◆ _dgels()

void _dgels	(	char	trans,
		int	m,
		int	n,
		int	nrhs,
		int	lda,
		double	a[],
		int	ldb,
		double	b[],
		double	work[],
		int	lwork,
		int *	info
	)

Solution to overdetermined or underdetermined linear equations Ax = b (Full rank)

Purpose

This routine solves overdetermined or underdetermined real linear systems involving an m x n matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

If trans = 'N' and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minimize || B - A*X ||.
If trans = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B.
If trans = 'T' and m >= n: find the minimum norm solution of an underdetermined system A^T * X = B.
If trans = 'T' and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minimize || B - A^T*X ||.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m x nrhs right hand side matrix B and the n x nrhs solution matrix X.

Parameters

[in]	trans	= 'N': The linear system involves A. = 'T': The linear system involves A^T.
[in]	m	Number of rows of the matrix A. (m >= 0) (If m = 0, returns zero vectors in b[][])
[in]	n	Number of columns of the matrix A. (n >= 0) (If n = 0, returns zero vectors in b[][])
[in]	nrhs	Number of right hand sides, i.e., number of columns of the matrices B and X. (nrhs >= 0) (If nrhs = 0, returns without computation)
[in]	lda	Leading dimension of the two dimensional array a[][]. (lda >= max(1, m))
[in,out]	a[][]	Array a[la][lda] (la >= n) [in] m x n matrix A. [out] m >= n: a[][] is overwritten by details of its QR factorization as returned by dgeqrf. m < n: a[][] is overwritten by details of its LQ factorization as returned by dgelqf.
[in]	ldb	Leading dimension of the two dimensional array b[][]. (ldb >= max(1, m, n))
[in,out]	b[][]	Array b[lb][ldb] (lb >= nrhs) [in] Matrix B of right hand side vectors, stored columnwise; B is m x nrhs if trans = 'N', or n x nrhs if trans = 'T'. [out] If info = 0, b[][] is overwritten by the solution vectors, stored columnwise: trans = 'N' and m >= n: Rows 0 to n-1 of b[][] contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements n to m-1 in that column. trans = 'N' and m < n: Rows 0 to n-1 of b[][] contain the minimum norm solution vectors. trans = 'T' and m >= n: Rows 0 to m-1 of b[][] contain the minimum norm solution vectors. trans = 'T' and m < n: Rows 0 to m-1 of b[][] contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements m to n-1 in that column.
[out]	work[]	Array work[lwork] Work array. On exit, if info = 0, work[0] returns the optimal lwork.
[in]	lwork	The dimension of the array work[]. (lwork >= max(1, mn + max(mn, nrhs))) For optimal performance, lwork >= max(1, mn + max(mn, nrhs)*nb), where mn = min(m, n) and nb is the optimum block size. If lwork = -1, then a workspace query is assumed. The routine only calculates the optimal size of the work[] array, and returns the value in work[0].
[out]	info	= 0: Successful exit = -1: The argument trans had an illegal value (trans != 'T' nor 'N') = -2: The argument m had an illegal value (m < 0) = -3: The argument n had an illegal value (n < 0) = -4: The argument nrhs had an illegal value (nrhs < 0) = -5: The argument lda had an illegal value (lda < max(1, m)) = -7: The argument ldb had an illegal value (ldb < max(1, m, n)) = -10: The argument lwork had an illegal value (lwork too small) = i > 0: The i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank. The least squares solution could not be computed.

Reference: LAPACK