Uses QR or LQ factorization to solve a overdetermined or underdetermined linear system with full rank matrix.
Syntax

lapack_int LAPACKE_sgels (int matrix_layout, char trans, lapack_int m, lapack_int n, lapack_int nrhs, float* a, lapack_int lda, float* b, lapack_int ldb);
lapack_int LAPACKE_dgels (int matrix_layout, char trans, lapack_int m, lapack_int n, lapack_int nrhs, double* a, lapack_int lda, double* b, lapack_int ldb);
lapack_int LAPACKE_cgels (int matrix_layout, char trans, lapack_int m, lapack_int n, lapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb);
lapack_int LAPACKE_zgels (int matrix_layout, char trans, lapack_int m, lapack_int n, lapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb);
Include Files
 mkl.h
Description
The routine solves overdetermined or underdetermined real/ complex linear systems involving an mbyn matrix A, or its transpose/ conjugatetranspose, using a QR or LQ factorization of A. It is assumed that A has full rank.
The following options are provided:
1. If trans = 'N' and m≥n: find the least squares solution of an overdetermined system, that is, solve the least squares problem
minimize b  A*x_{2}
2. If trans = 'N' and m < n: find the minimum norm solution of an underdetermined system A*X = B.
3. If trans = 'T' or 'C' and m≥n: find the minimum norm solution of an undetermined system A^{H}*X = B.
4. If trans = 'T' or 'C' and m < n: find the least squares solution of an overdetermined system, that is, solve the least squares problem
minimize b  A^{H}*x_{2}
Several right hand side vectors b and solution vectors x can be handled in a single call; they are formed by the columns of the right hand side matrix B and the solution matrix X (when coefficient matrix is A, B is mbynrhs and X is nbynrhs; if the coefficient matrix is A^{T} or A^{H}, B isnbynrhs and X is mbynrhs.
Input Parameters
 matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).
 trans

Must be 'N', 'T', or 'C'.
If trans = 'N', the linear system involves matrix A;
If trans = 'T', the linear system involves the transposed matrix A^{T} (for real flavors only);
If trans = 'C', the linear system involves the conjugatetransposed matrix A^{H} (for complex flavors only).
 m

The number of rows of the matrix A (m≥ 0).
 n

The number of columns of the matrix A
(n≥ 0).
 nrhs

The number of righthand sides; the number of columns in B (nrhs≥ 0).
 a, b

Arrays:
a(size max(1, lda*n) for column major layout and max(1, lda*m) for row major layout) contains the mbyn matrix A.
b(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m, n)) for row major layout) contains the matrix B of right hand side vectors.
 lda

The leading dimension of a; at least max(1, m) for column major layout and at least max(1, n) for row major layout.
 ldb

The leading dimension of b; must be at least max(1, m, n) for column major layout if trans='N' and at least max(1, n) if trans='T' and at least max(1, nrhs) for row major layout regardless of the value of trans.
Output Parameters
 a

On exit, overwritten by the factorization data as follows:
if m≥n, array a contains the details of the QR factorization of the matrix A as returned by ?geqrf;
if m < n, array a contains the details of the LQ factorization of the matrix A as returned by ?gelqf.
 b

If info = 0, b overwritten by the solution vectors, stored columnwise:
if trans = 'N' and m≥n, rows 1 to n 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 modulus of elements n+1 to m in that column;
if trans = 'N' and m < n, rows 1 to n of b contain the minimum norm solution vectors;
if trans = 'T' or 'C' and m≥n, rows 1 to m of b contain the minimum norm solution vectors;
if trans = 'T' or 'C' and m < n, rows 1 to m 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 modulus of elements m+1 to n in that column.
Return Values
This function returns a value info.
If info=0, the execution is successful.
If info = i, the ith parameter had an illegal value.
If info = i, the ith 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.