The routine may be called by the names f08aaf, nagf_lapackeig_dgels or its LAPACK name dgels.
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 right-hand side matrix and the solution matrix .
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
1: – Character(1)Input
On entry: if , the linear system involves .
If , the linear system involves .
2: – IntegerInput
On entry: , the number of rows of the matrix .
3: – IntegerInput
On entry: , the number of columns of the matrix .
4: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
5: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: if , a is overwritten by details of its factorization, as returned by f08aef.
If , a is overwritten by details of its factorization, as returned by f08ahf.
6: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08aaf is called.
7: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
On entry: the matrix of right-hand side vectors, stored in columns; b is if , or if .
On exit: b is overwritten by 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.
8: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08aaf is called.
9: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimal performance.
10: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08aaf 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.
for optimal performance, , where is the optimal block size.
11: – 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.
Diagonal element of the triangular factor of is zero, so that does not have full rank; the least squares solution could not be computed.
f08aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aaf 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.
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.