naginterfaces.library.linsys.real_gen_sparse_lsqsol¶
- naginterfaces.library.linsys.real_gen_sparse_lsqsol(n, b, aprod, damp=0.0, atol=- 1.0, btol=- 1.0, conlim=0.0, itnlim=0, msglvl=0, data=None, io_manager=None)[source]¶
real_gen_sparse_lsqsol
solves sparse nonsymmetric equations, sparse linear least squares problems and sparse damped linear least squares problems, using a Lanczos algorithm.For full information please refer to the NAG Library document for f04qa
https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f04/f04qaf.html
- Parameters
- nint
, the number of columns of the matrix .
- bfloat, array-like, shape
The right-hand side vector .
- aprodcallable (mode, x, y) = aprod(mode, x, y, data=None)
must perform the operations and for given vectors and .
- Parameters
- modeint
Specifies which operation is to be performed.
must compute .
must compute .
- xfloat, ndarray, shape
The vector .
- yfloat, ndarray, shape
The vector .
- dataarbitrary, optional, modifiable in place
User-communication data for callback functions.
- Returns
- modeint
May be used as a flag to indicate a failure in the computation of or . If is negative on exit from ,
real_gen_sparse_lsqsol
will exit immediately with set to .- xfloat, array-like, shape
If , must be unchanged.
If , must contain .
- yfloat, array-like, shape
If , must contain .
If , must be unchanged.
- dampfloat, optional
The value . If either problem (1) or problem (2) is to be solved, must be supplied as zero.
- atolfloat, optional
The tolerance, , of the convergence criteria (6) and (7); it should be an estimate of the largest relative error in the elements of . For example, if the elements of are correct to about significant figures, then should be set to about . If is supplied as less than , where is the machine precision, the value is used instead of .
- btolfloat, optional
The tolerance, , of the convergence criterion (6); it should be an estimate of the largest relative error in the elements of . For example, if the elements of are correct to about significant figures, then should be set to about . If is supplied as less than , the value is used instead of .
- conlimfloat, optional
The value of equation (8); it should be an upper limit on the condition number of . should not normally be chosen much larger than . If is supplied as zero, the value is used instead of .
- itnlimint, optional
An upper limit on the number of iterations. If , the value is used in place of , but for ill-conditioned problems a higher value of is likely to be necessary.
- msglvlint, optional
The level of printing from
real_gen_sparse_lsqsol
. If , then no printing occurs, but otherwise messages will be output on the file object associated with the advisory I/O unit (seeFileObjManager
). A description of the printed output is given in Further Comments. The level of printing is determined as follows:No printing.
A brief summary is printed just prior to return from
real_gen_sparse_lsqsol
.A summary line is printed periodically to monitor the progress of
real_gen_sparse_lsqsol
, together with a brief summary just prior to return fromreal_gen_sparse_lsqsol
.- dataarbitrary, optional
User-communication data for callback functions.
- io_managerFileObjManager, optional
Manager for I/O in this routine.
- Returns
- bfloat, ndarray, shape
is overwritten.
- xfloat, ndarray, shape
The solution vector .
- sefloat, ndarray, shape
The estimates of the standard errors of the components of . Thus contains an estimate of , where is the th diagonal element of the estimated variance-covariance matrix . The estimates returned in will be the lower bounds on the actual estimated standard errors, but will usually be correct to at least one significant figure.
- itnlimint
Unchanged unless on entry, in which case it is set to .
- itnint
The number of iterations performed.
- anormfloat
An estimate of for the matrix of (4).
- acondfloat
An estimate of which is a lower bound.
- rnormfloat
An estimate of for the residual, , of (5) corresponding to the solution returned in . Note that is the function being minimized.
- arnormfloat
An estimate of the corresponding to the solution returned in .
- xnormfloat
An estimate of for the solution returned in .
- informint
The reason for termination of
real_gen_sparse_lsqsol
.The exact solution is . No iterations are performed in this case.
The termination criterion of (6) has been satisfied with and as the values supplied in and respectively.
The termination criterion of (7) has been satisfied with as the value supplied in .
The termination criterion of (6) has been satisfied with and/or as the value , where is the machine precision. One or both of the values supplied in and must have been less than and was too small for this machine.
The termination criterion of (7) has been satisfied with as the value . The value supplied in must have been less than and was too small for this machine.
The values , or correspond to failure with = 2, 3 and 4 respectively (see Exceptions) and when is negative will be set to the same negative value.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
Termination criteria not satisfied, but the condition number bound supplied in has been met.
- (errno )
Termination criteria not satisfied, but the condition number is bounded by /
machine.precision
.- (errno )
Maximum number of iterations reached.
- Warns
- NagAlgorithmicWarning
- (errno )
in has been set to: .
- Notes
No equivalent traditional C interface for this routine exists in the NAG Library.
real_gen_sparse_lsqsol
can be used to solve a system of linear equationswhere is an sparse nonsymmetric matrix, or can be used to solve linear least squares problems, so that
real_gen_sparse_lsqsol
minimizes the value given bywhere is an sparse matrix and denotes the Euclidean length of so that . A damping argument, , may be included in the least squares problem in which case
real_gen_sparse_lsqsol
minimizes the value given byis supplied as the argument and should of course be zero if the solution to problems (1) and (2) is required. Minimizing in (3) is often called ridge regression.
real_gen_sparse_lsqsol
is based upon algorithm LSQR (see Paige and Saunders (1982a) and Paige and Saunders (1982b)) and solves the problems by an algorithm based upon the Lanczos process. The function does not require explicitly, but is specified via which must perform the operations and for a given -element vector and element vector . An argument to specifies which of the two operations is required on a given entry.The function also returns estimates of the standard errors of the sample regression coefficients ( , for ) given by the diagonal elements of the estimated variance-covariance matrix . When problem (2) is being solved and is of full rank, then is given by
and when problem (3) is being solved then is given by
Let denote the matrix
let denote the residual vector
corresponding to an iterate , so that is the function being minimized, and let denote the Frobenius (Euclidean) norm of . Then the function accepts as a solution if it is estimated that one of the following two conditions is satisfied:
where and are user-supplied tolerances which estimate the relative errors in and respectively. Condition (6) is appropriate for compatible problems where, in theory, we expect the residual to be zero and will be satisfied by an acceptable solution to a compatible problem. Condition (7) is appropriate for incompatible systems where we do not expect the residual to be zero and is based on the observation that, in theory,
when is a solution to the least squares problem, and so (7) will be satisfied by an acceptable solution to a linear least squares problem.
The function also includes a test to prevent convergence to solutions, , with unacceptably large elements. This can happen if is nearly singular or is nearly rank deficient. If we let the singular values of be
then the condition number of is defined as
where is the smallest nonzero singular value of and hence is the rank of . When , then is rank deficient, the least squares solution is not unique and
real_gen_sparse_lsqsol
will normally converge to the minimal length solution. In practice will not have exactly zero singular values, but may instead have small singular values that we wish to regard as zero.The function provides for this possibility by terminating if
where is a user-supplied limit on the condition number of . For problem (1) termination with this condition indicates that is nearly singular and for problem (2) indicates that is nearly rank deficient and so has near linear dependencies in its columns. In this case inspection of , and , which are all returned by the function, will indicate whether or not an acceptable solution has been found. Condition (8), perhaps in conjunction with , can be used to try and ‘regularize’ least squares solutions. A full discussion of the stopping criteria is given in Section 6 of Paige and Saunders (1982a).
Introduction of a nonzero damping argument tends to reduce the size of the computed solution and to make its components less sensitive to changes in the data, and
real_gen_sparse_lsqsol
is applicable when a value of is known a priori. To have an effect, should normally be at least where is the machine precision. For further discussion see Paige and Saunders (1982b) and the references given there.Whenever possible the matrix should be scaled so that the relative errors in the elements of are all of comparable size. Such a scaling helps to prevent the least squares problem from being unnecessarily sensitive to data errors and will normally reduce the number of iterations required. At the very least, in the absence of better information, the columns of should be scaled to have roughly equal column length.
- References
Paige, C C and Saunders, M A, 1982, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software (8), 43–71
Paige, C C and Saunders, M A, 1982, Algorithm 583 LSQR: Sparse linear equations and least squares problems, ACM Trans. Math. Software (8), 195–209