f08bnf computes the minimum norm solution to a complex linear least squares problem
using a complete orthogonal factorization of . is an matrix which may be rank-deficient. Several right-hand side vectors and solution vectors can be handled in a single call.
The routine may be called by the names f08bnf, nagf_lapackeig_zgelsy or its LAPACK name zgelsy.
3Description
The right-hand side vectors are stored as the columns of the matrix and the solution vectors in the matrix .
f08bnf first computes a factorization with column pivoting
with defined as the largest leading sub-matrix whose estimated condition number is less than . The order of , rank, is the effective rank of .
Then, is considered to be negligible, and is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5Arguments
1: – IntegerInput
On entry: , the number of rows of the matrix .
Constraint:
.
2: – IntegerInput
On entry: , the number of columns of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
Constraint:
.
4: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
.
On entry: the matrix .
On exit: a has been overwritten by details of its complete orthogonal factorization.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bnf is called.
Constraint:
.
6: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
.
On entry: the right-hand side matrix .
On exit: the solution matrix .
7: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08bnf is called.
Constraint:
.
8: – Integer arrayInput/Output
Note: the dimension of the array jpvt
must be at least
.
On entry: if , the th column of is permuted to the front of , otherwise column is a free column.
On exit: if , the th column of was the th column of .
9: – Real (Kind=nag_wp)Input
On entry: used to determine the effective rank of , which is defined as the order of the largest leading triangular sub-matrix in the factorization of , whose estimated condition number is .
Suggested value:
if the condition number of a is not known then (where is machine precision, see x02ajf) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective that could be larger than its actual rank, leading to meaningless results.
10: – IntegerOutput
On exit: the effective rank of , i.e., the order of the sub-matrix . This is the same as the order of the sub-matrix in the complete orthogonal factorization of .
11: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , the real part of contains the minimum value of lwork required for optimal performance.
12: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08bnf is called.
If , 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,
where and is the optimal block size.
Constraint:
or .
13: – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array rwork
must be at least
.
14: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
Background information to multithreading can be found in the Multithreading documentation.
f08bnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bnf 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.