sgelsy (l)  Linux Manuals
sgelsy: computes the minimumnorm solution to a real linear least squares problem
NAME
SGELSY  computes the minimumnorm solution to a real linear least squares problemSYNOPSIS
 SUBROUTINE SGELSY(
 M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO )
 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
 REAL RCOND
 INTEGER JPVT( * )
 REAL A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
SGELSY computes the minimumnorm solution to a real linear least squares problem:using a complete orthogonal factorization of A. A is an MbyN matrix which may be rankdeficient.
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 MbyNRHS right hand side matrix B and the NbyNRHS solution matrix X.
The routine first computes a QR factorization with column pivoting:
with R11 defined as the largest leading submatrix whose estimated condition number is less than 1/RCOND. The order of R11, RANK, is the effective rank of A.
Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:
A
The minimumnorm solution is then
X
where Q1 consists of the first RANK columns of Q.
This routine is basically identical to the original xGELSX except three differences:
ARGUMENTS
 M (input) INTEGER
 The number of rows of the matrix A. M >= 0.
 N (input) INTEGER
 The number of columns of the matrix A. N >= 0.
 NRHS (input) INTEGER
 The number of right hand sides, i.e., the number of columns of matrices B and X. NRHS >= 0.
 A (input/output) REAL array, dimension (LDA,N)
 On entry, the MbyN matrix A. On exit, A has been overwritten by details of its complete orthogonal factorization.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,M).
 B (input/output) REAL array, dimension (LDB,NRHS)
 On entry, the MbyNRHS right hand side matrix B. On exit, the NbyNRHS solution matrix X.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,M,N).
 JPVT (input/output) INTEGER array, dimension (N)
 On entry, if JPVT(i) .ne. 0, the ith column of A is permuted to the front of AP, otherwise column i is a free column. On exit, if JPVT(i) = k, then the ith column of AP was the kth column of A.
 RCOND (input) REAL
 RCOND is used to determine the effective rank of A, which is defined as the order of the largest leading triangular submatrix R11 in the QR factorization with pivoting of A, whose estimated condition number < 1/RCOND.
 RANK (output) INTEGER
 The effective rank of A, i.e., the order of the submatrix R11. This is the same as the order of the submatrix T11 in the complete orthogonal factorization of A.
 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. The unblocked strategy requires that: LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN = min( M, N ). The block algorithm requires that: LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper bound on the blocksize returned by ILAENV for the routines SGEQP3, STZRZF, STZRQF, SORMQR, and SORMRZ. If LWORK = 1, then 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 by XERBLA.
 INFO (output) INTEGER

= 0: successful exit
< 0: If INFO = i, the ith argument had an illegal value.
FURTHER DETAILS
Based on contributions by