-
On entry: if
TRANS='N', the linear system involves
A.
If TRANS='T', the linear system involves AT.
Constraint:
TRANS='N' or 'T'.
- 2: M – INTEGERInput
-
On entry:
m, the number of rows of the matrix A.
Constraint:
M≥0.
- 3: N – INTEGERInput
-
On entry:
n, the number of columns of the matrix A.
Constraint:
N≥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:
NRHS≥0.
- 5: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the m by n matrix A.
On exit: if
M≥N,
A is overwritten by details of its
QR factorization as returned by
F08AEF (DGEQRF).
If
M<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:
LDA≥max1,M.
- 7: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the matrix
B of right-hand side vectors, stored in columns;
B is
m by
r if
TRANS='N', or
n by
r if
TRANS='T'.
On exit:
B is overwritten by the solution vectors,
x, stored in columns:
- if TRANS='N' and m≥n, or
TRANS='T' and m<n, elements 1 to minm,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
minm,n+1
to maxm,n in that column;
- otherwise, elements 1 to maxm,n 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:
LDB≥max1,M,N.
- 9: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 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
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,
LWORK≥
minM,N
+
max1,M,N,NRHS
×
nb
, where nb is the optimal block size.
Constraint:
LWORK≥
minM,N
+
max1,M,N,NRHS
or LWORK=-1.
- 11: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
If 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
See Section 4.5 of
Anderson et al. (1999) for details of error bounds.
8 Further Comments
The total number of floating point operations required to factorize A is approximately
23
n2
3m-n
if m≥n and
23
m2
3n-m
otherwise. Following the factorization the solution for a single vector x requires
O
minm2,n2
operations.
The complex analogue of this routine is
F08ANF (ZGELS).
9 Example
This example solves the linear least squares problem
where
The square root of the residual sum of squares is also output.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
9.1 Program Text
Program Text (f08aafe.f90)
9.2 Program Data
Program Data (f08aafe.d)
9.3 Program Results
Program Results (f08aafe.r)