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---
:name: sgelsx
:md5sum: e4e07c42aeca28db21674ddeedfea311
:category: :subroutine
:arguments:
- m:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- nrhs:
:type: integer
:intent: input
- a:
:type: real
:intent: input/output
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- b:
:type: real
:intent: input/output
:dims:
- ldb
- nrhs
- ldb:
:type: integer
:intent: input
- jpvt:
:type: integer
:intent: input/output
:dims:
- n
- rcond:
:type: real
:intent: input
- rank:
:type: integer
:intent: output
- work:
:type: real
:intent: workspace
:dims:
- MAX((MIN(m,n))+3*n,2*(MIN(m,n))*nrhs)
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* This routine is deprecated and has been replaced by routine SGELSY.\n\
*\n\
* SGELSX computes the minimum-norm solution to a real linear least\n\
* squares problem:\n\
* minimize || A * X - B ||\n\
* using a complete orthogonal factorization of A. A is an M-by-N\n\
* matrix which may be rank-deficient.\n\
*\n\
* Several right hand side vectors b and solution vectors x can be \n\
* handled in a single call; they are stored as the columns of the\n\
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution\n\
* matrix X.\n\
*\n\
* The routine first computes a QR factorization with column pivoting:\n\
* A * P = Q * [ R11 R12 ]\n\
* [ 0 R22 ]\n\
* with R11 defined as the largest leading submatrix whose estimated\n\
* condition number is less than 1/RCOND. The order of R11, RANK,\n\
* is the effective rank of A.\n\
*\n\
* Then, R22 is considered to be negligible, and R12 is annihilated\n\
* by orthogonal transformations from the right, arriving at the\n\
* complete orthogonal factorization:\n\
* A * P = Q * [ T11 0 ] * Z\n\
* [ 0 0 ]\n\
* The minimum-norm solution is then\n\
* X = P * Z' [ inv(T11)*Q1'*B ]\n\
* [ 0 ]\n\
* where Q1 consists of the first RANK columns of Q.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* M (input) INTEGER\n\
* The number of rows of the matrix A. M >= 0.\n\
*\n\
* N (input) INTEGER\n\
* The number of columns of the matrix A. N >= 0.\n\
*\n\
* NRHS (input) INTEGER\n\
* The number of right hand sides, i.e., the number of\n\
* columns of matrices B and X. NRHS >= 0.\n\
*\n\
* A (input/output) REAL array, dimension (LDA,N)\n\
* On entry, the M-by-N matrix A.\n\
* On exit, A has been overwritten by details of its\n\
* complete orthogonal factorization.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,M).\n\
*\n\
* B (input/output) REAL array, dimension (LDB,NRHS)\n\
* On entry, the M-by-NRHS right hand side matrix B.\n\
* On exit, the N-by-NRHS solution matrix X.\n\
* If m >= n and RANK = n, the residual sum-of-squares for\n\
* the solution in the i-th column is given by the sum of\n\
* squares of elements N+1:M in that column.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the array B. LDB >= max(1,M,N).\n\
*\n\
* JPVT (input/output) INTEGER array, dimension (N)\n\
* On entry, if JPVT(i) .ne. 0, the i-th column of A is an\n\
* initial column, otherwise it is a free column. Before\n\
* the QR factorization of A, all initial columns are\n\
* permuted to the leading positions; only the remaining\n\
* free columns are moved as a result of column pivoting\n\
* during the factorization.\n\
* On exit, if JPVT(i) = k, then the i-th column of A*P\n\
* was the k-th column of A.\n\
*\n\
* RCOND (input) REAL\n\
* RCOND is used to determine the effective rank of A, which\n\
* is defined as the order of the largest leading triangular\n\
* submatrix R11 in the QR factorization with pivoting of A,\n\
* whose estimated condition number < 1/RCOND.\n\
*\n\
* RANK (output) INTEGER\n\
* The effective rank of A, i.e., the order of the submatrix\n\
* R11. This is the same as the order of the submatrix T11\n\
* in the complete orthogonal factorization of A.\n\
*\n\
* WORK (workspace) REAL array, dimension\n\
* (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value\n\
*\n\n\
* =====================================================================\n\
*\n"
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