NAG Library Routine Document
F08ANF (ZGELS)
1 Purpose
F08ANF (ZGELS) solves linear least squares problems of the form
where
is an
by
complex matrix of full rank, using a
or
factorization of
.
2 Specification
SUBROUTINE F08ANF ( |
TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO) |
INTEGER |
M, N, NRHS, LDA, LDB, LWORK, INFO |
COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
TRANS |
|
The routine may be called by its
LAPACK
name zgels.
3 Description
The following options are provided:
- If and : find the least squares solution of an overdetermined system, i.e., solve the least squares problem
- If and : find the minimum norm solution of an underdetermined system .
- If and : find the minimum norm solution of an undetermined system .
- If and : find the least squares solution of an overdetermined system, i.e., solve the least squares problem
Several right-hand side vectors and solution vectors can be handled in a single call; they are stored as the columns of the by right-hand side matrix and the by solution matrix .
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/lug
Golub 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
, the linear system involves
.
If , the linear system involves .
Constraint:
or .
- 2: M – INTEGERInput
On entry: , the number of rows of the matrix .
Constraint:
.
- 3: N – INTEGERInput
On entry: , the number of columns of the matrix .
Constraint:
.
- 4: NRHS – INTEGERInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
- 5: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the by matrix .
On exit: if
,
A is overwritten by details of its
factorization, as returned by
F08ASF (ZGEQRF).
If
,
A is overwritten by details of its
factorization, as returned by
F08AVF (ZGELQF).
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08ANF (ZGELS) is called.
Constraint:
.
- 7: B(LDB,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the matrix
of right-hand side vectors, stored in columns;
B is
by
if
, or
by
if
.
On exit:
B is overwritten by the solution vectors,
, stored in columns:
- if and , or and , elements to 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 to in that column;
- otherwise, elements to 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 F08ANF (ZGELS) is called.
Constraint:
.
- 9: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
, the real part of
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 F08ANF (ZGELS) is called.
If
, 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, , where is the optimal block size.
Constraint:
or .
- 11: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If , diagonal element of the triangular factor of is zero, so that 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.
The total number of floating point operations required to factorize is approximately if and otherwise. Following the factorization the solution for a single vector requires operations.
The real analogue of this routine is
F08AAF (DGELS).
9 Example
This example solves the linear least squares problem
where
and
The square root of the residual sum of squares is also output.
Note that the block size (NB) of 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 (f08anfe.f90)
9.2 Program Data
Program Data (f08anfe.d)
9.3 Program Results
Program Results (f08anfe.r)