F08ANF (ZGELS) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08ANF (ZGELS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08ANF (ZGELS) solves linear least squares problems of the form
minx b-Ax2   or   minx b-AHx2 ,
where A is an m by n complex matrix of full rank, using a QR or LQ factorization of A.

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:
  1. If TRANS='N' and mn: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-Ax2 .
  2. If TRANS='N' and m<n: find the minimum norm solution of an underdetermined system Ax=b.
  3. If TRANS='C' and mn: find the minimum norm solution of an undetermined system AHx=b.
  4. If TRANS='C' and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
    minx b-AHx2 .
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/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 TRANS='N', the linear system involves A.
If TRANS='C', the linear system involves AH.
Constraint: TRANS='N' or 'C'.
2:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
3:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
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: NRHS0.
5:     A(LDA,*) – COMPLEX (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 MN, A is overwritten by details of its QR factorization as returned by F08ASF (ZGEQRF).
If M<N, A is overwritten by details of its LQ 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: LDAmax1,M.
7:     B(LDB,*) – COMPLEX (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='C'.
On exit: B is overwritten by the solution vectors, x, stored in columns:
  • if TRANS='N' and mn, or TRANS='C' 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 F08ANF (ZGELS) is called.
Constraint: LDBmax1,M,N.
9:     WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
10:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08ANF (ZGELS) 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

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 83 n2 3m-n  if mn and 83 m2 3n-m  otherwise. Following the factorization the solution for a single vector x requires O minm2,n2  operations.
The real analogue of this routine is F08AAF (DGELS).

9  Example

This example solves the linear least squares problem
minx b-Ax2 ,
where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i
and
b = -2.09+1.93i 3.34-3.53i -4.94-2.04i 0.17+4.23i -5.19+3.63i 0.98+2.53i .
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 (f08anfe.f90)

9.2  Program Data

Program Data (f08anfe.d)

9.3  Program Results

Program Results (f08anfe.r)


F08ANF (ZGELS) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011