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NAG Toolbox: nag_lapack_zgels (f08an)
Purpose
nag_lapack_zgels (f08an) solves linear least squares problems of the form
where
is an
by
complex matrix of full rank, using a
or
factorization of
.
Syntax
[
a,
b,
info] = f08an(
trans,
a,
b, 'm',
m, 'n',
n, 'nrhs_p',
nrhs_p)
[
a,
b,
info] = nag_lapack_zgels(
trans,
a,
b, 'm',
m, 'n',
n, 'nrhs_p',
nrhs_p)
Description
The following options are provided:
1. |
If and : find the least squares solution of an overdetermined system, i.e., solve the least squares problem
|
2. |
If and : find the minimum norm solution of an underdetermined system . |
3. |
If and : find the minimum norm solution of an undetermined system . |
4. |
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 .
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
Parameters
Compulsory Input Parameters
- 1:
– string (length ≥ 1)
-
If
, the linear system involves
.
If , the linear system involves .
Constraint:
or .
- 2:
– complex array
-
The first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
The by matrix .
- 3:
– complex array
-
The first dimension of the array
b must be at least
.
The second dimension of the array
b must be at least
.
The matrix
of right-hand side vectors, stored in columns;
b is
by
if
, or
by
if
.
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
a.
, the number of rows of the matrix .
Constraint:
.
- 2:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
a.
, the number of columns of the matrix .
Constraint:
.
- 3:
– int64int32nag_int scalar
-
Default:
the second dimension of the array
b.
, the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
Output Parameters
- 1:
– complex array
-
The first dimension of the array
a will be
.
The second dimension of the array
a will be
.
If
,
a stores details of its
factorization, as returned by
nag_lapack_zgeqrf (f08as).
If
,
a stores details of its
factorization, as returned by
nag_lapack_zgelqf (f08av).
- 2:
– complex array
-
The first dimension of the array
b will be
.
The second dimension of the array
b will be
.
b stores 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.
- 3:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , parameter had an illegal value on entry. The parameters are numbered as follows:
1:
trans, 2:
m, 3:
n, 4:
nrhs_p, 5:
a, 6:
lda, 7:
b, 8:
ldb, 9:
work, 10:
lwork, 11:
info.
It is possible that
info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
-
-
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.
Accuracy
See Section 4.5 of
Anderson et al. (1999) for details of error bounds.
Further Comments
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 function is
nag_lapack_dgels (f08aa).
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.
Open in the MATLAB editor:
f08an_example
function f08an_example
fprintf('f08an example results\n\n');
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];
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];
[m,n] = size(a);
trans = 'No transpose';
[af, x, info] = f08an( ...
trans, a, b);
fprintf('\nLeast Squares Solution:\n');
disp(transpose(x(1:n)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(x(n+1:m),2));
f08an example results
Least Squares Solution:
-0.5044 - 1.2179i -2.4281 + 2.8574i 1.4872 - 2.1955i 0.4537 + 2.6904i
Square root of the residual sum of squares
0.0688
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