The routine may be called by the names f08kaf, nagf_lapackeig_dgelss or its LAPACK name dgelss.
f08kaf uses the singular value decomposition (SVD) of , where is an matrix which may be rank-deficient.
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 .
The effective rank of is determined by treating as zero those singular values which are less than rcond times the largest singular value.
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: – IntegerInput
On entry: , the number of rows of the matrix .
2: – IntegerInput
On entry: , the number of columns of the matrix .
3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrices and .
4: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the matrix .
On exit: the first rows of are overwritten with its right singular vectors, stored row-wise.
5: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kaf is called.
6: – Real (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: b is overwritten by the solution matrix . If and , the residual sum of squares for the solution in the th column is given by the sum of squares of elements in that column.
7: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08kaf is called.
8: – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array s
must be at least
On exit: the singular values of in decreasing order.
9: – Real (Kind=nag_wp)Input
On entry: used to determine the effective rank of . Singular values are treated as zero. If , machine precision is used instead.
10: – IntegerOutput
On exit: the effective rank of , i.e., the number of singular values which are greater than .
11: – Real (Kind=nag_wp) arrayWorkspace
On exit: if , 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 f08kaf 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, lwork should generally be larger. Consider increasing lwork by at least , where is the optimal block size.
or and .
13: – 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.
The algorithm for computing the SVD failed to converge; off-diagonal elements of an intermediate bidiagonal form did not converge to zero.
Background information to multithreading can be found in the Multithreading documentation.
f08kaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kaf 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.