SRC/sgelsd.f(3)

💅 NAME

SRC/sgelsd.f — okay bestie, this is the Library Functions Manual page for SRC/sgelsd.f(3), and it's all about the sgelsd subroutine. It's giving linear least squares with main character energy, no cap. ✨

✨ SYNOPSIS

🧚 Functions/Subroutines

subroutine sgelsd (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work,
           lwork, iwork, info)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

🌈 Function/Subroutine Documentation

subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension( lda,
     * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
     dimension( * ) s, real rcond, integer rank, real, dimension( * ) work,
     integer lwork, integer, dimension( * ) iwork, integer info)

SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

🎯 Purpose:

Okay so the source comment literally says: SGELSD computes the minimum-norm solution to a real linear least squares problem:

minimize 2-norm(| b - A*x |)

...using the singular value decomposition (SVD) of A. And A is an M-by-N matrix which may be rank-deficient. Very demure, very mindful. 🎀

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-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X. Slay, multi-RHS behavior. 💅

The problem is solved in three steps:

1. (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a (BLS).
2. (2) Solve the BLS using a divide and conquer approach.
3. (3) Apply back all the Householder transformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value. 🫶

🌸 Parameters

• M — M is INTEGER. The number of rows of A. M >= 0.
• N — N is INTEGER. The number of columns of A. N >= 0.
• NRHS — NRHS is INTEGER. The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.
• A — A is REAL array, dimension (LDA,N). On entry, the M-by-N matrix A. On exit, A has been destroyed. 🚨 Yes bestie, it gets destroyed — that's the lore.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• B — B is REAL array, dimension (LDB,NRHS). On entry, the M-by-NRHS right hand side matrix B. On exit, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column.
• LDB — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,max(M,N)).
• S — S is REAL array, dimension (min(M,N)). The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)).
• RCOND — RCOND is REAL. RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead.
• RANK — RANK is INTEGER. The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
• WORK — WORK is REAL array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER. The dimension of the array WORK. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the array WORK and the minimum size of the array IWORK, and returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA. 🫡

• IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
• INFO — INFO is INTEGER
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

👑 Author

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🌟 Contributors:

• Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
• Osni Marques, LBNL/NERSC, USA

Definition at line 202 of file sgelsd.f.

🫶 Author

Generated automatically by Doxygen for LAPACK from the source code.

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Signed, sealed, slayed: LAPACK Version 3.12.0 — page SRC/sgelsd.f(3). 💖