naginterfaces.library.opt.bnd_lin_lsq¶
- naginterfaces.library.opt.bnd_lin_lsq(a, b, bl, bu, itype=1, tol=0.0)[source]¶
bnd_lin_lsq
solves a linear least squares problem subject to fixed lower and upper bounds on the variables.For full information please refer to the NAG Library document for e04pc
https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/e04/e04pcf.html
- Parameters
- afloat, array-like, shape
The matrix .
- bfloat, array-like, shape
The right-hand side vector .
- blfloat, array-like, shape
and must specify the lower and upper bounds, and respectively, to be imposed on the solution vector .
- bufloat, array-like, shape
and must specify the lower and upper bounds, and respectively, to be imposed on the solution vector .
- itypeint, optional
Provides the choice of returning a regularized solution if the matrix is not of full rank.
Specifies that a regularized solution is to be computed.
Specifies that no regularization is to take place.
- tolfloat, optional
specifies a parameter used to determine the relative linear dependence of a column vector for a variable moved from its initial value. It determines the computational rank of the matrix. Increasing its value from will increase the likelihood of additional elements of being set to zero. It may be worth experimenting with increasing values of to determine whether the nature of the solution, , changes significantly. In practice a value of is recommended (see
machine.precision
).If on entry , is used.
- Returns
- afloat, ndarray, shape
If , contains the product matrix , where is an orthogonal matrix generated by
bnd_lin_lsq
; otherwise, is unchanged.- bfloat, ndarray, shape
If , the product of times the original vector , where is as described in argument ; otherwise, is unchanged.
- xfloat, ndarray, shape
The solution vector .
- rnormfloat
The Euclidean norm of the residual vector .
- nfreeint
Indicates the number of components of the solution vector that are not at one of the constraints.
- wfloat, ndarray, shape
Contains the dual solution vector. The magnitude of gives a measure of the improvement in the objective value if the corresponding bound were to be relaxed so that could take different values.
A value of equal to the special value is indicative of the matrix not having full rank.
It is only likely to occur when .
However a matrix may have less than full rank without being set to .
If , then the values contained in (other than those set to ) may be unreliable; the corresponding values in may likewise be unreliable.
If you have any doubts set .
Otherwise, the values of have the following meaning:
if is unconstrained.
if is constrained by its lower bound.
if is constrained by its upper bound.
may be any value if .
- indxint, ndarray, shape
The contents of this array describe the components of the solution vector as follows:
, for
These elements of the solution have not hit a constraint; i.e., .
, for
These elements of the solution have been constrained by either the lower or upper bound.
, for
These elements of the solution are fixed by the bounds; i.e., .
Here is determined from and the number of fixed components. (Often the latter will be , so will be .)
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, when , and .
Constraint: .
- (errno )
On entry, and .
Constraint: .
- Warns
- NagAlgorithmicMajorWarning
- (errno )
The function failed to converge in iterations. This is not expected. Please contact NAG.
- Notes
Given an matrix , an -vector of lower bounds, an -vector of upper bounds, and an -vector ,
bnd_lin_lsq
computes an -vector that solves the least squares problem subject to satisfying .A facility is provided to return a ‘regularized’ solution, which will closely approximate a minimal length solution whenever is not of full rank. A minimal length solution is the solution to the problem which has the smallest Euclidean norm.
The algorithm works by applying orthogonal transformations to the matrix and to the right-hand side to obtain within the matrix an upper triangular matrix . In general the elements of corresponding to the columns of will be the candidate nonzero solutions. If a diagonal element of is small compared to the other members of then this is undesirable. will be nearly singular and the equations for thus ill-conditioned. You may specify the tolerance used to determine the relative linear dependence of a column vector for a variable moved from its initial value.
- References
Lawson, C L and Hanson, R J, 1974, Solving Least Squares Problems, Prentice–Hall