f04amf calculates the accurate least squares solution of a set of linear equations in unknowns, and rank , with multiple right-hand sides, , using a factorization and iterative refinement.
The routine may be called by the names f04amf or nagf_linsys_real_gen_lsqsol.
3Description
To compute the least squares solution to a set of linear equations in unknowns , f04amf first computes a factorization of with column pivoting, , where is upper triangular, is an orthogonal matrix, and is a permutation matrix. is applied to the right-hand side matrix to give , and the solution matrix is calculated, to a first approximation, by back-substitution in . The residual matrix is calculated using additional precision, and a correction to is computed as the least squares solution to . is replaced by and this iterative refinement of the solution is repeated until full machine accuracy has been obtained.
4References
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
5Arguments
1: – Real (Kind=nag_wp) arrayInput
On entry: the matrix .
2: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f04amf is called.
Constraint:
.
3: – Real (Kind=nag_wp) arrayOutput
On exit: the solution matrix .
4: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f04amf is called.
Constraint:
.
5: – Real (Kind=nag_wp) arrayInput
On entry: the right-hand side matrix .
6: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04amf is called.
Constraint:
.
7: – IntegerInput
On entry: , the number of rows of the matrix , i.e., the number of equations.
Constraint:
.
8: – IntegerInput
On entry: , the number of columns of the matrix , i.e., the number of unknowns.
Constraint:
.
9: – IntegerInput
On entry: , the number of right-hand sides.
10: – Real (Kind=nag_wp)Input
On entry: must be set to the value of the machine precision.
11: – Real (Kind=nag_wp) arrayOutput
On exit: details of the factorization.
12: – IntegerInput
On entry: the first dimension of the array qr as declared in the (sub)program from which f04amf is called.
Constraint:
.
13: – Real (Kind=nag_wp) arrayOutput
On exit: the diagonal elements of the upper triangular matrix .
14: – Real (Kind=nag_wp) arrayWorkspace
15: – Real (Kind=nag_wp) arrayWorkspace
16: – Real (Kind=nag_wp) arrayWorkspace
17: – Real (Kind=nag_wp) arrayWorkspace
18: – Integer arrayOutput
On exit: details of the column interchanges.
19: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
The rank of a is less than n. The problem does not have a unique solution.
The iterative refinement fails to converge. The matrix is too ill-conditioned.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Although the correction process is continued until the solution has converged to full machine accuracy, all the figures in the final solution may not be correct since the correction to is itself the solution to a linear least squares problem. For a detailed error analysis see page 116 of Wilkinson and Reinsch (1971).
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f04amf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The time taken by f04amf is approximately proportional to , provided is small compared with .
10Example
This example calculates the accurate least squares solution of the equations